The Analysis of Matter
{"WorkMasterId":5233,"WpPageId":252992,"ParentWpPageId":189742,"Slug":"the-analysis-of-matter","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/the-analysis-of-matter/","RelativeUrl":"theos/humanities/philosophy/philosophers/bertrand-russell/the-analysis-of-matter/","HasFullText":true,"RawHtmlLength":1315959,"CleanHtmlLength":1259849,"Kicker":"Philosophy Work","Title":"The Analysis of Matter","Deck":"Russell examines physics, perception, causal structure, matter, space-time, and scientific knowledge after relativity.","BackLink":{"Text":"Back to Bertrand Russell","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/"},"AuthorCard":{"Label":"Author","Title":"Bertrand Russell","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/","MediaHref":"","ImageSrc":"https://chrisdeasy.com/wp-content/uploads/bertrand-russell-01-1954-portrait-2.jpg","ImageAlt":"Bertrand Russell Portrait, 1954","FilterTerra":"Western Europe","ClickText":"Bertrand Russell","ClickHref":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/","Copies":["1872 CE – 1970 CE","Trellech, Monmouthshire","British analytic philosopher, logician, mathematician, social critic, and Nobel laureate from Trellech whose logicism, theory of descriptions, logical atomism, epistemology, philosophy of language, ethics, pacifism, secular critique, and political writing shaped analytic philosophy and twentieth-century public reason."]},"ContextCards":[{"Label":"Period","Key":"Period:5","Title":"Contemporary History","DateText":"1945 CE – 2065 CE","Url":"https://chrisdeasy.com/theos/humanities/philosophy/eras-of-thought/philosophers-of-contemporary-history/"},{"Label":"Era","Key":"Era:12","Title":"World War Era","DateText":"1914 CE – 1944 CE","Url":"https://chrisdeasy.com/theos/humanities/philosophy/eras-of-thought/philosophers-of-modern-history/philosophers-of-the-world-war-era/"},{"Label":"Composition","Title":"1927 CE","Url":"","DateText":""}],"DateNote":"Displayed year is the publication year.","GeoCards":[{"Label":"Region","Key":"Region:1"},{"Label":"Terra Avita","Key":"TerraAvita:1"},{"Label":"Terra Avita Region","Key":"TerraAvitaRegion:2"},{"Label":"Modern Country","Key":"Country:GBR:1"}],"OriginalTitle":"","Language":"English","DisciplineCards":[{"Label":"Primary Discipline","Key":"Discipline:metaphysics"},{"Label":"Secondary Discipline","Key":"Discipline:philosophy-of-science"}],"Tradition":"Analytic philosophy, logicism, British empiricism, social criticism, secular humanism, and twentieth-century public reason","FullText":{"Title":"Full Text","Copy":"Public-domain full text from Project Gutenberg eBook #77427 .","Url":"","Label":"","Kicker":"","Cards":[]},"CoreThesis":["Russell examines physics, perception, causal structure, matter, space-time, and scientific knowledge after relativity."],"Classification":{"AlternateTitles":"","KeyConcepts":"The Analysis of Matter; Bertrand Russell; logicism; descriptions; logical atomism; knowledge; language; science; ethics; politics; religion; public reason","Methodology":"Logical analysis, formal argument, empiricist reconstruction, linguistic analysis, public criticism, historical explanation, and social-philosophical argument.","Structure":"Accepted work page for Russell under the Core Major scope; minor journalism, duplicate anthologies, individual letters, source/testimony pages, and works merely about Russell are excluded."},"Arguments":["Connects Russell\u0027s technical work in logic and language with his epistemology, philosophy of science, ethics, politics, secular criticism, and public writing."],"Influence":{"InfluencedBy":"Frege, Peano, Leibniz, Hume, Mill, Moore, Whitehead, Cantor, Cambridge mathematics, British empiricism, and anti-idealism.","InfluenceOn":""},"Significance":["Part of the Core Major Russell corpus that made him central to analytic philosophy, mathematical logic, public ethics, secular critique, and twentieth-century intellectual life.","Used in debates about reference, logic, mathematics, science, knowledge, mind, language, liberalism, religion, education, power, and public responsibility."],"EvidenceNote":["Accepted as a major work in Russell\u0027s philosophy of science and metaphysics."],"MainSections":[{"Kind":"RawSection","Title":"Full Versions","BodyHtml":"\u003cdiv class=\"dz-philo__full-version-grid\"\u003e\n \u003carticle class=\"dz-philo__full-version-card\"\u003e\n \u003cp class=\"dz-philo__full-version-provider\"\u003eProject Gutenberg\u003c/p\u003e\n \u003ch3 class=\"dz-philo__full-version-title\"\u003eProject Gutenberg eBook #77427\u003c/h3\u003e\n \u003cp class=\"dz-philo__full-version-meta\"\u003eHtmlText · Imported\u003c/p\u003e\n \u003ca class=\"dz-philo__full-version-link\" href=\"https://www.gutenberg.org/ebooks/77427\"\u003eOpen full version\u003c/a\u003e\n \u003c/article\u003e\n \u003c/div\u003e"},{"Kind":"TextSection","Title":"Core Thesis","Paragraphs":["Russell examines physics, perception, causal structure, matter, space-time, and scientific knowledge after relativity."]},{"Kind":"FieldSection","Title":"Classification","Fields":[{"Label":"Alternate Titles","Value":""},{"Label":"Key Concepts","Value":"The Analysis of Matter; Bertrand Russell; logicism; descriptions; logical atomism; knowledge; language; science; ethics; politics; religion; public reason"},{"Label":"Methodology","Value":"Logical analysis, formal argument, empiricist reconstruction, linguistic analysis, public criticism, historical explanation, and social-philosophical argument."},{"Label":"Structure","Value":"Accepted work page for Russell under the Core Major scope; minor journalism, duplicate anthologies, individual letters, source/testimony pages, and works merely about Russell are excluded."}]},{"Kind":"TextSection","Title":"Arguments","Paragraphs":["Connects Russell\u0027s technical work in logic and language with his epistemology, philosophy of science, ethics, politics, secular criticism, and public writing."]},{"Kind":"FieldSection","Title":"Influence","Fields":[{"Label":"Influenced By","Value":"Frege, Peano, Leibniz, Hume, Mill, Moore, Whitehead, Cantor, Cambridge mathematics, British empiricism, and anti-idealism."},{"Label":"Influence On","Value":"Analytic philosophy, mathematical logic, philosophy of language, logical atomism, logical positivism, secular humanism, public philosophy, peace activism, and twentieth-century liberal thought."}]},{"Kind":"TextSection","Title":"Significance","Paragraphs":["Part of the Core Major Russell corpus that made him central to analytic philosophy, mathematical logic, public ethics, secular critique, and twentieth-century intellectual life.","Used in debates about reference, logic, mathematics, science, knowledge, mind, language, liberalism, religion, education, power, and public responsibility."]},{"Kind":"TextSection","Title":"Evidence Note","Paragraphs":["Accepted as a major work in Russell\u0027s philosophy of science and metaphysics."]},{"Kind":"RawSection","Title":"Full Text","BodyHtml":"\u003cp class=\"dz-philo__section-copy dz-philo__full-text-source\"\u003ePublic-domain full text from \u003ca href=\"https://www.gutenberg.org/ebooks/77427\"\u003eProject Gutenberg eBook #77427\u003c/a\u003e.\u003c/p\u003e\n \u003carticle class=\"dz-philo__full-text-body\"\u003e\r\n\u003cfigure class=\"figcenter width500\" id=\"cover\" style=\"width: 1600px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-cover.jpg\" width=\"1600\" height=\"2741\" alt=\"Russell\nexplores the relationship between physics and perception, arguing that\nmatter should be understood through its mathematical structure rather\nthan its intrinsic qualities.\"\u003e\n\u003c/figure\u003e\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003ch1\u003eThe Analysis of\u003cbr\u003e\nMatter\u003c/h1\u003e\n\n\n\u003cp class=\"nindc space-above2 space-below2\"\u003e\u003cspan class=\"large\"\u003e\u003cspan class=\"allsmcap\"\u003eBY\u003c/span\u003e\u003c/span\u003e\u003cbr\u003e\n\u003cbr\u003e\nBERTRAND RUSSELL\u003cbr\u003e\nF.R.S.\u003c/p\u003e\n\n\n\u003cfigure class=\"figcenter width500\" id=\"i_001\" style=\"width: 100px;\"\u003e\n \u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-001.jpg\" width=\"100\" height=\"90\" alt=\"decorative\"\u003e\n\u003c/figure\u003e\n\n\n\u003cp class=\"nindc space-above2 space-below2\"\u003eNEW YORK\u003cbr\u003e\nHARCOURT, BRACE Sc COMPANY, INC.\u003cbr\u003e\nLONDON: KEGAN PAUL, TRENCH, TRUBNER \u0026amp; CO., LTD.\u003cbr\u003e\n1927\n\u003c/p\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp class=\"nindc space-above2 space-below2\"\u003e\u003cspan class=\"allsmcap\"\u003ePRINTED IN GREAT BRITAIN BY\u003cbr\u003e\nBILLING AND SONS, LTD., GUILDFORD AND ESHER\u003c/span\u003e\u003c/p\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_v\"\u003e[Pg v]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CONTENTS\"\u003eCONTENTS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003ctable class=\"autotable\"\u003e\n\u003ctbody\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003ePREFACE\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_vii\"\u003evii\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eCHAPTER\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u0026nbsp;\u0026nbsp;\u0026nbsp;\u003cspan class=\"allsmcap\"\u003ePAGE\u003c/span\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eI. THE NATURE OF THE PROBLEM\u003c/span\u003e\u003c/td\u003e\n\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_1\"\u003e1\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdc\"\u003ePART I\u003cbr\u003e\nTHE LOGICAL ANALYSIS OF PHYSICS\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eII. PRE-RELATIVITY PHYSICS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eIII. ELECTRONS AND PROTONS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eIV. THE THEORY OF QUANTA\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eV. THE SPECIAL THEORY OF RELATIVITY\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_48\"\u003e48\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eVI. THE GENERAL THEORY OF RELATIVITY\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eVII. THE METHOD OF TENSORS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_63\"\u003e63\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eVIII. GEODESICS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eIX. INVARIANTS AND THEIR PHYSICAL INTERPRETATION\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eX. WEYL\u0027S THEORY\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_95\"\u003e95\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXI. THE PRINCIPLE OF DIFFERENTIAL LAWS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXII. MEASUREMENT\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_109\"\u003e109\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXIII. MATTER AND SPACE\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_121\"\u003e121\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXIV. THE ABSTRACTNESS OF PHYSICS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_130\"\u003e130\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdc\"\u003ePART II\u003cbr\u003e\nPHYSICS AND PERCEPTION\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXV. FROM PRIMITIVE PERCEPTION TO COMMON SENSE\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXVI. FROM COMMON SENSE TO PHYSICS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXVII. WHAT IS AN EMPIRICAL SCIENCE?\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXVIII. OUR KNOWLEDGE OF PARTICULAR MATTERS OF FACT\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_178\"\u003e178\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXIX. DATA, INFERENCES, HYPOTHESES, AND THEORIES\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXX. THE CAUSAL THEORY OF PERCEPTION\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_197\"\u003e197\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXI. PERCEPTION AND OBJECTIVITY\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXII. THE BELIEF IN GENERAL LAWS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_229\"\u003e229\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXIII. SUBSTANCE\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXIV. IMPORTANCE OF STRUCTURE IN SCIENTIFIC INFERENCE\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXV. PERCEPTION FROM THE STANDPOINT OF PHYSICS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_257\"\u003e257\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXVI. NON-MENTAL ANALOGUES TO PERCEPTION\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e\u003cspan class=\"pagenum\" id=\"Page_vi\"\u003e[Pg vi]\u003c/span\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdc\"\u003ePART III\u003cbr\u003e\nTHE STRUCTURE OF THE PHYSICAL WORLD\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXVII. PARTICULARS AND EVENTS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_275\"\u003e275\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXVIII. THE CONSTRUCTION OF POINTS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXIX. SPACE-TIME ORDER\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXX. CAUSAL LINES\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_313\"\u003e313\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXI. EXTRINSIC CAUSAL LAWS\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXII. PHYSICAL AND PERCEPTUAL SPACE-TIME\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_333\"\u003e333\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXIII. PERIODICITY AND QUALITATIVE SERIES\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXIV. TYPES OF PHYSICAL OCCURRENCES\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXV. CAUSALITY AND INTERVAL\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_367\"\u003e367\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXVI. THE GENESIS OF SPACE-TIME\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_376\"\u003e376\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXVII. PHYSICS AND NEUTRAL MONISM\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_382\"\u003e382\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003e\u003cspan class=\"allsmcap\"\u003eXXXVIII. SUMMARY AND CONCLUSION\u003c/span\u003e\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\u003ctr\u003e\n\u003ctd class=\"tdl\"\u003eINDEX\u003c/td\u003e\n\u003ctd class=\"tdr\"\u003e\u003ca href=\"#Page_403\"\u003e403\u003c/a\u003e\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_vii\"\u003e[Pg vii]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"PREFACE\"\u003ePREFACE\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE attempt to discover the philosophical outcome of modern physics is\none which, at the present moment, is beset with great difficulties.\nFor, while the theory of relativity has achieved, at least temporarily,\na stable form, the theory of quanta and of atomic structure is\ndeveloping with such rapidity that it is impossible to guess what\nform it will take a few years hence. In these circumstances, it is\nnecessary to exercise judgment as to the parts of the theory which are\ndefinitively established and the parts which are likely to be modified\nin the near future. For one who, like the present author, is not a\nprofessional physicist, the exercise of such judgment is difficult,\nand is likely to be occasionally at fault. The subject of the relation\nof \"matter\" to what exists, and generally of the interpretation of\nphysics in terms of what exists, is, however, not one of physics alone.\nPsychology, physiology, mathematical logic, and philosophy are all\nrequired, in addition to physics, for the adequate discussion of the\ntheme with which this volume deals. Consequently certain shortcomings\non the part of a single author, however regrettable they may be, are\nperhaps scarcely avoidable.\u003c/p\u003e\n\n\u003cp\u003eI am indebted to Mr R. H. Fowler, F.R.S., Mr M. H. A. Newman of St.\nJohn\u0027s College, Cambridge, and Mr F. P. Ramsey of King\u0027s College,\nCambridge, for valuable help in regard to certain portions of the work;\nalso to Dr D. M. Wrinch for kindly reading the whole in typescript and\nsupplying many valuable criticisms and suggestions.\u003c/p\u003e\n\n\u003cp\u003eCertain portions of the book were delivered as the Tarner Lectures\nin Trinity College, Cambridge, during the Michaelmas Term, 1926. The\nbook was, however, in preparation before\u003cspan class=\"pagenum\" id=\"Page_viii\"\u003e[Pg viii]\u003c/span\u003e the invitation to give these\nlectures was received, and contains a good deal of material for which\nthere seemed no place in the lectures.\u003c/p\u003e\n\n\u003cp\u003eSince the purpose of the book is philosophical, it has been my\nendeavour to avoid physical and mathematical technicalities as far as\npossible. Some modern doctrines, however, perhaps because they are\nstill recent, I have not succeeded in translating into non-mathematical\nlanguage. In regard to them, I must beg the indulgence of the\nnon-mathematical reader if he finds too many symbols, and of the\nmathematical reader if he finds too few.\u003c/p\u003e\n\n\u003cp class=\"right\"\u003eB. R.\u003c/p\u003e\n\n\u003cp\u003e\u003ci\u003eJanuary\u003c/i\u003e, 1927.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003ch2 class=\"nobreak\" id=\"THE_ANALYSIS_OF_MATTER\"\u003eTHE ANALYSIS OF MATTER\u003c/h2\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_1\"\u003e[Pg 1]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_I\"\u003eCHAPTER I\u003cbr\u003e\nTHE NATURE OF THE PROBLEM\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nAPART from pure mathematics, the most advanced of the sciences is\nphysics. Certain parts of theoretical physics have reached the point\nwhich makes it possible to exhibit a logical chain from certain assumed\npremisses to consequences apparently very remote, by means of purely\nmathematical deductions. This is true especially of everything that\nbelongs to the general theory of relativity. It cannot be said that\nphysics as a whole has yet reached this stage, since quantum phenomena,\nand the existence of electrons and protons, remain, for the moment,\nbrute facts. But perhaps this state of affairs will not last long; it\nis not chimerical to hope that a unified treatment of the whole of\nphysics may be possible before many years have passed.\u003c/p\u003e\n\n\u003cp\u003eIn spite, however, of the extraordinary successes of physics considered\nas a science, the philosophical outcome is much less dear than it\nseemed to be when less was known. The purpose of the present chapter is\nto discuss what is meant by the \"philosophical outcome\" of physics, and\nwhat methods exist for determining its nature.\u003c/p\u003e\n\n\u003cp\u003eThere are three kinds of questions which we may ask concerning\nphysics or, indeed, concerning any science. The first is: What is its\nlogical structure, considered as a deductive system? What ways exist\nof defining the entities of physics and deducing the propositions\nfrom an initial apparatus of entities and propositions? This is a\nproblem in pure mathematics, for which, in its fundamental portions,\nmathematical\u003cspan class=\"pagenum\" id=\"Page_2\"\u003e[Pg 2]\u003c/span\u003e logic is the proper instrument. It is not quite correct\nto speak, as we did just now, of \"initial entities and propositions.\"\nWhat we really have to begin with, in this treatment, is hypotheses\ncontaining variables. In geometry, this procedure has become familiar.\nInstead of \"axioms,\" supposed to be \"true,\" we have the hypothesis\nthat a set of entities (otherwise undefined) has certain enumerated\nproperties. We proceed to prove that such a set of entities has the\nproperties which constitute the propositions of Euclidean geometry, or\nof whatever other geometry may be occupying our attention. Generally it\nwill be possible to choose many different sets of initial hypotheses\nwhich will all yield the same body of propositions; the choice between\nthese sets is logically irrelevant, and can be guided only by æsthetic\nconsiderations. There is, however, considerable utility in the\ndiscovery of a few simple hypotheses which will yield the whole of some\ndeductive system, since it enables us to know what tests are necessary\nand sufficient in deciding whether some given set of entities satisfies\nthe deductive system. Moreover, the word \"entities,\" which we have\nbeen using, is too narrow if used with any metaphysical implication.\nThe \"entities\" concerned may, in a given application of a deductive\nsystem, be complicated logical structures. Of this we have examples in\npure mathematics in the definitions of cardinal numbers, ratios, real\nnumbers, etc. We must be prepared for the possibility of a similar\nresult in physics, in the definition of a \"point\" of space-time, and\neven in the definition of an electron or a proton.\u003c/p\u003e\n\n\u003cp\u003eThe logical analysis of a deductive system is not such a\ndefinite and limited undertaking as it appears at first sight.\nThis is due to the circumstance just mentioned—namely, that\nwhat we took at first as primitive entities may be replaced by\ncomplicated logical structures. As this circumstance has an\nimportant bearing upon the philosophy of physics, it will be\u003cspan class=\"pagenum\" id=\"Page_3\"\u003e[Pg 3]\u003c/span\u003e\nworth while to illustrate its effect by examples from other\nfields.\u003c/p\u003e\n\n\u003cp\u003eOne of the best examples is the theory of finite integers. Weierstrass\nand others had shown that the whole of analysis was reducible to\npropositions about finite integers, when Peano showed that these\npropositions were all deducible from five initial propositions\ninvolving three undefined ideas.\u003ca id=\"FNanchor_1\" href=\"#Footnote_1\" class=\"fnanchor\"\u003e[1]\u003c/a\u003e The five initial propositions\nmight be regarded as assigning certain properties to the group of\nthree undefined ideas, the properties in question being of a logical,\nnot specifically arithmetical, character. What was proved by Peano\nwas this: Given any triad having the five properties in question,\nevery proposition of arithmetic and analysis is true of this triad,\nprovided the interpretation appropriate to this triad is adopted. But\nit appeared further that there is one such triad corresponding to each\ninfinite series \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e, … \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, …, in which\nthere is just one term corresponding to each finite integer. Such\nseries can be defined without mentioning integers. Any such series\ncould be taken, instead of the series of finite integers, as the\nbasis of arithmetic and analysis. Every proposition of arithmetic and\nanalysis will remain true for any such series, but for each series it\nwill be a different proposition from what it is for any other series.\u003c/p\u003e\n\n\u003cp\u003eTake, in illustration, some simple proposition of arithmetic, say:\n\"The sum of the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e odd numbers is \u003cimg style=\"vertical-align: -0.025ex; width: 2.345ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-6.png\" alt=\"\" data-tex=\"\\(n^{2}\\)\"\u003e.\" Suppose we wish\nto interpret this proposition as applying to the progression \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-7.png\" alt=\"\" data-tex=\"\\(x_0\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e,… \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, … In this progression, let \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e be the\nrelation of each term to its successor. Then \"odd numbers\" will mean\n\"terms having to \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e a relation which is a power of \u003cimg style=\"vertical-align: -0.048ex; width: 2.705ex; height: 1.934ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-9.png\" alt=\"\" data-tex=\"\\(R^{2}\\)\"\u003e,\" where\n\u003cimg style=\"vertical-align: -0.048ex; width: 2.705ex; height: 1.934ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-9.png\" alt=\"\" data-tex=\"\\(R^{2}\\)\"\u003e is the relation of an \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e to the next \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e but one.\u003ca id=\"FNanchor_2\" href=\"#Footnote_2\" class=\"fnanchor\"\u003e[2]\u003c/a\u003e We\ncan now define as meaning that power of \u003cimg style=\"vertical-align: -0.048ex; width: 3.632ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-11.png\" alt=\"\" data-tex=\"\\(R^{{x}_{n}}\\)\"\u003e which relates\n\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-7.png\" alt=\"\" data-tex=\"\\(x_0\\)\"\u003e to \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, and we can further define \u003cimg style=\"vertical-align: -0.357ex; width: 8.094ex; height: 1.676ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-12.png\" alt=\"\" data-tex=\"\\(x_m + x_n\\)\"\u003e as meaning\u003cspan class=\"pagenum\" id=\"Page_4\"\u003e[Pg 4]\u003c/span\u003e\nthat \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e to which \u003cimg style=\"vertical-align: -0.357ex; width: 2.887ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-13.png\" alt=\"\" data-tex=\"\\(x_m\\)\"\u003e has the relation \u003cimg style=\"vertical-align: -0.048ex; width: 3.632ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-11.png\" alt=\"\" data-tex=\"\\(R^{{x}_{n}}\\)\"\u003e. This\ndecides the interpretation of \"the sum of the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e odd numbers.\"\nTo define \u003cimg style=\"vertical-align: -0.025ex; width: 2.345ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-6.png\" alt=\"\" data-tex=\"\\(n^{2}\\)\"\u003e it will be best to define multiplication. We have\ndefined \u003cimg style=\"vertical-align: -0.048ex; width: 3.632ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-11.png\" alt=\"\" data-tex=\"\\(R^{{x}_{n}}\\)\"\u003e; consider the relation formed by the relative\nproduct of the converse of \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e together with \u003cimg style=\"vertical-align: -0.048ex; width: 3.632ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-11.png\" alt=\"\" data-tex=\"\\(R^{{x}_{n}}\\)\"\u003e. This\nrelation relates \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e to \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e; its square relates \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e to\n\u003cimg style=\"vertical-align: -0.357ex; width: 3.242ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-14.png\" alt=\"\" data-tex=\"\\(x_{2n}\\)\"\u003e; its cube relates \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e to \u003cimg style=\"vertical-align: -0.375ex; width: 3.242ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-15.png\" alt=\"\" data-tex=\"\\(x_{3n}\\)\"\u003e, etc. Any power\nof this relation can be shown to be equivalent to a certain power\nof the converse of \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e multiplied relatively by a certain power\nof \u003cimg style=\"vertical-align: -0.048ex; width: 3.632ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-11.png\" alt=\"\" data-tex=\"\\(R^{{x}_{n}}\\)\"\u003e. There is thus one power of this relation which\nis equivalent to moving backward from \u003cimg style=\"vertical-align: -0.357ex; width: 2.887ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-13.png\" alt=\"\" data-tex=\"\\(x_m\\)\"\u003e to \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-7.png\" alt=\"\" data-tex=\"\\(x_0\\)\"\u003e, and then\nforward; the term to which the forward movement takes us is defined\nas \u003cimg style=\"vertical-align: -0.357ex; width: 8.094ex; height: 1.468ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-16.png\" alt=\"\" data-tex=\"\\(x_m \\times x_n\\)\"\u003e. Thus we can now interpret \u003cimg style=\"vertical-align: -0.576ex; width: 2.442ex; height: 2.463ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-17.png\" alt=\"\" data-tex=\"\\(x_n^{2}\\)\"\u003e. It will\nbe found that the proposition from which we started is true with this\ninterpretation.\u003c/p\u003e\n\n\u003cp\u003eIt follows from the above that, if we start from Peano\u0027s undefined\nideas and initial propositions, arithmetic and analysis are not\nconcerned with definite logical objects called numbers, but with the\nterms of any progression. We may call the terms of any progression 0,\n1, 2, 3,…, in which case, with a suitable interpretation of + and\n\u003cimg style=\"vertical-align: 0.02ex; width: 1.76ex; height: 1.09ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-18.png\" alt=\"\" data-tex=\"\\(\\times\\)\"\u003e, all the propositions of arithmetic will be true of these\nterms. Thus 0, 1, 2, 3,…, become \"variables.\" To make them constants,\nwe must choose some one definite progression; the natural one to choose\nis the progression of finite cardinal numbers as defined by Frege.\nWhat were, in Peano\u0027s methods, primitive terms are thus replaced by\nlogical structures, concerning which it is necessary to prove that they\nsatisfy Peano\u0027s five primitive propositions. This process is essential\nin connecting arithmetic with pure logic. We shall find that a process\nsimilar in some respects, though very different in others, is required\nfor connecting physics with perception.\u003c/p\u003e\n\n\u003cp\u003eThe general process of which the above is an instance will be called\nthe process of \"interpretation.\" It frequently happens that we have a\ndeductive mathematical system,\u003cspan class=\"pagenum\" id=\"Page_5\"\u003e[Pg 5]\u003c/span\u003e starting from hypotheses concerning\nundefined objects, and that we have reason to believe that there are\nobjects fulfilling these hypotheses, although, initially, we are unable\nto point out any such objects with certainty. Usually, in such cases,\nalthough many different sets of objects are abstractly available as\nfulfilling the hypotheses, there is one such set which is much more\nimportant than the others. In the above instance, this set was the\ncardinal numbers. The substitution of such a set for the undefined\nobjects is \"interpretation.\" This process is essential in discovering\nthe philosophical import of physics.\u003c/p\u003e\n\n\u003cp\u003eThe difference between an important and an unimportant interpretation\nmay be made clear by the case of geometry. Any geometry, Euclidean or\nnon-Euclidean, in which every point has co-ordinates which are real\nnumbers, can be interpreted as applying to a system of sets of real\nnumbers—\u003ci\u003ei.e.\u003c/i\u003e a point can be taken to be the series of its\nco-ordinates. This interpretation is legitimate, and is convenient\nwhen we are studying geometry as a branch of pure mathematics. But it\nis not the \u003ci\u003eimportant\u003c/i\u003e interpretation. Geometry is important,\nunlike arithmetic and analysis, because it can be interpreted so as to\nbe part of applied mathematics—in fact, so as to be part of physics.\nIt is this interpretation which is the really interesting one, and\nwe cannot therefore rest content with the interpretation which makes\ngeometry part of the study of real numbers, and so, ultimately, part\nof the study of finite integers. Geometry, as we shall consider it in\nthe present work, will be always treated as part of physics, and will\nbe regarded as dealing with objects which are not either mere variables\nor definable in purely logical terms. We shall not regard a geometry as\nsatisfactorily interpreted until its initial objects have been defined\nin terms of entities forming part of the empirical world, as opposed to\nthe world of logical necessity. It is, of course, possible, and even\nlikely, that various different geometries,\u003cspan class=\"pagenum\" id=\"Page_6\"\u003e[Pg 6]\u003c/span\u003e which would be incompatible\nif applied to the same set of objects, may all be applicable to the\nempirical world by means of different interpretations.\u003c/p\u003e\n\n\u003cp\u003eSo far, we have been considering the logical analysis of physics, which\nwill form the topic of Part I. But in relation to the interpretation\nof geometry we have already been brought into contact with a very\ndifferent problem—namely, that of the application of physics to the\nempirical world. This is, of course, the vital problem: although\nphysics can be pursued as pure mathematics, it is not as pure\nmathematics that physics is important. What is to be said about the\nlogical analysis of physics is therefore only a necessary preliminary\nto our main theme. The laws of physics are believed to be at least\napproximately true, although they are not logically necessary; the\nevidence for them is empirical. All empirical evidence consists, in\nthe last analysis, of perceptions; thus the world of physics must be,\nin some sense, continuous with the world of our perceptions, since it\nis the latter which supplies the evidence for the laws of physics.\nIn the time of Galileo, this fact did not seem to raise any very\ndifficult problems, since the world of physics had not yet become so\nabstract and remote as subsequent research has made it. But already in\nthe philosophy of Descartes the modern problem is implicit, and with\nBerkeley it becomes explicit. The problem arises because the world\nof physics is, \u003ci\u003eprima facie\u003c/i\u003e, so different from the world of\nperception that it is difficult to see how the one can afford evidence\nfor the other; moreover, physics and physiology themselves seem to\ngive grounds for supposing that perception cannot give very accurate\ninformation as to the external world, and thus weaken the props upon\nwhich they are built.\u003c/p\u003e\n\n\u003cp\u003eThis difficulty has led, especially in the works of Dr Whitehead, to\na new interpretation of physics, which is to make the world of matter\nless remote from the world of our experience. The principles which\ninspire Dr Whitehead\u0027s work appear to\u003cspan class=\"pagenum\" id=\"Page_7\"\u003e[Pg 7]\u003c/span\u003e me essential to a right solution\nof the problem, although in the detail I should sometimes incline\nto a somewhat more conservative attitude. We may state the problem\nabstractly as follows:\u003c/p\u003e\n\n\u003cp\u003eThe evidence for the truth of physics is that perceptions occur as the\nlaws of physics would lead us to expect—\u003ci\u003ee.g.\u003c/i\u003e we see an eclipse\nwhen the astronomers say there will be an eclipse. But physics itself\nnever says anything about perceptions; it does not say that we shall\nsee an eclipse, but says something about the sun and moon. The passage\nfrom what physics asserts to the expected perception is left vague and\ncasual; it has none of the mathematical precision belonging to physics\nitself. We must therefore find an interpretation of physics which gives\na due place to perceptions; if not, we have no right to appeal to the\nempirical evidence.\u003c/p\u003e\n\n\u003cp\u003eThis problem has two parts: to assimilate the physical world to the\nworld of perceptions, and to assimilate the world of perceptions\nto the physical world. Physics must be interpreted in a way which\ntends towards idealism, and perception in a way which tends towards\nmaterialism. I believe that matter is less material, and mind less\nmental, than is commonly supposed, and that, when this is realized,\nthe difficulties raised by Berkeley largely disappear. Some of the\ndifficulties raised by Hume, it is true, have not yet been disposed\nof; but they concern scientific method in general, more particularly\ninduction. On these matters I do not propose to say anything in the\npresent volume, which will throughout assume the general validity of\nscientific method properly conducted.\u003c/p\u003e\n\n\u003cp\u003eThe problems which arise in attempting to bridge the gulf between\nphysics (as commonly interpreted) and perception are of two kinds.\nThere is first the epistemological problem: what facts and entities do\nwe know of that are relevant to physics, and may serve as its empirical\nfoundation? This demands a discussion of what, exactly, is to be\nlearnt from a\u003cspan class=\"pagenum\" id=\"Page_8\"\u003e[Pg 8]\u003c/span\u003e perception, and also of the generally assumed physical\ncausation of perceptions—\u003ci\u003ee.g.\u003c/i\u003e by light-waves or sound-waves.\nIn connection with this latter question, it is necessary to consider\nhow far, and in what way, a perception can be supposed to resemble its\nexternal cause, or, at least, to allow inferences as to characteristics\nof that cause. This, in turn, demands a careful consideration of\ncausal laws, which, however, is in any case a necessary part of the\nphilosophical analysis of physics. Throughout this inquiry, we are\nasking ourselves what grounds exist for supposing that physics is\n\"true.\" But the meaning of this question requires some elucidation in\nconnection with what has already been said about interpretation.\u003c/p\u003e\n\n\u003cp\u003eApart altogether from the general philosophical problem of the meaning\nof \"truth,\" there is a certain degree of vagueness about the question\nwhether physics is \"true.\" In the narrowest sense, we may say that\nphysics is \"true\" if we have the perceptions which it leads us to\nexpect. In this sense, a solipsist might say that physics is true; for,\nalthough he would suppose that the sun and moon, for instance, are\nmerely certain series of perceptions of his own, yet these perceptions\ncould be foreseen by assuming the generally received laws of astronomy.\nSo, for example, Leibniz says:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"Although the whole of this life were said to be nothing but a dream,\nand the visible world nothing but a phantasm, I should call this dream\nor phantasm real enough, if, using reason well, we were never deceived\nby it.\"\u003ca id=\"FNanchor_3\" href=\"#Footnote_3\" class=\"fnanchor\"\u003e[3]\u003c/a\u003e\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eA man who, without being a solipsist, believes that whatever is real\nis mental, need have no difficulty in declaring that physics is \"true\"\nin the above sense, and may even go further, and allow the truth of\nphysics in a much wider sense. This wider sense, which I regard as the\nmore important, is as follows: Given physics as a deductive system,\nderived from certain hypotheses as to undefined terms, do there exist\nparticulars,\u003cspan class=\"pagenum\" id=\"Page_9\"\u003e[Pg 9]\u003c/span\u003e or logical structures composed of particulars, which\nsatisfy these hypotheses? If the answer is in the affirmative, then\nphysics is completely \"true.\" We shall find, if I am not mistaken, that\nno conclusive reason can be given for a fully affirmative answer, but\nthat such an answer emerges naturally if we adopt the view that all\nour perceptions are causally related to antecedents which may not be\nperceptions. This is the view of common sense, and has always been,\nat least in practice, the view of physicists. We start, in physics,\nwith a vague mass of common-sense beliefs, which we can subject to\nprogressive refinements without destroying the truth of physics (in\nour present sense of \"truth\"); but if we attempt, like Descartes, to\ndoubt all common-sense beliefs, we shall be unable to demonstrate that\nany absurdity results from the rejection of the above hypothesis as to\nthe causes of perceptions, and we shall therefore be left uncertain\nas to whether physics is fully \"true\" or not. In these circumstances,\nit would seem to be a matter of individual taste whether we adopt or\nreject what may be called the realist hypothesis.\u003c/p\u003e\n\n\u003cp\u003eThe epistemological problem, which we have just been stating in\noutline, will occupy Part II. of the present work. Part III. will\nbe occupied with the outcome for ontology—\u003ci\u003ei.e.\u003c/i\u003e with the\nquestion: What are the ultimate existents in terms of which physics\nis true (assuming that there are such)? And what is their general\nstructure? And what are the relations of space-time, causality, and\nqualitative series respectively? (By \"qualitative series\" I mean such\nas are formed by the colours of the rainbow, or by notes of various\npitches.) We shall find, if I am not mistaken, that the objects which\nare mathematically primitive in physics, such as electrons, protons,\nand points in space-time, are all logically complex structures composed\nof entities which are metaphysically more primitive, which may be\nconveniently called \"events.\" It is a matter for mathematical logic\nto show how to construct, out of these, the objects required by the\n\u003cspan class=\"pagenum\" id=\"Page_10\"\u003e[Pg 10]\u003c/span\u003e\nmathematical physicist. It belongs also to this part of our subject\nto inquire whether there is anything in the known world that is not\npart of this metaphysically primitive material of physics. Here we\nderive great assistance from our earlier epistemological inquiries,\nsince these enable us to see how physics and psychology can be included\nin one science, more concrete than the former and more comprehensive\nthan the latter. Physics, in itself, is exceedingly abstract, and\nreveals only certain mathematical characteristics of the material\nwith which it deals. It does not tell us anything as to the intrinsic\ncharacter of this material. Psychology is preferable in this respect,\nbut is not causally autonomous: if we assume that psychical events are\nsubject, completely, to causal laws, we are compelled to postulate\napparently extra-psychical causes for some of them. But by bringing\nphysics and perception together, we are able to include psychical\nevents in the material of physics, and to give to physics the greater\nconcreteness which results from our more intimate acquaintance with\nthe subject-matter of our own experience. To show that the traditional\nseparation between physics and psychology, mind and matter, is not\nmetaphysically defensible, will be one of the purposes of this work;\nbut the two will be brought together, not by subordinating either to\nthe other, but by displaying each as a logical structure composed of\nwhat, following Dr H. M. Sheffer,\u003ca id=\"FNanchor_4\" href=\"#Footnote_4\" class=\"fnanchor\"\u003e[4]\u003c/a\u003e we shall call \"neutral stuff.\"\nWe shall not contend that there are demonstrative grounds in favour\nof this construction, but only that it is recommended by the usual\nscientific grounds of economy and comprehensiveness of theoretical\nexplanation.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_1\" href=\"#FNanchor_1\" class=\"label\"\u003e[1]\u003c/a\u003e\nOn this subject, cf. \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e,\nchap. \u003cspan class=\"allsmcap\"\u003eXIV\u003c/span\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_2\" href=\"#FNanchor_2\" class=\"label\"\u003e[2]\u003c/a\u003e\nThe definition of powers of a relation, in a form not\ninvolving numbers, is set forth in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, *91.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_3\" href=\"#FNanchor_3\" class=\"label\"\u003e[3]\u003c/a\u003e\nPhilosophische Werke, Gerhardt\u0027s edition, vol.\n\u003cspan class=\"allsmcap\"\u003eVII.\u003c/span\u003e, p. 320.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_4\" href=\"#FNanchor_4\" class=\"label\"\u003e[4]\u003c/a\u003e\nSee Preface to Holt\u0027s \u003ci\u003eConcept of Consciousness\u003c/i\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_11\"\u003e[Pg 11]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"PART_I\"\u003ePART I\u003cbr\u003e\nTHE LOGICAL ANALYSIS OF PHYSICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_13\"\u003e[Pg 13]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_II\"\u003eCHAPTER II\u003cbr\u003e\nPRE-RELATIVITY PHYSICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE physics of Newton, considered as a deductive system, had a\nperfection which is absent from the physics of the present day. Science\nhas two purposes, each of which tends to conflict with the other. On\nthe one hand, there is a desire to know as much as possible of the\nfacts in the region concerned; on the other hand, there is the attempt\nto embrace all the known facts in the smallest possible number of\ngeneral laws. The law of gravitation accounted for all the facts about\nthe motions of the planets and their satellites which were known in\nNewton\u0027s day; at the time, it exhibited the ideal of science. But facts\nand theories seem destined to conflict sooner or later. When this\nhappens, there is a tendency either to deny the facts or to despair\nof theory. Thanks to Einstein, the minute facts which have been found\nincompatible with the natural philosophy of Newton have been fitted\ninto a new natural philosophy; but there is not yet the complete\ntheoretical harmony that existed while Newton was undisputed.\u003c/p\u003e\n\n\u003cp\u003eIt is necessary to say something about the Newtonian system, since\neverything subsequent has arisen as an amendment to it, not as a\nfresh start. Most of the fundamental concepts of this system are due\nto Galileo, but the complete structure appears first in Newton\u0027s\n\u003ci\u003ePrincipia\u003c/i\u003e. The theory is simple and mathematical; indeed, one\nof its main differences from modern theories is its belief (perhaps\ntraceable to Greek geometry) that Nature is convenient for the\nmathematician, and requires little manipulation before his concepts\nbecome applicable.\u003c/p\u003e\n\n\u003cp\u003eThe Newtonian system, stated with schematic simplicity, as, \u003ci\u003ee.g.\u003c/i\u003e\nby Boscovitch, is as follows. There is an absolute\u003cspan class=\"pagenum\" id=\"Page_14\"\u003e[Pg 14]\u003c/span\u003e space, composed\nof points, and an absolute time, composed of instants; there are\nparticles of matter, each of which persists through all time and\noccupies a point at each instant. Each particle exerts forces on other\nparticles, the effect of which is to produce accelerations. Each\nparticle is associated with a certain quantity, its \"mass,\" which is\ninversely proportional to the acceleration produced in the particle\nby a given force. The laws of physics are conceived, on the analogy\nof the law of gravitation, as formulæ giving the force exerted by one\nparticle on another in a given relative situation. This system is\nlogically faultless. It was criticized on the ground that absolute\nspace and time were meaningless, and on the ground that action at a\ndistance was inconceivable. This latter objection was sanctioned by\nNewton, who was not a strict Newtonian. But in fact neither objection\nhad any force from a logical point of view. Kant\u0027s antinomies, and\nthe supposed difficulties of infinity and continuity, were finally\ndisposed of by Georg Cantor. There was no valid \u003ci\u003ea priori\u003c/i\u003e reason\nfor supposing that Nature was not such as the Newtonians averred, and\ntheir scientific successes afforded empirical, or at least pragmatic,\narguments in their favour. It is no wonder, therefore, that, throughout\nthe eighteenth century, the system of ideas which had led to the law of\ngravitation dominated all scientific thought.\u003c/p\u003e\n\n\u003cp\u003eBefore physics itself had made any breaches in this edifice, there\nwere, however, certain objections of an epistemological order. It will\nbe worth while to consider these, since it is urged that the theory of\nrelativity is not open to them, though I believe this claim to be only\npartially justified.\u003c/p\u003e\n\n\u003cp\u003eThe most formidable and persistent attack was upon absolute space and\ntime. This attack was initiated by Leibniz in the lifetime of Newton,\nespecially in his controversy with Clarke, who represented Newton.\nIn time, most physicists came to disbelieve in absolute space and\ntime, while retaining the Newtonian\u003cspan class=\"pagenum\" id=\"Page_15\"\u003e[Pg 15]\u003c/span\u003e technique, which assumed their\nexistence. In Clerk Maxwell\u0027s \u003ci\u003eMatter and Motion\u003c/i\u003e, absolute\nmotion is asserted in one passage and denied in another, with hardly\nany attempt to reconcile these two opinions. But at the end of the\nnineteenth century the prevalent view was certainly that of Mach, who\nvigorously denied absolute space and time. Although this denial has\nnow been proved to be right, I cannot think that before Einstein and\nMinkowski it had any conclusive arguments in its favour. In spite of\nthe fact that the whole question is now ancient history, it may be\ninstructive to consider the arguments briefly.\u003c/p\u003e\n\n\u003cp\u003eThe important reasons for rejecting absolute space and time were two.\nFirst, that everything we can observe has to do only with the relative\npositions of bodies and events; secondly, that points and instants\nare an unnecessary hypothesis, and are therefore to be rejected in\naccordance with the principle of economy, which is the same thing as\nOccam\u0027s razor. It appears to me that the first of these arguments has\nno force, while the second was false until the advent of the theory of\nrelativity. My reasons are as follows:\u003c/p\u003e\n\n\u003cp\u003eThat we can only observe relative positions is, of course, true; but\nscience assumes many things that cannot be observed, for the sake of\nsimplicity and continuity in causal laws. Leibniz assumed that there\nare infinitesimals, although everything that we can observe exceeds\na certain minimum size. We all think that the earth has an inside,\nand the moon a side which we cannot see. But, it will be said, these\nthings are like what we observe, and circumstances can be imagined\nunder which we should observe them, whereas absolute space and time\nare different in kind from anything directly known, and could not be\ndirectly known in any conceivable conditions. Unfortunately, however,\nthis applies equally to physical bodies. The relative positions which\nwe see are relative positions of parts of the visual field; but the\nthings in the visual field are\u003cspan class=\"pagenum\" id=\"Page_16\"\u003e[Pg 16]\u003c/span\u003e not bodies as conceived in traditional\nphysics, which is dominated by the Cartesian dualism of mind and\nmatter, and places the visual field in the former. This argument is\nnot valid as against Mach, who argued that our sensations are actually\npart of the physical world, and thus inaugurated the movement towards\nneutral monism, which denies the ultimate validity of the mind-matter\ndualism. But it is valid as against all those for whom matter is a\nsort of \u003ci\u003eDing-an-sich\u003c/i\u003e, essentially different from anything that\nenters into our experience. For them, it should be as illegitimate to\ninfer matter from our perceptions as to infer absolute space and time.\nThe one, like the other, is part of our naive beliefs, as is shown\nby the Copernican controversy, which would have been impossible for\nmen who rejected absolute space and time. And the remoteness from our\nperceptions is as much a discovery due to reflection in the one case as\nin the other.\u003c/p\u003e\n\n\u003cp\u003eIt is impossible to lay down a hard-and-fast rule that we can\nnever validly infer something radically different from what we\nobserve—unless, indeed, we take up the position that nothing\nunobserved can ever be validly inferred. This view, which is advocated\nby Wittgenstein in his \u003ci\u003eTractatus Logico-Philosophicus\u003c/i\u003e, has much\nin its favour, from the standpoint of a strict logic; but it puts an\nend to physics, and therefore to the problem with which this work\nis concerned. I shall accordingly assume that scientific inference,\nconducted with due care, may be valid, provided it is recognized as\ngiving only probability, not certainty. Given this assumption, I see\nno possible ground for rejecting an inference to absolute space and\ntime, if the facts seem to call for it. It may be admitted that it\nis better, if possible, to avoid inferring anything very different\nfrom what we know to exist. Such a principle will have to be based on\ngrounds of probability. It may be said that all inferences to something\nunobserved are only probable, and that their probability depends, in\npart, upon the \u003ci\u003ea priori\u003c/i\u003e probability\u003cspan class=\"pagenum\" id=\"Page_17\"\u003e[Pg 17]\u003c/span\u003e of the hypothesis; this may\nbe supposed greater when we infer something similar to what we know\nthan when we infer something dissimilar. But it seems questionable\nwhether there is much force in this argument. Everything that we\nperceive directly is subject to certain conditions, more especially\nphysiological conditions; it would seem \u003ci\u003ea priori\u003c/i\u003e probable that\nwhere these conditions are absent things would be different from\nanything that we can experience. If we suppose as we well may—that\nwhat we experience has certain characteristics connected with our\nexperiencing, there can be no \u003ci\u003ea priori\u003c/i\u003e objection to the\nhypothesis that some of the things we do not experience are lacking\nin some characteristics which are universal in our experience. The\ninference to absolute space and time must, therefore, be treated as on\na level with any other inductive inference.\u003c/p\u003e\n\n\u003cp\u003eThe second argument against absolute space and time—namely, that they\nare unnecessary hypotheses—has turned out to be valid; but it is only\nin quite recent times that Newton\u0027s argument to the contrary has been\nrefuted. The argument, as everyone knows, was concerned with absolute\nrotation. It is urged that, for \"absolute rotation,\" we may substitute\n\"rotation relatively to the fixed stars.\" This is formally correct,\nbut the influence attributed to the fixed stars savours of astrology,\nand is scientifically incredible. Apart from this special argument,\nthe whole of the Newtonian technique is based upon the assumption\nthat there is such a quantity as absolute acceleration; without this,\nthe system collapses. That is one reason why the law of gravitation\ncannot enter unchanged into the general theory of relativity. There\nare, of course, two distinct elements in the theory of relativity: one\nof them the merging of space and time into space-time—is wholly new,\nwhile the other—the substitution of relative for absolute motion—has\nbeen attempted ever since the time of Leibniz. But this older problem\ncould not be solved by\u003cspan class=\"pagenum\" id=\"Page_18\"\u003e[Pg 18]\u003c/span\u003e itself, because of the necessity for absolute\nacceleration in Newtonian dynamics. Only the method of tensors, and the\nnew law of gravitation obtained in accordance with this method, have\nmade it possible to answer Newton\u0027s arguments for absolute space and\ntime. While, therefore, the contention that these are unnecessary would\nalways have been a valid ground for rejecting them if it had been known\nto be true, it is only now that we can be confident of its correctness,\nsince it is only now that we possess a mathematical technique which is\nin accordance with it.\u003c/p\u003e\n\n\u003cp\u003eSomewhat similar considerations apply to action at a distance,\nwhich was also considered incredible by Newton\u0027s critics, from\nLeibniz onwards, and even by Newton himself. There is one theory,\nwhich may well be true, according to which action at a distance is\nself-contradictory: this is the theory which derives spatio-temporal\nseparation from causal separation. I shall say no more about this\npossibility at present, since it was not suggested by any of the\nopponents of action at a distance, all of whom considered spatial and\ntemporal relations totally distinct from causal relations. From their\npoint of view, therefore, the objection to action at a distance seems\nto have been little more than a prejudice. The source of the prejudice\nwas, I think, twofold: first, that the notion of \"force,\" which was the\ndynamical form of \"cause,\" was derived from the sensations of pushing\nand pulling; secondly, that people falsely supposed themselves in\ncontact with things when they pushed and pulled them, or were pushed\nand pulled by them. I do not mean that such crude notions would have\nbeen explicitly defended, but that they dominated the imaginative\npicture of the physical world, and made Newtonian dynamics seem what\nis absurdly called \"intelligible.\" Apart from such mistakes, it should\nhave been regarded as a purely empirical question whether there is\naction at a distance or not. It was in fact so regarded throughout the\nlatter half or three-quarters\u003cspan class=\"pagenum\" id=\"Page_19\"\u003e[Pg 19]\u003c/span\u003e of the eighteenth century, and it was\ngenerally held that the empirical arguments in favour of action at a\ndistance were overwhelming.\u003c/p\u003e\n\n\u003cp\u003eNot wholly unconnected with the question of action at a distance was\nthe question of the rôle of \"force\" in dynamics. In Newton, \"force\"\nplays a great part, and there seems no doubt that he regarded it as a\n\u003ci\u003evera causa\u003c/i\u003e. If there was action at a distance, the use of the\nwords \"central forces\" seemed to make it somehow more \"intelligible.\"\nBut gradually it was increasingly realized that \"force\" is merely\na connecting link between configurations and accelerations; that,\nin fact, causal laws of the sort leading to differential equations\nare what we need, and that \"force\" is by no means necessary for the\nenunciation of such laws. Kirchoff and Mach developed a mechanics\nwhich dispensed with \"force,\" and Hertz perfected their views in a\ntreatise\u003ca id=\"FNanchor_5\" href=\"#Footnote_5\" class=\"fnanchor\"\u003e[5]\u003c/a\u003e comparable to Euclid from the point of view of logical\nbeauty, leading to the result that there is only one law of motion, to\nthe effect that, in a certain defined sense, every particle describes a\ngeodesic. Although the whole of this development involved no essential\ndeparture from Newton, it paved the way for relativity dynamics, and\nprovided much of the necessary mathematical apparatus, particularly in\nthe use of the principle of least action.\u003c/p\u003e\n\n\u003cp\u003eThe first physical theory to be developed on lines definitely\ndifferent from those of Newtonian astronomy was the undulatory theory\nof light. Not that there was anything to contradict Newton, but that\nthe framework of ideas was different. Transmission through a medium\nhad been made fashionable by Descartes, and unfashionable by the\nNewtonians; in the case of the transmission of light it was found\nnecessary to revert to the older point of view. Moreover, the æther\nwas never so comfortably material as \"gross\" matter. It could vibrate,\nbut it did not seem to consist of little bits\u003cspan class=\"pagenum\" id=\"Page_20\"\u003e[Pg 20]\u003c/span\u003e each with its own\nindividuality, or to be subject to any discoverable molar motions. No\none knew whether it was a jelly or a gas. Its properties could not be\ninferred from those of billiard balls, but were merely those demanded\nby its functions. In fact, like a painfully good boy, it only did what\nit was told, and might therefore be expected to die young.\u003c/p\u003e\n\n\u003cp\u003eA more serious change was introduced by Faraday and Maxwell. Light\nhad never been treated on the analogy of gravitation, but electricity\nappeared to consist of central forces varying inversely as the square\nof the distance, and was therefore confidently fitted into the\nNewtonian scheme. Faraday experimentally and Maxwell theoretically\ndisplayed the inadequacy of this view; Maxwell, moreover, demonstrated\nthe identity of light and electromagnetism. The æther required for\nthe two kinds of phenomena was therefore the same, which gave it a\nmuch better claim to be supposed to exist. Maxwell\u0027s proof, it is\ntrue, was not conclusive, but it was made so by Hertz when he produced\nelectromagnetic waves artificially and studied their properties\nexperimentally. It thus became clear that Maxwell\u0027s equations, which\ncontained practically the whole of his system, must take their place\nbeside the law of gravitation as affording the mathematical formula for\na vast range of phenomena. The concepts required for these equations\nwere, at first, not definitely contradictory to the Newtonian dynamics;\nbut by the help of subsequent experimental results contradictions\nemerged which were only removed by the theory of relativity. Of this,\nhowever, we shall speak in a later chapter.\u003c/p\u003e\n\n\u003cp\u003eAnother breach in the orthodox system, of which the importance has\nonly become fully manifest since the publication of the general theory\nof relativity, was the invention of non-Euclidean geometry. In the\nwork of Lobatchevsky and Bolyai, although the philosophical challenge\nto Euclid was already complete, and the consequent argument against\nKant\u0027s\u003cspan class=\"pagenum\" id=\"Page_21\"\u003e[Pg 21]\u003c/span\u003e transcendental æsthetic very powerful, there were not yet, at\nleast obviously, the far-reaching physical implications of Riemann\u0027s\ninaugural dissertation \"Ueber die Hypothesen, welche der Geometric\nzu Grunde liegen.\" A few words on this topic are unavoidable at this\nstage, although the full discussion will come later.\u003c/p\u003e\n\n\u003cp\u003eOne broad result of non-Euclidean geometry, even in its earliest\nform, was that the geometry of actual space is, at least in part, an\nempirical study, not a branch of pure mathematics. It may be said that\nempiricists, such as J. S. Mill, always based geometry upon empirical\nobservation. But they did the same with arithmetic, in which they\nwere certainly mistaken. No one before the non-Euclideans perceived\nthat arithmetic and geometry stand on a quite different footing, the\nformer being continuous with pure logic and independent of experience,\nthe latter being continuous with physics and dependent upon physical\ndata. Geometry can, it is true, be still studied as a branch of pure\nmathematics, but it is then hypothetical, and cannot claim that its\ninitial hypotheses (which replace the axioms) are true in fact, since\nthis is a question outside the scope of pure mathematics. The geometry\nwhich is required by the engineer or the astronomer is not a branch\nof pure mathematics, but a branch of physics. Indeed, in the hands of\nEinstein geometry has become identical with the whole of the general\npart of theoretical physics: the two are united in the general theory\nof relativity.\u003c/p\u003e\n\n\u003cp\u003eRiemann, who was logically the immediate predecessor of Einstein,\nbrought in a new idea of which the importance was not perceived for\nhalf a century. He considered that geometry ought to start from the\ninfinitesimal, and depend upon integration for statements about finite\nlengths, areas, or volumes. This requires, \u003ci\u003einter alia\u003c/i\u003e, the\nreplacement of the straight line by the geodesic: the latter has a\ndefinition depending upon infinitesimal distances, while the former has\nnot. The traditional\u003cspan class=\"pagenum\" id=\"Page_22\"\u003e[Pg 22]\u003c/span\u003e view was that, while the length of a curve could,\nin general, only be defined by integration, the length of the straight\nline between two points could be defined as a whole, not as the limit\nof a sum of little bits. Riemann\u0027s view was that a straight line does\nnot differ from a curve in this respect. Moreover, measurement, being\nperformed by means of bodies, is a physical operation, and its results\ndepend for their interpretation upon the laws of physics. This point of\nview has turned out to be of very great importance. Its scope has been\nextended by the theory of relativity, but in essence it is to be found\nin Riemann\u0027s dissertation.\u003c/p\u003e\n\n\u003cp\u003eRiemann\u0027s work, as well as that of Faraday and Maxwell, belongs,\nlike the theory of relativity, to the development of the view of the\nphysical world as a continuous medium, which has, from the earliest\ntimes, contested the mastery with the atomic view. Just as Newton\ncaused absolute space and time to be embedded in the technique of\ndynamics, so Pythagoras caused spatial atomism to be embedded in the\ntechnique of geometry. Ever since Greek times, those who did not\nbelieve in the reality of \"points\" were faced with the difficulty\nthat a geometry based on points works, while no other way of starting\ngeometry was known. This difficulty, as Dr Whitehead has shown, exists\nno longer. It is now possible, as we shall see at a later stage, to\ninterpret geometry and physics with material all of which is of a\nfinite size—it is even possible to demand that none of the material\nshall be smaller than an assigned finite size. The fact that this\nhypothesis can be reconciled with mathematical continuity is a novel\ndiscovery of considerable importance; until recently, atomism and\ncontinuity appeared incompatible. There are, however, forms of atomism\nwhich have not hitherto been found easy to reconcile with continuity;\nand, as it happens, there is powerful experimental evidence in their\nfavour. Just at the moment when Maxwell, supplemented by Hertz,\nappeared to have reduced everything to continuity, the new evidence\nfor an atomic view of Nature began to accumulate. There is still an\nunreconciled conflict, one set of facts pointing in one direction, and\nanother in another; but it is legitimate to hope that the conflict will\nbe resolved before long modern atomism, however, demands a new chapter.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_5\" href=\"#FNanchor_5\" class=\"label\"\u003e[5]\u003c/a\u003e\n\u003ci\u003ePrinzipien der Mechanik.\u003c/i\u003e\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_24\"\u003e[Pg 24]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_III\"\u003eCHAPTER III\u003cbr\u003e\nELECTRONS AND PROTONS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nPHYSICS, at the present time, is divisible into two parts, the one\ndealing with the propagation of energy in matter or in regions\nwhere there is no matter, the other with the interchanges of energy\nbetween these regions and matter. The former is found to require\ncontinuity, the latter discontinuity. But before considering this\napparent conflict, it will be advisable to deal in outline with the\ndiscontinuous characteristics of matter and energy as they appear in\nthe theory of quanta and in the structure of atoms. It is necessary,\nhowever, for philosophical purposes, to deal only with the most general\naspects of modern theories, since the subject is developing rapidly,\nand any statement runs a risk of being out of date before it can be\nprinted. The topics considered in this chapter and the next have been\ntreated in an entirely new way by the theory initiated by Heisenberg\nin 1925. I shall, however, postpone the consideration of this theory\nuntil after that of the Rutherford-Bohr atom and the theory of quanta\nconnected with it.\u003c/p\u003e\n\n\u003cp\u003eIt appears that both matter and electricity are concentrated\nexclusively in certain finite units, called electrons and protons. It\nis possible that the helium nucleus may be a third independent unit,\nbut this seems improbable.\u003ca id=\"FNanchor_6\" href=\"#Footnote_6\" class=\"fnanchor\"\u003e[6]\u003c/a\u003e The net positive charge of a helium\nnucleus is double-that of a proton, and its mass is slightly less than\nfour times that of a proton. These facts are explicable (including the\nslight deficiency of mass) if the\u003cspan class=\"pagenum\" id=\"Page_25\"\u003e[Pg 25]\u003c/span\u003e helium nucleus consists of four\nprotons and two electrons; otherwise, they seem an almost incredible\ncoincidence. We may therefore assume that electrons and protons are\nthe sole constituents of matter; if it should turn out that the helium\nnucleus must be added, that would make little difference to the\nphilosophical analysis of matter, which is our task in this volume.\u003c/p\u003e\n\n\u003cp\u003eProtons all have the same mass and the same amount of positive\nelectricity. Electrons all have the same mass, about of the mass of a\nproton. The amount of negative electricity on an electron is always\nthe same, and is such as to balance exactly the amount on a proton, so\nthat one electron and one proton together constitute an electrically\nneutral system. An atom consists, when unelectrified, of a nucleus\nsurrounded by planetary electrons: the number of these electrons is\nthe atomic number of the element concerned. The nucleus consists of\nprotons and electrons: the number of the former is the atomic weight\nof the element, the number of the latter is such as to make the whole\nelectrically neutral—\u003ci\u003ei.e.\u003c/i\u003e it is the difference between the\nnumber of protons in the nucleus and the number of planetary electrons.\nEvery item in this complicated structure is supposed, at normal\ntimes, to be engaged in motions which result, on Newtonian principles\n(modified slightly by relativity considerations), from the attractions\nbetween electrons and protons and the repulsions between protons and\nprotons as well as between electrons and electrons. But of all the\nmotions which should be possible on the analogy of the solar system,\nit is held that only an infinitesimal proportion are in fact possible;\nthis depends upon the theory of quanta, in ways which we shall consider\nlater.\u003c/p\u003e\n\n\u003cp\u003eThe calculation of the orbits of planetary electrons, on Newtonian\nprinciples, is only possible in the two simplest cases: that of\nhydrogen, which consists (when unelectrified) of one proton and one\nelectron; and that of positively electrified\u003cspan class=\"pagenum\" id=\"Page_26\"\u003e[Pg 26]\u003c/span\u003e helium, which has lost\none, but not both, of its planetary electrons. In these two cases the\nmathematical theory is practically complete. In all other cases which\nactually occur, although the mathematics required is of a sort which\nhas been investigated ever since the time of Newton, it is impossible\nto obtain exact solutions, or even good approximations. The case is\nstill worse as regards nuclei. The nucleus of hydrogen is a single\nproton, but that of the next element, helium, is held to consist of\nfour protons and two electrons. The combination must be extraordinarily\nstable, both because no known process disintegrates the helium nucleus,\nand because of the loss of mass involved. (If the mass of the helium\natom is taken as 4, that of a hydrogen atom is not 1, but 1·008.) This\nlatter argument depends upon considerations connected with relativity,\nand must therefore be discussed at a later stage. Various suggestions\nhave been made as to the way in which the protons and electrons are\narranged in the helium nucleus, but none, so far, has yielded the\nnecessary stability. What we may call the geometry of nuclei is\ntherefore still unknown. It may be that, at the very small distances\ninvolved, the law of force is not the inverse square, although this\nlaw is found perfectly satisfactory in dealing with the motions of\nthe planetary electron in the two cases in which the mathematics is\nfeasible. This, however, is merely a speculation; for the present we\nmust be content with ignorance as regards the arrangement of protons\nand electrons in nuclei other than that of hydrogen (which contains no\nelectron in the nucleus).\u003c/p\u003e\n\n\u003cp\u003eSo long as an atom remains in a state of steady motion, it gives no\nevidence of its existence to the outside world. A material system\ndisplays its existence to outsiders by radiating or absorbing energy,\nand in no other way; and an atom does not absorb or part with energy\nexcept when it undergoes sudden revolutionary changes of the sort\nconsidered by the theory of quanta. This is of importance from our\npoint of view, since\u003cspan class=\"pagenum\" id=\"Page_27\"\u003e[Pg 27]\u003c/span\u003e it shows that no empirical evidence can decide\nbetween two theories of the atom which yield the same result as regards\nthe interchanges of energy between the atom and the surrounding medium.\nIt may be that the whole Rutherford-Bohr theory is too concrete and\npictorial; the analogy with the solar system may be much less close\nthan it is represented as being. A theory which accounts for all the\nknown facts is not thereby shown to be true, this would require\na proof that no other theory would do the same. Such a proof is\nvery seldom possible; certainly it is not possible in the case of\nthe structure of the atom. What may be taken as firm ground is the\nnumerical part of the theory. Certain quantities, and certain whole\nnumbers, are clearly involved; but it would be rash to say that such\nand such an interpretation of these quantities and whole numbers is the\nonly one possible. It is proper and right to use a pictorial theory as\na help in investigation; but what can count as definite knowledge is\nsomething much more abstract. And it is quite possible that the truth\ndoes not lend itself to pictorial statement, but only to expression in\nmathematical formulæ. This, as we shall see, is the view taken by what\nwe may call the Heisenberg theory.\u003c/p\u003e\n\n\u003cp\u003eIt may be worth while to linger a moment over this question of the\nnature of our read knowledge concerning atoms. In the last analysis,\nall our knowledge of matter is derived from perceptions, which are\nthemselves causally dependent upon effects on our body. In sight, for\nexample, we depend upon light-waves which impinge upon the eye. Given\nthe waves, we shall have the visual perception, assuming no defect in\nthe eye. Therefore nothing in visual perception alone can enable us to\ndistinguish between two theories which give the same result as regards\nthe light-waves which reach human eyes. This, as stated, seems to\nintroduce psychological considerations. But we may put the matter in\na way that makes its physical significance clearer. Consider an oval\nsurface, which is liable\u003cspan class=\"pagenum\" id=\"Page_28\"\u003e[Pg 28]\u003c/span\u003e to continuous motion and change of shape,\nbut persists throughout time; and let us suppose that no human being\nhas ever been inside this surface. In illustration, we might take a\nsphere surrounding the sun, or a little box surrounding an electron\nwhich never forms part of a human body. Energy will cross this surface,\nsometimes inward, sometimes outward. Two views which lead to the same\nresults as to the flow of energy across the boundary are empirically\nindistinguishable, since everything that we know independently of\nphysical theory lies outside the surface. We may enlarge our oval\nsurface until its \"inside\" consists of everything outside the body\nof the physicist concerned—to wit, ourselves. What we hear, and\nwhat we read in books, comes to us entirely through a flow of energy\nacross the boundary of our body. It may well be maintained that our\ndirect knowledge is less than this statement would imply, but it is\ncertainly not greater. Two universes which give the same results for\nthe flow of energy across the boundary of A\u0027s body will be totally\nindistinguishable for A.\u003c/p\u003e\n\n\u003cp\u003eMy object in bringing up these considerations is partly to give a new\nturn to the argument about solipsism. As a rule, solipsism is taken\nas a form of idealism—namely, the view that nothing exists except my\nmind and my mental events. I think, however, that it would be just as\nrational, or just as irrational, to say that nothing exists outside\nmy body, or that nothing exists outside a certain closed surface\nwhich includes my body. Neither of these is the general form of the\nargument. The general form is that first given above—namely, that,\ngiven any region not containing myself, two physical theories which\ngive the same boundary conditions all over this region are empirically\nindistinguishable. Electrons and protons, in particular, are only known\nby their effects elsewhere, and so long as these effects are unchanged\nwe may alter our views of electrons and protons as much as we please\nwithout making a difference in anything verifiable. The question of\nthe validity of the inference to things outside ourselves is logically\nquite distinct from the question whether the stuff of the world is\nmental, material, or neutral. I might be a solipsist, and hold at the\nsame time that I am my body; I might, conversely, allow inferences to\nthings other than myself, but maintain that these things were minds or\nmental events. In physics, the question is not that of solipsism, but\nthe much more definite question: Given the physical conditions at the\nbounding surface of some volume, without any direct knowledge of the\ninterior, how much can we legitimately infer as to what happens in the\ninterior? Is there good ground for supposing that we can infer as much\nas physicists usually assume? Or can we perhaps infer much less than\nis generally supposed? I do not propose as yet to attempt an answer to\nthis question; I have raised it at this stage in order to suggest a\ndoubt as to the completeness of our knowledge concerning the structure\nof the atom.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_6\" href=\"#FNanchor_6\" class=\"label\"\u003e[6]\u003c/a\u003e\nProfessors F. Paneth and K. Peters claim to have\ntransformed hydrogen into helium. If this claim is substantiated, it\ndisposes definitively of the possibility that the helium nucleus is an\nindependent unit. See \u003ci\u003eNature\u003c/i\u003e, October 9, 1926, p. 526.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_30\"\u003e[Pg 30]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_IV\"\u003eCHAPTER IV\u003cbr\u003e\nTHE THEORY OF QUANTA\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE atomicity of matter is a hypothesis as old as the Greeks, and\nin no way repugnant to our mental habits. The theory that matter is\ncomposed of electrons and protons is beautiful through its successful\nsimplicity, but is not difficult to imagine or believe. It is otherwise\nwith the form of atomicity introduced by the theory of quanta. This\nmight possibly not have surprised Pythagoras, but it would most\ncertainly have astonished every later man of science, as it has\nastonished those of our own day. It is necessary to understand the\ngeneral principles of the theory before attempting a modern philosophy\nof matter; but unfortunately there are still unsolved physical problems\nconnected with it, which make it improbable that a satisfactory\nphilosophy of the subject can yet be constructed. Nevertheless, we must\ndo what we can.\u003c/p\u003e\n\n\u003cp\u003eAs everyone knows, the quantum was first introduced by Planck in\n1900 in his study of black-body radiation. Planck showed that, when\nwe consider the vibrations which constitute the heat in a body,\nthese are not distributed among all possible values according to\nthe usual law of frequency which governs chance distributions, but\non the contrary are tied down by a certain law. If \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-19.png\" alt=\"\" data-tex=\"\\(\\epsilon\\)\"\u003e is\nthe energy of a vibration, and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e its frequency, then there is\na certain constant \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e,\u003ca id=\"FNanchor_7\" href=\"#Footnote_7\" class=\"fnanchor\"\u003e[7]\u003c/a\u003e known as Planck\u0027s constant, such that\n\u003cimg style=\"vertical-align: -0.781ex; width: 1.843ex; height: 2.361ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-22.png\" alt=\"\" data-tex=\"\\(\\frac{\\epsilon}{\\nu}\\)\"\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 2.434ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-23.png\" alt=\"\" data-tex=\"\\(2h\\)\"\u003e, or \u003cimg style=\"vertical-align: -0.05ex; width: 2.434ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-24.png\" alt=\"\" data-tex=\"\\(3h\\)\"\u003e, or some other\nsmall integral multiple of h. Vibrations with other amounts of energy\ndo not occur. No reason is known for their non-occurrence, which\nremains so far of the nature of a brute\u003cspan class=\"pagenum\" id=\"Page_31\"\u003e[Pg 31]\u003c/span\u003e fact. At first, it was an\nisolated fact. But now Planck\u0027s constant has been found to be involved\nin various other kinds of phenomena; in fact, wherever observation\nis sufficiently minute to make it possible to discover whether it is\ninvolved or not.\u003c/p\u003e\n\n\u003cp\u003eA second field for the quantum theory was found in the photo-electric\neffect. This effect is described as follows by Jeans:\u003ca id=\"FNanchor_8\" href=\"#Footnote_8\" class=\"fnanchor\"\u003e[8]\u003c/a\u003e\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The general features of the phenomenon are well known. For some time\nit has been known that the incidence of high-frequency light on to\nthe surface of a negatively charged conductor tended to precipitate a\ndischarge, while Hertz showed that the incidence of the light on an\nuncharged conductor resulted in its acquiring a positive charge. These\nphenomena have been shown quite conclusively to depend on the emission\nof electrons from the surface of the metal, the electrons being set\nfree in some way by the incidence of the light.\u003c/p\u003e\n\n\u003cp\u003e\"In any particular experiment, the velocities with which individual\nelectrons leave the metal have all values from zero up to a certain\nmaximum velocity \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e, which depends on the conditions of the\nparticular experiment. No electron is found to leave the metal with a\nvelocity greater than this maximum \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e. It seems probable that in any\none experiment all the electrons are initially shot off with the same\nvelocity \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e, but that those which come from a small distance below\nthe surface lose part of their velocity in fighting their way out to\nthe surface.\u003c/p\u003e\n\n\u003cp\u003e\"Leaving out of account such disturbing influences as films of\nimpurities on the metallic surface, it appears to be a general law that\nthe maximum velocity \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e depends only on the nature of the metal and\non the \u003ci\u003efrequency\u003c/i\u003e of the incident light. It does not depend on\nthe intensity of the light, and within the range of temperature within\nwhich experiments are possible it does not depend on the temperature\nof the metal…. For a given metal this maximum velocity increases\nregularly as the frequency of the light is increased, but there is a\ncertain frequency below which no emission takes place at all.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_32\"\u003e[Pg 32]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe explanation of this phenomenon in terms of the quantum was first\ngiven by Einstein\u003ca id=\"FNanchor_9\" href=\"#Footnote_9\" class=\"fnanchor\"\u003e[9]\u003c/a\u003e in 1905. When light of frequency \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e falls on\nthe conductor, it is found that the amount of energy absorbed by an\nelectron which the light separates from its atom is about five-sixths\nof \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e, where \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e is Planck\u0027s constant. It may be supposed that\nthe other one-sixth is absorbed by the atom, so that atom and electron\ntogether absorb exactly one quantum \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. When the light is of such\nlow frequency that \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e is not enough to liberate an electron, the\nphoto-electric effect does not take place. Explanations not involving\nthe quantum have been attempted, but none seem able to account for the\ndata.\u003c/p\u003e\n\n\u003cp\u003eAnother field in which the quantum hypothesis has been found necessary\nis the specific heat of solids at low temperatures. According to\nprevious theories, the specific heat (at constant volume) multiplied\nby the atomic weight ought to have the constant value 5·95. In fact,\nthis is found to be very approximately correct for high temperatures,\nbut for low temperatures there is a falling off which increases as the\ntemperature falls. The explanation of this fact offered by Debye is\nclosely analogous to Planck\u0027s explanation of the facts of black-body\nradiation; and as in that case, it seems definitely impossible to\nobtain a satisfactory theory without invoking the quantum.\u003ca id=\"FNanchor_10\" href=\"#Footnote_10\" class=\"fnanchor\"\u003e[10]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe most interesting application of quantum theory is Bohr\u0027s\nexplanation of the line spectra of elements. It had been found\nempirically that the lines in the hydrogen spectrum which were known\nhad frequencies obtained from the difference of two \"terms,\" according\nto the formula:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.148ex; width: 27.331ex; height: 5.428ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-27.png\" alt=\"\" data-tex=\"\\[\n\\nu = R\\left(\\frac{1}{n^{2}} – \\frac{1}{k^{2}} \\right),\\qquad \\text{(1)}\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e is the frequency, \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e is \"Rydberg\u0027s constant,\"\n\u003cspan class=\"pagenum\" id=\"Page_33\"\u003e[Pg 33]\u003c/span\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-28.png\" alt=\"\" data-tex=\"\\(k\\)\"\u003e are small integers, \u003cimg style=\"vertical-align: -1.654ex; width: 3.341ex; height: 4.728ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-29.png\" alt=\"\" data-tex=\"\\(\\dfrac{R}{n^{2}}\\)\"\u003e and\n\u003cimg style=\"vertical-align: -1.654ex; width: 3.162ex; height: 4.728ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-30.png\" alt=\"\" data-tex=\"\\(\\dfrac{R}{k^{2}}\\)\"\u003e are what are called \"terms.\" After the formula\nhad been discovered, new lines agreeing with it were sought and found.\nCertain lines formerly attributed to hydrogen, and not agreeing with\nthe above formula, were attributed by Bohr to ionized helium; they are\ngiven by the formula:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.148ex; width: 40.257ex; height: 5.428ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-31.png\" alt=\"\" data-tex=\"\\[\n\\nu = 4R\\left(\\frac{1}{3^{2}} – \\frac{1}{k^{2}}\\right)\\\\\n\\nu = 4R\\left(\\frac{1}{4^{3}} – \\frac{1}{k^{2}}\\right).\n\\]\"\u003e\u003c/span\u003e\nBohr\u0027s theoretical grounds for attributing these lines to helium were\nafterwards confirmed experimentally by Fowler. It will be seen that\nthey fit into the formula (1) when \u003cimg style=\"vertical-align: -0.048ex; width: 2.848ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-32.png\" alt=\"\" data-tex=\"\\(4R\\)\"\u003e is substituted for \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e, a\nfact which Bohr\u0027s theory explains, as well as the more delicate fact\nthat, to make the formula exact, we have to substitute, not exactly\n\u003cimg style=\"vertical-align: -0.048ex; width: 2.848ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-32.png\" alt=\"\" data-tex=\"\\(4R\\)\"\u003e, but a slightly smaller quantity.\u003c/p\u003e\n\n\u003cp\u003eThe form of the equation (1) suggested to Bohr that a line of the\nhydrogen spectrum is not to be regarded as something which the atom\nemits when it is in a state of periodic vibration, but as produced by\na change from a state connected with one integer to a state connected\nwith another. This would be explained if the orbit of the electron\nwere not just any orbit possible on Newtonian principles, but only an\norbit connected with an integral \"quantum number\"—\u003ci\u003ei.e.\u003c/i\u003e with a\nmultiple of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eThe way in which Bohr achieved a theory on these lines is as follows.\nHe supposed that the electron can only revolve round the nucleus in\ncertain circles, these being such that, if is the moment of momentum in\nany orbit, we shall have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 17.282ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-33.png\" alt=\"\" data-tex=\"\\[\n2 \\pi p= nh \\qquad \\text{(2)},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e is, as always, Planck\u0027s constant, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e is a small\nwhole number. (In theory \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e might be any whole number, but in\u003cspan class=\"pagenum\" id=\"Page_34\"\u003e[Pg 34]\u003c/span\u003e\npractice it is never found to be much larger than 30, and that only\nin certain very tenuous nebulæ.) The reason why the quantum principle\nassumes just this form will be explained presently.\u003c/p\u003e\n\n\u003cp\u003eNow if \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e is the mass of the electron, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e the radius of its\norbit, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-36.png\" alt=\"\" data-tex=\"\\(\\omega\\)\"\u003e its angular velocity, we have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.339ex; width: 29.953ex; height: 5.81ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-37.png\" alt=\"\" data-tex=\"\\[\n\\begin{array}{l}\n\u0026p = ma^{2}\\omega.\\\\\n\\text{Hence} \u00262 \\pi ma^{2}\\omega = nh \\qquad \\text{(3)}.\n\\end{array}\n\\]\"\u003e\u003c/span\u003e\nBut, on grounds of the usual theory, since the radial acceleration of\nthe electron is \u003cimg style=\"vertical-align: -0.025ex; width: 3.592ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-38.png\" alt=\"\" data-tex=\"\\(a \\omega^{2}\\)\"\u003e and the force attracting it to the\nnucleus is \u003cimg style=\"vertical-align: -1.651ex; width: 3.18ex; height: 5.068ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-39.png\" alt=\"\" data-tex=\"\\(\\dfrac{e^{2}}{a^{2}}\\)\"\u003e we have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.661ex; width: 24.384ex; height: 6.453ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-40.png\" alt=\"\" data-tex=\"\\[\n\\begin{array}{l}\n\u0026ma^{2}\\omega = \\frac{e^{2}}{a^{2}}.\\\\\n\\textit{I.e.} \u0026ma^{2}\\omega = e^{2} \\qquad \\text{(4)}.\n\\end{array}\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eFrom equations (3) and (4) we obtain:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.654ex; width: 33.548ex; height: 5.087ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-41.png\" alt=\"\" data-tex=\"\\[\na = \\frac{n^{2} h^{2}}{4\\pi^{2} m e^{2}}\\omega = \\frac{8\\pi^{2} m e^{4}}{n^{2} h^{2}}\\qquad \\text{(5)}.\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe possible orbits for the electron are obtained by putting \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e = 1,\n2, 3, 4, … in the above formulæ for \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e. Thus the smallest possible\norbit is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.654ex; width: 21.679ex; height: 5.07ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-42.png\" alt=\"\" data-tex=\"\\[\na_1 = \\frac{h^{2}}{4\\pi^{2} m e^{2}} \\qquad \\text{(6)};\n\\]\"\u003e\u003c/span\u003e\nand the other possible orbits are \u003cimg style=\"vertical-align: -0.339ex; width: 3.316ex; height: 1.871ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-43.png\" alt=\"\" data-tex=\"\\(4a_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 3.316ex; height: 1.846ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-44.png\" alt=\"\" data-tex=\"\\(9a_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 4.447ex; height: 1.846ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-45.png\" alt=\"\" data-tex=\"\\(16a_1\\)\"\u003e, etc.\u003c/p\u003e\n\n\u003cp\u003eFor the energy in an orbit of radius \u003cimg style=\"vertical-align: -0.339ex; width: 4.53ex; height: 2.226ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-46.png\" alt=\"\" data-tex=\"\\(n^{2} a_1\\)\"\u003e we have, since the\npotential energy is double the kinetic energy with its sign changed:\u003ca id=\"FNanchor_11\" href=\"#Footnote_11\" class=\"fnanchor\"\u003e[11]\u003c/a\u003e\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.654ex; width: 28.719ex; height: 5.087ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-47.png\" alt=\"\" data-tex=\"\\[\nW = – \\tfrac{1}{2} m a^{2} \\omega^{2} = -\\frac{2\\pi^{2} m e^{4}}{2 n^{2} h^{2}}\n\\]\"\u003e\u003c/span\u003e\nin virtue of (5). Thus when the electron falls from an orbit whose\nradius is \u003cimg style=\"vertical-align: -0.339ex; width: 4.351ex; height: 2.226ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-48.png\" alt=\"\" data-tex=\"\\(k^{2} a_1\\)\"\u003e to one whose radius is \u003cimg style=\"vertical-align: -0.339ex; width: 4.53ex; height: 2.226ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-46.png\" alt=\"\" data-tex=\"\\(n^{2} a_1\\)\"\u003e \u003cimg style=\"vertical-align: -0.566ex; width: 7.313ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-49.png\" alt=\"\" data-tex=\"\\((k \\gt n)\\)\"\u003e,\nthere is a loss of energy:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.148ex; width: 21.66ex; height: 5.582ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-50.png\" alt=\"\" data-tex=\"\\[\n\\frac{2\\pi^{2} m e^{4}}{2h^{2}} \\left( \\frac{1}{n^{2}} – \\frac{1}{k^{2}} \\right).\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_35\"\u003e[Pg 35]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt is assumed that this energy is radiated out in a light-wave whose\nenergy is one quantum of energy \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e, where \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e is its\nfrequency. Hence we obtain the frequency of the emitted light by the\nequation:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -5.356ex; width: 38.15ex; height: 11.843ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-51.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\nh v \u0026 =\\frac{2 \\pi^{2} m e^{4}}{h^{2}}\\left(\\frac{1}{n^{2}}-\\frac{1}{k^{2}}\\right); \\\\\n\\textit{i.e.}\\qquad\\qquad\\nu \u0026 =\\frac{2 \\pi^{2} m e^{4}}{h^{3}}\\left(\\frac{1}{n^{2}}-\\frac{1}{k^{2}}\\right).\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nThis agrees exactly with the observed lines if [see equation (1)]:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.652ex; width: 13.796ex; height: 5.086ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-52.png\" alt=\"\" data-tex=\"\\[\nR=\\frac{2 \\pi^{2} m e^{4}}{h^{3}},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e is Rydberg\u0027s constant. On inserting numerical values, it is\nfound that this equation is verified. This striking success was, from\nthe first, a powerful argument in favour of Bohr\u0027s theory.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_36\"\u003e[Pg 36]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eBohr\u0027s theory has been generalized by Wilson\u003ca id=\"FNanchor_12\" href=\"#Footnote_12\" class=\"fnanchor\"\u003e[12]\u003c/a\u003e and Sommerfeld so as\nto allow also elliptic orbits: these have two quantum numbers, one\ncorresponding, as before, to angular momentum or the moment of momentum\n(which is constant, by Kepler\u0027s second law), the other depending upon\nthe eccentricity. Only certain eccentricities are possible; in fact,\nthe ratio of the minor to the major axis is always rational, and has\nas its denominator the quantum number corresponding to the moment of\nmomentum. In order to explain the Zeeman effect (which arises in a\nmagnetic field) we used a third quantum number, corresponding to the\nangle between the plane of the magnetic field and the plane of the\nelectron\u0027s orbit. In all cases, however, there is a general principle,\nwhich must now be explained. This will show, also, why, in Bohr\u0027s\ntheory, the quantum equation (2) takes the form it does.\u003ca id=\"FNanchor_13\" href=\"#Footnote_13\" class=\"fnanchor\"\u003e[13]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe first thing to observe is that the quantum principle is really\nconcerned with atoms of action, not of energy: action is energy\nmultiplied by time. Suppose now that we have a system depending\nupon several co-ordinates, and periodic in respect of each. It is\nnot necessary to suppose that each co-ordinate has the same period:\nit is only necessary to suppose that the system is \"conditionally\nperiodic\"—\u003ci\u003ei.e.\u003c/i\u003e that each co-ordinate separately is periodic.\nWe must further assume that our co-ordinates are so chosen as to\nallow \"separation of variables\" (as to which, see Sommerfeld, \u003ci\u003eop.\ncit.\u003c/i\u003e, pp. 559-60). We then define the \"momentum\" (in a generalized\nsense) associated with the co-ordinate \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e as the partial\ndifferential of the kinetic energy with respect to \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e—\u003ci\u003ei.e.\u003c/i\u003e\ncalling the generalized momentum \u003cimg style=\"vertical-align: -0.439ex; width: 2.159ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-54.png\" alt=\"\" data-tex=\"\\(p_k\\)\"\u003e, we put:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.084ex; width: 12.284ex; height: 5.231ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-55.png\" alt=\"\" data-tex=\"\\[\np_{k}=\\frac{\\partial E_{k i n}}{\\partial \\dot{q}_{k}},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.339ex; width: 4.436ex; height: 1.878ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-56.png\" alt=\"\" data-tex=\"\\(E_{\\text {kin }}\\)\"\u003e is the kinetic energy. The quantum condition\nis to apply to the integral of \u003cimg style=\"vertical-align: -0.439ex; width: 2.159ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-54.png\" alt=\"\" data-tex=\"\\(p_k\\)\"\u003e over a complete period of\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e—\u003ci\u003ei.e.\u003c/i\u003e we are to have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.948ex; width: 15.207ex; height: 5.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-57.png\" alt=\"\" data-tex=\"\\[\n\\int p_{k} d q_{k}=n_{k} h,\n\\]\"\u003e\u003c/span\u003e\nwhere the integration is taken through one complete period of \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e.\nHere will be the quantum number associated with the co-ordinate\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e. The above is a general formula of which all known cases of\nquantum phenomena are special cases. This is its sole justification.\u003c/p\u003e\n\n\u003cp\u003eThe above principle is exceedingly complicated—more so, even, than it\nappears in our summary account, which has omitted various difficulties.\nIt is possible that its complication may be due to the fact that\nquantum dynamics has had to force its way through the obstacles which\nthe classical system put in its way; it is possible also that quantum\nphenomena may turn out to be deducible from classical principles. But\nbefore pursuing this line of thought, it may be well to say\u003cspan class=\"pagenum\" id=\"Page_37\"\u003e[Pg 37]\u003c/span\u003e a few\nwords about the developments of Bohr\u0027s theory by Sommerfeld and others.\u003c/p\u003e\n\n\u003cp\u003eIn its original form, in which circular orbits were assumed, Bohr\u0027s\ntheory accounted for the main facts concerning the line spectra of\nhydrogen and ionized helium. But there were a number of more delicate\nfacts which required the hypothesis of elliptic orbits: with this\nhypothesis, together with some niceties derived from relativity, the\nmost minute agreement has been obtained between theory and observation.\nBut perhaps this great success has made people think that more was\nproved than really was proved. The great advantage obtained from\nadmitting elliptic orbits is that they provide a second quantum number.\nIn the emission of light by atoms, what we have is essentially as\nfollows. The atom is capable of various states, characterized by whole\nnumbers (the quantum numbers). There may be more or fewer quantum\nnumbers, according to the degrees of freedom of the system. The loss\nor gain of energy when an atom passes from a state characterized by\none set of values of the quantum numbers to a state characterized by\nanother set is known. When energy is lost (without the loss of an\nelectron or of any part of the nucleus of the atom), it passes out as a\nlight-wave, whose energy is equal to what the atom has lost, and whose\nenergy multiplied by the time of one vibration is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. Energy is what\nis conserved, but action is what is quantized.\u003c/p\u003e\n\n\u003cp\u003eLet us revert, in illustration, to the circular orbits of Bohr\u0027s\noriginal theory, which remain possible, though not universal, in the\nnewer theory. If we call \u003cimg style=\"vertical-align: -0.357ex; width: 4.774ex; height: 1.895ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-58.png\" alt=\"\" data-tex=\"\\(E_{min}\\)\"\u003e the kinetic energy when the\nelectron is in the smallest possible orbit, the kinetic energy in the\n\u003cimg style=\"vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-59.png\" alt=\"\" data-tex=\"\\(n^{th}\\)\"\u003e orbit is \u003cimg style=\"vertical-align: -1.654ex; width: 5.769ex; height: 4.722ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-60.png\" alt=\"\" data-tex=\"\\(\\dfrac{E_{min}}{n^{2}}\\)\"\u003e. (The measure of the total\nenergy is the kinetic energy with its sign changed.) We do not know\nwhat determines the electron to jump from one orbit to another; on this\npoint, our knowledge is merely statistical.\u003cspan class=\"pagenum\" id=\"Page_38\"\u003e[Pg 38]\u003c/span\u003e We know, of course, that\nwhen the atom is not in a position to absorb energy the electron can\nonly jump from a larger to a smaller orbit, while the converse jump\noccurs when the atom absorbs energy from incident light. We know also,\nfrom the comparative intensities of different lines in the spectrum,\nthe comparative frequencies of different possible jumps, and on this\nsubject a theory exists. But we do not know in the least why, of a\nnumber of atoms whose electrons are not in minimum orbits, some jump at\none time and some at another, just as we do not know why some atoms of\nradio-active substances break down while others do not. Nature seems\nto be full of revolutionary occurrences as to which we can say that,\n\u003ci\u003eif\u003c/i\u003e they take place, they will be of one of several possible\nkinds, but we cannot say that they will take place at all, or, if they\nwill, at what time. So far as quantum theory can say at present, atoms\nmight as well be possessed of free will, limited, however, to one of\nseveral possible choices.\u003ca id=\"FNanchor_14\" href=\"#Footnote_14\" class=\"fnanchor\"\u003e[14]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eHowever this may be, it is clear that what we know is the changes\nof energy when an atom emits light, and we know that in the case\nof hydrogen or ionized helium these changes are measured by\n\u003cimg style=\"vertical-align: -1.654ex; width: 9.268ex; height: 4.69ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-61.png\" alt=\"\" data-tex=\"\\(\\dfrac{1}{n^{2}} – \\dfrac{1}{k^{2}}\\)\"\u003e. It seems almost unavoidable to\ninfer that the previous state of the atom was characterized by the\ninteger \u003cimg style=\"vertical-align: -0.025ex; width: 1.179ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-28.png\" alt=\"\" data-tex=\"\\(k\\)\"\u003e and the later one by the integer \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. But to assume\norbits and so on, though proper as a help to the imagination, is hardly\nsufficiently justified by the analogy of large-scale processes, since\nthe quantum principle itself shows the danger of relying upon this\nanalogy. In large-scale occurrences there is nothing to suggest the\nquantum, and perhaps other familiar features of such occurrences may\nresult merely from statistical averaging.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_39\"\u003e[Pg 39]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt may be worth while to consider briefly the elliptical orbits which\nare possible.\u003ca id=\"FNanchor_15\" href=\"#Footnote_15\" class=\"fnanchor\"\u003e[15]\u003c/a\u003e This will also illustrate the application of the\nquantum principle to systems with more than one co-ordinate.\u003c/p\u003e\n\n\u003cp\u003eTaking polar co-ordinates, the kinetic energy is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.552ex; width: 16.651ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-62.png\" alt=\"\" data-tex=\"\\[\n\\frac{1}{2} m\\left(\\dot{r}^{2}+r^{2} \\dot{\\theta}^{2}\\right).\n\\]\"\u003e\u003c/span\u003e\nThe two generalized momenta are therefore:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 19.855ex; height: 2.769ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-63.png\" alt=\"\" data-tex=\"\\[\np_{\\theta}=m r^{2} \\dot{\\theta}, p_{r}=m \\dot{r}.\n\\]\"\u003e\u003c/span\u003e\nWe have thus two quantum conditions:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -5.443ex; width: 31.745ex; height: 12.018ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-64.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n\u0026 \\int_{0}^{2 \\pi} 2 m r^{2} \\dot{\\theta} d \\theta=n h\\\\\n\\text{and}\\qquad \\qquad\n\u0026 \\int_{\\theta=0}^{\\theta=2 \\pi} m \\dot{r} d r=n h.\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nBy Kepler\u0027s second law, \u003cimg style=\"vertical-align: -0.025ex; width: 5.056ex; height: 2.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-65.png\" alt=\"\" data-tex=\"\\(m r^{2} \\dot{\\theta}\\)\"\u003e is constant; call it\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e. Thus:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 9.866ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-67.png\" alt=\"\" data-tex=\"\\[\n2 \\pi p=n h.\n\\]\"\u003e\u003c/span\u003e\nThe other integration is more troublesome, but we arrive at the result\nthat, if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e are the major and minor axes of the ellipse,\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.577ex; width: 12.555ex; height: 4.823ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-69.png\" alt=\"\" data-tex=\"\\[\n\\frac{a – b}{a}=\\frac{n\u0027}{n}.\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eA little further calculation leads to the result that the energy\nin the orbit which has the quantum numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.985ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-70.png\" alt=\"\" data-tex=\"\\(n\u0027\\)\"\u003e is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.62ex; width: 21.679ex; height: 6.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-71.png\" alt=\"\" data-tex=\"\\[\n-\\frac{2 \\pi^{2} m e^{4}}{h^{2}} \\cdot \\frac{1}{\\left(n+n\u0027\\right)^{2}}\n\\]\"\u003e\u003c/span\u003e\nThis is exactly the same as in the case of circular orbits, except\nthat \u003cimg style=\"vertical-align: -0.186ex; width: 6.108ex; height: 1.903ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-72.png\" alt=\"\" data-tex=\"\\(n + n\u0027\\)\"\u003e replaces \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. If this were all, the line spectrum of\nhydrogen would be exactly the same whether elliptic orbits occurred or\nnot, and there would be no empirical means of deciding the question.\u003c/p\u003e\n\n\u003cp\u003eHowever, by introducing considerations derived from the special\ntheory of relativity we are able to distinguish between the results\nto be expected from circular and elliptic orbits\u003cspan class=\"pagenum\" id=\"Page_40\"\u003e[Pg 40]\u003c/span\u003e respectively, and\nto show that the latter must occur to account for observed facts.\nThe crucial point is the variation of mass with velocity: the faster\na body is moving, the greater is its mass. Therefore in an elliptic\norbit the electron will have a greater mass at the perihelion than\nat the aphelion. From this it is found to follow that an elliptic\norbit will not be accurately elliptic, but that the perihelion will\nadvance slightly with each revolution.\u003ca id=\"FNanchor_16\" href=\"#Footnote_16\" class=\"fnanchor\"\u003e[16]\u003c/a\u003e That is to say, taking polar\nco-ordinates \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e, the co-ordinate \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e increases\nby slightly more than \u003cimg style=\"vertical-align: -0.025ex; width: 2.421ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-75.png\" alt=\"\" data-tex=\"\\(2\\pi\\)\"\u003e between one minimum of \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e and the next.\nThe system is thus \"conditionally periodic\"—\u003ci\u003ei.e.\u003c/i\u003e each separate\nco-ordinate changes periodically, but the periods of the two do not\ncoincide. The result\u003ca id=\"FNanchor_17\" href=\"#Footnote_17\" class=\"fnanchor\"\u003e[17]\u003c/a\u003e is that the equation \u003cimg style=\"vertical-align: -1.577ex; width: 11.926ex; height: 4.823ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-76.png\" alt=\"\" data-tex=\"\\(\\dfrac{a – b}{a} =\\dfrac{n\u0027}{n}\\)\"\u003e\nis replaced by:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -5.168ex; width: 29.801ex; height: 11.467ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-77.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n\\frac{a-b}{a} \u0026= \\frac{n\u0027}{n \\gamma\u0027}\\\\\n\\text{where}\\qquad\\qquad\n\\gamma^{2} \u0026=1-\\frac{e^{4}}{c^{2} p^{2}},\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e being the velocity of light, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e, as before, the angular\nmomentum. It will be seen that \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e is very nearly 1, because\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e is large.\u003c/p\u003e\n\n\u003cp\u003eThe formula for the energy associated with the quantum numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.985ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-70.png\" alt=\"\" data-tex=\"\\(n\u0027\\)\"\u003e now becomes much more complicated; its great merit is that it\naccounts for the fine structure of the hydrogen line spectrum. It\nmust be felt that this minuteness of agreement between theory and\nobservation is very remarkable. But it is still the case that the\nonly empirical evidence concerns differences of energy in connection\nwith different quantum numbers, and that the theory of actual orbits,\nproceeding, during steady motion, according to Newtonian principles,\nmust inevitably remain a hypothesis—a hypothesis which,\u003cspan class=\"pagenum\" id=\"Page_41\"\u003e[Pg 41]\u003c/span\u003e as we shall\nsee, has disappeared from the latest form of the quantum theory.\u003c/p\u003e\n\n\u003cp\u003eThe fact of the existence of the quantum is as strange as it is\nundeniable, unless it should turn out to be deducible from classical\nprinciples. It seems to be the case that quantum principles regulate\nall interchange of energy between matter and the surrounding medium.\nThere are grave difficulties in reconciling the quantum theory with\nthe undulatory theory of light, but we shall not consider these until\na later stage. What is much to be wished is some way of formulating\nthe quantum principle which shall be less strange and \u003ci\u003ead hoc\u003c/i\u003e\nthan that due to Wilson and Sommerfeld. For practical purposes, it\namounts to something like this: that a periodic process of frequency\n\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e has an amount of energy which is a multiple of \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e,\nand, conversely, if a given amount of energy is expended in starting\na periodic process, it will start a process with a frequency \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e\nsuch that the given amount of energy shall be a multiple of \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e. When a\nprocess has a frequency \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e and an energy \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-26.png\" alt=\"\" data-tex=\"\\(h \\nu\\)\"\u003e, the amount of\n\"action\" during one period is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. But we cannot say: In any periodic\nprocess the amount of action in one period is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e or a multiple of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. Nevertheless, some formulation analogous to this might in time\nturn out to be possible. As has appeared from the theory of relativity,\n\"action\" is more fundamental than energy in physical theory; it is\ntherefore perhaps not surprising that action should be found to play\nan important part. But the whole theory of the interaction of matter\nand the surrounding medium, at present, rests upon the conservation\nof energy. Perhaps a theory giving more prominence to action may\nbe possible, and may facilitate a simpler statement of the quantum\nprinciple.\u003c/p\u003e\n\n\u003cp\u003eIn Bohr\u0027s theory and its developments, there is a lacuna and there is\na difficulty. The lacuna has already been mentioned: we do not know in\nthe least why an electron chooses one moment rather than another to\njump from a larger to a\u003cspan class=\"pagenum\" id=\"Page_42\"\u003e[Pg 42]\u003c/span\u003e smaller orbit. The difficulty is that the jump\nis usually regarded as sudden and discontinuous: it is suggested that\nif it were continuous, the experimental facts in the regions concerned\nwould become inexplicable. Possibly this difficulty may be overcome,\nand it may be found that the transition from one orbit to another can\nbe continuous. But it is as well to consider the other possibility,\nthat the transition is really discontinuous. I have emphasized how\nlittle we really \u003ci\u003eknow\u003c/i\u003e about what goes on in the atom, because I\nwished to keep open the possibility of something quite different from\nwhat is usually supposed. Have we any good reason for thinking that\nspace-time is continuous? Do we know that, between one orbit and the\nnext, other orbits are \u003ci\u003egeometrically\u003c/i\u003e possible? Einstein has led\nus to think that the neighbourhood of matter makes space non-Euclidean;\nmight it not also make it discontinuous? It is certainly rash to\nassume that the minute structure of the world resembles that which is\nfound to suit large-scale phenomena, which may be only statistical\naverages. These considerations may serve as an introduction to the\nmost modern theory of quantum mechanics, to which we must now turn our\nattention.\u003ca id=\"FNanchor_18\" href=\"#Footnote_18\" class=\"fnanchor\"\u003e[18]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eIn the new theory inaugurated by Heisenberg, we no longer have the\nsimplicity of the Rutherford-Bohr atom, in which electrons revolve\nabout a nucleus like separate planets.\u003cspan class=\"pagenum\" id=\"Page_43\"\u003e[Pg 43]\u003c/span\u003e Heisenberg points out that in\nthis theory there are many quantities which are not even theoretically\nobservable—namely, those representing processes supposed to be\noccurring while the atom is in a steady state. In the new theory, as\nDirac says: \"The variable quantities associated with a stationary state\non Bohr\u0027s theory, the amplitudes and frequencies of orbital motion,\nhave no physical meaning and are of no physical importance\" (4, p.\n652). Heisenberg, in first introducing his theory, pointed out that\nthe ordinary quantum theory uses unobservable quantities, such as the\nposition and time of revolution of an electron (1, p. 879), and that\nthe electron ought to be represented by measurable quantities such\nas the frequencies of its radiation (1, p. 880). Now the observable\nfrequencies are always differences between two \"terms,\" each of which\nis represented by an integer. We thus arrive at a representation of the\nstate of an atom by means of an infinite array of numbers—\u003ci\u003ei.e.\u003c/i\u003e\nby a matrix. If \u003cimg style=\"vertical-align: -0.357ex; width: 2.469ex; height: 1.889ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-80.png\" alt=\"\" data-tex=\"\\(T_n\\)\"\u003e and \u003cimg style=\"vertical-align: -0.357ex; width: 2.914ex; height: 1.889ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-81.png\" alt=\"\" data-tex=\"\\(T_m\\)\"\u003e are two \"terms,\" an observable\nfrequency (in theory) is \u003cimg style=\"vertical-align: -0.357ex; width: 3.67ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-82.png\" alt=\"\" data-tex=\"\\(\\nu_{nm}\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.357ex; width: 15.464ex; height: 1.889ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-83.png\" alt=\"\" data-tex=\"\\[\n\\nu_{nm} = T_n – T_m.\n\\]\"\u003e\u003c/span\u003e\nIt is such numbers as \u003cimg style=\"vertical-align: -0.357ex; width: 3.67ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-82.png\" alt=\"\" data-tex=\"\\(\\nu_{nm}\\)\"\u003e (of which there is a doubly infinite\nseries) that characterize the atom, so far as it is observable.\u003c/p\u003e\n\n\u003cp\u003eHeisenberg sets out this view as follows (5, p. 685). In the classical\ntheory, given an electron with one degree of freedom, in harmonic\noscillation, the elongation \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e at time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e can be represented by a\nFourier series:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.623ex; width: 33.305ex; height: 4.772ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-85.png\" alt=\"\" data-tex=\"\\[\nx=x(n, t)=\\sum_{\\tau} x(n)_{\\tau} \\cdot e^{2 \\pi i \\nu(n) \\cdot \\tau t},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e is a constant and \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e is the number of the harmonic. The\nsingle terms of this series, namely:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 14.018ex; height: 2.7ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-86.png\" alt=\"\" data-tex=\"\\[\nx(n)_{\\tau} e^{2 \\pi i \\nu(n) \\tau t},\n\\]\"\u003e\u003c/span\u003e\nwould contain the quantities which have been signalized as directly\nobservable—namely, frequency, amplitude, and phase.\u003cspan class=\"pagenum\" id=\"Page_44\"\u003e[Pg 44]\u003c/span\u003e But in virtue of\nthe fact that, in atoms, frequencies are found to be the differences of\n\"terms\" we shall have to replace the above by:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 16.395ex; height: 2.7ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-87.png\" alt=\"\" data-tex=\"\\[\nx(nm)e^{2 \\pi i \\nu(nm)\\tau t};\n\\]\"\u003e\u003c/span\u003e\nand the collection (not the sum) of such terms represents what was\nformerly the elongation \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. The sum of all these terms has no longer\nany physical significance. Thus the atom comes to be represented by the\nnumbers \u003cimg style=\"vertical-align: -0.566ex; width: 6.303ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-88.png\" alt=\"\" data-tex=\"\\(\\nu(nm)\\)\"\u003e, arranged in an infinite rectangle or \"matrix.\"\u003c/p\u003e\n\n\u003cp\u003eIt is possible to construct an algebra of matrices, which differs\nformally from ordinary algebra in only one respect, namely, that\nmultiplication is not commutative.\u003c/p\u003e\n\n\u003cp\u003eA new operation is defined which, when the quantum numbers become\nlarge, approximates to differentiation. By using this operation,\nHamilton\u0027s equations of motion can be preserved in a form which is\napplicable equally to periodic and to unperiodic motions, so that it\nis no longer necessary to distinguish a certain sphere of quantum\nphenomena, to which different laws are applied from those applied\nto the phenomena amenable to classical dynamics: \"A distinction\nbetween \u0027quantized\u0027 and \u0027unquantized\u0027 motions loses all meaning in\nthis theory, since in it there is no question of a quantum condition\nwhich selects certain motions from a great number of possible ones;\nin place of this condition appears a quantum-mechanical fundamental\nequation … which is valid for all possible motions, and is necessary\nin order to give a definite meaning to the problem of motion\" (3, p.\n558). The fundamental equation alluded to in the above is as follows:\nLet \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e be a Hamiltonian co-ordinate, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e the corresponding\n(generalized) momentum, both being matrices. It will be remembered that\nmultiplication is not commutative for matrices; in fact, we have as the\nfundamental equation in question (2, p. 871):\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.577ex; width: 17.417ex; height: 4.676ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-90.png\" alt=\"\" data-tex=\"\\[\npq – qp = \\frac{h}{2 \\pi i} \\cdot \\text{I},\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_45\"\u003e[Pg 45]\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: 0; width: 0.817ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-91.png\" alt=\"\" data-tex=\"\\(\\text{I}\\)\"\u003e represents the matrix whose diagonal consists of\n\u003cimg style=\"vertical-align: 0; width: 0.817ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-91.png\" alt=\"\" data-tex=\"\\(\\text{I}\\)\"\u003e\u0027s, and whose other terms are all zero. The above is the\nsole fundamental equation containing \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e (Planck\u0027s constant), and it\nis true for \u003ci\u003eall\u003c/i\u003e motions.\u003c/p\u003e\n\n\u003cp\u003eHeisenberg does not claim that the new theory solves all difficulties.\nOn the contrary, he says (5, p. 705):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The theory here described must be regarded as still incomplete. The\nreal geometrical or kinematical meaning of the fundamental assumption\n(5)\u003ca id=\"FNanchor_19\" href=\"#Footnote_19\" class=\"fnanchor\"\u003e[19]\u003c/a\u003e has not yet been made completely clear. In particular,\nthere is a serious difficulty in the fact that the time apparently\nhas a different rôle from the space co-ordinates, and is formally\ndifferently treated. The formal character of the time co-ordinate in\nthe mathematical structure of the theory is made particularly evident\nby the fact that in the theory hitherto the question of the temporal\ncourse of a process has no immediate meaning, and that the concept of\nearlier and later can hardly be defined exactly. Nevertheless, we need\nnot consider these difficulties as an objection to the theory, since\nthe appearance of just such difficulties was to be expected from the\nnature of the space-time relations that hold for atomic systems.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eIn a more or less popular exposition (6), Heisenberg has set forth some\nof the consequences of his theory. Electrons and atoms, he says, do not\nhave \"the degree of immediate reality of objects of sense,\" but only\nthe sort of reality which one naturally ascribes to light quanta. The\ntroubles of the quantum theory have come, he thinks, from trying to\nmake models of atoms and picture them as in ordinary space. If we are\nto retain the corpuscular theory, we can only do it by not assigning\na definite point of space at each time to the electron or atom. We\nsubstitute a well-defined physical group of quantities which represent\nwhat \u003ci\u003ewas\u003c/i\u003e the place of the electron.\u003cspan class=\"pagenum\" id=\"Page_46\"\u003e[Pg 46]\u003c/span\u003e They are the observable\nradiation quantities, each of which is associated with two \"terms,\" so\nthat we obtain a matrix. The distinction of inner and outer electrons\nin an atom becomes meaningless. \"It is, moreover, in principle\nimpossible to identify again a particular corpuscle among a series of\nsimilar corpuscles\" (p. 993).\u003c/p\u003e\n\n\u003cp\u003eThe matrix theory of the electron is too new to be amenable, as yet, to\nthe kind of logical analysis which it is our purpose to undertake in\nthis Part. It is clear, however, that it affects a scientific economy\nby substituting for the merely hypothetical steady motions of Bohr\u0027s\natoms a set of quantities representing what we really know—namely, the\nradiations that come out of the region in which the atom is supposed\nto be. It is clear, also, that there is an immense logical progress in\nthe construction of a dynamic which destroys the distinction between\nquantized and unquantized motions, and treats all motions by means\nof a uniform set of principles. And the greater abstractness of the\nHeisenberg atom as compared with the Bohr atom makes it logically\npreferable, since the pictorial elements in a physical theory are those\nupon which least reliance can be placed.\u003c/p\u003e\n\n\u003cp\u003eAn apparently different quantum theory, due to de Broglie\u003ca id=\"FNanchor_20\" href=\"#Footnote_20\" class=\"fnanchor\"\u003e[20]\u003c/a\u003e and\nSchrödinger,\u003ca id=\"FNanchor_21\" href=\"#Footnote_21\" class=\"fnanchor\"\u003e[21]\u003c/a\u003e has been found to be formally the same as Heisinger\u0027s\ntheory, although at first sight very different. This is described by\nde Broglie as \"the new wave theory of matter,\" in which \"the material\npoint is conceived as a singularity in a wave.\"\u003ca id=\"FNanchor_22\" href=\"#Footnote_22\" class=\"fnanchor\"\u003e[22]\u003c/a\u003e Here, also, the\nradiations which we think of as coming out of the atom have more\nphysical \"reality\" than the atom itself. One of the merits of the\ntheory is that it diminishes the difficulties hitherto\u003cspan class=\"pagenum\" id=\"Page_47\"\u003e[Pg 47]\u003c/span\u003e existing in the\nway of a reconciliation of the facts of interference and dispersion\nwith the facts which led to the hypothesis of light quanta.\u003c/p\u003e\n\n\u003cp\u003eMeanwhile, there remains the possibility that all the quantum phenomena\nmay be deducible from classical principles, and that the apparent\ndiscontinuities may be only a question of sharp maxima or minima. The\nmost successful theory known to me on these lines is that of L. V.\nKing.\u003ca id=\"FNanchor_23\" href=\"#Footnote_23\" class=\"fnanchor\"\u003e[23]\u003c/a\u003e He assumes that electrons rotate with a certain fixed angular\nvelocity, the same for all; he makes a similar assumption as regards\nprotons. Consequently there is a magnetic field which introduces\nconditions that are absent if electrons and protons have no spin. There\nwill be electromagnetic radiation of frequency \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.552ex; width: 13.334ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-92.png\" alt=\"\" data-tex=\"\\[\nh \\nu = \\dfrac{1}{2}m_{0}v^{2}.\n\\]\"\u003e\u003c/span\u003e\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e being Planck\u0027s constant, \u003cimg style=\"vertical-align: -0.375ex; width: 2.974ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-93.png\" alt=\"\" data-tex=\"\\(m_0\\)\"\u003e the invariant mass of the\nelectron, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e its velocity. (The identity of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e with Planck\u0027s\nconstant is obtained by adjusting the hypothetical constants.) From\nthis formula he deduces many of the phenomena upon which the quantum\ntheory is based, and promises to deduce others in a later paper. An\narticle by Mr R. H. Fowler (\"Spinning Electrons,\" \u003ci\u003eNature\u003c/i\u003e, Jan.\n15, 1927) discusses Mr King\u0027s theory without arriving at a verdict for\nor against. Presumably it will not be long before a definite answer as\nto the adequacy of Mr King\u0027s theory is possible. If it is adequate,\nthe quantum theory ceases to concern the philosopher, since what\nremains valid in it becomes a deduction from more fundamental laws and\nprocesses which are continuous and involve no atomicity of action.\nFor the moment, until the physicists have arrived at a decision, the\nphilosopher must be content to investigate both hypotheses impartially.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_7\" href=\"#FNanchor_7\" class=\"label\"\u003e[7]\u003c/a\u003e\nThe numerical value of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e is \u003cimg style=\"vertical-align: -0.466ex; width: 21.539ex; height: 2.437ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-94.png\" alt=\"\" data-tex=\"\\(6.55 \\times 10^{-27}\n\\, \\text{erg} \\cdot \\text{secs.}\\)\"\u003e, and its dimensions are those of\n\"action\"—\u003ci\u003ei.e.\u003c/i\u003e energy x time.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_8\" href=\"#FNanchor_8\" class=\"label\"\u003e[8]\u003c/a\u003e\n\u003ci\u003eReport on Radiation and the Quantum Theory\u003c/i\u003e,\nPhysical Society of London, 1914, p. 58.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_9\" href=\"#FNanchor_9\" class=\"label\"\u003e[9]\u003c/a\u003e\n\u003ci\u003eAnnalen der Physik\u003c/i\u003e, vol. \u003cspan class=\"allsmcap\"\u003eXVII.\u003c/span\u003e, p. 146.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_10\" href=\"#FNanchor_10\" class=\"label\"\u003e[10]\u003c/a\u003e\nSee Jeans, \u003ci\u003eloc. cit.\u003c/i\u003e, chap. \u003cspan class=\"allsmcap\"\u003eVI\u003c/span\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_11\" href=\"#FNanchor_11\" class=\"label\"\u003e[11]\u003c/a\u003e\nSee Sommerfeld, \u003ci\u003eAtomic Structure and Spectral\nLines\u003c/i\u003e, pp. 547 ff.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_12\" href=\"#FNanchor_12\" class=\"label\"\u003e[12]\u003c/a\u003e\nW. Wilson, \u003ci\u003eThe Quantum Theory of Radiation and Line\nSpectra\u003c/i\u003e Phil. Mag., June, 1915.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_13\" href=\"#FNanchor_13\" class=\"label\"\u003e[13]\u003c/a\u003e\nWhat follows is taken from Note 7 (pp. 555 ff.) in\nSommerfeld\u0027s \u003ci\u003eAtomic Structure and Spectral Lines\u003c/i\u003e, translated\nfrom the third German edition by Henry L. Brose, M.A., 1923. See also\nNote 4 (pp. 541 ff.).\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_14\" href=\"#FNanchor_14\" class=\"label\"\u003e[14]\u003c/a\u003e\nThis, however, is probably a temporary state of affairs.\nCertain Pasons for quantum transitions are already known. See J. Franck\nand P. Jordan, \u003ci\u003eAnregung von Quaniensprüngen durch Stösse\u003c/i\u003e,\nBerlin, 1926; also P. Jordan, \u003ci\u003eKausalität und Statistik in der\nmodernen Physik\u003c/i\u003e, \u003ci\u003eNaturwissenschaften\u003c/i\u003e, Feb. 4, 1927.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_15\" href=\"#FNanchor_15\" class=\"label\"\u003e[15]\u003c/a\u003e\nSee Sommerfeld, \u003ci\u003eop. cit.\u003c/i\u003e, pp. 232 ff.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_16\" href=\"#FNanchor_16\" class=\"label\"\u003e[16]\u003c/a\u003e\nThis is not the same phenomenon as in the case of the\norbit of Mercury. The latter depends upon the general theory of\nrelativity, the former upon the special theory.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_17\" href=\"#FNanchor_17\" class=\"label\"\u003e[17]\u003c/a\u003e\nSommerfeld, \u003ci\u003eop. cit.\u003c/i\u003e, pp. 467 ff.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_18\" href=\"#FNanchor_18\" class=\"label\"\u003e[18]\u003c/a\u003e\nThe principal papers setting forth this theory are:\u003c/p\u003e\n\n\u003cp\u003e1. W. Heisenberg, \u003ci\u003eUeber quantentheoretische Umdeutung kinematischer\nund mechanischer Beziehungen\u003c/i\u003e. Zeitschrift für Physik, 33, pp.\n879-893, 1925.\u003c/p\u003e\n\n\u003cp\u003e2. M. Born and P. Jordan, \u003ci\u003eZur Quantenmechanik\u003c/i\u003e. Ibid. 34, pp.\n858-888, 1925.\u003c/p\u003e\n\n\u003cp\u003e3. M. Born, W. Heisenberg, and P. Jordan, \u003ci\u003eZur Quantenmechanik\nII\u003c/i\u003e. \u003ci\u003eIbid.\u003c/i\u003e 35, pp. 557-615, 1926.\u003c/p\u003e\n\n\u003cp\u003e4. P. A. M. Dirac, \u003ci\u003eThe Fundamental Equations of Quantum\nMechanics.\u003c/i\u003e Proc. Royal Soc., Series A, vol. 109, No. A752, pp.\n642-653, 1925.\u003c/p\u003e\n\n\u003cp\u003e5. W. Heisenberg, \u003ci\u003eUeber quantentheoretische Kinematik und\nMechanik\u003c/i\u003e. \u003ci\u003eMathematische Annalen\u003c/i\u003e, 95, pp. 683-705, 1926.\u003c/p\u003e\n\n\u003cp\u003e6. W. Heisenberg, \u003ci\u003eQuantenmechanik\u003c/i\u003e. Naturwissenschaften, 14\nJahrgang. Heft 45, pp. 989-994.\u003c/p\u003e\n\n\u003cp\u003eI shall quote these papers by the above numbers. I am much indebted in\nthis matter to Mr R. H. Fowler, F.R.S.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_19\" href=\"#FNanchor_19\" class=\"label\"\u003e[19]\u003c/a\u003e\nThis is the assumption, mentioned above, that an atom or\nelectron at time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e can be represented by a collection of terms of\nthe form:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 15.567ex; height: 2.7ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-95.png\" alt=\"\" data-tex=\"\\[\nx(nm)e^{2 \\pi i \\nu(nm)t}.\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_20\" href=\"#FNanchor_20\" class=\"label\"\u003e[20]\u003c/a\u003e\n\u003ci\u003eAnnales de Physique\u003c/i\u003e, 3, 22, 1925.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_21\" href=\"#FNanchor_21\" class=\"label\"\u003e[21]\u003c/a\u003e\n\u003ci\u003eAnnalen der Physik\u003c/i\u003e, 1926. Four papers, 79, pp.\n361, 489, 734; 80, p. 437.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_22\" href=\"#FNanchor_22\" class=\"label\"\u003e[22]\u003c/a\u003e\n\u003ci\u003eNature\u003c/i\u003e, Sp. 25, 1926, p. 441. See also Fowler,\n\"Matrix and Wave Mechanics.\" \u003ci\u003eib.\u003c/i\u003e Feb. 12, 1927.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_23\" href=\"#FNanchor_23\" class=\"label\"\u003e[23]\u003c/a\u003e\n\u003ci\u003eGyromagnetic Electrons and a Classical Theory of\nAtomic Structure and Radiation\u003c/i\u003e. By Louis Vessot King. F.R.S.,\nMacdonald Professor of Physics, McGill University. Louis Carrier,\nMercury Press, 1926.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_48\"\u003e[Pg 48]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_V\"\u003eCHAPTER V\u003cbr\u003e\nTHE SPECIAL THEORY OF RELATIVITY\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE theory of relativity has resulted from a combination of the\nthree elements which were called for in a reconstruction of physics;\nfirst, delicate experiment; secondly, logical analysis; and thirdly,\nepistemological considerations. These last played a greater part in the\nearly stages of the theory than in its finished form, and perhaps this\nis fortunate, since their scope and validity may be open to question,\nor at least would be but for the successes to which they have led. One\nmay say, broadly, that relativity, like earlier physics, has assumed\nthat when different observers are doing what is called \"observing the\nsame phenomenon,\" those respects in which their observations differ do\nnot belong to the phenomenon, but only those respects in which their\nobservations agree. This is a principle which common sense teaches at\nan early age. A young child, seeing a ship sailing away, thinks that\nthe ship is continually growing smaller; but before long he comes to\nrecognize that the diminution in size is only \"apparent,\" and that the\nship \"really\" remains of the same size throughout its voyage. In so\nfar as relativity has been inspired by epistemological considerations,\nthey have been of this common-sense kind, and the apparent paradoxes\nhave resulted from the discovery of unexpected differences between our\nobservations and those of other hypothetical observers. Relativity\nphysics, like all physics, assumes the realistic hypothesis, that there\nare occurrences which different people can observe. For the present, we\nmay ignore epistemology, and proceed to consider relativity simply as\ntheoretical physics. We may also ignore the experimental evidence, and\nregard the whole theory\u003cspan class=\"pagenum\" id=\"Page_49\"\u003e[Pg 49]\u003c/span\u003e as a deductive system, since that is the point\nof view with which we are concerned in Part I.\u003c/p\u003e\n\n\u003cp\u003eThe most remarkable feature of the theory of relativity, from a\nphilosopher\u0027s standpoint, was already present in the special theory:\nI mean the merging of space and time into space-time. The special\ntheory has now become only an approximation, which is not exactly true\nin the neighbourhood of matter. But it remains worth understanding,\nas a stage towards the general theory. Moreover, it does not demand\nthe abandonment of nearly such a large proportion of our common-sense\nnotions as is discarded by the general theory.\u003c/p\u003e\n\n\u003cp\u003eTechnically, the whole of the special theory is contained in the\nLorentz transformation. This transformation has the advantage that\nit makes the velocity of light the same with respect to any two\nbodies which are moving uniformly relatively to each other, and,\nmore generally, that it makes the laws of electromagnetic phenomena\n(Maxwell\u0027s equations) the same with respect to any two such bodies. It\nwas for the sake of this advantage that it was originally introduced;\nbut it was afterwards found to have wider bearings and a more general\njustification. In fact, it may be said that, given sufficient logical\nacumen, it could have been discovered at any time after it was known\nthat light is not propagated instantaneously. It has grown by this\ntime very familiar—so familiar that I have even seen it quoted (quite\ncorrectly) in an advertisement of Fortnum and Mason\u0027s. Nevertheless, it\nis, I suppose, desirable to set it forth. In its simplest form it is as\nfollows:\u003c/p\u003e\n\n\u003cp\u003eSuppose two bodies, one of which (\u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e) is moving relatively to the\nother (\u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e) with velocity v parallel to the \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e-axis. Suppose\nthat an observer on \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e observes an event which he judges to have\ntaken place at time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e, by his clocks, and in the place whose\nco-ordinates, for him, are \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e. (Each observer takes\nhimself as origin.) Suppose that an observer on \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e judges that\nthe\u003cspan class=\"pagenum\" id=\"Page_50\"\u003e[Pg 50]\u003c/span\u003e event occurs at time \u003cimg style=\"vertical-align: -0.025ex; width: 1.444ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-100.png\" alt=\"\" data-tex=\"\\(t\u0027\\)\"\u003e and that its co-ordinates are \u003cimg style=\"vertical-align: -0.025ex; width: 1.922ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-101.png\" alt=\"\" data-tex=\"\\(x\u0027\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.736ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-102.png\" alt=\"\" data-tex=\"\\(y\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.68ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-103.png\" alt=\"\" data-tex=\"\\(z\u0027\\)\"\u003e. We suppose that at the time when \u003cimg style=\"vertical-align: -0.186ex; width: 4.965ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-104.png\" alt=\"\" data-tex=\"\\(t = 0\\)\"\u003e the two\nobservers are at the same place, and also \u003cimg style=\"vertical-align: -0.186ex; width: 5.593ex; height: 1.903ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-105.png\" alt=\"\" data-tex=\"\\(t\u0027= 0\\)\"\u003e. It would formerly\nhave seemed axiomatic that we should have \u003cimg style=\"vertical-align: -0.186ex; width: 5.278ex; height: 1.903ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-106.png\" alt=\"\" data-tex=\"\\(t = t\u0027\\)\"\u003e. Both observers\nare supposed to employ faultless chronometers, and, of course, to\nallow for the velocity of light in estimating the time when the event\noccurs. It would be thought, therefore, that they would arrive at the\nsame estimate as to the time of the occurrence. It would also have been\nthought that we should have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 11.542ex; height: 2.016ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-107.png\" alt=\"\" data-tex=\"\\[\nx\u0027 =x – vt.\n\\]\"\u003e\u003c/span\u003e\nNeither of these, however, is correct. To obtain the correct\ntransformation, put:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.308ex; width: 14.041ex; height: 4.837ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-108.png\" alt=\"\" data-tex=\"\\[\n\\beta=\\frac{c}{\\sqrt{c^{2}-v^{2}}}\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e is, as always, the velocity of light. Then:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.099ex; width: 25.193ex; height: 7.329ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-109.png\" alt=\"\" data-tex=\"\\[\n\\begin{align*}\nx^{\\prime} \u0026 =\\beta(x-v t) \\\\\nt^{\\prime} \u0026 \\left.=\\beta\\left(t-\\frac{v x}{c^{2}}\\right)\\right\\} \\qquad \\text{(1)}.\n\\end{align*}\n\\]\"\u003e\u003c/span\u003e\nFor the other co-ordinates \u003cimg style=\"vertical-align: -0.464ex; width: 1.736ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-102.png\" alt=\"\" data-tex=\"\\(y\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.68ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-103.png\" alt=\"\" data-tex=\"\\(z\u0027\\)\"\u003e, we still have, as before:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 15.508ex; height: 2.294ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-110.png\" alt=\"\" data-tex=\"\\[\ny^{\\prime}=y,\\quad z^{\\prime}=z.\n\\]\"\u003e\u003c/span\u003e\nIt is the formulæ for \u003cimg style=\"vertical-align: -0.025ex; width: 1.922ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-101.png\" alt=\"\" data-tex=\"\\(x\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.444ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-100.png\" alt=\"\" data-tex=\"\\(t\u0027\\)\"\u003e that are peculiar. These\nformulæ contain, implicitly, the whole of the special theory of\nrelativity.\u003c/p\u003e\n\n\u003cp\u003eThe formula for \u003cimg style=\"vertical-align: -0.025ex; width: 1.922ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-101.png\" alt=\"\" data-tex=\"\\(x\u0027\\)\"\u003e embodies the FitzGerald contraction. Lengths on\neither body, as estimated by an observer on the other, will be shorter\nthan as estimated by an observer on the body on which the lengths are:\nthe longer length will have to the shorter the ratio \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e. More\ninteresting, however, is the effect as regards time. Suppose that an\nobserver on the body \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e judges two events at \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e to\nbe simultaneous, and both at time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e. Then an observer on \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e\nwill judge that they occur at times \u003cimg style=\"vertical-align: -0.339ex; width: 2.432ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-112.png\" alt=\"\" data-tex=\"\\(t_{1}{ }^{\\prime}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.432ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-113.png\" alt=\"\" data-tex=\"\\(t_{2}{}^{\\prime}\\)\"\u003e where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -4.028ex; width: 17.766ex; height: 9.187ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-114.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n\u0026 t_{1}^{\\prime}=\\beta\\left(t-\\frac{v x_{1}}{c^{2}}\\right) \\\\\n\u0026 t_{2}^{\\prime}=\\beta\\left(t-\\frac{v x_{2}}{c^{2}}\\right),\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_51\"\u003e[Pg 51]\u003c/span\u003e\nand therefore:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.654ex; width: 22.86ex; height: 4.957ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-115.png\" alt=\"\" data-tex=\"\\[\nt_{1}^{\\prime}-t_{2}^{\\prime}=\\beta\\frac{{v\\left(x_{2}-x_{1}\\right)}}{c^{2}}.\n\\]\"\u003e\u003c/span\u003e\nThis is not zero unless \u003cimg style=\"vertical-align: -0.339ex; width: 7.581ex; height: 1.658ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-116.png\" alt=\"\" data-tex=\"\\(x_1 = x_2\\)\"\u003e; thus in general events which\nare simultaneous for one observer are not simultaneous for the other.\nWe cannot therefore regard space and time as independent, as has\nalways been done in the past. Even the order of events in time is not\ndefinite: in one system of co-ordinates an event A may precede an event\nB, while in another B may precede A. This, however, is only possible\nif the events are so separated that, no matter how we choose our\nco-ordinates, light starting from either could not reach the place of\nthe other until after the other had occurred.\u003c/p\u003e\n\n\u003cp\u003eThe Lorentz transformation yields the result that:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 22.164ex; height: 2.185ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-117.png\" alt=\"\" data-tex=\"\\[\nc^{2} t^{2}-x^{2}=c^{2} t^{\\prime 2}-x^{\\prime 2}.\n\\]\"\u003e\u003c/span\u003e\nSince \u003cimg style=\"vertical-align: -0.464ex; width: 5.862ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-118.png\" alt=\"\" data-tex=\"\\(y = y\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: -0.186ex; width: 5.749ex; height: 1.903ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-119.png\" alt=\"\" data-tex=\"\\(z = z\u0027\\)\"\u003e, we have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.791ex; width: 46.524ex; height: 2.791ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-120.png\" alt=\"\" data-tex=\"\\[\nc^{2} t^{2}-\\left(x^{2}+y^{2}+z^{2}\\right)=c^{2} t^{\\prime 2}-\\left(x^{\\prime 2}+y^{\\prime 2}+z^{\\prime 2}\\right);\n\\]\"\u003e\u003c/span\u003e\nor, putting \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.648ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-121.png\" alt=\"\" data-tex=\"\\(r\u0027\\)\"\u003e for the distances of the event from the two\nobservers:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 28.404ex; height: 2.565ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-122.png\" alt=\"\" data-tex=\"\\[\n\\begin{equation*}\nc^{2} t^{2}-r^{2}=c^{2} t^{\\prime 2}-r^{\\prime 2} \\qquad \\text{(2)}\n\\end{equation*}\n\\]\"\u003e\u003c/span\u003e\nThis result is general—\u003ci\u003ei.e.\u003c/i\u003e given any two reference-bodies in\nuniform relative motion, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e is the distance between two events\naccording to one system, \u003cimg style=\"vertical-align: -0.025ex; width: 1.648ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-121.png\" alt=\"\" data-tex=\"\\(r\u0027\\)\"\u003e the distance according to the other,\nand if \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.444ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-100.png\" alt=\"\" data-tex=\"\\(t\u0027\\)\"\u003e are the corresponding time-intervals between the\nevents, equation (2) will always hold. Thus\n\u003cimg style=\"vertical-align: -0.186ex; width: 8.545ex; height: 2.072ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-123.png\" alt=\"\" data-tex=\"\\(c^{2}t^{2}-r^{2}\\)\"\u003e represents a physical quantity, independent of\nthe choice of co-ordinates; it is called the square of the \"interval\"\nbetween the two events. There are two cases, according as it is\npositive or negative. When it is positive, the interval between the\nevents is called \"time-like\"; when negative, \"space-like.\" In the\nintermediate case in which it is zero, the events are such that one\nlight-ray can be present at each. In this case, one event might be the\nseeing of the other. The time-order of two events\u003cspan class=\"pagenum\" id=\"Page_52\"\u003e[Pg 52]\u003c/span\u003e will be different in\ndifferent reference-systems when their interval is space-like, but when\nit is time-like the time-order is the same in all systems, though the\nmagnitude of the time-interval varies.\u003c/p\u003e\n\n\u003cp\u003eWhen the interval between two events is time-like, it is possible\nfor a body to move in such a way as to be present at both events.\nIn that case, the interval is what clocks on that body will show as\nthe time. When the interval between two events is space-like, it is\npossible for a body to move in such a way that, by its clocks, the two\nevents will be simultaneous; in that case, the interval is what, in\nrelation to that body, appears as their distance. (In these remarks,\nwe are taking the velocity of light as the unit of velocity, which is\nconvenient in relativity theory.) Both these are consequences of the\nLorentz transformation. From the first of them it follows that, if two\nevents both happen to me, the time between them as measured by my watch\n(assuming it to be a good watch) is the \"interval\" between them, and\nhas still a physical significance. Thus the time that is concerned in\npsychology is unaffected by relativity, assuming that everything that\npsychology is concerned with happens, from a physical point of view, in\nthe body of the person whose mental events are being considered. This\nis an assumption for which grounds will be given at a later stage.\u003c/p\u003e\n\n\u003cp\u003eIt follows from the ambiguity of simultaneity between distant events\nthat we cannot speak unambiguously of \"\u003ci\u003ethe\u003c/i\u003e distance between two\nbodies at a given time.\" If the two bodies are in relative motion,\na \"given time\" will be different for the two bodies and different\nagain for other reference-bodies. It follows that such a conception\ncannot enter into the correct statement of a physical law. On this\nground alone, we can conclude that the Newtonian form of the law of\ngravitation cannot be quite right. Fortunately, Einstein has supplied\nthe necessary correction.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_53\"\u003e[Pg 53]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt will be observed that, as a consequence of the Lorentz\ntransformation, the mass of a body will not be the same when it is\nin motion relatively to the reference-body as when it is at rest\nrelatively to it. The mass of a body is inversely proportional to the\nacceleration produced in it by a given force, and two reference-bodies\nin uniform relative motion will give different results for the\nacceleration of a third body. This is obvious as a consequence of\nthe FitzGerald contraction. The increase of mass with rapid motion\nwas known experimentally before the special theory of relativity had\nexplained it; it is very marked for velocities such as those attained\nby \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e-particles (electrons) emitted by radio-active bodies,\nsince these velocities may be as great as 99 per cent, of the velocity\nof light. This change of mass, like the FitzGerald contraction, seemed\nstrange and anomalous until the special theory of relativity explained\nit.\u003c/p\u003e\n\n\u003cp\u003eOne more point is important as showing how easily what seems axiomatic\nmay be false: it concerns the composition of velocities. Suppose three\nbodies moving uniformly in the same direction: the velocity of the\nsecond relatively to the first is \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e, that of the third relatively\nto the second is \u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-124.png\" alt=\"\" data-tex=\"\\(w\\)\"\u003e. What is the velocity of the third relatively to\nthe first? One would have thought it must be \u003cimg style=\"vertical-align: -0.186ex; width: 5.483ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-125.png\" alt=\"\" data-tex=\"\\(v + w\\)\"\u003e, but in fact it\nis:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.423ex; width: 7.809ex; height: 5.272ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-126.png\" alt=\"\" data-tex=\"\\[\n\\frac{v + w}{1 + \\tfrac{vw}{c^{2}}}\n\\]\"\u003e\u003c/span\u003e\nIt will be seen that this \u003cimg style=\"vertical-align: -0.312ex; width: 3.368ex; height: 1.751ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-127.png\" alt=\"\" data-tex=\"\\(\\leq c\\)\"\u003e; if \u003cimg style=\"vertical-align: -0.186ex; width: 5.094ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-128.png\" alt=\"\" data-tex=\"\\(v = c\\)\"\u003e or \u003cimg style=\"vertical-align: -0.186ex; width: 5.617ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-129.png\" alt=\"\" data-tex=\"\\(w = c\\)\"\u003e, it\nis \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, otherwise it is less than \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e. This is an illustration of\nthe way in which the velocity of light plays the part of infinity in\nrelation to material motions.\u003c/p\u003e\n\n\u003cp\u003eThe special theory set itself the task of making the laws of physics\nthe same relatively to any two co-ordinate systems in uniform\nrectilinear relative motion. There were two sets of equations to be\n\u003cspan class=\"pagenum\" id=\"Page_54\"\u003e[Pg 54]\u003c/span\u003e\nconsidered: those of Newtonian dynamics, and Maxwell\u0027s equations.\nThe latter are unaltered by a Lorentz transformation, but the former\nrequire certain adaptations. These, however, are such as experimental\nresults had already suggested. Thus the solution of the problem in hand\nwas complete, but of course it was obvious from the first that the\nreal problem was more general. There could be no reason for confining\nourselves to two co-ordinate systems in uniform rectilinear motion; the\nproblem ought to be solved for any two co-ordinate systems, no matter\nwhat the nature of their relative motion. This is the problem which has\nbeen solved by the general theory of relativity.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_55\"\u003e[Pg 55]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_VI\"\u003eCHAPTER VI\u003cbr\u003e\nTHE GENERAL THEORY OF RELATIVITY\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE general theory of relativity has a much wider sweep than the\nspecial theory, and a greater philosophical interest, apart from the\none matter of the substitution of space-time for space and time. The\ngeneral theory demands an abandonment of all direct relations between\ndistant events, the relations upon which space-time depends being\nprimarily confined to very small regions, and only extended, where\nthey can be extended, by means of integration. All the old apparatus\nof geometry—straight lines, circles, ellipses, etc.—is gone. What\nbelongs to \u003ci\u003eanalysis situs\u003c/i\u003e remains, with certain modifications;\nand there is a new geometry of geodesics, which has come from Gauss\u0027s\nstudy of surfaces by way of Riemann\u0027s inaugural dissertation. Geometry\nand physics are no longer distinct, so long as we are not considering\nthe parts of physics which introduce atomicity, such as electrons,\nprotons, and quanta. Perhaps even this exception may not long remain.\nThere are parts of physics which, so far, lie outside the general\ntheory of relativity, but there are no parts of physics to which it\nis not in some degree relevant. And its importance to philosophy is\nperhaps even greater than its importance to physics. It has, of course,\nbeen seized upon by philosophers of different schools as affording\nsupport to their respective nostrums; St. Thomas, Kant, and Hegel\nare claimed to have anticipated it. But I do not think that any of\nthe philosophers who make these suggestions have taken the trouble\nto understand the theory. For my part, I do not profess to know\nexactly what its philosophical consequences will prove to be, but I am\nconvinced that they are far-reaching, and quite different from\u003cspan class=\"pagenum\" id=\"Page_56\"\u003e[Pg 56]\u003c/span\u003e what\nthey seem to philosophers who are ignorant of mathematics.\u003c/p\u003e\n\n\u003cp\u003eIn the present chapter, I wish to consider Einstein\u0027s theory without\nany regard to its philosophical implications, simply as a logical\nsystem. The system starts by assuming a four-dimensional manifold\nhaving a definite order. The form which this assumption takes is\nsomewhat technical: it is assumed that, when we have what might be\ncalled an ordinary set of co-ordinates—\u003ci\u003ee.g.\u003c/i\u003e those which would\nnaturally be employed in Newtonian astronomy—there are certain\ntransformations of these co-ordinates which are legitimate, and certain\nothers which are not. Those which are legitimate are those which\ntransform infinitesimal distances into infinitesimal distances. This\nmeans to say that the transformations must be continuous. Perhaps what\nis assumed may be stated as follows: Given a set of points \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-130.png\" alt=\"\" data-tex=\"\\(p_1\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-131.png\" alt=\"\" data-tex=\"\\(p_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-132.png\" alt=\"\" data-tex=\"\\(p_3\\)\"\u003e,… whose co-ordinates tend towards a limiting set\nwhich is the co-ordinates of a point \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e, then in any new legitimate\nco-ordinate system those points \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-130.png\" alt=\"\" data-tex=\"\\(p_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-131.png\" alt=\"\" data-tex=\"\\(p_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-132.png\" alt=\"\" data-tex=\"\\(p_3\\)\"\u003e,… must\nhave co-ordinates tending to a limiting set which is the co-ordinates\nof \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e in the new system. This means that certain relations of\norder among the co-ordinates represent properties of the points of\nspace-time, and are presupposed in the assignment of co-ordinates.\nThe accurate statement of what is involved can only be made in\nterms of limits, but the correct meaning is conveyed by saying that\nneighbouring points must have neighbouring co-ordinates. The exact\nnature of the ordinal presuppositions of a relativistic co-ordinate\nsystem will occupy us in a later chapter; for the present I merely\nwish to emphasize that the space-time manifold, in the general theory\nof relativity, has an order which is not arbitrary, and which is\nreproduced in any legitimate co-ordinate system. This order, it is\nimportant to realize, is \u003ci\u003epurely\u003c/i\u003e ordinal, and does not involve\nany metrical elements Nor is it derivable from the metrical relations\nof points\u003cspan class=\"pagenum\" id=\"Page_57\"\u003e[Pg 57]\u003c/span\u003e which are afterwards introduced in the theory—\u003ci\u003ei.e.\u003c/i\u003e\nfrom \"intervals.\"\u003c/p\u003e\n\n\u003cp\u003eThe points of space-time have, of course, no duration as well as no\nspatial extension. It is generally assumed that several events may\noccupy the same point; this is involved in the conception of the\nintersection of world-lines. I think it may also be assumed that\none event may extend over a finite extent of space-time, but on\nthis point the theory is silent, so far as I know. I shall myself,\nin a later chapter, deal with the construction of points as systems\nof events, each of which events has a finite extension; this is a\nsubject which has been especially treated by Dr Whitehead, but I\nshall suggest a method somewhat different from his. So long as we\nconfine ourselves to the theory of relativity, it is not necessary to\nconsider whether events have a finite extension, though I think it is\nnecessary to assume that two events may both occupy the same point of\nspace-time. Even on this, however, there is a certain vagueness in the\nauthoritative expositions, which is due mainly to the large scale of\nthe phenomena with which the theory is principally concerned. Sometimes\nit would seem as if the whole earth counted as a point; certainly one\nphysical laboratory does so in the practice of writers on relativity.\nOn occasion, Professor Eddington considers an area of \u003cimg style=\"vertical-align: -0.05ex; width: 7.947ex; height: 2.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-133.png\" alt=\"\" data-tex=\"\\(9 \\times\n10^{23}\\)\"\u003e square kilometres to be an infinitesimal of the second order.\nThe fact that such a view is appropriate in discussions of relativity\nmakes it unnecessary to be precise as to what is meant by saying that\ntwo events occupy the same point, or that two world-lines intersect.\nFor the present I shall assume that this is possible in a strict sense;\nmy reasons will be given in a later chapter.\u003c/p\u003e\n\n\u003cp\u003eIt is assumed that every point of space-time can have four real numbers\nassigned to it, and conversely that any four real numbers (at any rate\nwithin certain limits) are the co-ordinates of a point. This amounts to\n\u003cspan class=\"pagenum\" id=\"Page_58\"\u003e[Pg 58]\u003c/span\u003ethe assumption that the number of points is \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-134.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_{0}}\\)\"\u003e, where\n\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-135.png\" alt=\"\" data-tex=\"\\(\\aleph_{0}\\)\"\u003e is the number of finite integers; that is to say, the\nnumber of points is the number of the Cantorian continuum. Every class\nof \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-134.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_{0}}\\)\"\u003e terms is the field of various multiple relations\nwhich arrange the class in a four-dimensional continuum—or an\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e-dimensional continuum, for that matter. But we require a little\nmore than this. Of all the ways of arranging the points of space-time\nin a four-dimensional continuum, there is only one that has physical\nsignificance; the others exist only for mathematical logic. That means\nthat there must be among points relations derivable from an empirical\nbasis, which generate a four-dimensional continuum. These will be the\nordinal relations spoken of in the last paragraph but one. We assume,\ntherefore, that these ordinal relations generate a continuum, and that\nco-ordinates are so assigned that neighbouring points have neighbouring\nco-ordinates. More exactly the co-ordinates of the limit of a set of\npoints are the limits of the co-ordinates of the set. This is not a law\nof nature, but a prescription as to the manner in which co-ordinates\nare assigned. It leaves great latitude, but not complete latitude. It\nallows any system of co-ordinates to be replaced by another system in\nwhich the new co-ordinates are any continuous functions of the old\nco-ordinates, but it excludes discontinuous functions.\u003c/p\u003e\n\n\u003cp\u003eWe now assume that any two neighbouring points have a metrical\nrelation, called their \"interval,\" whose square is a quadratic function\nof the differences of their co-ordinates. This is a generalization of\nthe theorem of Pythagoras, which has come by way of Gauss and Riemann.\nIt will be worth while to consider the historical development for a\nmoment.\u003c/p\u003e\n\n\u003cp\u003eBy the theorem of Pythagoras, if two points in a plane have\nco-ordinates (\u003cimg style=\"vertical-align: -0.464ex; width: 5.384ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-136.png\" alt=\"\" data-tex=\"\\(x_1, y_1\\)\"\u003e), (\u003cimg style=\"vertical-align: -0.464ex; width: 5.384ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-137.png\" alt=\"\" data-tex=\"\\(x_2, y_2\\)\"\u003e) and \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-138.png\" alt=\"\" data-tex=\"\\(s\\)\"\u003e is their distance\napart:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 27.615ex; height: 2.565ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-139.png\" alt=\"\" data-tex=\"\\[\ns^{2}= (x_2 – x_1)^{2} + (y_2 – y_1)^{2}\n\\]\"\u003e\u003c/span\u003e\nBy an immediately obvious extension, if two points in space\u003cspan class=\"pagenum\" id=\"Page_59\"\u003e[Pg 59]\u003c/span\u003e have\nco-ordinates (\u003cimg style=\"vertical-align: -0.464ex; width: 8.43ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-140.png\" alt=\"\" data-tex=\"\\(x_1, y_1, z_1\\)\"\u003e), (\u003cimg style=\"vertical-align: -0.464ex; width: 8.43ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-141.png\" alt=\"\" data-tex=\"\\(x_2, y_2, z_2\\)\"\u003e), their distance\napart is \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-138.png\" alt=\"\" data-tex=\"\\(s\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 40.603ex; height: 2.565ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-142.png\" alt=\"\" data-tex=\"\\[\ns^{2}= (x_2 – x_1)^{2} + (y_2 – y_1)^{2} + (z_2 – z_1)^{2}.\n\\]\"\u003e\u003c/span\u003e\nIf the distance apart is small, we write \u003cimg style=\"vertical-align: -0.025ex; width: 2.471ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-143.png\" alt=\"\" data-tex=\"\\(dx\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.285ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-144.png\" alt=\"\" data-tex=\"\\(dy\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 2.229ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-145.png\" alt=\"\" data-tex=\"\\(dz\\)\"\u003e for\n\u003cimg style=\"vertical-align: -0.339ex; width: 7.329ex; height: 1.658ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-146.png\" alt=\"\" data-tex=\"\\(x_2 – x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 6.958ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-147.png\" alt=\"\" data-tex=\"\\(y_2 – y_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 6.845ex; height: 1.658ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-148.png\" alt=\"\" data-tex=\"\\(z_2 – z_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e for \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-138.png\" alt=\"\" data-tex=\"\\(s\\)\"\u003e; thus:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 22.35ex; height: 2.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-150.png\" alt=\"\" data-tex=\"\\[\nds^{2} = dx^{2} + dy^{2} + dz^{2}.\n\\]\"\u003e\u003c/span\u003e\nGauss considered a problem concerned with surfaces, which arises\nnaturally out of the above. On a surface, the position of a point\ncan be fixed by two co-ordinates, which need not involve reference\nto anything outside the surface. Thus on the earth position is fixed\nby latitude and longitude. Suppose \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-151.png\" alt=\"\" data-tex=\"\\(u\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e are two such\nco-ordinates which fix position on a surface. Then in general we shall\nnot have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 15.728ex; height: 2.185ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-152.png\" alt=\"\" data-tex=\"\\[\nds^{2} = du^{2} + dv^{2}\n\\]\"\u003e\u003c/span\u003e\nfor the distance between neighbouring points; in general, we cannot\nget a formula of this kind however we may define \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-151.png\" alt=\"\" data-tex=\"\\(u\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e. We\ncan get a formula of this kind on a cylinder or a cone, and generally\non what are called \"developable\" surfaces, but not, \u003ci\u003ee.g.\u003c/i\u003e, on a\nsphere. The general formula takes the shape:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 30.199ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-153.png\" alt=\"\" data-tex=\"\\[\nds^{2} = E du^{2} + 2F du dv + G dv^{2},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.695ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-155.png\" alt=\"\" data-tex=\"\\(F\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-156.png\" alt=\"\" data-tex=\"\\(G\\)\"\u003e are in general functions of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-151.png\" alt=\"\" data-tex=\"\\(u\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e,\nnot constants. Gauss showed that there are certain functions of \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 1.695ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-155.png\" alt=\"\" data-tex=\"\\(F\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-156.png\" alt=\"\" data-tex=\"\\(G\\)\"\u003e which have the same value however the co-ordinates \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-151.png\" alt=\"\" data-tex=\"\\(u\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e may be defined; these functions express properties of the\nsurface, which can theoretically be discovered by measurements carried\nout on the surface, without reference to external space.\u003c/p\u003e\n\n\u003cp\u003eRiemann extended this method to space. He supposed that the theorem\nof Pythagoras may be not exact, and that the correct formula for the\ndistance between two points may be such as results from Gauss\u0027s formula\nby adding another variable. He showed that this supposition could\nbe made the basis of non-Euclidean geometry. The whole subject of\u003cspan class=\"pagenum\" id=\"Page_60\"\u003e[Pg 60]\u003c/span\u003e\nnon-Euclidean geometry remained, however, without visible relevance\nto physics until it was utilized in Einstein\u0027s theory of gravitation,\nwhich results from the combination of Riemann\u0027s ideas with the\nsubstitution of space-time \"interval\" for distance in space and time,\nwhich had already been made in the special theory of relativity.\u003c/p\u003e\n\n\u003cp\u003eIn the special theory of relativity, as we saw, the interval between\ntwo space-time points, one of which is the origin, is \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-138.png\" alt=\"\" data-tex=\"\\(s\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 24.181ex; height: 2.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-157.png\" alt=\"\" data-tex=\"\\[\ns^{2} = x^{2} + y^{2} + z^{2} – c^{2} t^{2},\n\\]\"\u003e\u003c/span\u003e\nif the interval is space-like, and:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 24.181ex; height: 2.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-158.png\" alt=\"\" data-tex=\"\\[\ns^{2} = c^{2} t^{2} – x^{2} – y^{2} – z^{2},\n\\]\"\u003e\u003c/span\u003e\nif the interval is time-like. In practice, the latter form is always\ntaken. Any system of co-ordinates allowed by the special theory\ngives the same value for the interval between two given space-time\npoints. But we are now allowing much greater latitude in the choice of\nco-ordinates, and we are assuming that the special theory represents\nonly an approximation, being not strictly true except in the absence\nof a gravitational field. We still assume that, for small distances,\nthere is a quadratic function of the co-ordinate differences which\nhas a physical significance, and has the same value however the\nco-ordinates may be assigned, subject to the condition of continuity\nalready explained. That is, if \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-159.png\" alt=\"\" data-tex=\"\\(x_4\\)\"\u003e are\nthe co-ordinates of a point, and \u003cimg style=\"vertical-align: -0.339ex; width: 8.506ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-160.png\" alt=\"\" data-tex=\"\\(x_1 + dx_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 8.506ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-161.png\" alt=\"\" data-tex=\"\\(x_2 + dx_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 8.506ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-162.png\" alt=\"\" data-tex=\"\\(x_3+ dx_3\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.339ex; width: 8.506ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-163.png\" alt=\"\" data-tex=\"\\(x_4 + dx_4\\)\"\u003e are the co-ordinates of a neighbouring point,\nwe assume that there is a quadratic function:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.685ex; width: 30.623ex; height: 2.382ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-164.png\" alt=\"\" data-tex=\"\\[\n\\Sigma g_{\\mu \\nu} d x_{\\mu} d x_{\\nu}\\quad (\\mu, \\nu = 1, 2, 3, 4),\n\\]\"\u003e\u003c/span\u003e\nwhich has the same value however the co-ordinates may be assigned;\nwe then define \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e as the \"interval\" between the two neighbouring\npoints. The \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-165.png\" alt=\"\" data-tex=\"\\(g_{\\mu \\nu}\\)\"\u003e\u0027s will be functions of the co-ordinates\n(in general not constants), and for convenience we take \u003cimg style=\"vertical-align: -0.685ex; width: 9.176ex; height: 2.004ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-166.png\" alt=\"\" data-tex=\"\\(g_{\\mu \\nu} = g_{\\nu \\mu}\\)\"\u003e.\nJust as Gauss was able to deduce the\u003cspan class=\"pagenum\" id=\"Page_61\"\u003e[Pg 61]\u003c/span\u003e geometry of a\nsurface from his formula, so we can deduce the geometry of space-time\nfrom our formula. But as we include time, our geometry is not merely\ngeometry, but physics; in other words, it combines history with\ngeography.\u003c/p\u003e\n\n\u003cp\u003eAt a great distance from matter, the special theory will still be\ntrue, and therefore space will be Euclidean, since, if we put \u003cimg style=\"vertical-align: -0.186ex; width: 6.142ex; height: 1.756ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-167.png\" alt=\"\" data-tex=\"\\(dt\n= 0\\)\"\u003e, the special theory gives the Euclidean formula for distance.\nThe neighbourhood of gravitating matter is shown by a non-Euclidean\ncharacter of the region concerned. This, however, requires some\npreliminary explanations, more especially an explanation of the method\nof tensors, which will form the subject of the next chapter.\u003c/p\u003e\n\n\u003cp\u003eEverything in the general theory of relativity is dependent upon\nthe existence of the above formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e. The formula itself\nis of the nature of an empirical generalization; no \u003ci\u003ea priori\u003c/i\u003e\njustification for it is suggested. It is a generalization of the\ntheorem of Pythagoras, which could formerly be proved. But the proof\nrested upon Euclid\u0027s axioms, which there is no reason to regard as\nexactly true. More than that, there is difficulty in assigning a\nmeaning to his fundamental concepts, such as the \"straight\" line. The\nold geometry assumed a static space, which it could do because space\nand time were supposed to be separable. It is natural to think of\nmotion as following a path in space which is there before and after\nthe motion: a tram moves along pre-existing tram-lines. This view of\nmotion, however, is no longer tenable. A moving point is a series\nof positions in space-time; a later moving point cannot pursue the\n\"same\" course, since its time co-ordinate is different, which means\nthat, in another equally legitimate system of co-ordinates, its space\nco-ordinates also will be different. We think of a tram as performing\nthe same journey every day, because we think of the earth as fixed; but\nfrom the sun\u0027s point of view, the tram never repeats a former journey.\n\"We cannot step twice into the same rivers,\" as\u003cspan class=\"pagenum\" id=\"Page_62\"\u003e[Pg 62]\u003c/span\u003e Heraclitus says.\nIt is thus obvious that, in place of Euclid\u0027s static straight line,\nwe shall have to substitute a movement having some special property\ndefined in terms of space-time, not of space. The movement required is\na \"geodesic,\" concerning which we shall have more to say later.\u003c/p\u003e\n\n\u003cp\u003eIn relativity theory, distant space-time points have only such\nrelations as can be obtained by integration from the relations of\nneighbouring points. Since the distance between two points is always\nfinite, what we call a relation between neighbouring points is not\nreally a relation between points at all, but is a limit, like a\nvelocity. Only the language of the calculus can express accurately\nwhat is meant. One might say, speaking pictorially, that the notion of\n\"interval\" is concerned with what, at each point, is \u003ci\u003etending\u003c/i\u003e to\nhappen, although we cannot say that this will actually happen, because\nbefore any assigned point is reached something may have occurred to\ncause a diversion. This is, of course, the case with velocity. From the\nfact that, at a given instant, a body is moving in a given direction\nwith a given velocity, we can infer nothing whatever as to where the\nbody will be at another assigned instant, however near to the first.\nTo infer the path of a body from its velocity, we must know its\nvelocity throughout a finite time. Similarly the formula for interval\ncharacterizes each separate point of space-time. To obtain the interval\nbetween one point and another, however near together, we must specify a\nroute, and integrate along that route. As we shall see, however, there\nare routes which may be called \"natural\"—namely, geodesics. It is only\nby means of them that the notion of interval can be profitably extended\nto the relations of points at a finite distance from each other.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_63\"\u003e[Pg 63]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_VII\"\u003eCHAPTER VII\u003cbr\u003e\nTHE METHOD OF TENSORS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE method of tensors contains the answer to a question which is\nrendered urgent by the arbitrary character of our co-ordinates. How\ncan we know whether a formula expressed in terms of our co-ordinates\nexpresses something which describes the physical occurrences, and\nnot merely the particular co-ordinate system which we happen to be\nemploying? A striking example of the mistakes that are possible in this\nrespect is afforded by simultaneity. Suppose we have two events, whose\nco-ordinates, in the system we are employing, are (\u003cimg style=\"vertical-align: -0.464ex; width: 7.29ex; height: 1.88ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-169.png\" alt=\"\" data-tex=\"\\(x, y, z, t\\)\"\u003e) and\n(\u003cimg style=\"vertical-align: -0.464ex; width: 9.8ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-170.png\" alt=\"\" data-tex=\"\\(x\u0027, y\u0027, z\u0027, t\u0027\\)\"\u003e)—\u003ci\u003ei.e.\u003c/i\u003e their time co-ordinates are the same.\nBefore the special theory of relativity everybody would have asserted\nthat this represented a physical fact about the two events—namely,\nthat they are simultaneous. Now we know that the fact concerned is one\nwhich also involves mention of the co-ordinate system—that is to say,\nit is not a relation between the two events only, but between them and\nthe body of reference. But this is to speak the language of the special\ntheory. In the general theory, our co-ordinates may have no important\nphysical significance, and a pair of events which have one co-ordinate\nidentical need not have any intrinsic physical property not possessed\nby other pairs of events. In practice, there must be \u003ci\u003esome\u003c/i\u003e\nprinciple on which co-ordinates are assigned, and this principle must\nhave some physical significance. But we might, for instance, measure\ntime by the worst clock ever made, provided it only went wrong and\ndid not actually stop. And we might use a certain worm as our unit\nof length, disregarding the \"FitzGerald contraction\" to which motion\nsubjects him.\u003c/p\u003e\n\n\u003cp\u003eIn that case, if we say that there was unit distance between\u003cspan class=\"pagenum\" id=\"Page_64\"\u003e[Pg 64]\u003c/span\u003e two\nevents which both occurred at a certain instant, we shall be making a\ncomplicated comparison between the events, a bad clock, and a certain\nworm—that is to say, we shall be making a statement which depends\nupon our co-ordinate system. We want to discover a sufficient, if not\nnecessary, condition which, if fulfilled, insures that a statement\nin terms of co-ordinates has a meaning independent of co-ordinates.\nThe difference is more or less analogous to that, in ordinary\nlanguage, between linguistic statements and statements which (as is\nusually the case) are about what words mean. If I say \"strength is\na desirable quality,\" my statement can be put into French or German\nwithout change of meaning. But if I say \"strength is a word containing\nseven consonants and only one vowel,\" my statement becomes false if\ntranslated into French or German. Now in physics co-ordinates are\nanalogous to words, with the difference that it is much harder to\ndistinguish \"linguistic\" statements from others. This is what the\nmethod of tensors undertakes to do.\u003c/p\u003e\n\n\u003cp\u003eIt does not seem possible to state the method of tensors in untechnical\nlanguage; I am afraid that those philosophers who have not thought it\nworth while to learn the calculus cannot hope to understand it. Perhaps\nin time some simple way of explaining it may be found, but none has\nbeen found so far.\u003ca id=\"FNanchor_24\" href=\"#Footnote_24\" class=\"fnanchor\"\u003e[24]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eSuppose we have a vector quantity whose components are \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-171.png\" alt=\"\" data-tex=\"\\(A^{1}\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-172.png\" alt=\"\" data-tex=\"\\(A^{2}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.885ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-173.png\" alt=\"\" data-tex=\"\\(A^{3}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.904ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-174.png\" alt=\"\" data-tex=\"\\(A^{4}\\)\"\u003e. (Here 1, 2, 3, 4 play the part of suffixes,\nnot of exponents denoting powers.) It happens in certain cases that, if\nwe transform to any other co-ordinates \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-175.png\" alt=\"\" data-tex=\"\\(x\u0027_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-176.png\" alt=\"\" data-tex=\"\\(x\u0027_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.616ex; width: 2.282ex; height: 2.333ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-177.png\" alt=\"\" data-tex=\"\\(x\u0027_3\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.6ex; width: 2.282ex; height: 2.317ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-178.png\" alt=\"\" data-tex=\"\\(x\u0027_4\\)\"\u003e, which are continuous functions of the old co-ordinates\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-159.png\" alt=\"\" data-tex=\"\\(x_4\\)\"\u003e, we shall have, as the components\nof the vector in the new co-ordinates, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-179.png\" alt=\"\" data-tex=\"\\(A\u0027^{1}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-180.png\" alt=\"\" data-tex=\"\\(A\u0027^{2}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.885ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-181.png\" alt=\"\" data-tex=\"\\(A\u0027^{3}\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.904ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-182.png\" alt=\"\" data-tex=\"\\(A\u0027^{4}\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.927ex; width: 43.408ex; height: 5.267ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-183.png\" alt=\"\" data-tex=\"\\[\nA^{\\prime 1}=\\frac{\\partial x_{1}^{\\prime}}{\\partial x_{1}} A^{1}+\\frac{\\partial x_{1}^{\\prime}}{\\partial x_{2}} A^{2}+\\frac{\\partial\n x_{1}^{\\prime}}{\\partial x_{3}} A^{3}+\\frac{\\partial x_{1}^{\\prime}}{\\partial x_{3}} A^{4}\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_65\"\u003e[Pg 65]\u003c/span\u003e\nwith similar formulæ for \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-180.png\" alt=\"\" data-tex=\"\\(A\u0027^{2}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.885ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-181.png\" alt=\"\" data-tex=\"\\(A\u0027^{3}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.904ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-182.png\" alt=\"\" data-tex=\"\\(A\u0027^{4}\\)\"\u003e. When this\nhappens, the vector in question is called \u003ci\u003econtravariant\u003c/i\u003e. The\nsimplest example is (\u003cimg style=\"vertical-align: -0.439ex; width: 16.851ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-184.png\" alt=\"\" data-tex=\"\\(dx_1, dx_2, dx_3, dx_4\\)\"\u003e). Except in this one\ncase, the \"contravariant\" property is symbolized by the upper position\nof the suffix.\u003c/p\u003e\n\n\u003cp\u003eAgain we may have a vector, whose components are \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-171.png\" alt=\"\" data-tex=\"\\(A^{1}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-172.png\" alt=\"\" data-tex=\"\\(A^{2}\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.885ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-173.png\" alt=\"\" data-tex=\"\\(A^{3}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 2.685ex; height: 1.904ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-174.png\" alt=\"\" data-tex=\"\\(A^{4}\\)\"\u003e, which is transformed according to the law:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.225ex; width: 42.968ex; height: 5.372ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-185.png\" alt=\"\" data-tex=\"\\[\nA_{1}^{\\prime}=\\frac{\\partial x_{1}}{\\partial x_{1}} A_{1}+\\frac{\\partial x_{2}}{\\partial x_{1}^{\\prime}} A_{2}+\\frac{\\partial\n x_{3}}{\\partial x_{1}^{\\prime}} A_{3}+\\frac{\\partial x_{4}}{\\partial x_{1}^{\\prime}} A_{4}\n\\]\"\u003e\u003c/span\u003e\nwith similar formulæ for \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-180.png\" alt=\"\" data-tex=\"\\(A\u0027^{2}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.885ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-181.png\" alt=\"\" data-tex=\"\\(A\u0027^{3}\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.124ex; height: 1.904ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-182.png\" alt=\"\" data-tex=\"\\(A\u0027^{4}\\)\"\u003e. Such a vector\nis called \u003ci\u003ecovariant\u003c/i\u003e. The simplest example is the vector whose\ncomponents are:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.927ex; width: 21.878ex; height: 5.078ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-186.png\" alt=\"\" data-tex=\"\\[\n\\frac{\\partial \\varphi}{\\partial x_{1}}, \\frac{\\partial \\varphi}{\\partial x_{2}}, \\frac{\\partial \\varphi}{\\partial x_{3}}, \\frac{\\partial \\varphi}{\\partial x_{4}},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e is some function which has a fixed value at each\npoint, independently of the co-ordinate system.\u003c/p\u003e\n\n\u003cp\u003eIt is obvious that, if we have two contravariant vectors \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e whose components are equal in one system of co-ordinates, then\ntheir components are equal in any system of co-ordinates; and the same\napplies to two covariant vectors \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e. This follows at\nonce from the above rules of transformation. Thus an equality of two\ncontravariant vectors, or of two covariant vectors, when it occurs, is\na fact independent of the co-ordinate system. It is, in fact, a tensor\nequation of the simplest kind.\u003c/p\u003e\n\n\u003cp\u003eThe general definition of a \"tensor\" is a generalization of those of\ncontravariant and covariant vectors. Instead of a vector with only four\ncomponents, we may have a quantity with sixteen components:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 62.112ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-190.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\nA_{11}, A_{12}, A_{13}, A_{14}, A_{21}, A_{22}, \u0026 A_{23}, A_{24}, \\\\\n\u0026 A_{31}, A_{32}, A_{33}, A_{34}, A_{41}, A_{42}, A_{43}, A_{44}.\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nSuch a quantity may be denoted by \"\u003cimg style=\"vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-191.png\" alt=\"\" data-tex=\"\\(A_{\\mu \\nu}\\)\"\u003e\" where it is\nunderstood that \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e can each take all values from 1\nto 4.\u003cspan class=\"pagenum\" id=\"Page_66\"\u003e[Pg 66]\u003c/span\u003e Similarly we may have a quantity with sixty-four components,\n\u003cimg style=\"vertical-align: -0.339ex; width: 4.284ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-193.png\" alt=\"\" data-tex=\"\\(A_{111}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 4.284ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-194.png\" alt=\"\" data-tex=\"\\(A_{112}\\)\"\u003e, etc.; such a quantity may be denoted by\n\"\u003cimg style=\"vertical-align: -0.685ex; width: 4.611ex; height: 2.305ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-195.png\" alt=\"\" data-tex=\"\\(A_{\\mu\\nu\\sigma}\\)\"\u003e\" where \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e can each take\nall values from 1 to 4. Such quantities are called \"tensors\" if they\nobey laws of transformation analogous to those of contravariant and\ncovariant vectors. Thus a contravariant tensor with sixteen components,\nwhich is written \"\u003cimg style=\"vertical-align: 0; width: 3.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-197.png\" alt=\"\" data-tex=\"\\(A^{\\mu \\nu}\\)\"\u003e,\" is one which satisfies the rule:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.148ex; width: 60.628ex; height: 5.937ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-198.png\" alt=\"\" data-tex=\"\\[\nA\u0027^{11}=\\left(\\frac{\\partial x\u0027_{1}}{\\partial x_{1}}\\right)^{2} A^{11}+\\left(\\frac{\\partial x\u0027_{1}}{\\partial x_{2}}\\right)^{2} A^{22}+\\ldots+\\frac{\\partial x\u0027_{1}}{\\partial x_{1}}\\frac{\\partial x_{1}} {\\partial x_{2}}A^{12}+\\ldots\n\\]\"\u003e\u003c/span\u003e\nwith similar equations for the other components—\u003ci\u003ee.g.\u003c/i\u003e:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.891ex; width: 45.742ex; height: 5.232ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-199.png\" alt=\"\" data-tex=\"\\[\nA\u0027^{12}=\\frac{\\partial x\u0027_{1}}{\\partial x_{1}}\\frac{\\partial x\u0027_{2}}{\\partial x_{1}} A^{11}+ \\ldots + \\frac{\\partial x\u0027_{1}}{\\partial x_{1}}\\frac{\\partial x\u0027_{2}}{\\partial x_{1}} A^{12}+\\ldots\n\\]\"\u003e\u003c/span\u003e\nThese equations are comprised in:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.06ex; width: 20.872ex; height: 6.723ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-200.png\" alt=\"\" data-tex=\"\\[\nA\u0027^{\\mu\\nu}=\\sum_{\\alpha,\\beta} \\frac{\\partial x\u0027_{\\mu}}{\\partial x_{\\alpha}} \\frac{\\partial x\u0027_{\\nu}}{\\partial x_{\\beta}},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are to take all values from 1 to 4.\nSimilarly a covariant tensor with sixteen components, written \"\u003cimg style=\"vertical-align: 0; width: 3.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-202.png\" alt=\"\" data-tex=\"\\(A^{\\mu\\nu}\\)\"\u003e,\"\nis one which is transformed according to the rule:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.06ex; width: 24.627ex; height: 6.368ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-203.png\" alt=\"\" data-tex=\"\\[\nA\u0027_{\\mu \\nu}=\\sum_{\\alpha, \\beta} \\frac{\\partial x_{a}}{\\partial x\u0027^{\\mu}} \\frac{\\partial x_{\\beta}}{\\partial x\u0027_{\\nu}}A_{\\alpha \\beta},\n\\]\"\u003e\u003c/span\u003e\nand a mixed tensor, written \u003cimg style=\"vertical-align: -0.904ex; width: 2.849ex; height: 2.524ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-204.png\" alt=\"\" data-tex=\"\\(A^{\\nu}_{\\mu}\\)\"\u003e, is one which satisfies\nthe rule:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.06ex; width: 22.761ex; height: 6.377ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-205.png\" alt=\"\" data-tex=\"\\[\nA_{\\mu}^{\u0027\\nu}=\\sum_{\\alpha,\\beta} \n\\frac{\\partial x_{a}}{\\partial x\u0027_{\\mu}} \\frac{\\partial x\u0027_{\\nu}}{\\partial x_{\\beta}} A_{\\alpha}^{\\beta}.\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThere is no difficulty in extending these definitions to any number of\nsuffixes. It is obvious, as in the case of contravariant and covariant\nvectors, that if two tensors of the same kind are equal in one system\nof co-ordinates they are equal in any system of co-ordinates, so that\ntensor equations express conditions which are independent of the choice\nof co-ordinates. For this reason it is necessary to express all the\ngeneral laws of physics as tensor equations; if this cannot be done,\nthe\u003cspan class=\"pagenum\" id=\"Page_67\"\u003e[Pg 67]\u003c/span\u003e law concerned must be wrong, and must require such correction\nas will enable it to be expressed as a tensor equation. The law of\ngravitation is the most noteworthy example of this; but perhaps the\nconservation of energy is scarcely less noteworthy.\u003ca id=\"FNanchor_25\" href=\"#Footnote_25\" class=\"fnanchor\"\u003e[25]\u003c/a\u003e It seems\nnatural to suppose that it would be possible to develop a less indirect\nmethod of expressing physical laws than that afforded by the method of\ntensors, which is perhaps a consequence of the historical development\nof physics. Originally, in physics, the co-ordinates were intended to\nexpress physical relations between the event concerned and the origin.\nThree of the co-ordinates were lengths, which, it was thought, could\nbe ascertained by measurement with a rigid rod. The fourth was a time,\nwhich could be measured by a chronometer. There were difficulties,\nhowever, which the progress of physics made increasingly evident.\nSo long as the earth could be regarded as motionless, axes fixed\nrelatively to the earth and clocks which remained on the surface of the\nearth seemed to suffice. It was possible to disregard the facts that\nno body is quite rigid and no clock quite accurate, because the system\nof physical laws suggested by the choice of the most rigid bodies and\nthe most accurate clocks could be used to estimate the departure of\nthese instruments from strict constancy, and the results were on the\nwhole self-consistent. But in astronomical problems, including that of\nthe tides, the earth could not be treated as fixed. It was necessary\nto Newtonian dynamics that the axes should not have any acceleration,\nbut it resulted from the law of gravitation that any material axes must\nhave some acceleration. The axes, therefore, became ideal structures\nin absolute space; actual measurements with actual rods could only\napproximate to the results which would have followed if we could have\nused unaccelerated axes. This difficulty was not the most serious: the\nworst trouble was concerned with absolute\u003cspan class=\"pagenum\" id=\"Page_68\"\u003e[Pg 68]\u003c/span\u003e acceleration. Then came the\nexperimental discovery of the facts which led to the special theory\nof relativity: the variation of length and mass with velocity, and\nthe constancy of the velocity of light \u003ci\u003ein vacuo\u003c/i\u003e no matter what\nbody was used to define the co-ordinates. This set of difficulties\nwas solved by the special theory of relativity, which showed that\nequivalent results come from employing as reference-body any one of\na set of bodies in uniform rectilinear motion. This, however, only\nachieved what Galileo and Newton thought they had achieved. It included\nelectromagnetic phenomena within the scope of relativity as regards\nvelocities, but it was clearly necessary to extend relativity to\naccelerations, and when this was done, co-ordinates ceased to have the\nclear physical meaning they had formerly possessed. It is true that,\neven in the general theory, a co-ordinate, in any system which can\nactually be used, will always have some physicals significance, but its\nsignificance is trivial and complicated, not, as before, important and\nsimple.\u003c/p\u003e\n\n\u003cp\u003eIt is natural to ask: Could we not dispense with co-ordinates\naltogether, since they have become little more than conventional\nnames systematically assigned? Perhaps this will become possible in\ntime, but at present the necessary mathematics is lacking. We wish,\nfor example, to be able to differentiate, and we cannot differentiate\na function unless its arguments and values are numbers. This is not\ndue to what might seem the more difficult parts of the definition of\na differential. We can define for a non-numerical function the limit\n(if it exists) of a function for a given argument, and also the four\nlimits which exist more frequently—viz. the maximum and minimum for\napproaches from above and below; we can also define a \"continuous\"\nnon-numerical function. (See \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, *230—*234.)\nWhat, so far, has not been defined, except for numbers, is a fraction.\nNow \u003cimg style=\"vertical-align: -1.577ex; width: 3.466ex; height: 4.676ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-206.png\" alt=\"\" data-tex=\"\\(\\dfrac{dy}{dx}\\)\"\u003e is\u003cspan class=\"pagenum\" id=\"Page_69\"\u003e[Pg 69]\u003c/span\u003e the limit of a fraction; thus, although we\ncan generalize the notion of a limit, we cannot at present generalize\n\u003cimg style=\"vertical-align: -1.577ex; width: 3.466ex; height: 4.676ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-206.png\" alt=\"\" data-tex=\"\\(\\dfrac{dy}{dx}\\)\"\u003e, because we cannot generalize the notion of a\nfraction. It seems clear \u003ci\u003ea priori\u003c/i\u003e that, since differentiation\nof co-ordinates is physically useful even when the quantitative value\nof the co-ordinates is conventional, there must be some process,\nof which differentiation is a special numerical form, which can be\napplied wherever we have continuous functions, even when they are\nnon-numerical. To define such a process is a problem in mathematical\nlogic, probably soluble, but hitherto unsolved. If it were solved, it\nmight become possible to avoid the elaborate and round-about process of\nassigning co-ordinates and then treating almost all their properties\nas irrelevant, which is what is done when the method of tensors is\nemployed.\u003c/p\u003e\n\n\u003cp\u003eThere are, it is true, certain numbers which are important in the\nnew geometry: they are those giving the measure of intervals. But,\nas we have already seen, two points at a finite distance apart do\nnot have an unambiguous interval; and any two points are at a finite\ndistance apart. The numbers involved in the notion of interval\nare not finite distances, but numbers derivable from the sixteen\ncoefficients \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-207.png\" alt=\"\" data-tex=\"\\(g_{\\mu\\nu}\\)\"\u003e involved in the formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e in\nthe previous chapter. These coefficients themselves depend upon the\nco-ordinate system, but \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e does not. We cannot develop this\ntheme until we have considered geodesics; it is from them that we must\nderive the numbers which have, in the new geometry, the same sort of\nphysical importance as co-ordinates were supposed to have in the old.\nThese numbers will be the integrals of \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e taken along certain\ngeodesics. But, unlike lengths in the old metrical geometry, they are\ngeometrically insufficient. To avoid irrelevant complications, we may\nillustrate this insufficiency by considering the special theory.\u003c/p\u003e\n\n\u003cp\u003eThe most obvious example of the failure of interval to\u003cspan class=\"pagenum\" id=\"Page_70\"\u003e[Pg 70]\u003c/span\u003e constitute a\ngeometry is derived from consideration of light-rays. The interval\nbetween two events which are parts of the same light-ray is zero.\nSuppose now that a light-ray starts from an event \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, and arrives at\nan event \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e at the moment when it reaches \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, another light-ray\nstarts from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and reaches \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e. Then the interval between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nand \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is zero, that between \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e is zero, but that\nbetween \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e may have any time-like magnitude. Euclid proved\nthat two sides of a triangle are together greater than the third side,\nand was criticized on the ground that this proposition was evident even\nto asses. But in relativity geometry this proposition is false. In our\ntriangle \u003cimg style=\"vertical-align: -0.05ex; width: 5.133ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-209.png\" alt=\"\" data-tex=\"\\(ABC\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e are zero, while \u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e may have any\nfinite magnitude.\u003c/p\u003e\n\n\u003cp\u003eAgain, the events which are parts of a single light-ray have a definite\ntime-order, in spite of the fact that the interval between any two of\nthem is zero. This appears as follows. Suppose a light-ray proceeds\nfrom the sun to the moon and is thence reflected to the earth: it\nreaches the earth later than a direct ray which left the sun at the\nsame time. There is therefore a definite sense in saying that the ray\nreached the moon later than it left the sun—\u003ci\u003ei.e.\u003c/i\u003e we can say\nthat the ray went from the sun to the moon, not from the moon to the\nsun. Generalizing, we may say: If \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e are part of one\nlight-ray, and light-rays from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, distinct from the\nprevious light-ray, contain events \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e whose interval is\ntime-like, then the time-order of \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 2.465ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-213.png\" alt=\"\" data-tex=\"\\(C\u0027\\)\"\u003e is the same whatever\nthese new light-rays may be—\u003ci\u003ei.e.\u003c/i\u003e we shall have always \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e\nbefore \u003cimg style=\"vertical-align: -0.05ex; width: 2.465ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-213.png\" alt=\"\" data-tex=\"\\(C\u0027\\)\"\u003e, or always \u003cimg style=\"vertical-align: -0.05ex; width: 2.465ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-213.png\" alt=\"\" data-tex=\"\\(C\u0027\\)\"\u003e before \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e. In the first case, we say\nthat the \"sense\" of the ray is from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e in the second, from\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. This illustrates the difficulties which would arise if\nwe were to attempt to found our geometry on interval alone. We must\nalso take account of the purely ordinal properties of the space-time\nmanifold. These properties give a wide separation between the departure\nof a light-ray from the sun and its\u003cspan class=\"pagenum\" id=\"Page_71\"\u003e[Pg 71]\u003c/span\u003e arrival on the earth, although the\n\"interval\" between these two events is zero.\u003c/p\u003e\n\n\u003cp\u003eReverting now to the method of tensors and its possible eventual\nsimplification, it seems probable that we have an example of a general\ntendency to over-emphasize numbers, which has existed in mathematics\never since the time of Pythagoras, though it was temporarily less\nprominent in later Greek geometry as exemplified in Euclid. Euclid\u0027s\ntheory of proportion does not, of course, dispense with numbers, since\nit uses \"equimultiples\"; but at any rate it requires only integers,\nnot irrationals. Owing to the fact that arithmetic is easy, Greek\nmethods in geometry have been in the background since Descartes, and\nco-ordinates have come to seem indispensable. But mathematical logic\nhas shown that number is logically irrelevant in many problems where\nit formerly seemed essential, notably mathematical induction, limits,\nand continuity. A new technique, which seems difficult because it is\nunfamiliar, is required when numbers are not used; but there is a\ncompensating gain in logical purity. It should be possible to apply\na similar process of purification to physics. The method of tensors\nfirst assigns co-ordinates, and then shows how to obtain results which,\nthough expressed in terms of co-ordinates, do not really depend upon\nthem. There must be a less indirect technique possible, in which we\nuse no more apparatus than is logically necessary, and have a language\nwhich will only express such facts as are now expressed in the language\nof tensors, not such as depend upon the choice of co-ordinates. I\ndo not say that such a method, if discovered, would be preferable\nin practice, but I do say that it would give a better expression\nof the essential relations, and greatly facilitate the task of the\nphilosopher. In the meantime, the method of tensors is technically\ndelightful, and suffices for mathematical needs.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_24\" href=\"#FNanchor_24\" class=\"label\"\u003e[24]\u003c/a\u003e\nFor what follows see Eddington, \u003ci\u003eMathematical Theory of\nRelativity\u003c/i\u003e, chap. \u003cspan class=\"allsmcap\"\u003eII.\u003c/span\u003e, Cambridge, 1924.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_25\" href=\"#FNanchor_25\" class=\"label\"\u003e[25]\u003c/a\u003e\nSee Eddington, \u003ci\u003eop. cit.\u003c/i\u003e, p. 134.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_72\"\u003e[Pg 72]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_VIII\"\u003eCHAPTER VIII\u003cbr\u003e\nGEODESICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE importance of geodesics arises through the law that, in the general\ntheory of relativity, a particle not subject to constraints moves in a\ngeodesic. But let us first consider what a geodesic is.\u003c/p\u003e\n\n\u003cp\u003eAn adventurous pedestrian in the Alps may wish to go from a place in\none valley to a place in another by the shortest route—\u003ci\u003ei.e.\u003c/i\u003e the\nshortest compatible with remaining all the time on the earth\u0027s surface.\nHe cannot determine the shortest route by looking at a large-scale map\nand drawing a straight line between the two places, for if this line\ninvolves a greater average gradient than another it may be longer, in\ndistance as well as in time, than another route which slopes gradually\nto the head of a pass and then down again. What the traveller is\nseeking is a \"geodesic\"—\u003ci\u003ei.e.\u003c/i\u003e the shortest line that can be\ndrawn on the earth\u0027s surface between the two points. In the absence\nof hills—\u003ci\u003ee.g.\u003c/i\u003e on the sea—the shortest route is by a great\ncircle. On complicated surfaces, geodesics may become very complicated\ncurves. The definition is not exactly \"the shortest route between two\npoints.\" The definition is that the distance along a geodesic from\nany one of its points to any other must be \"stationary\"—\u003ci\u003ei.e.\u003c/i\u003e\nsuch that either all very slightly different paths are longer, or\nall very slightly different paths are shorter. This means that, for\nsmall variations of path, the first-order change of length is zero. In\neffect, in the ordinary geometry of surfaces the geodesic distance is a\nminimum, and in relativity theory it is a maximum. This is not so great\na difference as it may seem to the non-mathematical reader, since the\ngeodesic\u003cspan class=\"pagenum\" id=\"Page_73\"\u003e[Pg 73]\u003c/span\u003e distance concerned in relativity theory is more analogous\nto what would ordinarily count as lapse of time than to what would\nordinarily count as distance in space.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_74\"\u003e[Pg 74]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eLet us try to make the matter a little more concrete. The earth, in\nits annual revolution, travels from place to place in space-time;\nbetween the positions of Greenwich Observatory on two occasions six\nmonths apart, there is a certain interval. From the point of view of\nan observer in the sun, the interval would formerly have been divided\ninto two parts—namely, six months and about 186,000,000 miles. But\nfrom the point of view of the observer at Greenwich there is only one\ninterval—namely, time—since the place concerned is the same on both\noccasions. Given a clock which travels without constraint from one\npoint of space-time to another, the interval between these two points\nis what that clock registers as the time between them. I say that if\na clock were constrained to travel by some other slightly different\nroute, so as to be present at Greenwich Observatory on two occasions\nsix months apart, but absent from the earth in the meantime, the time\nwhich that clock would register as having been taken by its journey\nwould be less than six months. The interval between distant points\nis not, like distance in geometry, something which can be defined\nindependently of the route chosen. The interval must be obtained by\nintegration along a specified route, and a geodesic route is one which\nmakes the interval greater than it is by any slightly different route.\nThe time between two given events at which a man is present seems less\nif he has spent the intervening time in rapid travel than if he has\nlet himself drift passively; this is a sort of law of cosmic boredom.\nAll bodies, left to themselves, choose the course which is at each\nmoment the most boring, in the sense that it makes the time between two\ngiven events seem longest. However, it is time to have done with these\nirrelevancies, and return to seriousness.\u003c/p\u003e\n\n\u003cp\u003eSince the small interval \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e is independent of the co-ordinates,\na geodesic also is independent of them. We can easily obtain the\ndifferential equations which a geodesic must satisfy, and these\nequations must be satisfied by the same lines whatever system of\nco-ordinates we are employing. From a given point, geodesics start in\nall directions. Some of these are the paths of freely moving particles;\nothers are not. The law that the path of a particle is a geodesic\ndoes not tell us quite as much as it seems to do, since it is only by\nobservation of the motions of bodies that we discover what paths are\ngeodesics. Assuming that the orbit of the earth is a geodesic, we can\ndraw inferences as to the nature of the formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e in the\nsun\u0027s gravitational field. For we have no \u003ci\u003ea priori\u003c/i\u003e knowledge\nabout the coefficients \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-207.png\" alt=\"\" data-tex=\"\\(g_{\\mu\\nu}\\)\"\u003e which appear in the formula\nfor \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e; their values are to be deduced from observation. What\nwe can say is that it is possible, compatibly with observed facts,\nso to determine the \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-207.png\" alt=\"\" data-tex=\"\\(g_{\\mu\\nu}\\)\"\u003e that the path of a body in a\ngravitational field shall be a geodesic. In fact, we get in this way\na more accurate representation of the facts than we got from the\nNewtonian law, but the observable differences between the two are few\nand minute.\u003c/p\u003e\n\n\u003cp\u003eAlthough the new law of gravitation and the old do not lead to very\ndifferent results—as, indeed, they could not, since the old law\naccorded closely with observed facts—yet the difference in the ideas\ninvolved is very great. A planet, in the new theory, is moving freely,\nwhereas in the old theory it was subject to a central force directed\ntowards the sun. In the old theory, the planet moved in an ellipse;\nin the new theory, it moves in the nearest possible approach to a\nstraight line—to wit, a geodesic. In the old theory, the sun was like\na despotic government, emitting decrees from the metropolis; in the\nnew, the solar system is like the society of Kropotkin\u0027s dreams, in\nwhich everybody does what he prefers at each moment, and the result is\nperfect order. The odd thing is\u003cspan class=\"pagenum\" id=\"Page_75\"\u003e[Pg 75]\u003c/span\u003e that, as far as observation goes, the\ndifference between these two theories is exceedingly minute. To the\nplain man, it would seem impossible to reconcile the statement that\nthe earth moves in an ellipse with the statement that it moves in a\nsort of straight line, however queer the sort may be. And yet almost\nthe whole of the difference between these two statements is a matter\nof convention. It is possible to adhere to Euclidean space even now;\nthis requires a different way of stating Einstein\u0027s law of gravitation,\nbut does not demand the rejection of anything that has been proved\ntrue. Dr Whitehead considers this plan preferable to Einstein\u0027s. What\nmay be called the new orthodoxy, per contra, is set forth by Professor\nEddington. It will be worth while to consider the point at issue\nbetween them.\u003c/p\u003e\n\n\u003cp\u003eProfessor Eddington says (\u003ci\u003eop. cit.\u003c/i\u003e, p. 37):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"Suppose that an observer has chosen a definite system of space\nco-ordinates and of time-reckoning (\u003cimg style=\"vertical-align: -0.439ex; width: 12.145ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-214.png\" alt=\"\" data-tex=\"\\(x_1, x_2, x_3, x_4\\)\"\u003e), and that\nthe geometry of these is given by:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 51.063ex; height: 2.565ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-215.png\" alt=\"\" data-tex=\"\\[\n\\begin{equation*}\nd s^{2}=g_{11} d x_{1}^{2}+g_{22} d x_{2}^{2}+\\ldots+2 g_{12} d x_{1} d x_{2} \\qquad \\text{(16·1)}\n\\end{equation*}\n\\]\"\u003e\u003c/span\u003e\nLet him be under the mistaken impression that the geometry\nis:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.594ex; width: 40.44ex; height: 2.594ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-216.png\" alt=\"\" data-tex=\"\\[\n\\begin{equation*}\nd s_{0}^{2}=-d x_{1}^{3}-d x_{2}^{2}-d x_{3}^{2}+d x_{4}^{2} \\qquad \\text{(16·2)}\n\\end{equation*}\n\\]\"\u003e\u003c/span\u003e\n—that being the geometry with which he is most familiar in pure\nmathematics. We use \u003cimg style=\"vertical-align: -0.375ex; width: 3.225ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-217.png\" alt=\"\" data-tex=\"\\(ds_0\\)\"\u003e to distinguish his mistaken value of the\ninterval. Since intervals can be compared by experimental methods, he\nought soon to discover that his \u003cimg style=\"vertical-align: -0.375ex; width: 3.225ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-217.png\" alt=\"\" data-tex=\"\\(ds_0\\)\"\u003e cannot be reconciled with\nobservational results, and so realize his mistake. But the mind does\nnot so readily get rid of an obsession. It is more likely that our\nobserver will continue in his opinion, and attribute the discrepancy\nof the observations to some influence which is present and affects\nthe behaviour of his test-bodies. He will, so to speak, introduce a\nsupernatural agency which he can blame for the consequences of his\nmistake. Let us examine what name he would apply to this agency.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_76\"\u003e[Pg 76]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003e\"Of the four test-bodies considered the moving particle is in general\nthe most sensitive to small changes of geometry, and it would be by\nthis test that the observer would first discover discrepancies. The\npath laid down for it by our observer is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.948ex; width: 18.969ex; height: 5.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-218.png\" alt=\"\" data-tex=\"\\[\n\\int d s_{0} \\text { is stationary }\n\\]\"\u003e\u003c/span\u003e\n—\u003ci\u003ei.e.\u003c/i\u003e a straight line in the co-ordinates (\u003cimg style=\"vertical-align: -0.439ex; width: 12.145ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-214.png\" alt=\"\" data-tex=\"\\(x_1, x_2, x_3, x_4\\)\"\u003e).\nThe particle, of course, pays no heed to this, and moves in the\ndifferent track:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.948ex; width: 18.61ex; height: 5.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-219.png\" alt=\"\" data-tex=\"\\[\n\\int d s \\text { is stationary. }\n\\]\"\u003e\u003c/span\u003e\nAlthough apparently undisturbed it deviates from \u0027uniform motion in a\nstraight line.\u0027 The name given to any agency which causes deviation\nfrom uniform motion in a straight line is \u003ci\u003eforce\u003c/i\u003e according to the\nNewtonian definition of force. Hence the agency invoked through our\nobserver\u0027s mistake is described as a \u0027field of force.\u0027\u003c/p\u003e\n\n\u003cp\u003e\"The field of force is not always introduced by inadvertence, as in the\nforegoing illustration. It is sometimes introduced deliberately by the\nmathematician—\u003ci\u003ee.g.\u003c/i\u003e when he introduces the centrifugal force.\nThere would be little advantage and many disadvantages in banishing\nthe phrase \u0027field of force\u0027 from our vocabulary. We shall therefore\nregularize the procedure which our observer has adopted. We call (16·2)\nthe \u003ci\u003eabstract geometry\u003c/i\u003e of the system of co-ordinates\n(\u003cimg style=\"vertical-align: -0.439ex; width: 12.145ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-214.png\" alt=\"\" data-tex=\"\\(x_1, x_2, x_3, x_4\\)\"\u003e); it may be chosen arbitrarily by the observer. The\n\u003ci\u003enatural geometry\u003c/i\u003e is (16·1).\u003c/p\u003e\n\n\u003cp\u003e\"\u003ci\u003eA field of force represents the discrepancy between the natural\ngeometry of a co-ordinate system and the abstract geometry arbitrarily\nascribed to it.\u003c/i\u003e\u003c/p\u003e\n\n\u003cp\u003e\"A field of force thus arises from an attitude of mind. If we do not\ntake our co-ordinate system to be something different from that which\nit really is, there is no field of force.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eIt is not quite clear why the man who uses forces with a conventional\ngeometry should be regarded as making a \"mistake,\" while the man who\nsays that free particles travel in geodesics, and to justify himself\nhas a queer geometry, is thought to be saying something substantially\nmore accurate.\u003cspan class=\"pagenum\" id=\"Page_77\"\u003e[Pg 77]\u003c/span\u003e It is true that we must not conceive \"force\" as an\nactual agency, as the older mechanics did; it is merely part of\nthe method of describing how bodies move. But as soon as this is\nrecognized, it is a mere question of convenience whether we speak of\nforces or not. Let it be conceded that the method of the general theory\nof relativity is better from a logico-æsthetic point of view; I do not\nsee, however, why we should regard it as any more \"true.\" I am not\nconsidering, at the moment, the fact that Einstein\u0027s law of gravitation\ngives a slightly more accurate picture of the phenomena than Newton\u0027s,\nsince this is not really relevant to the particular point at issue.\u003c/p\u003e\n\n\u003cp\u003eLet us now consider Dr Whitehead\u0027s view, which is, on this point, the\nopposite of Professor Eddington\u0027s. In the Preface to The \u003ci\u003ePrinciple\nof Relativity\u003c/i\u003e,[1] he says:\u003c/p\u003e\n\n\u003cp\u003e\"As the result of a consideration of the character of our knowledge in\ngeneral, and of our knowledge of nature in particular, … I deduce\nthat our experience requires and exhibits a basis of uniformity, and\nthat in the case of nature this basis exhibits itself as the uniformity\nof spatio-temporal relations. This conclusion entirely cuts away the\ncasual heterogeneity of these relations which is the essential of\nEinstein\u0027s later theory. It is this uniformity which is essential to\nmy outlook, and not the Euclidean geometry which I adopt as lending\nitself to the simplest exposition of the facts of nature. I should be\nvery willing to believe that each permanent space is either uniformly\nelliptic or uniformly hyperbolic, if any observations are more simply\nexplained by such a hypothesis. It is inherent in my theory to maintain\nthe old division between physics and geometry. Physics is the science\nof the contingent relations of nature, and geometry expresses its\nuniform relatedness.\"\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_78\"\u003e[Pg 78]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eAgain, in discussing the structure of space-time, he says (\u003ci\u003eib.\u003c/i\u003e,\np. 29):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The structure is uniform because of the necessity for knowledge\nthat there be a system of uniform relatedness, in terms of which the\ncontingent relations of natural factors can be expressed. Otherwise we\ncan know nothing until we know every thing.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eAnd on p. 64:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"Though the character of time and space is not in any sense \u003ci\u003ea\npriori\u003c/i\u003e, the essential relatedness of any perceived field of events\nto all other events requires that this relatedness of all events\nshould conform to the ascertained disclosure derived from the limited\nfield. For we can only know that distant events are spatio-temporally\nconnected with the events immediately perceived by knowing what\nthese relations are. In other words, these relations must possess a\nsystematic uniformity in order that we may know of nature as extending\nbeyond isolated cases subjected to the direct examination of individual\nperception…. This doctrine leads to the rejection of Einstein\u0027s\ninterpretation of his formulæ, as expressing a casual heterogeneity of\nspatio-temporal warping, dependent upon contingent adjectives.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eThus whereas Eddington seems to regard it as necessary to adopt\nEinstein\u0027s variable space, Whitehead regards it as necessary to reject\nit. For my part, I do not see why we should agree with either view: the\nmatter seems to be one of convenience in the interpretation of formulæ.\nNevertheless, Dr Whitehead\u0027s arguments deserve careful examination.\u003c/p\u003e\n\n\u003cp\u003eThe main force of the above passages is epistemological: the question\ninvolved is the Kantian one, How is knowledge possible? I do not wish\nto deal with this question in its general form. But without going\ninto theory of knowledge, there is what may be called a common-sense\nanswer. Einstein enables us to predict what in fact can be predicted\nabout astronomical occurrences, and that seems all that ought to be\ndemanded of him. Dr Whitehead objects to the \"casual\" heterogeneity\nof space-time in Einstein\u0027s system. In a sense, this adjective is\njustified, since the character of space-time\u003cspan class=\"pagenum\" id=\"Page_79\"\u003e[Pg 79]\u003c/span\u003e in any region depends\nupon circumstances which can only be ascertained empirically—namely,\nthe distribution of matter in the neighbourhood. But in another\nsense the adjective is not justified, since Einstein\u0027s law of\ngravitation gives the rule according to which space-time is affected\nby the neighbourhood of matter. To say that we cannot, by the help\nof this rule, know in advance the geometry of a region we have not\nexplored, seems an insufficient objection, since we also cannot know\nwhat astronomical occurrences will take place unless we know the\ndistribution of matter. Einstein, like other people, assumes the\npermanence of matter; this is a point to be considered in another\nconnection, but it has no particular relevance to the present issue.\nThe way the heavenly bodies move depends upon the distribution of\nmatter in their neighbourhood, which is, in Dr Whitehead\u0027s phrase,\n\"casual.\" Even by assuming Euclidean geometry we cannot make\nastronomical predictions unless we assume that we know the important\nfacts about the distribution of matter in the region concerned. Whether\nwe put the consequences of these facts into our geometry or not does\nnot seem to make any real difference to the possibility of physical\nknowledge. In all theoretical physics, there is a certain admixture\nof facts and calculations; so long as the combination is such as to\ngive results which observation confirms, I cannot see that we can have\nany \u003ci\u003ea priori\u003c/i\u003e objection. Dr Whitehead\u0027s view seems to rest upon\nthe assumption that the principles of scientific inference ought to\nbe in some sense \"reasonable.\" Perhaps we all make this assumption\nin one form or another. But for my part I should prefer to infer\n\"reasonableness\" from success, rather than set up in advance a standard\nof what can be regarded as credible.\u003c/p\u003e\n\n\u003cp\u003eI do not therefore see any ground for rejecting a variable geometry\nsuch as Einstein\u0027s. But equally I see no ground for supposing that\nthe facts necessitate it. The question is, to\u003cspan class=\"pagenum\" id=\"Page_80\"\u003e[Pg 80]\u003c/span\u003e my mind, merely one of\nlogical simplicity and comprehensiveness. From this point of view,\nI prefer the variable space in which bodies move in geodesics to a\nEuclidean space with a field of force. But I cannot regard the question\nas one concerning the facts.\u003c/p\u003e\n\n\u003cp\u003eThe conclusion would seem to be, therefore, that, when physics is\nconsidered, as we are now considering it, as a deductive system, we\ndo well to adopt the Einsteinian interpretation: free particles move\nin geodesics, and the law of gravitation is a law as to how geodesics\nare shaped in the l neighbourhood of matter. This view is essentially\nsimple, though it leads to complicated mathematics. It accords with\nthe facts, and it puts the law of gravitation in a recognizable place\namong physical principles, instead of leaving it, as heretofore, an\nisolated and unrelated law. I propose, therefore, to continue to adopt\nEinstein\u0027s view as to the best way of interpreting the principles of\nphysics, without suggesting that no other way is logically possible.\u003c/p\u003e\n\n\u003cp\u003eThere is one matter of great theoretical importance, which is not very\nclear in the usual accounts of relativity. How do we know whether\ntwo events are to be regarded as happening to the same piece of\nmatter? An electron or a proton is supposed to preserve its identity\nthroughout time; but our fundamental continuum is a continuum of\n\u003ci\u003eevents\u003c/i\u003e. One must therefore suppose that one unit of matter is a\nseries of events, or a series of sets of events. It is not clear what\nis the theoretical criterion for determining whether two events both\nbelong to one such series. We may assume, I suppose, that two events\nwhich overlap—\u003ci\u003ei.e.\u003c/i\u003e which are both present at some point of\nspace-time—must belong to one unit of matter. (It is not to be assumed\nthat an event which belongs to one unit of matter belongs to no other.)\nWe may also assume that two events which have a space-like interval,\nor have a zero interval without overlapping, do not belong to one unit\nof\u003cspan class=\"pagenum\" id=\"Page_81\"\u003e[Pg 81]\u003c/span\u003e matter. But when two events have a time-like interval, there is no\nobvious criterion. Any two such events can be connected by a geodesic\nin which any two points have a time-like separation; therefore, so far\nas the laws of dynamics are concerned, they \u003ci\u003emight\u003c/i\u003e both belong to\nthe same material unit. Yet sometimes we think they do, and sometimes\nwe think they do not. It is evidently part of the business of physics\nto tell us how we are to decide this question in a given case. What can\nwe say about it?\u003ca id=\"FNanchor_26\" href=\"#Footnote_26\" class=\"fnanchor\"\u003e[26]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe decision must depend upon intermediate history—\u003ci\u003ei.e.\u003c/i\u003e upon\nthe existence of some series of intermediate events (or sets of events)\nfollowing each other according to some law. If there exists any law\nwhich is in fact obeyed by strings of events, such a law can be used\nto define what we mean by one material unit. We know that there are\nsuch laws, but their importance in this connection is not emphasized,\nbecause it has hardly been realized that there is a problem owing to\nthe substitution of events for bits of matter as the fundamental stuff\nof physics. For common sense, there is a more or less vague law of what\nmay be called qualitative continuity. If you look persistently in a\ngiven direction, what you see, as a rule, alters gradually; there are\nexceptions, such as explosions, but they are rare. (I am not talking\nof a theoretical gradualness, but of one that is obvious to untrained\nperception.) If you see, say, a well-defined red patch, whose shape\nand tint do not alter greatly while you are looking, you conclude\nthat there is a material object there, especially if you can touch it\nwhenever you choose. Common sense achieves in this way a considerable\nmeasure of constancy in its objects. More is achieved by reducing\nmatter to molecules, more still by reducing it to atoms, and yet more\nby reducing it to protons and electrons. But physicists would not feel\npleased with\u003cspan class=\"pagenum\" id=\"Page_82\"\u003e[Pg 82]\u003c/span\u003e electrons and protons but for the fact that their tables\nand chairs, their laboratories and their books, consist, on the whole,\nof the same electrons and protons on different occasions. Qualitative\ncontinuity remains the basis of the whole proceeding. Suppose, one\nevening, you were to say to an astronomer: How do you know that that\nwhite patch in the sky is the moon? He would stare at you, and think\nyou mad. He would \u003ci\u003enot\u003c/i\u003e reply: because the course and phases of\nthe moon have been worked out by astronomical theory, and that is where\nthe moon ought to be, and the shape it ought to have, at the present\nmoment in this latitude and longitude. What he would say is: Why, can\u0027t\nyou \u003ci\u003esee\u003c/i\u003e it\u0027s the moon? To which the right answer would be: Yes,\n\u003ci\u003eI\u003c/i\u003e can, but I didn\u0027t suppose \u003ci\u003eyou\u003c/i\u003e could, because you ought\nto have got beyond such a crude criterion.\u003c/p\u003e\n\n\u003cp\u003eMoreover, there are identities in physics which are not material. A\nwave has a certain identity; if this were not the case, our visual\nperceptions would not have the intimate connection they in fact do have\nwith physical objects. Suppose we see several lamps simultaneously:\nwe are able to distinguish them because each sends out its own\nlight-waves, which preserve their individuality until they reach the\neye. Our chief reason for not regarding a wave as a physical object\nseems to be that it is not indestructible. But this is not our only\nreason, since, if it were, we might regard the energy of a wave as a\nphysical object. We do not regard energy as a \"thing,\" because it is\nnot connected with the qualitative continuity of common-sense objects:\nit may appear as light or heat or sound or what not. But now that\nenergy and mass have turned out to be identical, our refusal to regard\nenergy as a \"thing\" should incline us to the view that what possesses\nmass need not be a \"thing.\" We seem driven, therefore, to the view\nadvocated by Eddington, that there are certain invariants, and that\n(with some degree of inaccuracy) our senses and our common sense have\nsingled them out as deserving names. The correct theoretical definition\nof a single piece of matter will thus depend upon the mathematical\ninvariants resulting from our formula for interval. This topic,\nhowever, demands a new chapter.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_26\" href=\"#FNanchor_26\" class=\"label\"\u003e[26]\u003c/a\u003e\nThis subject is considered again in Chap. XIV. from a\nsomewhat different standpoint.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_84\"\u003e[Pg 84]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_IX\"\u003eCHAPTER IX\u003cbr\u003e\nINVARIANTS AND THEIR PHYSICAL INTERPRETATION\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHERE is a point of view specially associated with Professor Eddington,\nwhich it is necessary to consider at this stage, since it arises\nnaturally in the attempt to develop physics as a self-contained\ndeductive system. According to this view, practically all theoretical\nphysics is a vast tautology or convention, the only part excepted, so\nfar, being the part which involves quantum-theory. This is not the\nwhole of Professor Eddington\u0027s view on the subject, as he has shown\nwhen not writing simply as a technical physicist;\u003ca id=\"FNanchor_27\" href=\"#Footnote_27\" class=\"fnanchor\"\u003e[27]\u003c/a\u003e but it is what we\nmay call his \"professional\" view.\u003ca id=\"FNanchor_28\" href=\"#Footnote_28\" class=\"fnanchor\"\u003e[28]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eLet us begin with the conservation of momentum and of energy (or mass).\nHere we start from a proposition of pure mathematics. To explain this\nproposition will require certain preliminaries. It will be remembered\nthat we had:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.685ex; width: 18.085ex; height: 2.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-220.png\" alt=\"\" data-tex=\"\\[\nd s^{2}=\\Sigma g_{\\mu \\nu} d x_{\\mu} d x_{\\nu}\n\\]\"\u003e\u003c/span\u003e\nWe put:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -5.317ex; width: 23.857ex; height: 11.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-221.png\" alt=\"\" data-tex=\"\\[\ng=\\left|\\begin{array}{llll}\ng_{11} \u0026 g_{12} \u0026 g_{13} \u0026 g_{14} \\\\\ng_{21} \u0026 g_{22} \u0026 g_{23} \u0026 g_{24} \\\\\ng_{31} \u0026 g_{32} \u0026 g_{33} \u0026 g_{34} \\\\\ng_{41} \u0026 g_{42} \u0026 g_{43} \u0026 g_{44}\n\\end{array}\\right|\n\\]\"\u003e\u003c/span\u003e\nAnd we write \u003cimg style=\"vertical-align: -0.464ex; width: 3.08ex; height: 1.992ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-222.png\" alt=\"\" data-tex=\"\\(g^{\\mu\\nu}\\)\"\u003e for the minor of in this determinant,\ndivided by \u003cimg style=\"vertical-align: -0.464ex; width: 1.079ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-223.png\" alt=\"\" data-tex=\"\\(g\\)\"\u003e. Also:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.601ex; width: 13.685ex; height: 3.243ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-224.png\" alt=\"\" data-tex=\"\\[\ng_{\\mu}^{\\nu}=\\underset{\\sigma}{\\Sigma} g_{\\mu \\sigma} g^{\\nu \\sigma},\n\\]\"\u003e\u003c/span\u003e\nwhich = 0 if \u003cimg style=\"vertical-align: -0.489ex; width: 5.58ex; height: 2.109ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-225.png\" alt=\"\" data-tex=\"\\(\\mu \\neq \\nu\\)\"\u003e and =1 if \u003cimg style=\"vertical-align: -0.489ex; width: 5.58ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-226.png\" alt=\"\" data-tex=\"\\(\\mu = \\nu\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eThe next step is the definition of the \"three-index symbols,\"\nwhich are:\u003cspan class=\"pagenum\" id=\"Page_85\"\u003e[Pg 85]\u003c/span\u003e\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -5.611ex; width: 44.76ex; height: 12.354ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-227.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n\u0026 {[\\mu \\nu, \\sigma]=\\frac{1}{2}\\left(\\frac{\\partial g_{\\mu \\sigma}}{\\partial x_{\\nu}}+\\frac{\\partial g_{\\nu \\sigma}}{\\partial x_{\\mu}}-\\frac{\\partial\n g_{\\mu \\nu}}{\\partial x_{\\sigma}}\\right)} \\\\\n\u0026 \\{\\mu \\nu, \\sigma\\}=\\frac{1}{2} \\sum_{\\lambda}g^{\\sigma \\lambda}\\left(\\frac{\\partial g_{\\mu \\lambda}}{\\partial x_{\\nu}}+\\frac{\\partial g_{\\nu \\lambda}}{\\partial\n x_{\\mu}}-\\frac{\\partial g_{\\mu \\nu}}{\\partial x_{\\lambda}}\\right)\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nWe can now define the tensor which Einstein uses for his law of\ngravitation. It is \u003cimg style=\"vertical-align: -0.685ex; width: 3.779ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-228.png\" alt=\"\" data-tex=\"\\(G_{\\mu\\nu}\\)\"\u003e, where:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.909ex; width: 67.87ex; height: 5.056ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-229.png\" alt=\"\" data-tex=\"\\[\nG_{\\mu \\nu}=\\{\\mu \\sigma, \\alpha\\}\\{\\alpha \\nu, \\sigma\\}-\\{\\mu \\nu, \\alpha\\}\\{\\alpha \\sigma, \\sigma\\}+\\frac{\\partial}{\\partial x_{\\nu}}\\{\\mu\n \\sigma, \\sigma\\}-\\frac{\\partial}{\\partial x_{\\sigma}}\\{\\mu \\nu, \\sigma\\}\n\\]\"\u003e\u003c/span\u003e\nsummed for all values of \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e from 1 to 4. Einstein\ntakes as the law of gravitation \u003cimg style=\"vertical-align: -0.685ex; width: 7.927ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-230.png\" alt=\"\" data-tex=\"\\(G_{\\mu\\nu} = 0\\)\"\u003e in empty space. For\nthe moment, we are not concerned with the law of gravitation, but with\ncertain identities. We put:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.685ex; width: 32.068ex; height: 2.382ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-231.png\" alt=\"\" data-tex=\"\\[\nG=\\Sigma g^{\\mu \\nu} G_{\\mu \\nu}\\quad(\\mu, \\nu=1, 2, 3, 4).\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eFurther, there is a rule for raising or lowering suffixes in any\ntensor, of which an illustration is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.602ex; width: 30.681ex; height: 4.751ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-232.png\" alt=\"\" data-tex=\"\\[\nA_{\\alpha}^{\\mu}=\\sum_{\\nu} g^{\\mu \\nu} A_{\\alpha \\nu}\\,\\,(\\nu= 1, 2, 3, 4)\n\\]\"\u003e\u003c/span\u003e\nso that—\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.904ex; width: 41.455ex; height: 2.922ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-233.png\" alt=\"\" data-tex=\"\\[\nG_{\\mu}^{\\nu}=g^{\\nu 1} G_{\\mu 1}+g^{\\nu 2} G_{\\mu 2}+g^{\\nu 3} G_{\\mu 8}+g^{\\nu 4} G_{\\mu 4}.\n\\]\"\u003e\u003c/span\u003e\nGeneralizing the notion of the \"divergence\" of a vector, we obtain a\ngeneral definition of the divergence of any tensor. Taking a tensor\nof the form for purposes of illustration, its \"divergence\" has four\ncomponents:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.02ex; width: 47.004ex; height: 3.037ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-234.png\" alt=\"\" data-tex=\"\\[\n\\left(A_{\\mu}^{1}\\right)_{1}+\\left(A_{\\mu}^{2}\\right)_{2}+\\left(A_{\\mu}^{3}\\right)_{3}+\\left(A_{\\mu}^{4}\\right)_{4}\\,\\,(\\mu= 1, 2, 3, 4),\n\\]\"\u003e\u003c/span\u003e\nwhere:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.619ex; width: 48.092ex; height: 5.766ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-235.png\" alt=\"\" data-tex=\"\\[\n\\left(A_{\\mu}^{1}\\right)_{1}=\\frac{\\partial}{\\partial x_{1}} A_{\\mu}^{1}+\\sum_{\\alpha}\\{\\alpha\n \\mathrm{I}, \\mathrm{I}\\} A_{\\mu}^{\\alpha}-\\sum_{\\alpha}\\{\\mu \\mathrm{I}, \\alpha\\} A_{\\alpha}^{1},\n\\]\"\u003e\u003c/span\u003e\nand similarly for \u003cimg style=\"vertical-align: -0.984ex; width: 5.909ex; height: 2.906ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-236.png\" alt=\"\" data-tex=\"\\(\\left(A_{\\mu}^{2}\\right)_{2}\\)\"\u003e, etc. These\ndefinitions have been given in order to enunciate the proposition:\u003ca id=\"FNanchor_29\" href=\"#Footnote_29\" class=\"fnanchor\"\u003e[29]\u003c/a\u003e\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.552ex; width: 49.439ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-237.png\" alt=\"\" data-tex=\"\\[\n\\textit{The divergence of}\\,\\, G_{\\mu}^{\\nu}-\\frac{1}{2} g_{\\mu}^{\\nu} G\\,\\, \\textit{is identically zero},\n\\]\"\u003e\u003c/span\u003e\n\u003cspan class=\"pagenum\" id=\"Page_86\"\u003e[Pg 86]\u003c/span\u003ewhich Eddington calls \"the fundamental theorem of mechanics.\"\u003c/p\u003e\n\n\u003cp\u003eIn order to see the use made of this proposition, we need to introduce\nthe \"material energy-tensor,\" defined as:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.575ex; width: 18.852ex; height: 4.918ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-238.png\" alt=\"\" data-tex=\"\\[\nT^{\\nu\\mu} = \\rho_0 \\frac{\\partial x_{\\mu}}{ds}\\frac{\\partial x_{\\nu}}{ds},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.489ex; width: 2.157ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-239.png\" alt=\"\" data-tex=\"\\(\\rho_{0}\\)\"\u003e is the \"proper density\" of the matter concerned—\u003ci\u003ei.e.\u003c/i\u003e\nits density relative to axes moving with the matter. From\nthis, by the usual rule for lowering a suffix, we obtain a tensor\n\u003cimg style=\"vertical-align: -0.904ex; width: 2.755ex; height: 2.436ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-240.png\" alt=\"\" data-tex=\"\\(T^{\\nu}_{\\mu}\\)\"\u003e. The principles of the conservation of mass and\nmomentum are contained in the statement that the divergence of\n\u003cimg style=\"vertical-align: -0.904ex; width: 2.755ex; height: 2.436ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-240.png\" alt=\"\" data-tex=\"\\(T^{\\nu}_{\\mu}\\)\"\u003e vanishes. This suggests the identification of\n\u003cimg style=\"vertical-align: -0.904ex; width: 2.755ex; height: 2.436ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-240.png\" alt=\"\" data-tex=\"\\(T^{\\nu}_{\\mu}\\)\"\u003e with \u003cimg style=\"vertical-align: -0.904ex; width: 11.502ex; height: 2.861ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-241.png\" alt=\"\" data-tex=\"\\(G^{\\nu}_{\\mu} – \\tfrac{1}{2}g^{\\nu}_{\\mu}G\\)\"\u003e,\nwhose divergence vanishes identically—apart from a numerical factor,\nwhich, for convenience, is taken as \u003cimg style=\"vertical-align: -0.186ex; width: 4.181ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-242.png\" alt=\"\" data-tex=\"\\(-8\\pi\\)\"\u003e. Thus Eddington puts:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.904ex; width: 31.763ex; height: 2.861ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-243.png\" alt=\"\" data-tex=\"\\[\nG^{\\nu}_{\\mu} – \\tfrac{1}{2}g^{\\nu}_{\\mu}G = -8\\pi T^{\\nu}_{\\mu} \\qquad \\text{(54·3)}\n\\]\"\u003e\u003c/span\u003e\nwhich is the law of gravitation for continuous matter.\u003c/p\u003e\n\n\u003cp\u003eIt has been necessary to make the above excursion into mathematical\nregions in order to be able to understand the observations which\nsucceed to the above in Eddington\u0027s exposition (\u003ci\u003eop. cit.\u003c/i\u003e p.\n119). He says:\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_87\"\u003e[Pg 87]\u003c/span\u003e\u003c/p\u003e\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"Appeal is now made to a Principle of Identification. Our deductive\ntheory starts with the interval …, from which the tensor is\nimmediately obtained. By pure mathematics we derive other tensors….\nThese constitute our world-building material; and the aim of the\ndeductive theory is to construct from this a world which functions\nin the same way as the known physical world. If we succeed, mass,\nmomentum, stress, etc., must be the vulgar names for certain analytical\nquantities in the deductive theory; and it is this stage of naming the\nanalytical tensors which is reached in (54·3). If the theory provides\na tensor \u003cimg style=\"vertical-align: -0.904ex; width: 11.502ex; height: 2.861ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-241.png\" alt=\"\" data-tex=\"\\(G^{\\nu}_{\\mu} – \\tfrac{1}{2}g^{\\nu}_{\\mu}G\\)\"\u003e, which behaves\nin exactly the same way as the tensor summarizing the mass, momentum\nand stress of matter is observed to behave, it is difficult to see how\nanything more could be required of it.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eThere are a number of other examples of the same method in Eddington\u0027s\nwork, but we may take the above as typical, since it is the simplest\nmathematically. It is worth while to consider the nature of the method,\napart from its technical embodiment. This is the more necessary, as it\nis not easy to be clear as to the logical and empirical elements in\ntheoretical physics as developed by the above method.\u003c/p\u003e\n\n\u003cp\u003eFundamentally, the method is the same as that which has always been\npursued when mathematics has been applied to the physical world. The\naim has been to obtain mathematical laws which gave correct results\nwherever they could be tested by observation. The fewer and more\ngeneral and more comprehensive the laws, the more scientific taste\nwas gratified. Newton\u0027s law of gravitation was better than Kepler\u0027s\nlaws, both because it was one law instead of three, and because it\ngave a larger number of correct deductions. But at every stage the\nsubject-matter of physics grows more abstract, and its connection with\nwhat we observe grows more remote. Eddington\u0027s ideal is to start with\nonly one fundamental law—namely, the formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e is—which,\nas generalized by Weyl, will give electromagnetic equations as well\nas gravitation. From this one fundamental law, by pure mathematics,\nwe deduce the existence of quantities behaving in certain ways.\nElementary theorizing from observation has led us to believe that\nthere are quantities connected with what we observe which behave in\nthese ways. We therefore identify the observed quantities with the\ndeduced quantities. This is, in essence, the same sort of thing as we\ndo when we associate what we see with light-waves. We may thus regard\nphysics from the two points of view, the inductive and the deductive.\nIn the latter, we start from the formula for interval (together with\ncertain other assumptions), and we deduce by mathematics a world having\ncertain mathematical characteristics. In the inductive view, the same\nmathematical characteristics\u003cspan class=\"pagenum\" id=\"Page_88\"\u003e[Pg 88]\u003c/span\u003e are arrived at, but they are now those\nwhich may be supposed to belong to the physical world in its entirety\nif we supplement observation by means of the postulate that everything\nhappens in accordance with simple general laws.\u003c/p\u003e\n\n\u003cp\u003eWe may thus say that the world of elementary physics is semi-abstract,\nwhile that of deductive relativity-theory is wholly abstract. The\nappearance of deducing actual phenomena from mathematics is delusive;\nwhat really happens is that the phenomena afford inductive verification\nof the general principles from which our mathematics starts. Every\nobserved fact retains its full evidential value; but now it confirms\nnot merely some particular law, but the general law from which the\ndeductive system starts. There is, however, no logical necessity for\none fact to follow given another, or a number of others, because there\nis no logical necessity about our fundamental principles.\u003c/p\u003e\n\n\u003cp\u003eThe question of interpretation, it must be admitted, is somewhat\ndifficult when physics is conceived in this very abstract manner.\nWhat, for example, is \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e? We start from a view which is, to a\ncertain extent, intelligible in terms of observation. In the case of a\ntime-like interval, it is the time which elapses between the two events\naccording to a clock, not subject to constraints, which is present at\nboth events. On the earth\u0027s surface, the time measured by a clock can\nbe inferred, with suitable precautions, from the visual perceptions of\na careful observer. In the case of a space-like interval, \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e is\nthe distance between two events as estimated by measurements carried\nout on a body which is present at both, and for which the two events\nare simultaneous. The elementary operation of measuring lengths is\nhere supposed possible. But when we pass from this initial view to the\nabstract view which is required by the general theory of relativity,\nthe interval can only be actually estimated by using rather elaborate\nphysics to make deductions from what can be\u003cspan class=\"pagenum\" id=\"Page_89\"\u003e[Pg 89]\u003c/span\u003e actually observed by\nmeans of clocks and footrules. For logical theory, the interval is\nprimitive, but from the point of view of empirical verification it is\na complicated function of empirical data, deduced by means of physics\nin its semi-abstract form. The unity and simplicity of the deductive\nedifice, therefore, must not blind us to the complexity of empirical\nphysics, or to the logical independence of its various portions.\u003c/p\u003e\n\n\u003cp\u003eIn particular, when the conservation of mass or of momentum appears\nas an identity, that is only true in the deductive system; in their\nempirical meaning, these laws are by no means logical necessities.\nThere might easily be a world in which they were false, and it might\nbe capable of a treatment as unified and mathematical as the general\ntheory of relativity; but, if so, the fundamental laws would be\ndifferent.\u003c/p\u003e\n\n\u003cp\u003eWhat is novel and interesting in the point of view we have been\nconsidering is the character of the relation between empirical and\ndeductive physics. But there is no real diminution of the need for\nempirical observation. I do not for a moment suggest that anything in\nthe above is a criticism of Professor Eddington; indeed, I imagine he\nwould regard it as a string of truisms. I have been concerned only to\nguard against a possible misunderstanding on the part of those who do\nnot feel for mathematics the contempt which is bred of familiarity.\u003c/p\u003e\n\n\u003cp\u003eIn the foregoing remarks, however, we have neglected one important\naspect of Eddington\u0027s theory. In addition to the fact that the\nwhole general theory of relativity can be deduced from a few simple\nassumptions, interest attaches to the manner of the deduction and the\nconsiderations by which the substantial import of mathematical formulæ\nis made less, or at least other, than would naturally be supposed.\nA good example is afforded by a paragraph headed \"Interpretation\nof Einstein\u0027s Law of Gravitation.\"\u003ca id=\"FNanchor_30\" href=\"#Footnote_30\" class=\"fnanchor\"\u003e[30]\u003c/a\u003e The law concerned is not\u003cspan class=\"pagenum\" id=\"Page_90\"\u003e[Pg 90]\u003c/span\u003e\n\u003cimg style=\"vertical-align: -0.685ex; width: 7.927ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-244.png\" alt=\"\" data-tex=\"\\(G_{\\nu\\mu} = 0\\)\"\u003e, which is not supposed to be quite accurate where\nstellar distances are concerned; it is the modified law:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.685ex; width: 11.823ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-245.png\" alt=\"\" data-tex=\"\\[\nG_{\\nu\\mu} = \\lambda g_{\\nu\\mu},\n\\]\"\u003e\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e must be very small, so small that within\nthe solar system the new law gives the same results, within\nthe limits of observation, as \u003cimg style=\"vertical-align: -0.685ex; width: 7.927ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-244.png\" alt=\"\" data-tex=\"\\(G_{\\nu\\mu} = 0\\)\"\u003e. The new law\nis shown to be equivalent to the assumption that, in empty\nspace, the radius of curvature in every direction is everywhere\n\u003cimg style=\"vertical-align: -1.306ex; width: 4.236ex; height: 4.208ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-247.png\" alt=\"\" data-tex=\"\\(\\sqrt{\\tfrac{3}{\\lambda}}\\)\"\u003e But this is interpreted as a law about\nour measuring rods—namely, that they adjust themselves to the radius\nof curvature at any place and in any direction. It is interpreted as\nmeaning:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The length of a specified material structure bears a constant ratio to\nthe radius of curvature of the world at the place and in the direction\nin which it lies.\" And the following gloss is added:\u003c/p\u003e\n\n\u003cp\u003e\"The law no longer appears to have any reference to the constitution\nof an empty continuum. It is a law of material structure showing what\ndimensions a specified collection of molecules must take up in order\nto adjust itself to equilibrium with the surrounding conditions of the\nworld.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eIn particular, electrons must make these adjustments, and it is\nsuggested elsewhere that the symmetry of an electron and its equality\nwith other electrons are not substantial facts, but consequences of the\nmethod of measurement (pp. 153-4). One cannot complain of an author\nfor not doing everything, but at this point most readers will feel a\ndesire for some discussion of the theory of measurement. The elementary\nmeaning of measurement of lengths is derived from superposition of a\nsupposedly rigid body. A rigid body, as Dr Whitehead has pointed out,\nis primarily one which \u003ci\u003eseems\u003c/i\u003e rigid, such as a steel bar in\ncontradistinction to a piece of putty. When I say that a body \"seems\"\nrigid, I mean that it looks and feels as if it were not altering its\nshape and size.\u003cspan class=\"pagenum\" id=\"Page_91\"\u003e[Pg 91]\u003c/span\u003e This, so far as it can be relied upon, implies some\nconstant relation to the human body: if the eye and the hand grew at\nthe same rate as the \"rigid\" body, it would look and feel as if it\nwere unchanging. But if other objects in our immediate environment did\nnot grow meanwhile, we should infer that we and our measure had grown.\nThere would, however, be no meaning in the supposition that all bodies\nare bigger in certain places than they are in certain others; at least,\nif we suppose the alteration to be in a fixed ratio. If we do not add\nthis proviso, there is a good meaning in the supposition; in fact, we\ndo actually believe that all bodies are bigger at the equator than at\nthe North Pole, except such as are too small to be visible or palpable.\nWhen we say that the length of an object at the equator is one metre,\nwe do not mean that its length is that which the standard metre would\nhave if moved from Paris to the equator. But the expansion of bodies\nwith temperature would have been difficult to discover if it had not\nbeen possible to bring bodies of different temperatures into the same\nneighbourhood and measure them before their temperatures had become\nequal; it would also have been difficult if all bodies had expanded\nequally when their temperatures rose. These elementary considerations,\nalong with many others, make rigidity an ideal, which actual bodies\napproach without attaining. Mere superposition thus ceases to give\na measure of length: it gives still a comparison of the two bodies\nconcerned, but not of either with the standard unit of length. To\nobtain the latter, we have to adjust the immediate results of the\noperation of measuring, by means of a mass of physical theory. If the\nmeasures which we obtain are mutually consistent, that is all we can\nask; but it is possible that a change in physical theory might have\ngiven other measures which would also have been mutually consistent.\u003c/p\u003e\n\n\u003cp\u003eProfessor Eddington, in the passage which we quoted partially in\nintroducing this discussion, is careful to say that\u003cspan class=\"pagenum\" id=\"Page_92\"\u003e[Pg 92]\u003c/span\u003e he is concerned\nwith measurement by direct comparison. He says:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The statement that the radius of curvature is a constant length\nrequires more consideration before its full significance is\nappreciated. Length is not absolute, and the result can only mean\n\u003ci\u003econstant relative to the material standards of length\u003c/i\u003e used in\nall our measurements and in particular in those measurements which\nverify \u003cimg style=\"vertical-align: -0.685ex; width: 11.194ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-248.png\" alt=\"\" data-tex=\"\\(G_{\\nu\\mu} = \\lambda g_{\\nu\\mu}\\)\"\u003e. In order to make a direct\ncomparison the material unit must be conveyed to the place and pointed\nin the direction of the length to be measured. It is true that we\noften use indirect methods, avoiding actual transfer or orientation;\nbut the justification of these indirect methods is that they give the\nsame result as a direct comparison, and their validity depends upon\nthe truth of the fundamental laws of nature. We are here discussing\nthe most fundamental of these laws, and to admit the validity of the\nindirect methods of comparisons at this stage would land us in a\nvicious circle.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eI confess that I am puzzled by this passage. Taken in its plain and\nobvious sense, it means that the standard metre is to be taken from\nParis, and used without any corrections for temperature, etc., because\nas soon as we introduce such corrections we are assuming a great deal\nof physics, and thus seem to be making ourselves liable to the vicious\ncircle which, we are told, is to be avoided. It is evident, however,\nthat this is not what Professor Eddington means, since he goes on at\nonce to speak of the electron as making the adjustments concerned.\nNow the electron may be, theoretically, a perfect spatial unit, but\nwe certainly cannot compare its size with that of larger bodies\n\u003ci\u003edirectly\u003c/i\u003e, without assuming any previous physical knowledge. It\nseems that Professor Eddington is postulating an ideal observer, who\ncan see electrons just as directly as (or, rather, much more directly\nthan) we can see a metre rod. In short, his \"direct measurement\" is an\noperation as abstract and theoretical as his mathematical symbolism.\u003cspan class=\"pagenum\" id=\"Page_93\"\u003e[Pg 93]\u003c/span\u003e\nThat being admitted, we may take the electron as our spatial unit, and\nask ourselves what our ideal observer could do with it. He could not\ntake a lot of electrons and place them end on in a row, with a view\nto measuring a given length, since an infinite force is required to\nmake two electrons touch. To measure ordinary lengths, he would have\nto take (say) hydrogen at a given temperature and pressure, enclosed\nin a balloon whose radius is the length to be measured; he could then\ncount the number of electrons in the balloon and take its cube root\nas a measure of the said length. But to ascertain the temperature and\npressure, he will have to make other measurements; moreover, he will\nhave to \u003ci\u003eassume\u003c/i\u003e that his balloon is spherical. Altogether, the\nmethod does not seem very practical.\u003c/p\u003e\n\n\u003cp\u003eI have no complete theory of physical measurements to offer, but it\nseemed desirable to illustrate how difficult it is to say precisely\nwhat measurement means in an advanced science such as physics. We\nhave certain postulates, such as \"lengths which are equal to the same\nlength are equal to one another,\" but actual measurements, when made\nwith sufficient accuracy, are not found to verify these postulates.\nTherefore we invent physical laws to save the postulates. With each\nfresh law it becomes more difficult to say exactly what we do mean\nwhen, \u003ci\u003ee.g.\u003c/i\u003e, we give the wave-length of a certain line in the\nspectrum of hydrogen in terms of the metre. (This is particularly\nodd in view of the fact that these wave-lengths are given to more\nsignificant figures than can be warranted by the operations applicable\nto the standard metre itself, whose length is only known, in comparison\nwith other lengths, to a very moderate degree of approximation.) In\nphysical theory, measurement should rest upon an integration of the\nformula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e. But in physical practice the \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-249.png\" alt=\"\" data-tex=\"\\(g_{\\nu\\mu}\\)\"\u003e of\nthat formula can only be determined by means of measurements. Thus\nthe only thing we seem warranted in saying\u003cspan class=\"pagenum\" id=\"Page_94\"\u003e[Pg 94]\u003c/span\u003e is this: It is possible\nto correct the results of actual measurements according to certain\nknown rules, in such a way that the corrected lengths shall satisfy\nsuch postulates as Euclid\u0027s first axiom; when this is done, we find,\nby means of physical theory, that all electrons have the same size.\nBut this is not, considered empirically, at all a simple fact. And\nconsidered as a statement in the deductive theory it probably has a\ngood meaning, but one which demands much elucidation. Until this is\nforthcoming, all use of numbers as measures of physical quantities in\ntheoretical physics raises problems, since we do not know what, in\ntheoretical physics, replaces the operation of measurement as conducted\nin the laboratory and in daily life.\u003c/p\u003e\n\n\u003cp\u003eThe theory of length-measurement raises problems which bring us\nnaturally to Weyl\u0027s relativistic theory of electromagnetism, which we\nmust now briefly consider.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_27\" href=\"#FNanchor_27\" class=\"label\"\u003e[27]\u003c/a\u003e\nSee his essay in \u003ci\u003eScience, Religion, and Reality\u003c/i\u003e,\nedited by Needham, 1925.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_28\" href=\"#FNanchor_28\" class=\"label\"\u003e[28]\u003c/a\u003e\nCf. \u003ci\u003eMathematical Theory of Relativity\u003c/i\u003e, §§ 52, 54,\n66.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_29\" href=\"#FNanchor_29\" class=\"label\"\u003e[29]\u003c/a\u003e\nEddington, \u003ci\u003eop. cit.\u003c/i\u003e p. 115.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_30\" href=\"#FNanchor_30\" class=\"label\"\u003e[30]\u003c/a\u003e\n\u003ci\u003eOp. cit.\u003c/i\u003e, § 66, pp. 152-155.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_95\"\u003e[Pg 95]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_X\"\u003eCHAPTER X\u003cbr\u003e\nWEYL\u0027S THEORY\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE theory to be considered in this chapter is, from a geometrical\npoint of view, a natural generalization of Einstein\u0027s arbitrariness of\nco-ordinates; from a physical point of view, it fits electromagnetism\ninto the deductive system, which Einstein\u0027s theory does not do. The\ntheory is due to Hermann Weyl, and will be found in his \u003ci\u003eSpace, Time,\nMatter\u003c/i\u003e (1922).\u003c/p\u003e\n\n\u003cp\u003eThe puzzles about measurement considered at the end of Chapter IX.\nnaturally suggest the point of view from which Weyl starts. As he\nsays: \"The same certainty that characterizes the relativity of motion\naccompanies the principle of the \u003ci\u003erelativity of magnitude\u003c/i\u003e\"\n(\u003ci\u003eop. cit.\u003c/i\u003e p. 283). Measurement is a comparison of lengths,\nand Weyl suggests that, when lengths in different places are to be\ncompared, the result may depend upon the route pursued in passing from\nthe one place to the other. Lengths at the same place (\u003ci\u003ei.e.\u003c/i\u003e\nhaving one end identical), if small, he regards as directly comparable;\nalso he assumes continuity in the changes accompanying transportation.\nThis is not the sum-total of his assumptions, nor the most general\nway of stating them; but before we can state them adequately certain\nexplanations are necessary.\u003c/p\u003e\n\n\u003cp\u003eReduced to its simplest terms, the conception used by Weyl may be\nexpressed as follows. Given a vector at a point, what are we to mean\nby the statement that a vector at another point is equal to it? There\nmust be some element of convention in our definition; let us therefore,\nas a first step, set up a unit of length in each place, and see what\nlimitations it is desirable to impose on our initial arbitrariness.\u003c/p\u003e\n\n\u003cp\u003eThere is, to begin with, an assumption which is made almost\u003cspan class=\"pagenum\" id=\"Page_96\"\u003e[Pg 96]\u003c/span\u003e\ntacitly, and that is, that we can recognize something in one\nplace as the \"same\" vector as something at another place.\nWe may perhaps take this sameness as being merely analytical:\nthe two are the same function of the co-ordinates at their\nrespective places. I do not think this is all that is meant, since\na vector is supposed to have some physical significance; but\nif more is meant, it is not clear how it is to be defined. We\nwill therefore assume that, given a function of the co-ordinates\nwhich is a vector, we shall regard the same function of other\nvalues of the co-ordinates as the \"same\" vector at another\nplace.\u003c/p\u003e\n\n\u003cp\u003eWe next have to define \"parallel displacement.\" This may be defined\nin various ways. Perhaps the most graphic description is to say that\nit is displacement along a geodesic (Eddington, \u003ci\u003eop. cit.\u003c/i\u003e p.\n71). Another definition is that it is a displacement such that the\n\"covariant derivative\" vanishes, the covariant derivative of a vector\n\u003cimg style=\"vertical-align: -0.685ex; width: 2.849ex; height: 2.305ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-250.png\" alt=\"\" data-tex=\"\\(A_{\\mu}\\)\"\u003e with respect to \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e being defined as \u003cimg style=\"vertical-align: -0.685ex; width: 3.697ex; height: 2.305ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-251.png\" alt=\"\" data-tex=\"\\(A_{\\mu\\nu}\\)\"\u003e,\nwhere:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.619ex; width: 44.72ex; height: 5.965ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-252.png\" alt=\"\" data-tex=\"\\[\nA_{\\mu\\nu} = \\frac{\\partial A_\\mu}{dx_{\\nu}} – \\sum_{\\alpha}\\{\\mu\\nu,\\alpha\\} A_{\\alpha} \\quad (\\alpha = 1, 2, 3, 4).\n\\]\"\u003e\u003c/span\u003e\nFor the definition of \u003cimg style=\"vertical-align: -0.566ex; width: 7.28ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-253.png\" alt=\"\" data-tex=\"\\(\\{\\mu\\nu,\\alpha\\}\\)\"\u003e, see the beginning of\nChapter IX. In the tensor calculus, covariant differentiation takes\nthe place of ordinary differentiation for many purposes, since the\ncovariant derivative of a tensor is a tensor, whereas the ordinary\nderivative is in general not a tensor. We assume that our units\nof length in different places are so chosen that, when a small\ndisplacement is moved to a neighbouring place by parallel displacement,\nthe change in the measure of its length is small, and is proportional\nto its length. We assume, in short, that the ratio of the increase of\nlength to the initial length for a change of co-ordinates (\u003cimg style=\"vertical-align: -0.439ex; width: 15.845ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-254.png\" alt=\"\" data-tex=\"\\(dx_1, dx_2, dx_3 dx_4\\)\"\u003e) is:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 30.796ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-255.png\" alt=\"\" data-tex=\"\\[\nk_1dx_1 + k_2dx_2 + k_3dx_3 + k_4dx_4\n\\]\"\u003e\u003c/span\u003e\n\u003cspan class=\"pagenum\" id=\"Page_97\"\u003e[Pg 97]\u003c/span\u003eSo that (\u003cimg style=\"vertical-align: -0.439ex; width: 11.684ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-256.png\" alt=\"\" data-tex=\"\\(k_1, k_2, k_3, k_4\\)\"\u003e) form a vector, \u003cimg style=\"vertical-align: -0.685ex; width: 2.331ex; height: 2.255ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-257.png\" alt=\"\" data-tex=\"\\(k_{\\mu}\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eNow it is possible to express Maxwell\u0027s equations in terms of a vector\nwhich may be identified with the above vector. Hence it is possible\nto regard electromagnetic phenomena as explained by the variation of\nwhat is taken as the unit as we pass from point to point. I shall not\nattempt to explain the theory, as it would in any case be necessary to\nread a full account in order to grasp its significance.\u003c/p\u003e\n\n\u003cp\u003eHere, perhaps even more than elsewhere in relativity theory, it is\ndifficult to disentangle the conventional elements from those having\nphysical significance. On the face of it, it might seem as though we\nwere attempting to account for actual physical phenomena by means of a\nmere convention as to choice of units. But this, of course, is not what\nis meant. The way the unit is assigned in different places is called by\nEddington the \"gauge-system\": this is only partially arbitrary, and is\nin part the representation of the physical state of the world. This has\nto do with the fact that vectors are not purely analytical expressions,\nbut also correspond to physical facts. It would seem, however, that\nthe theory has not yet been expressed with the logical purity that is\nto be desired, chiefly because it is not prefaced by any clear account\nof what is to be understood by \"measurement\"—or, what comes to much\nthe same thing from the standpoint of theory, what we are to mean when\nwe talk of \"moving\" a vector, whether by parallel displacement or in\nany other way. To \"move\" something, we must be able to recognize some\nidentity between things in different places. Perhaps all this is quite\nclear in the minds of competent exponents of the theory, but if so they\nhave not succeeded in conveying their thoughts without loss of clarity\nto readers who have not their background. When Eddington says: \"Take a\ndisplacement at \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and transfer it by parallel displacement to an\ninfinitely near point \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e\" (p. 200), I find myself wondering how,\nexactly, the displacement is to preserve its identity throughout the\u003cspan class=\"pagenum\" id=\"Page_98\"\u003e[Pg 98]\u003c/span\u003e\ntransfer, and the only answer suggested by the accompanying formulæ is\nthat the identity is that of an algebraic expression in terms of the\nco-ordinates. This, however, is clearly insufficient.\u003c/p\u003e\n\n\u003cp\u003eProfessor Eddington, after expounding Weyl\u0027s theory, proceeds to\ngeneralize it, and some of his accompanying elucidations are relevant\nto our present difficulties. Thus he says (p. 217):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"In Weyl\u0027s theory, a gauge-system is partly physical and partly\nconventional; lengths in different directions but at the same point are\nsupposed to be compared by experimental (optical) methods; but lengths\nat different points are not supposed to be comparable by physical\nmethods (transfer of clocks and rods), and the unit of length at each\npoint is laid down by a convention. I think this hybrid definition of\nlength is undesirable, and that length should be treated as a purely\nconventional or else a purely physical conception.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eHe proceeds to a generalized theory in which, at first, length is\n\u003ci\u003epurely\u003c/i\u003e conventional, for comparisons at a point as well as for\ncomparisons between different points. This generalized theory does not\nseem to involve the same kind of difficulties as those which have been\ntroubling us. The following passage, for example, states the matter\nwith great clearness (p. 226):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The relation of displacement, between point-events and the relation\nof \u0027equivalence\u0027 between displacements form part of one idea, which\nare only separated for convenience of mathematical manipulation.\nThat the relation of displacement between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e amounts to\nsuch-and-such a quantity conveys no absolute meaning; but that the\nrelation of displacement between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is \u0027equivalent\u0027 to the\nrelation of displacement between \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.873ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-260.png\" alt=\"\" data-tex=\"\\(D\\)\"\u003e is (or at any rate\nmay be) an absolute assertion. Thus four points is the minimum number\nfor which an assertion of absolute structural relation can be made.\nThe ultimate elements of structure are thus four-point elements. By\nadopting the condition of affine geometry, I have limited the possible\nassertion with regard to a four-point\u003cspan class=\"pagenum\" id=\"Page_99\"\u003e[Pg 99]\u003c/span\u003e element to the statement that\nthe four points do, or do not, form a parallelogram. The defence of\naffine geometry thus rests on the not implausible view that four-point\nelements are recognized to be differentiated from one another by a\nsingle character—viz. that they are or are not of a particular kind\nwhich is conventionally named parallelogramical. Then the analysis\nof the parallelogram property into a double equivalence of \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e to\n\u003cimg style=\"vertical-align: -0.05ex; width: 3.593ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-261.png\" alt=\"\" data-tex=\"\\(CD\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 3.59ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-262.png\" alt=\"\" data-tex=\"\\(BD\\)\"\u003e, is merely a definition of what is meant by\nthe equivalence of displacements.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eHere we have a logically satisfactory theoretical basis for a metric.\nWe may suppose that, as a matter of fact, there are important\nproperties of groups of four points which are \"parallelogramical,\"\nand that actual physical measurement is an approximate method of\ndiscovering which groups have this property. We shall find certain laws\napproximately fulfilled by rough-and-ready measurements, and fulfilled\nwith increasing accuracy as we introduce refinements into the process\nof measurement. Consider, for example, Euclid\u0027s first axiom: Things\nwhich are equal to the same thing are equal to one another. Presumably\nEuclid regarded this as a logically necessary proposition, and so do\npeople who are engaged in the practice of measurement. If two lengths\neach equal to a metre are found to be not equal to each other, the\nplain man assumes that there must be a mistake somewhere. We are\ntherefore continually redefining the actual operations of measurement\nwith a view to verifying Euclid\u0027s first axiom as nearly as possible.\nBut with the above-quoted definition of equality of length the first\naxiom becomes a substantial proposition, namely: If \u003cimg style=\"vertical-align: -0.05ex; width: 7.007ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-263.png\" alt=\"\" data-tex=\"\\(ABCD\\)\"\u003e is a\nparallelogram, and likewise \u003cimg style=\"vertical-align: -0.05ex; width: 7.016ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-264.png\" alt=\"\" data-tex=\"\\(DCEF\\)\"\u003e, then \u003cimg style=\"vertical-align: 0; width: 6.837ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-265.png\" alt=\"\" data-tex=\"\\(ABEF\\)\"\u003e is a parallelogram.\nIf this proposition is true, then it is theoretically possible to\ndefine measurement in such a way that two lengths each equal to a metre\nshall always be equal to each other. What is called \"accuracy\" is,\nspeaking generally, an attempt to obtain a result conformable with some\nideal standard supposed to be logical but in fact physical.\u003cspan class=\"pagenum\" id=\"Page_100\"\u003e[Pg 100]\u003c/span\u003e What do\nwe mean by saying that a length has been \"wrongly\" measured? Whatever\nresult we obtain from measuring a given length, the result represents a\nfact in the world. But in what we call a \"wrong\" measurement, the fact\nascertained is complex and of small universality. If the observer has\nsimply misread a scale, the fact ascertained involves reference to his\npsychology. If he has neglected a physical correction—\u003ci\u003ee.g.\u003c/i\u003e for\nthe temperature of his measure—the fact refers only to a measurement\ncarried out with that particular apparatus on that particular occasion.\nIn relativity theory we have another set of what might be called\n\"inaccurate\" measurements—\u003ci\u003ee.g.\u003c/i\u003e measurements of the masses of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-particles or \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e-particles emitted from radio-active\nbodies must be corrected for their motion relative to the observer\nbefore they acquire any general significance. It is always the search\nfor simple relations which enter into general laws that governs\nsuccessive refinements. But the existence of such relations (where\nthey do exist) is an empirical fact, so that much that seems \u003ci\u003eprima\nfacie\u003c/i\u003e to be logically necessary is really contingent. On the other\nhand, the number of premisses in a deductive system which has to agree\nwith an empirical science can, by logical skill, be diminished to an\nextent which may be astonishing. Of this, the theory of relativity is\na very remarkable example. The theory is a combination of two diverse\nelements: on the one hand, new experimental data; on the other, a new\nlogical method. It must be regarded as a happy accident that the two\nappeared together; if the right kind of theoretical genius had not\nhappened to be forthcoming, we might have had to be content for a long\ntime with patched-up hypotheses such as the FitzGerald contraction. As\nit is, the combination of experiment and theory has produced one of the\nsupreme triumphs of human genius.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_101\"\u003e[Pg 101]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XI\"\u003eCHAPTER XI\u003cbr\u003e\nTHE PRINCIPLE OF DIFFERENTIAL LAWS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHROUGHOUT the theory of relativity, there is an application, with\nincreasing stringency, of a principle which begins to make itself felt\nin physics with Galileo, in spite of the fact that he did not possess\nthe mathematical technique which it demands. The principle I mean is\nthat of \"differential laws,\" as it may be called. This means that any\nconnection which may exist between distant events is the result of\nintegration from a law giving a rate of change at every point of some\nroute from the one to the other. One may give a simple illustration of\na differential law from the \"curve of pursuit\": a man is walking along\na straight road, and his dog is in a field beside the road; the man\nwhistles to the dog, and the dog runs towards him. We suppose that at\neach moment the dog runs exactly towards where his master is at that\nmoment. To discover the curve described by the dog is a problem in\nintegration, which becomes definite given certain further data. The\nNewtonian law of gravitation gives a very similar type of law, except\nthat it is the acceleration of the planet, not its velocity, that is\ndirected towards the sun at each moment. It has long been a commonplace\nof physics that its causal laws should have this differential\ncharacter: they should tell primarily a tendency at each moment, not\nthe outcome after a finite time. In a word, its causal laws take the\nform of differential equations, usually of the second order.\u003c/p\u003e\n\n\u003cp\u003eThis view of causal laws is absent from quantum theory, from the ideas\nof savages and uneducated persons, and from the works of philosophers,\nincluding Bergson and J. S. Mill. In quantum theory, we have a discrete\nseries of possible\u003cspan class=\"pagenum\" id=\"Page_102\"\u003e[Pg 102]\u003c/span\u003e sudden changes, and a certain statistical knowledge\nof the proportion of cases in which each possibility is realized;\nbut we have no knowledge as to what determines the occurrence of a\nparticular change in a particular case. Moreover, the change is not\nof the sort that can be expressed by differential equations: it is a\nchange from a state expressed by one integer or set of integers to a\nstate expressed by another. This kind of change may turn out to be\nphysically ultimate, and to mark out at least a part of physics as\ngoverned by laws of a new sort. But we are not likely to find science\nreturning to the crude form of causality believed in by Fijians and\nphilosophers, of which the type is \"lightning causes thunder.\" It can\nnever be a law that, given \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e at one time, there is sure to be \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e\nat another time, because something might intervene to prevent n \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e.\nWe do not derive such laws from quantum phenomena, because we do not,\nin their case, know that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e will not continue throughout the time in\nquestion. The natural view to take at present is that quantum phenomena\nhave to do with the interchange of energy between matter and the\nsurrounding medium, while continuous change is found in all processes\nwhich involve no such interchange. There are, however, difficulties\nin any view at present, and it is not for a layman to venture an\nopinion. It seems not improbable that, as Heisenberg suggests, our\nviews of space-time may have to be modified profoundly before harmony\nis achieved between quantum phenomena and the laws of transmission\nof light \u003ci\u003ein vacuo\u003c/i\u003e. For the moment, however, I wish to confine\nmyself to the standpoint of relativity theory.\u003c/p\u003e\n\n\u003cp\u003eAlthough physics has worked with differential equations ever since the\ninvention of the calculus, geometry was supposed to be able to start\nwith laws applying to finite spaces. If we accept the Einsteinian point\nof view, there can no longer be any separation between geometry and\nphysics; every proposition of geometry will be to some extent causal.\nTake first\u003cspan class=\"pagenum\" id=\"Page_103\"\u003e[Pg 103]\u003c/span\u003e the special theory. Relatively to axes (\u003cimg style=\"vertical-align: -0.464ex; width: 7.29ex; height: 1.88ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-169.png\" alt=\"\" data-tex=\"\\(x, y, z, t\\)\"\u003e)\nwe can obtain propositions of geometry by keeping \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e constant;\nbut relatively to other axes these propositions will refer to events\nat different times. It is true that these events, in any system of\nco-ordinates, will have a space-like interval, and will have no direct\ncausal relations with each other; but they will have indirect causal\nrelations derived from a common ancestry. Let us take some example,\nsay: The sum of the angles of a triangle is two right angles. Our\ntriangle may be composed of rods or of light-rays. In either case,\nit must preserve a certain constancy while we measure it. Both rods\nand light-rays are complicated physical structures, and the physical\nlaws of their behaviour are involved in taking them as approximations\nto ideal straight lines. Nevertheless, so far as the special theory\nis concerned, all this might be allowed, and yet we might maintain a\ncertain distinction between geometry and physics, the former being\na set of laws supposed exact, and approximately verified, for the\nrelations of the \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e co-ordinates in any Galilean frame\nwhen \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e is kept constant.\u003c/p\u003e\n\n\u003cp\u003eBut in the general theory the intermixture of geometry and physics is\nmore intimate. We cannot accurately reduce \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e to the form:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.594ex; width: 22.759ex; height: 2.594ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-266.png\" alt=\"\" data-tex=\"\\[\ndx_4^{2} – dx_1^{2} – dx_2^{2} -dx_3^{2},\n\\]\"\u003e\u003c/span\u003e\nand therefore we cannot accurately distinguish one co-ordinate as\nrepresenting the time. We cannot therefore obtain a timeless geometry\nby putting \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-159.png\" alt=\"\" data-tex=\"\\(x_4\\)\"\u003e=constant. With this goes a change in our axioms.\nWe no longer have, as in Euclid, in Lobatchevsky and Bolyai, and in\nprojective geometry, axioms dealing with straight lines of finite\nlength. We have now only, as our initial apparatus, a geometry of the\ninfinitesimal, from which large-scale results must be obtained by\nintegration. From this point of view, Weyl\u0027s extension of Einstein\nappears natural. As we saw in the last chapter, quoting Eddington, the\nstatement that the distances \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e,\u003cspan class=\"pagenum\" id=\"Page_104\"\u003e[Pg 104]\u003c/span\u003e \u003cimg style=\"vertical-align: -0.05ex; width: 3.593ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-261.png\" alt=\"\" data-tex=\"\\(CD\\)\"\u003e are equal is the assertion\nof a relation between the four points. \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.873ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-260.png\" alt=\"\" data-tex=\"\\(D\\)\"\u003e. If\nall the relations which constitute our initial apparatus are to be\nconfined to the infinitesimal, so must this relation; if so, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.873ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-260.png\" alt=\"\" data-tex=\"\\(D\\)\"\u003e must all be close together, and Weyl\u0027s geometry\nresults.\u003c/p\u003e\n\n\u003cp\u003eAt this point, however, the pure mathematician is likely to feel a\ndifficulty which does not greatly trouble the physicist. The physicist\nthinks of his infinitesimals as actual small quantities, which\nmay—\u003ci\u003ee.g.\u003c/i\u003e in astronomical problems—be such as would be reckoned\nlarge in other problems. For him, therefore, a statement in terms of\ninfinitesimals is quite satisfactory.\u003c/p\u003e\n\n\u003cfigure class=\"figleft width500\" id=\"i_104\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-104.jpg\" width=\"200\" height=\"119\" alt=\" geometric\ndiagram showing a parallelogram divided into two equal parts by a\ndiagonal line, with vertices labeled a, b, c, d, e, and f. Simple line\ndrawing illustrating a mathematical concept.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eBut for the pure mathematician there are no infinitesimals, and all\nstatements in which they seem to occur must be expressible as limits\nof what happens to finite quantities. To take our particular case:\nWe must be able to say of a small finite quadrilateral that it is\n\u003ci\u003eapproximately\u003c/i\u003e a parallelogram, if we are to be able to assign\na meaning to the statement that an infinitesimal quadrilateral may be\n\u003ci\u003eaccurately\u003c/i\u003e a parallelogram. The case is exactly analogous to\nvelocity in elementary kinematics: we can assign a meaning to velocity\nonly because we can measure finite distances and times, and so form\nthe conception of the limit of their quotient. It is not wholly clear\nhow we are to satisfy this requirement in the case of Weyl\u0027s theory.\nI think, however, that there is not the slightest reason to suppose\nthat it cannot be satisfied. Let \"\u003cimg style=\"vertical-align: -0.566ex; width: 10.819ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-267.png\" alt=\"\" data-tex=\"\\(R(a, b, c, d)\\)\"\u003e\" mean \"\u003cimg style=\"vertical-align: -0.439ex; width: 7.342ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-268.png\" alt=\"\" data-tex=\"\\(a, b, c, d\\)\"\u003e\nform a parallelogram.\" We are supposed to have also \u003cimg style=\"vertical-align: -0.566ex; width: 7.801ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-269.png\" alt=\"\" data-tex=\"\\(R (abcd)\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.566ex; width: 7.801ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-270.png\" alt=\"\" data-tex=\"\\(R(bacd)\\)\"\u003e, etc., but not \u003cimg style=\"vertical-align: -0.566ex; width: 7.801ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-271.png\" alt=\"\" data-tex=\"\\(R(acbd)\\)\"\u003e etc. Also if we have \u003cimg style=\"vertical-align: -0.566ex; width: 7.801ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-272.png\" alt=\"\" data-tex=\"\\(R(abcd)\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.566ex; width: 7.932ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-273.png\" alt=\"\" data-tex=\"\\(R(cdef)\\)\"\u003e, we are to have \u003cimg style=\"vertical-align: -0.566ex; width: 7.943ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-274.png\" alt=\"\" data-tex=\"\\(R(abfe)\\)\"\u003e. But if we take \"\u003cimg style=\"vertical-align: -0.566ex; width: 10.561ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-275.png\" alt=\"\" data-tex=\"\\(S(a, b, c, d)\\)\"\u003e\"\nto mean \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e form an approximate\nparallelogram,\" we cannot (if there\u003cspan class=\"pagenum\" id=\"Page_105\"\u003e[Pg 105]\u003c/span\u003e is any way of specifying a\ndegree of approximation) argue from \u003cimg style=\"vertical-align: -0.566ex; width: 7.543ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-277.png\" alt=\"\" data-tex=\"\\(S(abcd)\\)\"\u003e and \u003cimg style=\"vertical-align: -0.566ex; width: 7.674ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-278.png\" alt=\"\" data-tex=\"\\(S(cdef)\\)\"\u003e to\n\u003cimg style=\"vertical-align: -0.566ex; width: 7.686ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-279.png\" alt=\"\" data-tex=\"\\(S(abfe)\\)\"\u003e. Now if we assume, as Weyl does, that lengths at a given\npoint are comparable, we can perhaps give the necessary definitions. We\nshall have to take \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e, not \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e, as our fundamental relation, since\nthe distance between any two points is finite, and it is assumed that\nno finite quadrilateral can be accurately a parallelogram. Or perhaps\nwe shall have to go a step further, and take as fundamental a relation\nof eight points, say\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 14.747ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-280.png\" alt=\"\" data-tex=\"\\[\n(abcd)T(efgh),\n\\]\"\u003e\u003c/span\u003e\nmeaning \"\u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e is more nearly a parallelogram than \u003cimg style=\"vertical-align: -0.464ex; width: 4.681ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-282.png\" alt=\"\" data-tex=\"\\(efgh\\)\"\u003e\" We\nshall then say that, given any four points, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-284.png\" alt=\"\" data-tex=\"\\(f\\)\"\u003e,\nit is possible to find points \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e nearer to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\nrespectively than \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-284.png\" alt=\"\" data-tex=\"\\(f\\)\"\u003e are, such that\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 14.532ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-285.png\" alt=\"\" data-tex=\"\\[\n(abcd)T(abfe).\n\\]\"\u003e\u003c/span\u003e\nFurther, we can say that, if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e are\nsufficiently near together, and\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 15.017ex; height: 2.396ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-286.png\" alt=\"\" data-tex=\"\\[\n(abcd\u0027)T(abcd),\n\\]\"\u003e\u003c/span\u003e\nthen the ratio of \u003cimg style=\"vertical-align: -0.023ex; width: 2.981ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-287.png\" alt=\"\" data-tex=\"\\(dd\u0027\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-288.png\" alt=\"\" data-tex=\"\\(dc\\)\"\u003e can be made to approach zero as a\nlimit by diminishing the size of \u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e in a purely ordinal sense.\n(Ordinal relations among points, as we saw earlier, are presupposed in\nthe theory of relativity.)\u003c/p\u003e\n\n\u003cp\u003eIt is highly probable that the above process can be simplified. It is,\nhowever, of no importance in itself; its only purpose is to show that\nthe derivatives required can be correctly defined, and that, however\nthe mathematical treatment may confine itself to infinitesimals,\nrelations between points whose distances are finite must be presupposed\nif the infinitesimal calculus is to be applicable.\u003c/p\u003e\n\n\u003cp\u003eThis last result, whose generality is obvious from the theory of\nlimits, is of some philosophical importance. Wherever mathematics works\nin a continuous medium with relations which may be loosely described as\nnext-to-next, there must be other\u003cspan class=\"pagenum\" id=\"Page_106\"\u003e[Pg 106]\u003c/span\u003e relations, holding between points at\nfinite distances from each other, and having the next-to-next relations\nas their limits. Thus, when we say that laws have to be expressed by\ndifferential equations, we are saying that the finite relations which\noccur cannot be brought under accurate laws, but only their limits as\ndistances are diminished. We are not saying that these limits are the\nphysical realities; on the contrary, the physical realities continue\nto be the finite relations. And if our theory is to be adequate, some\nway must be found of so defining the finite relations as to make the\npassage to the limit possible.\u003c/p\u003e\n\n\u003cp\u003eIt is considered a merit in the general theory of relativity,\nparticularly in Weyl\u0027s form (or the still more general form suggested\nby Eddington), that it dispenses with what we may call \"integrated\"\nrelations as regards its fundamentals. Thus Eddington, after pointing\nout that he is concerned with structure, not with substance, proceeds\n(p. 224):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"But structure can be described to some extent; and when reduced to\nultimate terms it seems to resolve itself into a complex of relations.\nAnd further these relations cannot be entirely devoid of comparability;\nfor if nothing in the world is comparable with anything else, all parts\nof it are alike in their unlikeness, and there cannot be even the\nrudiments of a structure.\u003c/p\u003e\n\n\u003cp\u003e\"The axiom of parallel displacement is the expression of this\ncomparability, and the comparability postulated seems to be\nalmost the minimum conceivable. Only relations which are close\ntogether—\u003ci\u003ei.e.\u003c/i\u003e interlocked in the relation-structure—are\nsupposed to be comparable, and the conception of equivalence is applied\nto only one type of relation. This comparable relation is called\ndisplacement. By representing this relation graphically we obtain the\nidea of location in space; the reason why it is natural for us to\nrepresent this particular relation graphically does not fall within the\nscope of physics.\u003c/p\u003e\n\n\u003cp\u003e\"Thus our axiom of parallel displacement is the geometrical garb\nof a principle which may be called \u0027the comparability of proximate\nrelations.\u0027\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_107\"\u003e[Pg 107]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt is obvious that, in the above passage, Eddington is \u003ci\u003eimagining\u003c/i\u003e\ndisplacements at a small finite distance from each other, not at\nan infinitesimal distance; he is not thinking of all the apparatus\ninvolved in a procedure which replaces infinitesimals by limits. One\nmight suggest that he is supposing, \u003ci\u003ee.g.\u003c/i\u003e, that a footrule will\nnot change much during the portion of a second required to transfer it\nfrom one part of a given page to another. But when we say that it will\nnot change \"much,\" we imply some standard of quantitative comparison\nother than the footrule; and this leads to the problems we have been\nconsidering.\u003c/p\u003e\n\n\u003cp\u003eI cannot but think that Eddington\u0027s point of view lends itself to\ndevelopment and further analysis by means of mathematical logic; in\nparticular, this applies to the conditions for the possibility of\nmeasurement, a subject which will be considered explicitly in the next\nchapter. But for the present my concern is with \"the comparability\nof proximate relations.\" In the first place, what is meant by\n\"comparability\"? A moment\u0027s reflection shows that what is wanted is\na symmetrical transitive relation which each of the relations in\nquestion has to some others, but not to all. (It is assumed, in the\nparticular case of Eddington\u0027s general geometry, that when there\nis such a relation of the interval \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e to the interval \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-290.png\" alt=\"\" data-tex=\"\\(cd\\)\"\u003e,\nthere is also such a relation of the interval \u003cimg style=\"vertical-align: -0.023ex; width: 2.373ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-291.png\" alt=\"\" data-tex=\"\\(ad\\)\"\u003e to the interval\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.95ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-292.png\" alt=\"\" data-tex=\"\\(bc\\)\"\u003e. But this, as he admits (p. 226), is not essential.) Now why\nshould we suppose that a transitive symmetrical relation of the above\nsort is more likely to exist between small intervals than between\nlarge ones? \u003ci\u003eI.e.\u003c/i\u003e, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.598ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-293.png\" alt=\"\" data-tex=\"\\(b\u0027\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, and\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.607ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-294.png\" alt=\"\" data-tex=\"\\(c\u0027\\)\"\u003e between \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, is it more likely that the relation\nin question will hold between \u003cimg style=\"vertical-align: -0.025ex; width: 2.795ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-295.png\" alt=\"\" data-tex=\"\\(ab\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 2.784ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-296.png\" alt=\"\" data-tex=\"\\(dc\u0027\\)\"\u003e than between\n\u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-288.png\" alt=\"\" data-tex=\"\\(dc\\)\"\u003e? I do not see why we should think so. And I think\nfurther that, with a correct interpretation of infinitesimals, the\nwhole belief that causation must always be from next-to-next becomes\nuntenable unless continuity is abandoned. Causal laws may all be\n\u003cspan class=\"pagenum\" id=\"Page_108\"\u003e[Pg 108]\u003c/span\u003e\ndifferential equations, but the grounds for thinking that they are\nmust be empirical, not \u003ci\u003ea priori\u003c/i\u003e. They cannot be derived from\nthe impossibility of action at a distance unless distance itself is a\nderivative from causality, which may well be the case, but does not\nrepresent any part of the views of those who are anxious to dispense\nwith action at a distance. It may well be, therefore, that there is\none department of physics—that included in the general theory of\nrelativity, as supplemented by Weyl—in which everything proceeds by\ndifferential equations, while there is another part—that dealt with by\nquantum theory—in which this whole apparatus is inapplicable. There\nis absolutely no \u003ci\u003ea priori\u003c/i\u003e reason why everything should go by\ndifferential equations, since, even then, causation does not really go\nfrom next-to-next: in a continuum there is no \"next.\" It is, at bottom,\nbecause \"next-to-next\" seems natural that we like a procedure of\ndifferential equations; but the two are logically incompatible, and our\npreference for the second on account of the first proceeds only from\nlogical confusion.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_109\"\u003e[Pg 109]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XII\"\u003eCHAPTER XII\u003cbr\u003e\nMEASUREMENT\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nREPEATEDLY, in previous discussions, we have come up against the\nproblem of measurement. It is time to consider it on its own account,\nboth how it is to be defined, and in what circumstances it is possible.\u003c/p\u003e\n\n\u003cp\u003eIn the first place, what do we mean by measurement? Clearly we do\n\u003ci\u003enot\u003c/i\u003e mean \u003ci\u003eany\u003c/i\u003e method of assigning numbers to a collection\nof objects; there must be properties of importance connected with\nthe numbers assigned. We do not say that the books in the British\nMuseum are \"measured\" by their press-marks. Given any collection whose\ncardinal number is less than or equal to \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-134.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_{0}}\\)\"\u003e, we can\nassign some or all of the real numbers as \"press-marks\" of the several\nmembers of the collection. Given any collection of \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-134.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_{0}}\\)\"\u003e\nterms, it can be arranged in a Euclidean or non-Euclidean space of\nany known sort with any finite number of dimensions, and when so\narranged it will be amenable to the whole of metrical geometry. But\nthe \"distance\" between two terms of the collection, when it is defined\nin this way, will, in general, be quite unimportant, in the sense that\nit will have only such properties as follow tautologically from its\ndefinition, not such further empirical properties as would make the\ndefinition valuable. So long as this is the case, there is no reason to\nprefer one to another of the various incompatible systems of distances\nwhich pure mathematics would allow us to assign.\u003c/p\u003e\n\n\u003cp\u003eLet us take an illustration. In projective geometry we start from\na set of axioms which say nothing about quantity, and do not even\n\u003ci\u003eobviously\u003c/i\u003e involve order. But it is found that they do lead to an\norder, and that, by means of the order,\u003cspan class=\"pagenum\" id=\"Page_110\"\u003e[Pg 110]\u003c/span\u003e co-ordinates can be assigned\nto points. These co-ordinates have a definite projective meaning:\nthey represent the series of quadrilateral constructions required to\nreach the point in question from certain given initial points, (I omit\ncomplications concerning limits; these are dealt with in the chapter\n\"Projective Geometry\" in \u003ci\u003eThe Principles of Mathematics\u003c/i\u003e.) In this\ncase, it may seem doubtful whether we have measurement or not. We have\nassigned co-ordinates in a manner which preserves the order-relations\nof points, and it turns out that the ordinary distance between two\npoints is a simple function of their projective co-ordinates, though\nthe function is somewhat different according as space is Euclidean,\nhyperbolic, or elliptic. It is just because of this difference that\nwe shall not say we have \"measured\" distances when we have introduced\nprojective co-ordinates. These co-ordinates, for example, will not tell\nus, even approximately, how long it would take to walk from one place\nto another, and this is the sort of thing that measurement ought to\ntell us.\u003c/p\u003e\n\n\u003cp\u003eWhat, then, \u003ci\u003eis\u003c/i\u003e meant when it is said that, in the theory of\nrelativity, there is a metrical relation of interval? Let us take up\nthe matter at the point where Eddington leaves it. He suggests that all\nthat is needed is \"comparability\" between two point-pairs, or, as he\nsays, between two \"displacements.\" (We may leave aside for the moment\nthe question whether this is only to hold for point-pairs which are\nvery near together.) This language seems somewhat vague; let us try to\ngive it precision.\u003c/p\u003e\n\n\u003cp\u003eSuppose that between two point-pairs there is sometimes, but not\nalways, a symmetrical transitive relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e. Then we can define as\n\"the distance between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\" the class of all point-pairs\nhaving the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e to (\u003cimg style=\"vertical-align: -0.464ex; width: 2.403ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-297.png\" alt=\"\" data-tex=\"\\(xy\\)\"\u003e). If now instead of \u003cimg style=\"vertical-align: -0.566ex; width: 9.174ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-298.png\" alt=\"\" data-tex=\"\\((xy)S(zw\\)\"\u003e)\nwe write \u003cimg style=\"vertical-align: -0.464ex; width: 8.092ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-299.png\" alt=\"\" data-tex=\"\\(xy = zw\\)\"\u003e, we shall have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 38.198ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-300.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n\\text{If}\\,\\, \u0026xy=zw,\\,\\, \\text{then}\\,\\, zw=xy;\\\\\n\\text{If}\\,\\, \u0026xy=zw\\,\\, \\text{and}\\,\\, zw=uv, \\text{then}\\,\\, xy=uv.\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_111\"\u003e[Pg 111]\u003c/span\u003e\nFrom these two it follows that every pair of objects \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e in\nthe field of \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e is such that\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 8.451ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-301.png\" alt=\"\" data-tex=\"\\[\nxy=xy.\n\\]\"\u003e\u003c/span\u003e\nThis seems to be as much as is strictly implied by Eddington\u0027s words,\nbut it is certainly not all that we need. Nor does it become sufficient\nif we add:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 25.239ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-302.png\" alt=\"\" data-tex=\"\\[\n\\text{If}\\,\\, xy=zw, \\text{then}\\,\\, xz=yw.\n\\]\"\u003e\u003c/span\u003e\nThere must be a connection between distances and ordinal relations,\nthere must be ways of adding distances, and there must be ways of\ninferring new distances from a certain number of data, as in\n\u003cimg style=\"vertical-align: -0.685ex; width: 19.217ex; height: 2.572ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-303.png\" alt=\"\" data-tex=\"\\(ds^{2} = \\sum{g_{\\mu\\nu}dx_{\\mu}dx_{\\nu}}\\)\"\u003e. If all these conditions are\nfulfilled, we can then proceed to ask whether our distances have any\nfurther important physical properties.\u003c/p\u003e\n\n\u003cp\u003eThe sort of relation that will not do is illustrated if we take \u003cimg style=\"vertical-align: -0.464ex; width: 8.092ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-304.png\" alt=\"\" data-tex=\"\\(xy =\nzw\\)\"\u003e to mean that \u003cimg style=\"vertical-align: -0.464ex; width: 2.403ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-297.png\" alt=\"\" data-tex=\"\\(xy\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 2.672ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-305.png\" alt=\"\" data-tex=\"\\(zw\\)\"\u003e have the same apparent dimensions\nin the visual field of a certain observer—\u003ci\u003ee.g.\u003c/i\u003e the diameters\nof the sun and moon will approximately have this relation, which is\nsymmetrical and transitive, but physically unimportant. Let us see\nwhat is necessary in order to get a definition of distance which will\nhave as many as possible of the properties possessed by distance in\nelementary geometry.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_112\"\u003e[Pg 112]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIf we confine ourselves to three dimensions, we can at once define a\nplane: it will consist of all points equidistant from two given points.\nThe points in this plane which are equidistant from two given points\nin it lie on a straight line; we may take this as the definition of\na straight line. Thus given two points, \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e, we can define\nthe middle point \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e of \u003cimg style=\"vertical-align: -0.439ex; width: 3.489ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-308.png\" alt=\"\" data-tex=\"\\(PQ\\)\"\u003e it is the point on \u003cimg style=\"vertical-align: -0.439ex; width: 3.489ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-308.png\" alt=\"\" data-tex=\"\\(PQ\\)\"\u003e which is\nequidistant from \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e. We shall need an axiom to the effect\nthat this point always exists and is always unique. Thus we can halve\ndistances and double them: we shall of course define \u003cimg style=\"vertical-align: 0; width: 4.077ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-309.png\" alt=\"\" data-tex=\"\\(PM\\)\"\u003e as half of\n\u003cimg style=\"vertical-align: -0.439ex; width: 3.489ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-308.png\" alt=\"\" data-tex=\"\\(PQ\\)\"\u003e. From this point onwards, the assignment of numerical measures\nto our distances offers no difficulty. It is therefore only necessary\nto scrutinize what has already been said.\u003c/p\u003e\n\n\u003cp\u003eIn ordinary Euclidean geometry, there is exactly one point on a plane\nwhich is equidistant from three given points on the plane; it is the\ncentre of the circumscribed circle. In three dimensions, there is one\npoint equidistant from four given points; in four, from five. This\nlast holds also in the special theory of relativity, and even in the\ngeneral theory so long as the distances concerned are small. If we take\na point (\u003cimg style=\"vertical-align: -0.439ex; width: 20.802ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-310.png\" alt=\"\" data-tex=\"\\(d_1x_1, d_1x_2, d_1x_3, d_1x_4\\)\"\u003e) near the origin, another\npoint (\u003cimg style=\"vertical-align: -0.439ex; width: 16.851ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-184.png\" alt=\"\" data-tex=\"\\(dx_1, dx_2, dx_3, dx_4\\)\"\u003e) is equidistant from this point and\nthe origin if \u003cimg style=\"vertical-align: -0.685ex; width: 22.369ex; height: 2.572ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-311.png\" alt=\"\" data-tex=\"\\(\\sum{g_{\\mu\\nu}dx_{\\mu}d_1dx_{\\nu}} = d_1s^{2}\\)\"\u003e (where\nthe \u003cimg style=\"vertical-align: -0.685ex; width: 3.08ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-207.png\" alt=\"\" data-tex=\"\\(g_{\\mu\\nu}\\)\"\u003e have their values at the origin), which is a simple\nequation in \u003cimg style=\"vertical-align: -0.685ex; width: 3.623ex; height: 2.255ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-312.png\" alt=\"\" data-tex=\"\\(dx_{\\mu}\\)\"\u003e. Four such equations give a unique set of\nvalues for (\u003cimg style=\"vertical-align: -0.439ex; width: 16.851ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-184.png\" alt=\"\" data-tex=\"\\(dx_1, dx_2, dx_3, dx_4\\)\"\u003e). Thus there is just one point\nequidistant from five given points close together. Moreover, a simple\nequation, which we may take to be that of the part of a plane near the\norigin, gives the locus of points near the origin and equidistant from\nit and a neighbouring point. In fact, as we should expect, for small\ndistances everything proceeds as in elementary geometry, given the\nformula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eBut the mere assumption that there is such a relation as \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e between\npoint-pairs does not yield these results, since it does not imply the\ninterrelation of distances which is given by the formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e.\nNevertheless, it does suffice theoretically as a basis of measurement,\nsince, as we have seen, it enables us to halve distances and double\nthem, and therefore to assign numbers to them. This shows that the\ngeometry of relativity, even in its most general and abstract form,\nassumes a good deal more than the mere possibility of measurement,\nwhich, in itself, is of very little value. In itself, it does not lead\nto a geometry; this only results when there is some interconnection\nbetween different measures.\u003c/p\u003e\n\n\u003cp\u003eIt may be asked whether, when the geometry of relativity is generalized\nto the utmost, any genuinely quantitative element remains in its\nformulæ. We start with an ordered four-dimensional manifold, and\nwe assign co-ordinates subject to\u003cspan class=\"pagenum\" id=\"Page_113\"\u003e[Pg 113]\u003c/span\u003e the sole restriction that their\norder-relations are to reproduce those of the given manifold. We then\nproceed to find formulæ (tensor-equations) which hold equally in all\nsystems of co-ordinates satisfying the above condition. It might seem\na possibility that such formulæ really express only ordinal relations,\nand that the sole advantage of co-ordinates lies in the fact that\nthey provide names for the terms of a manifold of the required sort.\n(They do not provide names for \u003ci\u003eall\u003c/i\u003e of them; the number of names\nis \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e, and therefore only a vanishing proportion of real\nnumbers can be named—\u003ci\u003ei.e.\u003c/i\u003e expressed by means of a formula of\nfinite complexity which employs integers.) This possibility requires\ninvestigation.\u003c/p\u003e\n\n\u003cp\u003eThe problem can be discussed equally well in two dimensions. In\nGauss\u0027s theory of surfaces, a sphere and an ellipsoid, \u003ci\u003ee.g.\u003c/i\u003e are\ndistinguishable by the fact that there is an irreducible difference\nbetween the formulæ for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e which hold for the two surfaces when\nexpressed in terms of two co-ordinates; this expresses the fact that\nthe measure of curvature is constant in the case of the sphere, but\nnot in the case of the ellipsoid. Yet from a purely ordinal point\nof view, such as that of \u003ci\u003eanalysis situs\u003c/i\u003e, the two figures are\nindistinguishable. What, exactly, is added to make the difference? This\nproblem is essentially the same as that which arises in the general\ntheory of relativity.\u003c/p\u003e\n\n\u003cp\u003eIn part, the answer in this case is simple. What is added is the\ncomparability of distances in different directions. So long as our\napparatus is purely ordinal, we can say of three points which have\nthe order \u003cimg style=\"vertical-align: -0.05ex; width: 5.133ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-209.png\" alt=\"\" data-tex=\"\\(ABC\\)\"\u003e that \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is nearer to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e than \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e is, but we\ncannot say anything analogous of three points which are not in a row—I\ndo not say \"in a straight line,\" because the concept involved is more\ngeneral, as will appear later. But although this is part of the answer,\nit does not seem to be the whole, since our relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e also enabled\nus to compare distances not having a common origin.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_114\"\u003e[Pg 114]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt seems that what distinguishes distance as required in geometry from\nsuch a relation as \"subtending a given angle at a given point\" is the\nabsence of reference to anything external. When the distance between\ntwo points is equal to the distance between two others, we are supposed\nto have a fact which does not demand reference to some other point\nor points. In fact, this is the reason why the \"interval\" has been\nsubstituted for distance: the latter, as hitherto conceived, was found\nto depend upon the motion of the co-ordinate frame, and thus to be not\nan intrinsic geometrical relation. The distance, if it is to serve its\npurpose, must be a function of the two points exclusively, and must\nnot involve any other geometrical data. Here, for relativity purposes,\n\"geometry\" includes \"kinematics.\" The angle which two points subtend\nat a given point becomes a function of \u003ci\u003ethree\u003c/i\u003e points as soon as\nthe given point is thought of as variable. There must be no such way of\nturning the distance between two points into a function involving other\nvariables also.\u003c/p\u003e\n\n\u003cp\u003eI am not sure, however, whether it is necessary to introduce this\nsomewhat difficult consideration. In ordinary geometry, the points at\na given distance from a given point lie on the surface of a sphere;\nbut if we define the distance \u003cimg style=\"vertical-align: -0.439ex; width: 3.489ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-308.png\" alt=\"\" data-tex=\"\\(PQ\\)\"\u003e as the angle \u003cimg style=\"vertical-align: -0.439ex; width: 5.215ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-314.png\" alt=\"\" data-tex=\"\\(POQ\\)\"\u003e, where \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e\nis a fixed point, the points at a given distance from \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e lie on\na cone. Now a sphere and a cone are distinguishable in \u003ci\u003eanalysis\nsitus\u003c/i\u003e. Thus the above undesirable definition could be excluded by\ninsisting that points at a given distance from a given point are to\nform an oval figure. In relativity theory, this is not true of points\nhaving zero interval from a given point; indeed, it is only true when\nthe interval concerned is space-like. But it is possible to specify the\ncharacteristics, for \u003ci\u003eanalysis situs\u003c/i\u003e, of the three-dimensional\nsurface of constant distance from a given point. These might be\nadded to the postulate that distance exists. Whether, in some such\nway, we could overcome the apparent necessity for\u003cspan class=\"pagenum\" id=\"Page_115\"\u003e[Pg 115]\u003c/span\u003e distinguishing\nbetween a sphere and an ellipsoid, making the difference relative to\nthe definition of distance, I do not feel sure, though obviously the\nquestion must be easily soluble.\u003c/p\u003e\n\n\u003cp\u003eEvery principle of measurement which is to be used in practice must be\nsuch that important empirical laws are connected with measures. There\nwill always be an infinite number of ways of correlating numbers with\nthe members of a class whose cardinal number is less than or equal to\n\u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-316.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_0}\\)\"\u003e. Some of these may be important, but most must be\nunimportant. Some conditions can be laid down. In the first place, the\nmembers of the class concerned may be obviously capable of an order\nwhich is causally important. If we take all the patches of colour that\never have been or will be perceived, they have in the first place\nan order in space-time, which is obviously important causally; in\nthis order, no two of them occupy the same position—\u003ci\u003ei.e.\u003c/i\u003e the\nrelations concerned are all asymmetrical. But they have also an order\nas shades of colour and as of varying brightness. In this order there\nare symmetrical transitive relations—\u003ci\u003ee.g.\u003c/i\u003e between two patches\nof exactly the same shade. Physics professes to correlate also these\nfurther characteristics of colours with spatio-temporal quantities such\nas wave-lengths. This would not be plausible if continuous alterations\nof quality were not correlated with continuous alterations in the\ncorrelated physical quantities. Whenever we notice a qualitative\nseries, such as that of colours of the rainbow, we assume that it must\nhave causal importance, and we insist that numbers used as measures\nshall have the same order as the qualities which they measure. The\nformer is a postulate, the latter a convention. Both have proved highly\nsuccessful, but neither is an \u003ci\u003ea priori\u003c/i\u003e necessity.\u003c/p\u003e\n\n\u003cp\u003eThere are orders which are obviously of no causal\nimportance—\u003ci\u003ee.g.\u003c/i\u003e alphabetical order among human beings.\nHuman beings, like colours, have various orders that are causally\nimportant—the space-time order, order of height.\u003cspan class=\"pagenum\" id=\"Page_116\"\u003e[Pg 116]\u003c/span\u003e weight, income,\nintelligence as measured by Professor X\u0027s tests, etc. But alphabetical\norder would never be thought important; no one would hope to found\na biometric calculus upon a system in which a human being had\nco-ordinates depending upon the alphabetical order of his name.\nGenerally speaking, it would seem that the simplest relations are the\nmost important. Here I am using a purely logical test of simplicity:\ntaking propositions in which the given relation occurs, there will be\nsome having the smallest number of constituents compatible with the\nmention of that relation; and again, a relation may be a molecular\ncompound of other relations—\u003ci\u003ei.e.\u003c/i\u003e a disjunction, conjunction,\nnegation, or complex of all these. A relation which is molecular has\nalways a certain definite number of atoms; a relation which is not\nmolecular is called atomic, and has then a definite number of terms\nin the simplest propositions in which it occurs. An atomic relation\nis simpler in proportion to the fewness of its terms; a molecular\nrelation, in proportion to the fewness of its atoms. There is much\nempirical reason to think that the laws of a science become more\nimportant and comprehensive as the relations involved become simpler.\nThe relation of a man to his name is of immense complexity, whereas we\nmay suppose that the relation upon which interval depends is fairly\nsimple. And the qualitative order of colours alluded to above is also\nsimple, so long as we are thinking of colours as given in perception,\nnot as interpreted in physics. Such simple relations should, as far as\npossible, be the basis for systems of measurement.\u003c/p\u003e\n\n\u003cp\u003eThere is a traditional distinction between extensive and intensive\nquantities, which is somewhat misleading when taken seriously. The\ntheory is that extensive quantities are composed of parts and intensive\nquantities are not. The only truly extensive quantities are numbers and\nclasses. Where finite classes are concerned, the number of their terms\nmay be taken\u003cspan class=\"pagenum\" id=\"Page_117\"\u003e[Pg 117]\u003c/span\u003e as a measure of them, and they have parts corresponding\nto all smaller numbers. But in geometry we are never concerned with\nquantities which have parts. The number of points in a volume, whether\nlarge or small, is always \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-316.png\" alt=\"\" data-tex=\"\\(2^{\\aleph_0}\\)\"\u003e in the usual kinds of\ngeometry; thus magnitude has nothing to do with number. Interval, as\nwe have seen, is a relation, and smaller intervals are not parts of\nit. If \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e are equal intervals in a straight line, we\nsay that the interval \u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e is double of each, and we think of it as\nthe \"sum\" of \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e. But it is only by a convention, though\nan almost irresistible one, that we assign as the measure of \u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e\na number double that which we assign as the measure of \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e or of\n\u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e. And to say that \u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e is the \"sum\" of \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e is to\nsay something very ambiguous, since the word \"sum\" has many meanings.\nWhen \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e are considered as vectors, we may say that\n\u003cimg style=\"vertical-align: -0.05ex; width: 3.416ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-212.png\" alt=\"\" data-tex=\"\\(AC\\)\"\u003e is their sum even when they are not in one straight line. Again,\ngiven suitable definitions, we may say that the points between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e are the sum (in the logical sense) of the points between\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and between \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e; this will only hold if\n\u003cimg style=\"vertical-align: -0.05ex; width: 5.133ex; height: 1.67ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-209.png\" alt=\"\" data-tex=\"\\(ABC\\)\"\u003e is a straight line. But the distance between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e,\nconsidered as a relation, is not properly the \"sum,\" in any recognized\nsense, of the distances \u003cimg style=\"vertical-align: 0; width: 3.414ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-210.png\" alt=\"\" data-tex=\"\\(AB\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 3.437ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-211.png\" alt=\"\" data-tex=\"\\(BC\\)\"\u003e. Thus all geometrical quantities\nare \"intensive.\" This shows that the distinction of intensive and\nextensive is unimportant.\u003c/p\u003e\n\n\u003cp\u003eIn connection with interval, it is worth while to compare its formal\ncharacteristics with those of similarity. We saw that, in the\ngeneralized geometry with which Eddington ends, we want a relation\nof four neighbouring points, expressing the fact that they form a\nparallelogram. But we met with certain difficulties owing to the\nfact that this is only supposed to be possible for an infinitesimal\nquadrilateral, which is a figment of the mathematical imagination, and\nthat it was not wholly easy to see how to substitute a procedure by\nmeans of limits. We were led to the suggestion that, instead of saying\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e\u003cspan class=\"pagenum\" id=\"Page_118\"\u003e[Pg 118]\u003c/span\u003e is a parallelogram,\" we should have to say \"\u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e is more\nnearly a parallelogram than \u003cimg style=\"vertical-align: -0.464ex; width: 4.681ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-282.png\" alt=\"\" data-tex=\"\\(efgh\\)\"\u003e.\" Perhaps this could be somewhat\nsimplified. Suppose we say: \"\u003cimg style=\"vertical-align: -0.025ex; width: 4.951ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-317.png\" alt=\"\" data-tex=\"\\(abcd\u0027\\)\"\u003e is more nearly a parallelogram\nthan \u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e.\" And perhaps this could be still further simplified so\nas to take the form: \"\u003cimg style=\"vertical-align: -0.025ex; width: 2.784ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-318.png\" alt=\"\" data-tex=\"\\(cd\u0027\\)\"\u003e is more like \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-319.png\" alt=\"\" data-tex=\"\\(ba\\)\"\u003e than \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-290.png\" alt=\"\" data-tex=\"\\(cd\\)\"\u003e is.\" We\nhere suppose that between \u003ci\u003eany\u003c/i\u003e two points there is a relation,\nwhich we will not call distance, but (say) \"separation,\" and that this\nrelation, like a shade of colour, is capable of a greater or less\nresemblance to another of the same kind. In a Euclidean space, two\nfinite separations finitely separated may be exactly similar in the\nrelevant respects; we then have a finite parallelogram.\u003c/p\u003e\n\n\u003cfigure class=\"figleft width500\" id=\"i_118\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-118.jpg\" width=\"200\" height=\"121\" alt=\"A geometric\ndiagram of a parallelogram with vertices labeled a, b, c, and d, with\nan additional point d\u0027 creating an extended line at the top right\ncorner. Simple line drawing illustrating a geometric principle.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eBut in the generalized geometry that we are considering, we shall say\nthat no two separations are \u003ci\u003eexactly\u003c/i\u003e alike, though they are\ncapable of indefinite approximation to exact likeness. Let us see how\nfar this will take us.\u003c/p\u003e\n\n\u003cp\u003eIn the case of similarity, we have a relation which is capable of\ndegrees, and may be called \"quasi-transitive\"—\u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nis very like \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is very like \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, then \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e must be\nrather like \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e. This is just the sort of thing required for Weyl\u0027s\ngeometry. Consider four points, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, and suppose\nthat \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e is rather like \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-290.png\" alt=\"\" data-tex=\"\\(cd\\)\"\u003e. Take a series of points forming a\ncontinuous route from \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, without loops; this can be done\nby purely ordinal methods to be explained later. Suppose that among\nthese points there are some, such as \u003cimg style=\"vertical-align: -0.023ex; width: 1.804ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-320.png\" alt=\"\" data-tex=\"\\(d\u0027\\)\"\u003e which make \u003cimg style=\"vertical-align: -0.025ex; width: 2.784ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-318.png\" alt=\"\" data-tex=\"\\(cd\u0027\\)\"\u003e more\nlike \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e than \u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-290.png\" alt=\"\" data-tex=\"\\(cd\\)\"\u003e is. We may suppose that these points have a\nlimit or last term, which we will call \u003cimg style=\"vertical-align: -0.023ex; width: 1.804ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-320.png\" alt=\"\" data-tex=\"\\(d\u0027\\)\"\u003e. We can then similarly\nproceed along \u003cimg style=\"vertical-align: -0.023ex; width: 3.001ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-321.png\" alt=\"\" data-tex=\"\\(ad\u0027\\)\"\u003e to a point \u003cimg style=\"vertical-align: -0.023ex; width: 2.244ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-322.png\" alt=\"\" data-tex=\"\\(d\u0027\u0027\\)\"\u003e which gives \u003cimg style=\"vertical-align: -0.025ex; width: 3.224ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-323.png\" alt=\"\" data-tex=\"\\(cd\u0027\u0027\\)\"\u003e more\nlike \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e than for any other point on \u003cimg style=\"vertical-align: -0.023ex; width: 3.001ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-321.png\" alt=\"\" data-tex=\"\\(ad\u0027\\)\"\u003e. We have then done\nnearly as well as possible, if not quite, with the three points \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e as starting-points. By means of suitable postulates,\nwe could\u003cspan class=\"pagenum\" id=\"Page_119\"\u003e[Pg 119]\u003c/span\u003e insure that a construction of the above sort, carried out\nrepeatedly without changing the points \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, should at\nlast end with a definite point \u003cimg style=\"vertical-align: -0.375ex; width: 2.164ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-324.png\" alt=\"\" data-tex=\"\\(d_0\\)\"\u003e such that \u003cimg style=\"vertical-align: -0.375ex; width: 3.144ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-325.png\" alt=\"\" data-tex=\"\\(cd_0\\)\"\u003e is more like\n\u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-289.png\" alt=\"\" data-tex=\"\\(ab\\)\"\u003e than any other distance from \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e is. We may call the figure\n\u003cimg style=\"vertical-align: -0.375ex; width: 5.311ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-326.png\" alt=\"\" data-tex=\"\\(abcd_0\\)\"\u003e a \"quasi-parallelogram.\" Now let \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e, …\n\u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, … be a series of points on a route from \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e. Then\nproceed to take points \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e, … between \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e\non some route, and form the quasi-parallelograms having one corner\nat \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, one corner at \u003cimg style=\"vertical-align: -0.357ex; width: 2.887ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-13.png\" alt=\"\" data-tex=\"\\(x_m\\)\"\u003e and one at \u003cimg style=\"vertical-align: -0.464ex; width: 2.256ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-329.png\" alt=\"\" data-tex=\"\\(y_n\\)\"\u003e, the fourth being\ncalled \u003cimg style=\"vertical-align: -0.357ex; width: 3.604ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-330.png\" alt=\"\" data-tex=\"\\(z_{mn}\\)\"\u003e.\u003c/p\u003e\n\n\u003cfigure class=\"figright width500\" id=\"i_119\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-119.jpg\" width=\"200\" height=\"151\" alt=\"A geometric\ndiagram showing a trapezoid divided into a grid pattern with multiple\ncells. Points are labeled a, b, c, d at corners, with subscripted\nlabels (x, y, z with numerical subscripts) marking interior divisions.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eIf, as Weyl assumes, infinitesimal distances which have one end in\ncommon are comparable, this must be taken to mean that two small\nfinite distances are capable of a resemblance which may be called\n\"quasi-equality,\" which grows more nearly complete resemblance as\nthe distance grows smaller. We may assume, as before, that, given a\npoint \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e and a definite route from \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, there will\nbe one definite point \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e on this route such that \u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e is\nmore nearly equal to \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e than is any other distance by on the\nroute in question. We shall then say that \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e are\n\"quasi-equal.\" Take also \u003cimg style=\"vertical-align: -0.339ex; width: 4.564ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-333.png\" alt=\"\" data-tex=\"\\(x_1x_2\\)\"\u003e … quasi-equal, and \u003cimg style=\"vertical-align: -0.464ex; width: 4.193ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-334.png\" alt=\"\" data-tex=\"\\(y_1y_2\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.464ex; width: 4.193ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-335.png\" alt=\"\" data-tex=\"\\(y_2y_3\\)\"\u003e … quasi-equal. In this way we can construct a co-ordinate\nmesh with axes \u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-319.png\" alt=\"\" data-tex=\"\\(ba\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.95ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-292.png\" alt=\"\" data-tex=\"\\(bc\\)\"\u003e. And we can now construct what will be\nin effect straight lines through \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e: take all the points \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e\nwhich are the corners opposite to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e of quasi-parallelograms\n\u003cimg style=\"vertical-align: -0.464ex; width: 7.165ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-336.png\" alt=\"\" data-tex=\"\\(bx_{m}zy_{n}\\)\"\u003e, for different initial points \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e subject\nto quasi-equality between \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e. These points may\nbe regarded as forming the quasi-straight line whose equation is\n\u003cimg style=\"vertical-align: -1.577ex; width: 8.352ex; height: 4.106ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-337.png\" alt=\"\" data-tex=\"\\(\\dfrac{x}{m} = \\dfrac{y}{n}\\)\"\u003e. (Irrationals can be dealt with by the\nusual methods.) This quasi-straight line will start from \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e in a\u003cspan class=\"pagenum\" id=\"Page_120\"\u003e[Pg 120]\u003c/span\u003e\ncertain direction, and may, for differential purposes, be regarded as\nreally a straight line. It is not worth while to proceed further, since\nit is obvious that we have the necessary material.\u003c/p\u003e\n\n\u003cp\u003eDegrees of similarity may be, in a sense, measured by\nquasi-transitiveness. Suppose that \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 4.936ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-338.png\" alt=\"\" data-tex=\"\\(y_1z_{11}\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.464ex; width: 4.936ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-339.png\" alt=\"\" data-tex=\"\\(y_2z_{12}\\)\"\u003e, … each have quasi-equality with the next. It may or\nmay not happen that \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e has quasi-equality with \u003cimg style=\"vertical-align: -0.464ex; width: 5.256ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-340.png\" alt=\"\" data-tex=\"\\(y_nz_{1n}\\)\"\u003e.\nOne may presume that this will happen if \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e are\nvery small and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e is not very large. Similarly, or rather \u003ci\u003ea\nfortiori\u003c/i\u003e, we cannot infer that \u003cimg style=\"vertical-align: -0.357ex; width: 3.857ex; height: 1.927ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-341.png\" alt=\"\" data-tex=\"\\(bx_m\\)\"\u003e has quasi-equality with\n\u003cimg style=\"vertical-align: -0.464ex; width: 3.672ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-342.png\" alt=\"\" data-tex=\"\\(by_m\\)\"\u003e. The larger the value of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e for which such an inference\nremains true, the closer is the resemblance between \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e or between \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 4.936ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-338.png\" alt=\"\" data-tex=\"\\(y_1z_{11}\\)\"\u003e. It is to be assumed\nthat, by continually diminishing \u003cimg style=\"vertical-align: -0.339ex; width: 3.252ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-332.png\" alt=\"\" data-tex=\"\\(bx_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 3.067ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-331.png\" alt=\"\" data-tex=\"\\(by_1\\)\"\u003e the number of\nsteps for which the inference is permitted can be increased without\nfinite limit.\u003c/p\u003e\n\n\u003cp\u003eIf the above is in any degree valid, it would seem that, if space-time\nis continuous, spatio-temporal measurement depends theoretically upon\nqualitative similarity, capable of varying degrees, between relations\nof pairs of points. It is not suggested that the analysis cannot be\ncarried further, but only that this is a valid stage in the process of\nexplaining what is meant by the quantitative character of intervals and\nby their measurement as numerical multiples of units.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_121\"\u003e[Pg 121]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XIII\"\u003eCHAPTER XIII\u003cbr\u003e\nMATTER AND SPACE\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nCOMMON sense starts with the notion that there is matter where we\ncan get sensations of touch, but not elsewhere. Then it gets puzzled\nby wind, breath, clouds, etc., whence it is led to the conception of\n\"spirit\"—I speak etymologically. After \"spirit\" has been replaced\nby \"gas,\" there is a further stage, that of the æther. Assuming the\ncontinuity of physical processes, there must be things happening\nbetween the earth and the sun when light travels from the sun to\nthe earth; assuming the mediæval metaphysic of \"substance,\" as all\nphysicists did until recently, what is happening between the earth and\nthe sun must be happening \"in\" or \"to\" a substance, which is called the\næther.\u003c/p\u003e\n\n\u003cp\u003eApart from metaphysical interpretations, what we may be said to know\n(using this word somewhat liberally) is that processes occur where\nthere is no gross matter, and that these processes proceed, at least\napproximately, in accordance with Maxwell\u0027s equations. There does not\nseem any necessity to interpret these processes in terms of substance;\nindeed, I shall argue that processes associated with gross matter\nshould also be interpreted so as not to involve substance. There must,\nhowever, remain a difference, expressible in physical terms, between\nregions where there is matter and other regions. In fact, we know\nthe difference. The law of gravitation is different, and the laws of\nelectromagnetism suffer a discontinuity when we reach the surface\nof an electron or proton. These differences, however, are not of a\nmetaphysical kind. To the philosopher, the difference between \"matter\"\nand \"empty space\" is, I believe, merely a difference as to the causal\nlaws governing successions of events, not a difference expressible\u003cspan class=\"pagenum\" id=\"Page_122\"\u003e[Pg 122]\u003c/span\u003e as\nthat between the presence or absence of substance, or as that between\none kind of substance and another.\u003c/p\u003e\n\n\u003cp\u003ePhysics, as such, should be satisfied when it has ascertained the\nequations according to which a process takes place, with just enough\ninterpretation to know what experimental evidence confirms or confutes\nthe equations. It is not necessary to the physicist to speculate as to\nthe concrete character of the processes with which he deals, though\nhypotheses (false as well as true) on this subject may sometimes be\na help to further valid generalizations. For the present, we are\nconfining ourselves to the standpoint of physics. Whether anything\nfurther can be known or fruitfully conjectured is a matter which we\nshall discuss at a later stage. We want, therefore, to consider the\ndifference in physical formulæ which is described as that between\nthe presence and absence of matter, and also to consider briefly the\ndifficulties as to the interchanges of energy between matter and empty\nspace. I say \"empty space\" or \"æther\" indifferently; the difference\nseems to be merely one of words.\u003c/p\u003e\n\n\u003cp\u003eOne way of approaching this subject is through the connection of mass\nwith energy.\u003ca id=\"FNanchor_31\" href=\"#Footnote_31\" class=\"fnanchor\"\u003e[31]\u003c/a\u003e In elementary dynamics, the two are quite distinct,\nbut nowadays they have become amalgamated. There axe two kinds of\nmass involved in physics, of which one may be called the \"invariant\"\nmass, the other the \"relative\" mass. The latter is the mass obtained\nby measurement, when the body concerned may be moving relatively to\nthe observer; the former is the mass obtained when the body is at rest\nrelatively to the observer. If we call the invariant mass \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e and the\nrelative mass \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e, then, taking the velocity of light as unity, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e\nis the velocity of the body relative to the observer, we have:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.308ex; width: 14.302ex; height: 4.837ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-343.png\" alt=\"\" data-tex=\"\\[\nM = \\frac{m}{\\sqrt{1 – v^{2}}}\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_123\"\u003e[Pg 123]\u003c/span\u003e\nThus \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e increases as \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e increases; if \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e is the velocity\nof light, \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e becomes infinite if \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e is finite. In fact, the\ninvariant mass of light is zero, and its relative mass is finite.\nWherever energy is associated with matter, there is a finite invariant\nmass \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e; but where energy is in \"empty space\", \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e is zero. This\nmight be regarded as a definition of the difference between matter and\nempty space.\u003c/p\u003e\n\n\u003cp\u003eIt will be seen that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e is small, so that \u003cimg style=\"vertical-align: -0.025ex; width: 2.085ex; height: 1.929ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-344.png\" alt=\"\" data-tex=\"\\(v^{4}\\)\"\u003e and higher\npowers can be neglected, the above equation becomes approximately\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.781ex; width: 16.643ex; height: 2.78ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-345.png\" alt=\"\" data-tex=\"\\[\nM = m + \\tfrac{1}{2}mv^{2}.\n\\]\"\u003e\u003c/span\u003e\nNow \u003cimg style=\"vertical-align: -0.781ex; width: 5.867ex; height: 2.737ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-346.png\" alt=\"\" data-tex=\"\\(\\tfrac{1}{2}mv^{2}\\)\"\u003e is the kinetic energy. Thus the change of\n\u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e with changes of motion is the same as the change of the kinetic\nenergy. But energy is fixed only to the extent of its changes, not in\nits absolute amount. Hence \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e may be identified with the energy. And\nthis suggests further that the usual definition of energy is only an\napproximation, which holds when \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e is small. The accurate formula for\nenergy is\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.308ex; width: 8.907ex; height: 4.837ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-347.png\" alt=\"\" data-tex=\"\\[\n\\frac{m}{\\sqrt{1 – v^{2}}}\n\\]\"\u003e\u003c/span\u003e\n—\u003ci\u003ei.e.\u003c/i\u003e accurately the same as \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eThe conservation of energy is the conservation of \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e, not of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e;\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e also is approximately conserved, but not exactly. \u003ci\u003eE.g.\u003c/i\u003e\nthere is a loss of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e when four protons and two electrons combine\nto form a helium nucleus. The term \"invariant\" refers to changes of\nco-ordinates, not to constancy throughout time.\u003c/p\u003e\n\n\u003cp\u003eIt is necessary to say something about the difficulties of reconciling\nthe laws governing the propagation of light with those governing\ninterchanges of energy between light and atoms. On this subject the\npresent position of physics is one of perplexity, aptly summarized by\nDr Jeans in \u003ci\u003eAtomicity and Quanta\u003c/i\u003e (Cambridge, 1926) and by Dr C.\nD. Ellis in \u003ci\u003eNature\u003c/i\u003e, June 26, 1926, pp. 895-7. The wave theory\nof light accounts adequately for all phenomena in which only light\nis concerned,\u003cspan class=\"pagenum\" id=\"Page_124\"\u003e[Pg 124]\u003c/span\u003e such as interference and diffraction; but it fails to\naccount for quantum phenomena such as the photo-electric effect (see\nChapter IV.). On the other hand, theories which account for the quantum\nphenomena seem unable to account for the very things which the wave\ntheory explains perfectly.\u003c/p\u003e\n\n\u003cp\u003eSome of the difficulties of the light-quantum theory are set forth as\nfollows by Dr Jeans (\u003ci\u003eop. cit.\u003c/i\u003e pp. 29, 30):\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"If, however, radiation is to be compared to rifle bullets, we know\nboth the number and size of these bullets. We know, for instance, how\nmuch energy there is in a cubic centimetre of bright sunlight, and if\nthis energy is the aggregate of the energies of individual quanta,\nwe know the energy of each quantum (since we know the frequency of\nthe light) and so can calculate the number of quanta in the cubic\ncentimetre. The number is found to be about ten millions. By a similar\ncalculation it is found that the light from a sixth magnitude star\ncomprises only about one quantum per cubic metre, and the light from a\nsixteenth magnitude star, only about one quantum per ten thousand cubic\nmetres. Thus if light travels in indivisible quanta like bullets, the\nquanta from a sixteenth magnitude star can only enter a terrestrial\ntelescope at comparatively rare intervals, and it will be exceedingly\nrare for two or more quanta to be inside the telescope at the same\ntime. A telescope of double the aperture ought to trap the quanta four\ntimes as frequently, but there should be no other difference. This, as\nLorentz pointed out in 1906, is quite at variance with our everyday\nexperience. When the light of a star passes through a telescope and\nimpresses an image on a photographic plate, this image is not confined\nto a single molecule or to a close cluster of molecules as it would\nbe if individual quanta left their marks like bullets on a target. An\nelaborate and extensive diffraction pattern is formed; the intensity of\nthe pattern depends on the number of quanta, but its design depends on\nthe diameter and also on the shape of the object-glass. Moreover, the\ndesign does not bear any resemblance whatever to the \u0027trial and error\u0027\ndesign which is observed on a target battered by bullets. It seems\nimpossible to reconcile this with the hypothesis that quanta travel\nlike\u003cspan class=\"pagenum\" id=\"Page_125\"\u003e[Pg 125]\u003c/span\u003e bullets directly from one atom of the star to one molecule of the\nphotographic plate.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eThe difficulties of the wave-theory, on the other hand, are illustrated\nby Dr Ellis as follows:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"To take a definite case, suppose X-rays are incident on a plate of\nsome material, then it is found that electrons are ejected from the\nplate with considerable velocities. The number of the electrons depends\non the intensity of the X-rays and diminishes in the usual way as the\nplate is moved farther from the source of X-rays. The velocity or\nenergy of each electron, however, does not vary, but depends only on\nthe frequency of the X-rays. The electrons are found to have the same\nenergy whether the material from which they come is close to the X-ray\nbulb or whether it is removed away to any distance.\u003c/p\u003e\n\n\u003cp\u003e\"This is a result which is quite incompatible with the ordinary\nwave-theory of radiation, because as the distance from the source\nincreases the radiation spreading out on all sides becomes weaker\nand weaker, the electric forces in the wave-front diminishing as the\ninverse square of the distance. The experimental result that the\nphoto-electron always picks up the same amount of energy from the\nradiation could only be accounted for by giving it the power either\nto collect energy from a large volume or to collect energy for a long\ntime. Both of these assumptions are unworkable, and the only conclusion\nis that the radiated energy must be localized in small bundles.\u003c/p\u003e\n\n\u003cp\u003e\"This is the basis of the light-quantum theory. Light of frequency\n\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-20.png\" alt=\"\" data-tex=\"\\(\\nu\\)\"\u003e is considered to consist of small bundles or quanta of energy\nall identical and of magnitude \u003cimg style=\"vertical-align: -0.025ex; width: 2.502ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-348.png\" alt=\"\" data-tex=\"\\(h\\nu\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e being Planck\u0027s constant.\nThese quanta travel through space, being unaffected by each other, and\npreserving their own individuality until they make a suitable collision\nwith an atom.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eAfter setting forth the difficulties encountered by this theory in\nregard to interference and diffraction, Dr Ellis proceeds to the very\ninteresting suggestion made by Professor G. N. Lewis in \u003ci\u003eNature\u003c/i\u003e,\nFebruary 13, 1926, p. 236. \"It is a striking fact,\" says Dr Ellis,\nsummarizing this suggestion, \"that while all the theories are directed\ntowards explaining the propagation of\u003cspan class=\"pagenum\" id=\"Page_126\"\u003e[Pg 126]\u003c/span\u003e light, one theory suggesting\nthat it occurs in the form of waves, the other in the form of\ncorpuscles, yet light has never been observed in empty space. It is\nquite impossible to observe light in the course of propagation; the\nonly events that can ever be detected are the emission and absorption\nof light. Until there is some atom to absorb the radiation we must be\nunaware of its existence. In other words, the difficulty of explaining\nthe propagation of light may be because we are endeavouring to explain\nsomething about which we have no experimental evidence. It might be\nmore correct to interpret the experimental facts quite directly and to\nsay that one atom can transfer energy to another atom although they\nmay be far apart, in a manner analogous to the transference of energy\nbetween two atoms which collide.\"\u003c/p\u003e\n\n\u003cp\u003eProfessor Lewis\u0027s theory suggests that we should take seriously the\nfact that the interval between two parts of a light-ray is zero, so\nthat its point of departure and its point of arrival may be regarded\nas, in some sense, in contact. In a passage quoted by Dr Ellis, he says:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"I shall make the contrary assumption that an atom never emits light\nexcept to another atom, and that in this process, which may rather be\ncalled a transmission than an emission, the atom which loses energy and\nthe atom which gains energy play co-ordinate and symmetrical parts.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eIn a later letter to \u003ci\u003eNature\u003c/i\u003e (December 18, 1926), Professor Lewis\nsuggests that light is carried by corpuscles of a new sort, which he\ncalls \"photons.\" He supposes that, when light radiates, what happens is\nthat a photon travels; but at other times the photon is a structural\nelement within an atom. The photon, he says, \"is not light, but plays\nan essential part in every process of radiation.\" He assigns to the\nphoton the following properties: \"(1) In any isolated system the total\nnumber of photons is constant. (2) All radiant energy is carried by\nphotons, the only difference between the radiation\u003cspan class=\"pagenum\" id=\"Page_127\"\u003e[Pg 127]\u003c/span\u003e from a wireless\nstation and from an X-ray tube being that the former emits a vastly\ngreater number of photons, each carrying a very much smaller amount\nof energy. (3) All photons are intrinsically identical…. (4) The\nenergy of an isolated photon, divided by the Planck constant, gives\nthe frequency of the photon…. (5) All photons are alike in one\nproperty which has the dimensions of action or of angular momentum,\nand is invariant to a relativity transformation. (6) The condition\nthat the frequency of a photon emitted by a certain system be equal to\nsome physical frequency existing within that system, is not in general\nfulfilled, but comes nearer to fulfilment the lower the frequency is.\"\nProfessor Lewis promises to deal with difficulties in the way of his\nhypothesis on a future occasion.\u003c/p\u003e\n\n\u003cp\u003eProfessor Lewis\u0027s view is perhaps less radical than the view which it\nsuggests—namely, that nothing whatever happens between the emission of\nlight by one atom and its absorption by another. Whether this view is\nProfessor Lewis\u0027s or not, it deserves to be considered, for although it\nis revolutionary, it may well prove to be right. If so, \"empty space\"\nis practically abolished. There will be need of a considerable labour\nif physics is to be re-written in accordance with this theory, but what\nis said about the necessary absence of evidence concerning light in\ntransit is a powerful consideration. It is common in science to find\nhypotheses which, from a theoretical point of view, are unnecessarily\ncomplicated, because people cannot sufficiently divest themselves of\ncommon-sense prejudices. Why should we suppose that anything at all\nhappens between the emission of light and its absorption? One might be\ninclined to attach weight to the fact that light travels with a certain\nvelocity. But relativity has made this argument less convincing than\nit once was. Everything that has to do with the velocity of light is\ncapable of being interpreted in a \"Pickwickian\" sense, and in any case\nour prejudices\u003cspan class=\"pagenum\" id=\"Page_128\"\u003e[Pg 128]\u003c/span\u003e must be shocked. It is of course premature to adopt\nsuch an hypothesis definitively, and I shall continue to suppose that\nlight does really travel across an intervening region. But it will be\nwise to remember the possibility, and to bear in mind the great changes\nin our imaginative picture of the world that are compatible with our\nexisting physical knowledge.\u003c/p\u003e\n\n\u003cp\u003eThe picture presented by this development of Professor Lewis\u0027s\nsuggestion would be something like this: the world contains bits of\nmatter (electrons and protons) possessing various amounts of energy.\nSometimes energy is transferred from one of these bits of matter to\nanother; usually this process has been thought to be casual, like\nthe wandering of thistledown, but it is found to be more like the\nparcels post, in the sense that the energy has a definite destination.\nIt is now suggested that there is no postman, because, if there\nwere, he would be as magical as Santa Claus; the alternative is to\nsuppose that the energy passes immediately from one piece of matter\nto another. It is true that, by the clock, there is a lapse of time\nbetween the departure of the energy from the source and its arrival\nat its destination. But there is no interval in the relativity sense,\nand the lapse of time will vary according to the co-ordinate system\nemployed—\u003ci\u003ei.e.\u003c/i\u003e according to the way in which the clock is\nmoving. I do not know how the view we are considering will account\nfor the time taken by a double journey to a reflector and back, which\nis not purely conventional. Nor do I know what will happen to the\nconservation of energy if light cannot be radiated into the void.\nThis latter argument, however, is not serious, since light which\nnever hits a piece of matter is in any case purely hypothetical. I am\nnot sure, either, that the theory is intended to be as radical as I\nhave suggested; perhaps it is only meant that light never starts on a\njourney without having a destination in view. In this form, however,\nthe theory would seem scarcely credible: we should have to suppose that\nmatter could exercise a mysterious\u003cspan class=\"pagenum\" id=\"Page_129\"\u003e[Pg 129]\u003c/span\u003e attraction from a distance, which\nwould undo the gain derived from Einstein\u0027s theory of gravitation.\nPerhaps the theory may have gained undue plausibility from a belief\nthat the whole geometry of space-time depended upon interval, whereas\nin fact there is a space-time order which is not derivable from\ninterval, and which, as presupposed in relativity theory, does not\nregard as contiguous parts of a light ray which would ordinarily be\nregarded as widely separated.\u003ca id=\"FNanchor_32\" href=\"#Footnote_32\" class=\"fnanchor\"\u003e[32]\u003c/a\u003e Perhaps it may be possible to avoid\nthese difficulties, but, if so, a very great theoretical reconstruction\nwill be necessary. Meanwhile it must be regarded as still possible that\nsome less revolutionary theory may solve the difficulties connected\nwith the interchange of energy between light and bodies.\u003c/p\u003e\n\n\u003cp\u003eThere are three papers by Einstein which discuss the possibility\nof obtaining quantum laws as consequences of a modified relativity\ntheory.\u003ca id=\"FNanchor_33\" href=\"#Footnote_33\" class=\"fnanchor\"\u003e[33]\u003c/a\u003e These papers do not arrive at any definite conclusion\nconfidently asserted; but they suffice to show that the problem of\ncombining quantum laws with those of gravitational and electromagnetic\nfields is not a hopeless one, a view which is strengthened by Mr L.\nV. King\u0027s theory alluded to above (Chapter IV.). So long as it is not\nknown to be hopeless, it is perhaps rash to fly to heroic solutions of\nthe problem. And it is as yet by no means universally admitted that\nthe wave-theory of light is inadequate in its own domain; Dr Jeans\n(\u003ci\u003eloc. cit.\u003c/i\u003e), for example, regards the hypothesis of light-quanta\nas unnecessary for reasons which demand serious consideration. We must\ntherefore await further knowledge before venturing upon a definite\nopinion.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_31\" href=\"#FNanchor_31\" class=\"label\"\u003e[31]\u003c/a\u003e\nSee Eddington, \u003ci\u003eop. cit.\u003c/i\u003e, §§ 10, 11, 12.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_32\" href=\"#FNanchor_32\" class=\"label\"\u003e[32]\u003c/a\u003e\nOn this matter, cf. Eddington, \u003ci\u003eop. cit.\u003c/i\u003e, § 98 (pp.\n224-6).\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_33\" href=\"#FNanchor_33\" class=\"label\"\u003e[33]\u003c/a\u003e\n\u003ci\u003eBietet die Feldtheorie Möglichkeiten für die Lösung\ndes Quantenproblems\u003c/i\u003e? Sitzungsberichte der preussischen Akademie der\nWissenschaften, 1923, pp. 359-64. \u003ci\u003eQuantentheorie des einatomigen\nidealen Gases\u003c/i\u003e. \u003ci\u003eIb.\u003c/i\u003e, 1924, pp. 261-7, and 1925, pp. 3-14.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_130\"\u003e[Pg 130]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XIV\"\u003eCHAPTER XIV\u003cbr\u003e\nTHE ABSTRACTNESS OF PHYSICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nBEFORE embarking upon the epistemological discussions which will\nconcern us in Part II., it will be well to draw some morals from\nour previous chapters. Throughout these chapters, I have carefully\nabstained from speculations which would have taken us outside the\ndomain of physics; in particular, I have not sought to interpret the\nmathematically fundamental notions of physics in terms of entities\nnot directly amenable to ordinary mathematical treatment. It seemed\ndesirable to be clear first as to what physics has to say, before\nundertaking either the epistemological criticism of the evidence or the\nmetaphysical interpretation of the logically primitive apparatus of\nphysics. This is the purpose of the present chapter.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_131\"\u003e[Pg 131]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003ePhysics started historically, and still starts in the education of the\nyoung, with matters that seem thoroughly concrete. Levers and pulleys,\nfalling bodies, collisions of billiard balls, etc., are all familiar in\neveryday life, and it is a pleasure to the scientifically-minded youth\nto find them amenable to mathematical treatment. But in proportion as\nphysics increases the scope and power of its methods, in that same\nproportion it robs its subject-matter of concreteness. The extent\nto which this is the case is not always realized, at any rate in\nunprofessional moments, even by the physicist himself; he may tell you\nthat he can \"see\" an electron hitting a screen, which is of course\na telescoped expression for a complicated inference. Dr Whitehead\nhas done more than any other author to show the need of undoing the\nabstractions of physics. For the moment, I am not concerned with this\nneed, but with the abstractions themselves.\u003c/p\u003e\n\n\u003cp\u003eLet us take space, time, light and matter as illustrative of the\ngradually increasing abstractness of physics. These four notions are\nall extracted from common sense. We see objects spread out in space,\nwe can feel their shapes with our fingers; we know what it is to walk\nto a neighbouring town or travel to a neighbouring country. All this\nmakes \"space\" seem something familiar and easy, until, in the course of\neducation, we learn the puzzles to which it has given rise. Time seems\nequally obvious: we remember past events in a time-order we notice\nday and night, summer and winter, youth and age, we know that history\nrelates events of previous epochs, we insure our lives in the confident\nexpectation that we shall die in the future. Light, again, seemed in no\nway mysterious to the author of Genesis, as, indeed, how should it to\nanyone who had experienced the difference between night and day? Matter\nwas equally obvious: it was primarily anything that we could touch,\nthough the first step towards mystification was taken when Empedocles\nincluded air. However, we are conscious of air in the form of wind and\nas something that fills our lungs, so that less effort was required to\nadmit air among the elements than to exclude fire.\u003c/p\u003e\n\n\u003cp\u003eFrom this happy familiarity with the everyday world physics has been\ngradually driven by its own triumphs, like a monarch who has grown\ntoo grand to converse with his subjects. The space-time of relativity\nis very far removed from the space and time of our unscientific\nexperience; yet even space-time is nearer to common sense than the\nconceptions towards which physics is tending. \"Space and time\" says\nEddington,\u003ca id=\"FNanchor_34\" href=\"#Footnote_34\" class=\"fnanchor\"\u003e[34]\u003c/a\u003e \"are only approximate conceptions, which must ultimately\ngive way to a more general conception of the ordering of events in\nnature not expressible in terms of a fourfold co-ordinate system. It is\nin this direction that some physicists hope to find a solution of the\ncontradictions of the quantum theory.\u003cspan class=\"pagenum\" id=\"Page_132\"\u003e[Pg 132]\u003c/span\u003e It is a fallacy to think that\nthe conception of location in space-time based on the observation of\nlarge-scale phenomena can be applied unmodified to the happenings which\ninvolve only a small number of quanta. Assuming that this is the right\nsolution it is useless to look for any means of introducing quantum\nphenomena into the later formulæ of our theory; these phenomena have\nbeen excluded at the outset by the adoption of a co-ordinate frame\nof reference.\" But even if space-time, as it appears in the general\ntheory of relativity, were the last word as regards the physical order\ncorresponding to our usual notions of space and time, it is evident\nthat we should have travelled very far from those notions, and have\narrived at a region in which pictorial imagination is useless.\u003c/p\u003e\n\n\u003cp\u003eThe view of Locke, that the secondary qualities are subjective but\nnot the primary qualities, was more or less compatible with physics\nuntil very recent times. There are spaces and times in our immediate\nexperience, and there seemed no insuperable obstacle to identifying\nthem with the spaces and times of the physical world. In regard to\ntime, at least, practically no one doubted the rightness of this\nidentification. There were doubts as regards space, but they came from\npsychologists rather than physicists. Now, however, both space and\ntime, as they occur in immediate experience, are recognized by writers\non relativity as something quite different from the space-time which\nphysics requires. Locke\u0027s half-way house has therefore been definitely\nabandoned.\u003c/p\u003e\n\n\u003cp\u003eI come now to the relation of light as experienced to light in physics.\nHere the cleavage is older than in the case of space and time;\nindeed, it is already admitted in Locke\u0027s theory. It is impossible to\nexaggerate the importance of this cleavage in separating the world of\nphysics from the world of common sense. With the exception of parts\nof our own body and bodies with which our own body is in contact, the\nobjects which, according to common sense, we perceive, are known\u003cspan class=\"pagenum\" id=\"Page_133\"\u003e[Pg 133]\u003c/span\u003e by\nmeans of light, sound, or odour. The last of these, though important to\nmany species of animals, is relatively subordinate in the perceptions\nof human beings. Sound is less important than light, and in any\ncase raises exactly the same problems in the present connection. We\nmay therefore concentrate upon light as a source of our knowledge\nconcerning the external world.\u003c/p\u003e\n\n\u003cp\u003eWhen we \"see\" an object, we \u003ci\u003eseem\u003c/i\u003e to have immediate knowledge of\nsomething external to our own body. But physics says that a complicated\nprocess starts from the external object, travels across the intervening\nregion, and at last reaches the eye. What goes on between the eye\nand the brain is a question for the physiologists, and what finally\nhappens when we \"see\" is a question for the psychologist. But without\ntroubling ourselves about what happens after the light reaches the eye,\nit is evident that what the physicist has to say is destructive of\nthe common-sense notion of \"seeing.\" It makes no difference, in this\nmatter, which of the possible theories we adopt as to the physical\ncharacter of light, since all equally make it something utterly\ndifferent from what we see. The data of sight, analyzed as much as\npossible, resolve themselves into coloured shapes. But the physical\nanalogue of a colour is a periodic process of a certain frequency\nrelative to the eye of the observer. The physical world, it seems\nnatural to infer, is destitute of colour. Moreover, the correspondence\nbetween colours and their physical counterparts is peculiar: colours\nare qualities, which are static while they last, whereas their\ncounterparts are periodic processes, which are in the medium between\nthe eye and the object which we say we \"see.\" What happens in the\nobject itself, if it shines by its own light, is the sort of thing\nconsidered in Bohr\u0027s theory: a sudden jump of an electron from one\norbit to another. This is very unlike a sensation of (say) red. And\nwhat looks to the eye like a continuous red surface is supposed to be\nreally a\u003cspan class=\"pagenum\" id=\"Page_134\"\u003e[Pg 134]\u003c/span\u003e volume whose apparent colour is due to the fact that some\nof the electrons in it are jumping in a certain way. When we say they\nare \"jumping,\" we are saying something too pictorial. What we mean is\nthat they possess an unknown quality called \"energy,\" which is a known\nfunction of a certain number of small integers, and that one or more of\nthese integers have suddenly changed their values. It may be claimed\nas a merit in such theories as Professor Lewis\u0027s, considered in the\npreceding chapter, that it makes the connection between this process\nand the eye rather less indirect than it appears on the undulatory\ntheory. But even then the sort of sudden transition contemplated by\nBohr is very unlike the perception of a red patch: it is \u003ci\u003eprima\nfacie\u003c/i\u003e quite dissimilar in structure, and unknown as regards its\nintrinsic properties.\u003c/p\u003e\n\n\u003cp\u003eI come now to the most serious of our questions: How is matter to be\nunderstood in modern physics? Educated common sense regards matter as\nthe cause of sensations; broadly speaking, sensations private to one\nperson are caused by the matter of that person\u0027s body—\u003ci\u003ee.g.\u003c/i\u003e\nheadaches and toothaches—while sensations common to several, or\nof a sort which is common to several in suitable circumstances,\nare attributed to causes external to the bodies of the persons\nexperiencing the sensations. (I am not at present attempting to make\nthese statements exact, but merely to interpret what common sense\nwould reply if questioned.) We recognize the \"same\" piece of matter\non different occasions by similarity in its qualities, though we\nadmit that this is a rough-and-ready test which may lead us astray.\nWe think, however, that, if we had observed closely and continuously,\nwe could have distinguished between two similar objects by means\nof continuity in their perceived spatial relations. The three-card\ntrick illustrates what I mean: if we watch the performer carefully,\nwe can tell which is the card we saw a moment ago, by means of the\nspatio-temporal continuity of its positions. What\u003cspan class=\"pagenum\" id=\"Page_135\"\u003e[Pg 135]\u003c/span\u003e common sense assumes\nmay be expressed, in language foreign to common sense, by saying: A\npiece of matter is manifested by sensible qualities whose variations\nare continuous, and whose sensible spatial relations to other such\ncontinuous series of qualities are continuous functions of the time. In\npractice, the changes of sensible quality are often so slow as to be\nnegligible, and this greatly facilitates the task of common sense in\nrecognizing the \"same\" object on two different occasions.\u003c/p\u003e\n\n\u003cp\u003eOn the common-sense level, there are difficulties in certain cases:\na drop, in a sensibly homogeneous fluid in which there is a current,\ncannot be distinguished at a later moment from another drop which was\nnear it at the earlier moment. Combustion also offers difficulties\nto common sense. Both these matters can, however, be dealt with on a\ncommon-sense basis. A small solid object floating in the water will\nshow which way the water is moving, and the smoke shows, more or less,\nwhat happens to an object which is burned. The elaboration immediately\nsuggested leads on naturally to elementary physics and chemistry, where\nit is still assumed, at least tacitly, that the objects concerned are\nof the same sort as sensible objects, but rather smaller. Often they\ncan actually be seen under the microscope. Imaginatively, we continue\nto attribute this continuity with sensible objects to our scientific\nobjects, our electrons and protons, thus concealing from ourselves the\nhighly abstract character of our assertions. At moments, we realize\nthis abstractness; but it does not make its due impression, because\nimagination reasserts itself as soon as we are off our guard.\u003c/p\u003e\n\n\u003cp\u003eIn theoretical physics, what is an electron, and how do we decide\nwhether two events belong to the history of the same electron? I am\nnot asking how we decide in practice, but what is our theoretical\ndefinition. Ever since Minkowski, people have spoken of \"world-lines,\"\nwhich are in fact the series of events constituting the history of\none unit of matter\u003cspan class=\"pagenum\" id=\"Page_136\"\u003e[Pg 136]\u003c/span\u003e but they have not always been as explicit as one\ncould wish in telling us the criterion by which, in theory, it is\ndecided that two events belong to one world-line. The test of identity\nbetween the parts of a world-line must obviously depend upon the\nlaws of physics. These laws say that a material unit will move in\nsuch-and-such a way; inverting this statement, they say that what has\nmoved in such-and-such a way is to count as one unit of matter. This\nis substantially the method pursued by Eddington. In Chapter IX. we\nconsidered the tensor\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.904ex; width: 12.131ex; height: 2.861ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-349.png\" alt=\"\" data-tex=\"\\[\nG_{\\mu}^{\\nu} – \\tfrac{1}{2}g_{\\mu}^{\\nu}G,\n\\]\"\u003e\u003c/span\u003e\nwhich, as Eddington shows (§ 52), has the property of\nconservation—\u003ci\u003ei.e.\u003c/i\u003e if the amount of it in any closed region\nvaries, it does so by a flux across the boundaries. He identifies this\nquantity with matter, because of its property of conservation: \"The\nquantity \u003cimg style=\"vertical-align: -0.904ex; width: 11.502ex; height: 2.861ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-350.png\" alt=\"\" data-tex=\"\\(G_{\\mu}^{\\nu} – \\tfrac{1}{2}g_{\\mu}^{\\nu}G\\)\"\u003e appearing\nin our theory is, on account of its property of conservation, now\nidentified with matter, or rather with the mechanical abstraction of\nmatter which comprises the measurable properties of mass, momentum\nand stress sufficing for all mechanical phenomena\" (p. 146). And the\nabove quantity, it will be remembered, is defined solely by means of\nthe formula for small intervals. It will be admitted that matter, so\ndefined, has become rather different from the matter in which common\nsense believes. If Dr Johnson had known Eddington\u0027s definition of\nmatter, he might have been less satisfied with his practical refutation\nof Berkeley.\u003c/p\u003e\n\n\u003cp\u003eThe exact form of Eddington\u0027s definition is not important for our\npresent purposes; indeed, he himself somewhat generalizes it in a later\npassage. The point is that it is the \u003ci\u003esort\u003c/i\u003e of definition to which\nmodern physics is bound to be led. Approximately, matter as conceived\nby common sense is conserved; wherever it appears to be destroyed or\ncreated, we can find ways of explaining away this appearance. Hence,\nas\u003cspan class=\"pagenum\" id=\"Page_137\"\u003e[Pg 137]\u003c/span\u003e an ideal suggested by empirical facts, we adopt the view that\nmatter is indestructible. We then turn round, and beginning from the\nformula for interval we construct a mathematical quantity which is\nindestructible. This, we say, we shall call \"matter\"; and no harm comes\nof our doing so. But whenever we take a step of this sort, we widen\nthe gulf between mathematical physics and observation, and increase\nthe problem of building a bridge between them. This problem has not\nbeen taken as seriously by physicists as it deserves to be taken. The\nreason is partly that it has arisen gradually. Physics and perception\nare like two people on opposite sides of a brook which slowly widens\nas they walk: at first it is easy to jump across, but imperceptibly\nit grows more difficult, and at last a vast labour is required to\nget from one side to the other. Another reason is that physiology\nand psychology, the two sciences concerned with perception, are less\nadvanced than physics. The man accustomed to the beauty and exactitude\nof physics is liable to feel a kind of intellectual nausea when he\nfinds himself among the uncertain and vague speculations of the less\nscientific sciences. He cannot be expected to admit that these sciences\nhave a part to play in providing the premisses for his own precise\nmathematical deductions. Perhaps he is right, but \u003ci\u003eprima facie\u003c/i\u003e\nphysics, as an empirical study, derives its facts from perception, and\ncannot remain indifferent to any argument which throws doubt on the\nvalidity of perception, least of all when that argument is derived\nfrom physics itself. An argument designed to prove that a proposition\nis \u003ci\u003efalse\u003c/i\u003e is not invalidated by having that proposition among\nits premisses. Hence \u003ci\u003eif\u003c/i\u003e modern physics invalidates perception\nas a source of knowledge about the external world, and yet depends\nupon perception, that is a valid argument against modern physics. I\ndo not say that physics in fact has this defect, but I do say that a\nconsiderable labour of interpretation is necessary in order to show\nthat it\u003cspan class=\"pagenum\" id=\"Page_138\"\u003e[Pg 138]\u003c/span\u003e can be absolved in this respect. And it is because of the\nabstractness of physics, as developed by mathematicians, that this\nlabour is required.\u003c/p\u003e\n\n\u003cp\u003eThe inevitable specialism which is forced upon men of science by\nthe very increase of scientific knowledge has had a good deal to do\nwith obscuring this problem. Few men have been both physicists and\nphysiologists. Helmholtz\u0027s researches concerning vision are a notable\nexample of the combination of these studies, but there are not many\nothers. Physiologists and psychologists are seldom well-informed in\nphysics, and are apt to assume an old-fashioned physics which makes\ntheir problems look easier than they are. Moreover, even when the\nproblem is realized, a man may not possess a mastery of the proper\ninstrument for its solution—namely, mathematical logic. It is by means\nof mathematical logic that Dr Whitehead has been enabled to make his\nimmense contribution to our problem. But, greatly as I admire his work,\nwhich I place far above anything else that has been written on the\nrelation of abstract physics to the sensible world, I think there are\npoints—and not unimportant points—where his methods break down for\nwant of due attention to psychology and physiology. Moreover, there\nseem to be premisses in his construction which are derived rather from\na metaphysic than from the actual needs of the problem. For these\nreasons, I venture to think that it is possible to obtain a solution\nless revolutionary than his, and somewhat simpler from a logical point\nof view. The solution, however, must wait until we have examined\nperception as a source of knowledge, which will be our topic in Part\nII. The metaphysic which reconciles the results of Part II. with the\nabstract physics which we have been considering in Part I. will be the\nsubject of Part III.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_34\" href=\"#FNanchor_34\" class=\"label\"\u003e[34]\u003c/a\u003e\n\u003ci\u003eOp. cit.\u003c/i\u003e, p. 225.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_139\"\u003e[Pg 139]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"PART_II\"\u003ePART II\u003cbr\u003e\n\u003cbr\u003e\nPHYSICS AND PERCEPTION\u003c/h2\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_141\"\u003e[Pg 141]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XV\"\u003eCHAPTER XV\u003cbr\u003e\nFROM PRIMITIVE PERCEPTION TO COMMON SENSE\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN this Part, the subject with which we are concerned is the evidence\nfor the truth of physics—not of this or that special result in\nphysics, but of the general structure of the science. It is to be\nexpected that the evidence will not be such as to give certainty, but\nat best such as to give probability; it is to be expected, also, that\nthis probability may be increased by a suitable interpretation of\nphysics, where \"interpretation\" is understood in the sense considered\nin Chapter I. We shall find it desirable to divide our problem into\nseveral parts, each of which will have an importance not confined to\nphysics. There is need, first, to be clear as to what we mean by an\nempirical science, and what is the degree of certainty to be expected\nof it at the best. There is need to discuss what can be meant by\n\"data,\" and to distinguish inferences, theories and hypotheses. We\nshall then discuss the causal theory of perception, and at the same\ntime the philosophy called \"phenomenalism.\" From these topics we shall\npass to general discussion, first of cause, then of substance. This\nwill lead us to the epistemological grounds for interpreting physics\nin accordance with neutral monism, and to the paramount importance of\nstructure in scientific inference. We shall conclude with a definition\nof perception considered as affording the empirical data for physics,\nand with the consideration of phenomena analogous to perception in\nthe non-mental world. But first of all it will be well to examine the\nhistorical development by means of which our problem has assumed its\npresent form—both the pre-scientific development leading to common\nsense, and the scientific development leading from common sense to\nphysics.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_142\"\u003e[Pg 142]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eCommon sense consists of a set of beliefs, or at least habits, which\nwork well in practice except in situations which rarely occur. A savage\nmay be puzzled by a box containing an unseen gyroscope, or by rails\ncarrying an electric current; common sense has not prepared him for\noddities of this sort. But a little familiarity enables a man to fit\nthem into his common-sense world, and a mechanic soon learns their\nways if he has occasion to do so. This illustrates the fact that\nthere is no sharp line between science and common sense: both involve\nexpectations, but those resulting from science are more accurate. It is\npossible to pursue science practically without any fundamental change\nfrom the metaphysic of common sense. But when theoretical science is\ntaken seriously, it is found to involve a quite changed metaphysic,\nwhose relation to that of common sense demands investigation. This will\nform the topic of the next chapter; in the present chapter, I shall\nconsider the genesis of common sense, not in the race, since that is\nundiscoverable, but in the individual.\u003c/p\u003e\n\n\u003cp\u003eIn studying infants, as in studying animals, we are compelled to\nconfine ourselves to behaviouristic methods, whatever our views may be\non the subject of behaviourism as a general principle in psychology. We\ncan observe the bodily acts of young infants, but they cannot tell us\ntheir thoughts. At a low mental level, however, it is hardly profitable\nto distinguish between a belief and a habit of action. Beliefs, in the\npsychological sense, seem to emerge out of previously existing habits,\nand to be, at first, little more than verbal representations of habits\nformed before words could be uttered. There is therefore no great loss\nin being confined to behaviouristic methods when we are considering\ninfants before the age at which they can speak.\u003c/p\u003e\n\n\u003cp\u003eIt is of course obvious and generally recognized that very young\ninfants do not possess the common-sense notion of an \"object.\"\nThis is by no means obvious with the young of\u003cspan class=\"pagenum\" id=\"Page_143\"\u003e[Pg 143]\u003c/span\u003e some other kinds of\nanimals—with chickens, for example. They possess, as instincts,\nuseful ways of behaviour which in the human young are only learnt by\nexperience; for example, they can pick up a grain which they see on\nthe ground. The human infant has no such innate skill; for several\nmonths, it makes no attempt to touch what it sees. The \"hand-eye\nco-ordination\" comes as a result of experience. Some native aptitudes,\nof course, a new-born child does possess; for example, it can turn its\neyes towards a bright light, though not very quickly or accurately. It\nhas a reflex connected with sucking, but not a very intelligent one;\nindeed, it hardly amounts to more than the practice of trying to suck\nanything that comes in contact with the lips. Even in this respect,\nthe human infant is inferior to the young of other mammals. We can say\nthat certain stimuli rouse certain reflexes, but these are only just\nsufficient to keep the infant alive with the help of maternal care.\u003c/p\u003e\n\n\u003cp\u003eIn this primitive condition, the infant obviously has no conception\nof an \"object.\" An \"object,\" for common sense, is something having\na certain degree of permanence, and connected with several kinds of\nsensation. This involves something like memory, to give rise to the\nidea of permanence, or rather, at first, to the feeling of recognition;\nand it involves experience, to give to one sensory stimulus a reaction\noriginally associated with another. In infants, the most important\nfactor in forming the common-sense notion of an object is the hand-eye\nco-ordination, the discovery that it is possible, often, to grasp what\nis seen. In this way, visual and tactual spaces become correlated,\nwhich is one of the most important steps in the mental growth of an\ninfant.\u003c/p\u003e\n\n\u003cp\u003eAt this point, it is important to be clear as to the difference between\n\"space\" in psychology and \"space\" in physics. There is undoubtedly a\nconnection between the two, which it will be part of our business to\nmake clear at a later stage. But\u003cspan class=\"pagenum\" id=\"Page_144\"\u003e[Pg 144]\u003c/span\u003e the connection is very round-about\nand inferential. At the outset, it is much more useful to realize the\ndifference between them than the connection, since much confusion of\nthought arises from supposing the connection to be closer than it is.\nIn physics there is only one space, while in psychology there are\nseveral for each individual; these can, it is true, be reduced by\nmanipulation to one for each individual, but they cannot be reduced\nfurther without introducing obscurities that it is impossible to\ndissipate. The space containing my visual objects has no point in\ncommon with the space containing yours, since no visual object in my\nworld is precisely identical with one in yours. And the amalgamation of\nthe spaces of my different senses into one space is a piece of early\nscience, performed by the infant at about the age of three months.\nDr Whitehead, who is anxious to bridge the gulf between perception\nand physics, seems to me to make his task too easy where space is\nconcerned. For example, he says:\u003ca id=\"FNanchor_35\" href=\"#Footnote_35\" class=\"fnanchor\"\u003e[35]\u003c/a\u003e\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"The current doctrine of different kinds of space—tactual space,\nvisual space, and so on—arises entirely from the error of deducing\nspace from the relations between figures. With such a procedure, since\nthere are different types of figures for different types of sense,\nevidently there must be different types of space for different types\nof sense. And the demand created the supply. If, however, the modern\nassimilation of space and time is to hold, we must go further and admit\ndifferent kinds of time for different kinds of sense—namely, a tactual\ntime, a visual time, and so on. If this be allowed, it is difficult\nto understand how the disjecta membra of our perceptual experience\nmanage to collect themselves into a common world. For example, it would\nrequire a pre-established harmony to secure that the visual newspaper\nwas delivered at the visual time of the visual breakfast in the visual\nroom, and also the tactual newspaper was delivered at the tactual\ntime of the tactual breakfast in the tactual room. It is difficult\nenough for the plain man—such as the present author—to accept the\nmiracle of getting the two\u003cspan class=\"pagenum\" id=\"Page_145\"\u003e[Pg 145]\u003c/span\u003e newspapers into the two rooms daily with\nsuch admirable exactitude at the same time. But the additional miracle\nintroduced by the two times is really incredible.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eThis passage is so pleasant that I hate to criticize it. But I do not\nknow how else to make clear where I differ from Dr Whitehead. There is\nfirst a purely verbal question to be cleared up. Dr Whitehead says it\nis an error to deduce space from the relations between figures. It is\ncertainly an error to deduce \u003ci\u003ephysical\u003c/i\u003e space in this way, but\nwith psychological space the matter is different. There certainly are\nperceived relations between figures, and these perceived relations are\npart of our perceptual data in physics. Whether they are to be said\nto constitute a space or not, is a verbal question. Psychologists, as\na rule, find it convenient to say so; but the matter is unimportant.\nWhen this question has been cleared away, however, there remain others\nwhich are vital to an understanding of the relation between physics and\nperception.\u003c/p\u003e\n\n\u003cp\u003eTake, first, the question of the two times. As will appear when we\ncome to the causal theory of perception, the whole of my perceptual\nworld is, from the standpoint of physics, in my head; any two events\nwhich I experience together overlap in physical space, and all of them\ntogether, in \u003ci\u003ephysical\u003c/i\u003e space, occupy a volume smaller than my\nhead, since it certainly does not include the hair, skull, teeth, etc.\nConsequently, on relativity principles, there is no question of two\ntimes, since this only arises for events which are spatially separated\nin physical space.\u003c/p\u003e\n\n\u003cp\u003eAs for the necessity of distinguishing tactual and visual space: there\nare perceived relations between objects seen simultaneously, and also\nbetween objects touched simultaneously, and these relations are part\nof the crude material out of which we construct our notion of space.\n\u003ci\u003eThese\u003c/i\u003e relations cannot hold between a visual and a tactual\npercept. But there are other relations which do hold—namely, those\nof correlation:\u003cspan class=\"pagenum\" id=\"Page_146\"\u003e[Pg 146]\u003c/span\u003e when I see my hand in contact with a visual object I\nfeel it in contact with a tactual object, and moreover the visual and\nthe tactual object have certain relations to each other—\u003ci\u003ee.g.\u003c/i\u003e\nwhere we see a corner we get a tactual sensation of sharpness. All\nthis, however, is learnt by experience; that is to say, we learn the\n\u003ci\u003elaws\u003c/i\u003e of the correlation by experience. The infant can be seen\nlearning them. One may call these laws \"pre-established harmonies,\"\nbut they are no more so than any other scientific laws. Unless we\nare going to say that all laws of nature must be demonstrable by\npure logic, which is hardly conceivable nowadays, we must admit that\nthere are co-existences and sequences which we expect on a basis of\npast experience, in spite of the fact that their failure would not\nbe logically impossible. And the correlation of visual and tactual\nsensations is a case of this sort.\u003c/p\u003e\n\n\u003cp\u003eIt is sometimes suggested, in such cases, that the correlated\noccurrences are merely different manifestations of one and the same\nentity. This is, in fact, the view of common sense, which holds that it\ncan both see and touch the same object. I have no objection whatever\nto this way of speaking, and I do not deny that, rightly interpreted,\nit may express a correct view. But it remains nevertheless true that\nthe entity said to be manifested is inferred from experience of a\ncorrelation, and that the percepts correlated are not \u003ci\u003elogically\u003c/i\u003e\ninterconnected, but only empirically. We have \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e, a visual percept,\nand at the same time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e, a tactual percept. Each rouses appropriate\nreflexes, and, owing to their frequently occurring together, it happens\nin time that each rouses also the reflexes appropriate to the other.\nThis practical induction occurs before the child has reflected that the\ntwo are correlated; indeed, unless he becomes a learned man he probably\nnever realizes the correlation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e. But as soon as we\nreflect upon the matter we can see that there is no \u003ci\u003enecessary\u003c/i\u003e\ncorrelation. It fails with blind men, and with men whose fingers have\nbeen anæsthetized.\u003cspan class=\"pagenum\" id=\"Page_147\"\u003e[Pg 147]\u003c/span\u003e In general, however, the correlation holds good.\nCommon sense explains it by regarding both touch and sight as ways of\ngetting to know an object which is at once tangible and visible. In\nthe language of the causal theory of perception, we say that \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e have a common cause, in general external to the body. I do not\nwish to deny this, but only to point out that, when we are considering\nthe grounds of our knowledge, we cannot say that we know of the\ncorrelation because we know of the common external cause. The order in\nknowledge is the opposite: we have evidence for the correlation in our\nexperience, and we infer\u003ca id=\"FNanchor_36\" href=\"#Footnote_36\" class=\"fnanchor\"\u003e[36]\u003c/a\u003e the common cause from the correlation, so\nthat the common cause cannot have more certainty than the correlation,\nwhich is its premiss. From a behaviouristic point of view, the infant\n\"knows\" the correlation when either stimulus calls out the response\noriginally appropriate to the other.\u003c/p\u003e\n\n\u003cp\u003eWe must here guard against a small possible misunderstanding. If \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e are invariably correlated, it may be said, it is impossible\nthat one should occur without the other, and therefore there can be no\nmeans of judging whether one alone would elicit the response belonging\nto the other. In fact, the matter is not quite so simple as we have\nbeen taking it to be. What we learn by infantile experience is not that\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e are \u003ci\u003ealways\u003c/i\u003e correlated; it is possible to touch\nin the dark, or with the eyes shut, and it is possible to see without\ntouching. What we learn is that the correlation can be brought about\neasily in many cases. Movements of the eye will usually give a visual\nsensation corresponding to a previously uncorrelated tactual sensation,\nand movements of the hand (or other part of the body) will, in a\ncertain proportion of cases, give a tactual sensation corresponding to\na previously uncorrelated visual sensation. Children practising the\nhand-eye co-ordination attempt to grasp objects not within their reach;\nit is only\u003cspan class=\"pagenum\" id=\"Page_148\"\u003e[Pg 148]\u003c/span\u003e gradually that distance comes to be judged more or less\ncorrectly. When objects are not within our grasp, a new correlation\ncomes into play—namely, between the visual sensation and the journey\nrequired to bring the object within our reach. Unfamiliar circumstances\nwill cause even adults to make mistakes—for example, that of\nunderestimating the depth of objects under water. Great distances\nremain permanently beyond the scope of common sense: only science can\nassure us that the sun is farther off than the moon.\u003c/p\u003e\n\n\u003cp\u003eWhat we can observe the infant learning is the bodily acts which will,\nin fact, reinforce a percept of one sense by a percept of another; more\nparticularly he learns to touch what he sees—\u003ci\u003ei.e.\u003c/i\u003e to procure\nfor himself a correlated pair \u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e, instead of the isolated\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.097ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-25.png\" alt=\"\" data-tex=\"\\(v\\)\"\u003e. Similarly he learns to look round when he hears a voice, and\nso on. All this implies that he has, so far as action is concerned,\nthe notion of a physical object, as something capable of affecting\nseveral senses simultaneously. The element of recognition is logically\nseparable, and arises somewhat earlier.\u003c/p\u003e\n\n\u003cp\u003eThese motor habits are essential in generating common-sense beliefs,\nwhich arise at a much later stage of mental growth. Common sense,\nin its more primitive form, is hardly aware that there is such an\noccurrence as perceiving; it is only aware of the perceived object.\nAnd by the time that even the most rudimentary reflection begins,\neach sense calls out responses connected with other senses, so that\neven when, from the standpoint of external stimulus, only one sense\nis affected, the experience has the massiveness of something in which\nseveral senses are involved. See, for example, the pictures in Kohler\u0027s\n\u003ci\u003eMentality of Apes\u003c/i\u003e: here we see chimpanzees which are watching\nothers with sympathetic movements of the arms that indicate stimulation\nof bodily feelings connected with balance, although the sole stimulus\nis visual. This accounts for the fact that common sense can so\nconfidently identify an object touched and not seen with an\u003cspan class=\"pagenum\" id=\"Page_149\"\u003e[Pg 149]\u003c/span\u003e object\nseen but not touched—\u003ci\u003ee.g.\u003c/i\u003e the cricket-ball now successfully\ncaught and the same ball as it flew through the air. The reason is\nthat the experience is always richer than the sensory stimulus alone\nwould warrant: it contains always responses arising from physiological\nexperience of past correlations. If an adult were to hear a donkey\u0027s\nbray for the first time, without having previously known that there\nwas an animal which made that noise, his experience would be amazingly\nunlike that of a normal adult in the same circumstances.\u003c/p\u003e\n\n\u003cp\u003eCommon sense does not initially distinguish as sharply as civilized\nnations do between persons, animals, and things. Primitive religion\naffords abundant evidence of this. A thing, like an animal, has a sort\nof power residing within it: it may fall on your head, roll over in the\nwind, and so on. It is only gradually that inanimate objects become\nsharply separated from people, through the observation that their\nactions have no purpose. But animals are not separable from people\non this ground, and are in fact thought by savages to be much more\nintelligent than they are.\u003c/p\u003e\n\n\u003cp\u003eCommon sense is, in most respects, naively realistic: it believes\nthat, as a rule, our perceptions show us objects as they really are.\nIt is able to hold this view because of the mass of experience which,\nin each individual, precedes the common-sense outlook. We do not\nthink a distant person smaller than a person near at hand; we do not\njudge circular objects seen sideways to be elliptic; and so on. All\nthis is, for common sense, part of the perception; it may be doubted\nwhether it is not so also for psychology. But it is certainly not part\nof the infant\u0027s initial perceptive apparatus: it is something which\nthe infant has to learn. Some of it is learnt after the beginnings of\nspeech have been acquired—particularly a right judgment as to the\nsize of distant objects. But at any rate by the time a child is three\nyears old he has acquired the common-sense outlook. That is to say,\nhis immediate reaction\u003cspan class=\"pagenum\" id=\"Page_150\"\u003e[Pg 150]\u003c/span\u003e to a sensory stimulus involves a great deal of\nprevious experience, and is such as to enable him to arrive, without\nany mental process, at a far more objective view of what he perceives\nthan was possible at birth. I mean here by \"objective\" not anything\nmetaphysical, but merely \"agreeing with the testimony of others.\" It\nwould be a complete mistake to suppose that, in an adult, there is\nfirst an experience corresponding to the bare sensory stimulus, and\nthen an inference to that of which it is a sign. This may occur in\ncertain cases, for example, if we watch a man drawing a face in an\napparently haphazard manner, and do not realize till the last moment\nthat a face is being intended. But such an experience is quite unlike\nnormal perception, where the \"inference,\" in the only sense in which it\ncan be said to exist, is physiological, or at any rate not discoverable\nby introspection. It is because the sensory stimulus is able to lead\nus, without any mental intermediary, to an object practically identical\nwith that perceived by others in our neighbourhood, that we are able\nto adopt the common-sense belief that we actually perceive external\nobjects.\u003c/p\u003e\n\n\u003cp\u003eThe notion of cause is part of the apparatus of common sense. I do\nnot think it would be true to say that common sense regards objects\nas the causes of our perceptions; it would not, unless challenged,\nthink of bringing in causation in this connection. It looks for causes\nwhen it is surprised, not when an occurrence seems perfectly natural.\nIt demands causes for a mirage, a reflexion, a dream, an earthquake,\na plague, and so on, but not for the ordinary course of nature. And\nthe cause which it looks for, wherever the event concerned has great\nemotional interest, is pretty sure to be animistic: the anger of the\ngods, or something analogous. The idea of universal causation, and of\ncausation divorced from purpose, belongs to a later stage of mental\ndevelopment, and marks the beginnings of philosophy and science.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_151\"\u003e[Pg 151]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eSubstance is a category which comes naturally to common sense,\nthough without the attribute of indestructibility added by the\nmetaphysicians—but as to this perhaps diverse opinions are possible.\nOne would be inclined to suppose that common sense regards fire as\ndestroying what it burns; but the Chinese, when they had made a solemn\ncovenant, used to burn it, in order that the gods might take cognisance\nof it through the smoke. (A copy was kept for terrestrial purposes.)\nAnd races that practise cremation do not, as a rule, suppose that\nthey are totally destroying the body. On the other hand, there has\nexisted a religious prejudice against cremation which implied the\nbelief that the body was thereby totally annihilated. I think one\nmust conclude, therefore, that the attitude of common sense as to\nthe indestructibility of substance is vacillating; on the whole, the\nsuccess of physics in providing immortal material units represents a\ntriumph of the philosopher over the plain man.\u003c/p\u003e\n\n\u003cp\u003eSubstance, whether indestructible or not, is of great importance in\nprimitive thought, and dominates syntax, through which it has dominated\nphilosophy down to our own day. At a primitive stage, there is no\ndistinction between \"substance\" and \"thing\"; both express, first in\nlanguage and then in thought, the emotion of recognition. To an infant,\nrecognition is a very strong emotion, particularly when connected with\nsomething agreeable or disagreeable. When the infant begins to use\nwords, it applies the same word to percepts on two occasions, if the\nsecond rouses the emotion of recognition associated with memory of the\nfirst, or perhaps merely with the word which was learnt in presence of\nthe first. (When I say that the infant uses the \"same\" word, I mean\nthat he makes closely similar noises.) Using a given word as a response\nto stimuli of a certain kind is a motor habit, like reaching for the\nbottle. Two percepts to which the same word applies are thought to be\nidentical, unless both can be present\u003cspan class=\"pagenum\" id=\"Page_152\"\u003e[Pg 152]\u003c/span\u003e at once; this characteristic\ndistinguishes general names from proper names. The basis of this whole\nprocess is the emotion of recognition. When the process, as a learning\nof motor habits, is complete, and reflection upon it begins, identity\nof name is taken to indicate identity of substance—in one sense\nin the case of proper names, in another sense in the case of names\napplicable to two or more simultaneous percepts—\u003ci\u003ei.e.\u003c/i\u003e general\nnames (Platonic ideas, universals). Throughout, language comes first\nand thought follows in its footsteps. And language is governed largely\nby physiological causation.\u003c/p\u003e\n\n\u003cp\u003eA substance or thing is supposed to be identical at different times,\nalthough its properties may change. John Jones is the same person\nthroughout his life, although he grows from childhood to manhood, is\nsometimes pleased and sometimes cross, sometimes awake and sometimes\nasleep. Primarily, he is considered to be the same person because he\nhas the same name. But the name, like the person, is not exactly the\nsame on different occasions; it may be spoken loud or soft, quickly\nor slowly. These differences, however, are too slight to prevent\nrecognition, except on rare occasions—\u003ci\u003ee.g.\u003c/i\u003e when the name is\npronounced very badly by a foreigner; one of the merits of names is\nthat they change less than the person named.\u003c/p\u003e\n\n\u003cp\u003eThe conception of substantial identity with varying properties is\nembedded in language, in common sense, and in metaphysic. To my mind,\nit is useful in practice, but harmful in theory. It is harmful, I mean,\nif taken as metaphysically ultimate: what appears as one substance\nwith changing states should, I maintain, be conceived as a series of\noccurrences linked together in some important way. I will not yet argue\nthis view. It would have been utterly foreign to physics until the\nsubstitution of space-time for space and time, with the corresponding\nsubstitution of a four-dimensional continuum of events for the older\nconception of persistent material units moving in a three-dimensional\nspace. But the older conception\u003cspan class=\"pagenum\" id=\"Page_153\"\u003e[Pg 153]\u003c/span\u003e still appears the natural one to apply\nto electrons and protons, so that physics may be said to have, at the\nmoment, two different points of view on this issue. For the present,\nI am not concerned to criticize the notion of substance, but only\nto show its genesis, which I take to be derived from the pre-human\nemotion which we reflectively call \"recognition,\" though it has not,\noriginally, the definite cognitive character attached to the word when\napplied to the mental processes of an adult human being.\u003c/p\u003e\n\n\u003cp\u003eInduction, like substance, plays a large part in common sense, and\nhas a basis which is primarily physiological. I am not at present\ndiscussing the validity of induction, but the cause of the practice of\ninduction among animals, children, and savages. Of course the validity\nof induction is really assumed in such a discussion, since, without it,\ncauses cannot be discovered. But we do not assume the validity of the\nprimitive inductions which we are discussing; we assume only that there\nis \u003ci\u003esome\u003c/i\u003e valid form of induction. Throughout genetic psychology\nwe assume the validity of ordinary scientific procedure. If this\nassumption were to lead us to views on genetic psychology which threw\ndoubt on the validity of scientific procedure, that would constitute\na \u003ci\u003ereductio ad absurdum\u003c/i\u003e, which would destroy genetic psychology\nalong with the rest. Therefore, whenever some obviously invalid process\nis said to be the psychological source of a method essential to\nscience, we must suppose, unless we are to embrace complete scepticism,\nthat there is some valid process which, in most of the cases to which\nthe invalid process is applied by unscientific people, gives rather\nsimilar results. All this has perhaps only a pragmatic justification,\nbut whether this is the case cannot be decided \u003ci\u003eab initio\u003c/i\u003e. The\nreal utility of investigating crude primitive forms of inference is\nthat the contrast between them and current scientific inference may\nsuggest directions in which the latter is capable of still further\u003cspan class=\"pagenum\" id=\"Page_154\"\u003e[Pg 154]\u003c/span\u003e\nimprovement. The direct logical importance of investigations into the\norigins of our mental processes is \u003ci\u003enil\u003c/i\u003e, but the importance as\na means of stimulating imagination in the formation of hypotheses may\nbe considerable. It is for this reason that the topics of the present\nchapter form a useful introduction to those which form our proper\nsubject-matter.\u003c/p\u003e\n\n\u003cp\u003eThe source of induction, speaking historically, is the general law\nof what Dr J. B. Watson calls \"learned reactions.\" In its schematic\nsimplicity, this law is as follows: If a stimulus \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e to a living\nbody of an animal produces a reaction \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e, and a stimulus \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e\nproduces a reaction \u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e, then if \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e are applied\ntogether, there is a tendency for \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e alone, afterwards, to produce\n\u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e as well as \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e. \u003ci\u003eE.g.\u003c/i\u003e if you expose a person frequently\nto a certain loud noise and a bright light simultaneously, after a\nwhile the loud noise alone will cause his pupils to contract. It is\nobvious that the \u003ci\u003epractice\u003c/i\u003e of induction is simply the application\nof this law to cognitive reactions. If you have frequently heard the\nwords \"there\u0027s Jones\" when you could see Jones, these words will in the\nend cause you to believe that Jones is present even if, for the moment,\nyou do not see him. This form of induction is involved in understanding\nspeech. It is obvious that, in its cruder forms, induction may give\nrise to false beliefs as well as to true ones; scientific methodology\nhas to seek a form of induction which shall make false inferences\nmuch rarer than true ones. If such a form can be found, a man may\ntrain himself, in his professional activity, to abstain from the\nmore primitive forms. But as an ordinary mortal he could not survive\nfor a day if he refused to trust to what we may call physiological\ninduction, which stores up in the body the lessons of past experience.\nIn practice, a nearly instantaneous method of inference which is\nright nine times out of ten is preferable to a slow method which is\nalways right. A man who subjected all his food to chemical\u003cspan class=\"pagenum\" id=\"Page_155\"\u003e[Pg 155]\u003c/span\u003e analysis\nbefore eating it would avoid being poisoned, but would also fail to be\nadequately nourished.\u003c/p\u003e\n\n\u003cp\u003eThroughout the development of theory, great intellectual changes have\nbeen repeatedly necessitated by errors which were very small from\nthe standpoint of practice. The theory of relativity is a remarkable\ninstance of this: an immense reconstruction has been made to meet\ndiscrepancies which could only be detected by the most delicate\nmeasurements. The further science advances, the more minute become the\nfacts which it cannot yet assimilate. Common sense does well enough\nfor most of the needs of a pre-industrial community, but not for the\nconstruction of a dynamo or a wireless station. For these, we have to\nadvance to the standpoint of pre-relativity physics. Machines involving\nrelativity physics do not yet exist, but presumably they will some\nday. This, however, is beside the point. The point is, that a small\ndiscrepancy between theory and observation may indicate a large error\nin theory. Take, \u003ci\u003ee.g.\u003c/i\u003e, naive realism and the velocity of light,\nthe latter from a pre-relativity point of view. The supposition of\ncommon sense and naive realism, that we see the actual physical object,\nis very hard to reconcile with the scientific view that our perception\noccurs somewhat later than the emission of light by the object; and\nthis difficulty is not overcome by the fact that the time involved,\nlike the notorious baby, is a very little one. We cannot therefore\nargue from the practical success of common sense to its approximate\ntheoretical accuracy, but only to a certain rough correspondence\nbetween its commoner inferences and those permitted by a correct\ntheory. If physics has had to desert common sense, that is no reason\nfor finding fault with physics.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_35\" href=\"#FNanchor_35\" class=\"label\"\u003e[35]\u003c/a\u003e\n\u003ci\u003eThe Principles of Natural Knowledge\u003c/i\u003e, pp. 193-4.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_36\" href=\"#FNanchor_36\" class=\"label\"\u003e[36]\u003c/a\u003e\nI am here using the word \"infer\" in a behaviouristic\nsense.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_156\"\u003e[Pg 156]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XVI\"\u003eCHAPTER XVI\u003cbr\u003e\nFROM COMMON SENSE TO PHYSICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIT was in the seventeenth century that the scientific outlook, as\nopposed to that of common sense, first became important. It had existed\nin individuals among the Greeks, but it had not been able to point to\nsufficiently great achievements to impress the general educated public.\nIt was in the seventeenth century that science began to win spectacular\nvictories, and to develop an outlook definitely different, in certain\nimportant respects, from that of common sense. The historical aspects\nof this change have been set forth by Dr Whitehead in his \u003ci\u003eScience\nand the Modern World\u003c/i\u003e, particularly in the chapter on \"The Century\nof Genius,\" so admirably that it would be foolish to attempt to cover\nthe ground again. I shall therefore select only certain topics which\nare important in relation to subsequent chapters.\u003c/p\u003e\n\n\u003cp\u003eThe chief thing that happened in the seventeenth century, from our\npoint of view, was the divorce between perception and matter, which\noccupied all the philosophers from Descartes to Berkeley, leading the\nlatter to deny matter, while it had, in effect, led Leibniz to deny\nperception.\u003c/p\u003e\n\n\u003cp\u003eCommon sense believes that there is interaction between mind and\nmatter: when a stone hits us our mind feels pain, and when we will to\nthrow a stone it moves. The development of physics made matter seem\ncausally self-contained: it appeared that there were always physical\ncauses for the movements of matter, so that volitions must be otiose.\nDescartes, believing in the conservation of \u003ci\u003evis viva\u003c/i\u003e, but\nignorant of the conservation of momentum, thought that the mind could\ninfluence the direction of the motion of the animal spirits, but\u003cspan class=\"pagenum\" id=\"Page_157\"\u003e[Pg 157]\u003c/span\u003e not\nits amount. This half-way house had to be abandoned by his followers,\nowing to the discovery of the conservation of momentum. They therefore\ndecided that mind can never influence matter. They also decided that\nmatter can never influence mind. This latter view was not based\ndirectly upon science, but upon the metaphysic which had been invented\nto explain away the apparent influences of mind on matter. To suppose\nthat the movement of my arm is not caused by my volition is to suppose\nsomething very odd; it is no odder to suppose that the perception of my\narm is not caused by my arm. The view that there were two substances,\nmind and matter, and that neither could act upon the other, explained\nthe causal independence of the physical world, and entailed that of the\nmental world. Thus mind and matter became very widely separated—much\nmore so than they had been before the rise of modern physics.\u003c/p\u003e\n\n\u003cp\u003eAll modern philosophy before Kant is dominated by this problem, for\nwhich a variety of solutions were offered. Spinoza held that there was\nonly one substance, whose only \u003ci\u003eknown\u003c/i\u003e attributes were thought and\nextension, which ran parallel without interaction, like the two perfect\nclocks of the occasionalists. Leibniz believed in an immense number of\nsubstances, all causally independent of each other, but all running\nparallel in virtue of a pre-established harmony; these substances were\nall minds, more or less developed, and matter was only a confused way\nof \"perceiving\" a number of substances. The word \"perceiving\" has, in\nLeibniz\u0027s philosophy, a peculiar meaning, derived from parallelism and\nfrom the notion of \"mirroring the universe.\" Without attempting to\nadhere closely to Leibniz\u0027s own words, we may set forth the view which\nis implied in his system, whether he held it in its entirety or not, as\nfollows: Each monad, at each moment, is in an infinitely complex state,\nwhich is capable of a one-one correspondence with the state of each\nother monad at that\u003cspan class=\"pagenum\" id=\"Page_158\"\u003e[Pg 158]\u003c/span\u003e moment. (This is the pre-established harmony.)\nThe differences between the states of different monads are like the\ndifferences between the aspects of a given object from different\nplaces, and are compared by Leibniz to differences of perspective\nor point of view. These differences are capable of arrangement in\na three-dimensional order, so that the monads form a pattern which\nchanges with the time. In addition to the one-one correspondences\nbetween the monads, there is a one-one correspondence between the state\nof each monad and the pattern formed by all the monads (mirroring the\nworld). It will be seen that the latter logically implies the former:\nif each monad always mirrors the world, each is always in harmony with\nevery other. Let us take a mathematical analogy: suppose the states of\nthe \u003cimg style=\"vertical-align: -0.025ex; width: 3.673ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-352.png\" alt=\"\" data-tex=\"\\(m^{th}\\)\"\u003e monad at a given moment are represented by the numbers:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.816ex; width: 24.637ex; height: 2.773ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-353.png\" alt=\"\" data-tex=\"\\[\nm – 1, m – \\tfrac{1}{2}, m – \\tfrac{1}{3}, …\n\\]\"\u003e\u003c/span\u003e\nthen there is a one-one correspondence between these states and those\nof the \u003cimg style=\"vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-59.png\" alt=\"\" data-tex=\"\\(n^{th}\\)\"\u003e monad, which are:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.781ex; width: 22.751ex; height: 2.737ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-354.png\" alt=\"\" data-tex=\"\\[\nn – 1, n – \\tfrac{1}{2}, n – \\tfrac{1}{2}, …\n\\]\"\u003e\u003c/span\u003e\nand there is also a one-one correspondence between the states of each\nmonad and the series:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 20.445ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-355.png\" alt=\"\" data-tex=\"\\[\n1, 2, 3, … m, … n, …\n\\]\"\u003e\u003c/span\u003e\nwhich may be taken to be the series of monads. Substitute three\ncontinuous co-ordinates for one discrete co-ordinate, and we get a\nmathematical representation of Leibniz\u0027s world.\u003c/p\u003e\n\n\u003cp\u003eThe obvious difficulty in this system was that no conceivable reason\ncould be given for supposing that a monad mirrored the world. Leibniz\nhimself was one monad, and, on his own theory, would have had exactly\nthe same life if he had been the only monad, since the monads were\n\"windowless.\" He could not therefore give any grounds against solipsism\nexcept some rather far-fetched arguments derived\u003cspan class=\"pagenum\" id=\"Page_159\"\u003e[Pg 159]\u003c/span\u003e from theology and\nGod\u0027s \"metaphysical perfection.\" This defect was due to his theory\nof causality, which was an outcome of the Cartesian denial that one\nsubstance could act upon another, which in turn was inspired by the\nsuccess of physics in establishing purely physical causal laws which\nseemed to account for all the motions of matter. In spite of this\nglaring defect, I have lingered on Leibniz\u0027s system, because I believe\nthat it contains hints for a metaphysic compatible with modern physics\nand with psychology, although of course it will require very serious\nmodifications.\u003c/p\u003e\n\n\u003cp\u003eThe problem of perception remained unsolved, although it was one of\nthe main pre-occupations of philosophers. Locke, important as he was,\ndid not contribute much on this question, except his theory that\nprimary qualities are objective and secondary qualities subjective;\nbut his \u003ci\u003eEssay\u003c/i\u003e led others to theories which have remained\nimportant. Berkeley discarded the material world, though he need not\nhave discarded physics, since the formulæ of physics may perfectly well\nbe applicable to collections of mental events, as Leibniz supposed.\nBerkeley does not seem to have been influenced by the argument which\naffected the Cartesians—namely, the supposed impossibility of\ninteraction between mind and matter. What influenced Berkeley was\nrather the epistemological argument, that everything with which we\nare acquainted is a mental event, and there is no valid reason for\ninferring that there are events of quite another kind. This type of\nargument is, I think, new in Berkeley, when regarded as a source of\nmetaphysic; in another form, it achieved fame through Kant. Hume\ncarried the same type of reasoning much further than Berkeley did,\nsince he was content to remain sceptical, whereas Berkeley employed\nscepticism about matter as a support of religion, and therefore had to\nlimit the scope of his criticism of what passed as knowledge. Hume\u0027s\ncriticism of the notion of cause cut at the root of science, and\ndemanded an answer imperatively.\u003cspan class=\"pagenum\" id=\"Page_160\"\u003e[Pg 160]\u003c/span\u003e Of course innumerable answers were\nforthcoming, but I cannot persuade myself that any of them were in any\ndegree valid, not even that of Kant. I do not wish, however, to discuss\nat this moment any philosophy which has still a more than historical\ninterest, as is the case with Berkeley, Hume, and Kant. Let us\ntherefore return from this excursus to topics more intimately connected\nwith science.\u003c/p\u003e\n\n\u003cp\u003eThe profound and lasting effect of Cartesianism upon the outlook of\nphilosophers and men of science was to widen the gulf between mind and\nmatter. Physicists were satisfied with the view that their science\ncould be pursued independently of considerations concerned with mind,\nand contentedly left the philosophers to wrangle, under the impression\nthat philosophy did not matter to them. For a time, from the point of\nview of the progress of science, there was much truth in this view;\nbut in the long run science cannot shut its eyes to problems which\nare logically relevant to its investigations. It may be admitted that\nmost of what has passed for philosophy would not have been very useful\nto the men of science; but that was chiefly because philosophy was no\nlonger being created by men like Descartes and Leibniz, who were of\nsupreme eminence in science as well. It may be hoped that this state of\naffairs is coming to an end.\u003c/p\u003e\n\n\u003cp\u003eThe \"matter\" of the Cartesians, owing to their denial of interaction\nbetween mind and matter, should have been just as abstract, and just\nas purely mathematical, as in the most modern physics. But in fact\nthis was not the case: the technique of the period still depended\nupon notions which had an immediate basis in our own experience. We\nmay perhaps distinguish three sorts of physics, in relation to the\nsense-experiences from which their ideas are derived: I will call them\nmuscular physics, touch physics, and sight physics respectively. Of\ncourse no one of them has ever existed in isolation: actual physics has\nalways been a mixture of the three.\u003cspan class=\"pagenum\" id=\"Page_161\"\u003e[Pg 161]\u003c/span\u003e But it will be a help in analysis\nto imagine a separation of each from the others, and ask ourselves\nwhich elements in actual physics belong to the first, which to the\nsecond, and which to the third. Broadly we may say that sight-physics\nhas more and more predominated, and has achieved an almost complete\nvictory over the others in the theory of relativity.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_162\"\u003e[Pg 162]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eMuscular physics is embodied in the idea of \"force.\" Newton evidently\nthought of force as a \u003ci\u003evera causa\u003c/i\u003e, not as a mere term in a\nmathematical equation. This was natural; we all know the experience\nof \"exerting force,\" and are aware that it is connected with setting\nbodies in motion. By a sort of unconscious animism, physicists supposed\nthat something analogous occurs whenever one body sets another in\nmotion. Unfortunately for dynamics we have the experience of \"exerting\nforce\" when we merely cause a body to preserve a constant velocity,\nas in dragging a weight along a road; this misled Aristotle into\nthinking that force was to be regarded as the cause of velocity,\nnot of acceleration, a mistake first corrected by Galileo—though\nLeonardo came very near seeing the truth. It may be said: if force\nis a mathematical fiction, how can it be more \"true\" to regard it as\nproportional to the acceleration than to regard it as proportional to\nthe velocity? The reason is that laws can be found connecting force\nwith the situation of a body relative to other bodies, if force is\ndefined as Galileo defined it, but not if it is defined as Aristotle\ndefined it. Galileo\u0027s discovery that falling bodies have a constant\nacceleration, which is the same for all (\u003ci\u003ein vacuo\u003c/i\u003e), is a very\nsimple instance. More generally we may say: The laws of physics are,\nas a rule, differential equations of the second order—with respect to\ntime in Newtonian physics, and with respect to interval in the physics\nof Einstein. This is a very different notion from that of force as\nderived from experience of muscular exertion; yet the one has led to\nthe other by an evolution containing many intermediate links.\u003c/p\u003e\n\n\u003cp\u003eTouch-physics has led to the passion for conceiving the world as\ncomposed of billiard balls—a passion which existed already in the\nGreek atomists. We know what it is to bump into people, or to have them\nbump into us; we know that when this happens motion is communicated\nwithout the exercise of volition. Billiard balls exhibit the phenomena\nconcerned in the best form for elementary mathematical manipulation.\nThe way billiard balls move when they hit each other is not at all\nsurprising; on the contrary, in a general way it is such as everyone\nwould expect. If all the world consisted of billiard balls, it would be\nwhat is called \"intelligible\"—\u003ci\u003ei.e.\u003c/i\u003e it would n never surprise\nus sufficiently to make us realize that we do not understand it.\nThe conservation of momentum, which is exemplified in the impacts\nof billiard balls, seemed to give an admirably simple view of the\nwhole occurrence. We can regard momentum as \"quantity of motion,\" and\nsay that in an impact a certain quantity of motion is interchanged\nbetween two bodies, just as nowadays electrons are exchanged when\none body becomes positively electrified and another negatively. This\nview was preferable to that which used force, because it did not seem\nto demand of matter anything even remotely analogous to volition; it\nwas therefore beloved of pre-Newtonian materialism. It has, however,\ncompletely disappeared from modern notions of the structure of matter.\nThe \"atoms\" which are believed to exist—electrons and protons—never\ncome into contact, but move as if they exerted attractions and\nrepulsions at a distance; these, however, are explained as due to\nsomething transmitted through the intervening medium. What has remained\nfrom touch-physics is an objection to \"action at a distance.\" But this\nobjection can hardly be now attributed to an \u003ci\u003ea priori\u003c/i\u003e prejudice;\nit is rather the outcome of experiment. We believe that, when one body\nseems to influence another at a distance, this is either capable of\nbeing explained away, or is attributed to the continuous passage of\nenergy\u003cspan class=\"pagenum\" id=\"Page_163\"\u003e[Pg 163]\u003c/span\u003e across the space between the two bodies; but we believe this\nbecause it is the view which fits best with known facts, not because\nit seems the only \"intelligible\" view. The latter opinion is no doubt\nwidely held, but is not required to justify existing physical theories.\u003c/p\u003e\n\n\u003cp\u003eSight-physics has inevitably been dominant in astronomy, owing to\nthe fact that sight is the only sense by means of which we have\ncognizance of the heavenly bodies. So long as we only see a motion,\nwe are not conscious of anything analogous to force. The fact that\ngravitation remained so long unexplained may have stimulated the\ndesire of theoretical physicists to develop their subject without the\nnotion of \"force\" since the \"force\" of gravitation remained totally\nobscure. Sight-physics also had the advantage that it dealt with a\nwider range of phenomena than were included in dynamics, since it\nincluded everything to do with light. Thus physics came more and more\nto use only such notions as were intelligible in terms of visual data.\nMass, it is true, remained from another order of ideas. Obviously\nthe sensational source of the idea of mass is the feeling of weight.\nBut even mass has gradually yielded. On the one hand, it is less\nfundamental than it formerly seemed; on the other hand, it can be\ninferred from optical data, by the deflection from a straight line\nwhich a body suffers in a known field of force. (Consider methods of\ndetermining the apparent masses of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e particles.)\nSight-physics also makes the relativity of motion much more evident\nthan either of the other kinds. A train exerts force, and a railway\nstation does not, so that, from this point of view, it seems natural\nand right to say that the train is \"really\" moving while the station is\n\"really\" at rest. But from a visual point of view the appearance of the\nstation from the train is exactly correlative to that of the train from\nthe station.\u003c/p\u003e\n\n\u003cp\u003eIn the visual world, quite independently of the velocity of light, a\nrapid movement can be produced by a very small\u003cspan class=\"pagenum\" id=\"Page_164\"\u003e[Pg 164]\u003c/span\u003e \"force\"—for instance,\nby rotating a mirror which is reflecting a bright light. Rotating\nlighthouses at night send out beams which can be seen travelling with\ngreat rapidity. A beam is not a \"thing,\" because it is not tangible,\nand yet, for common sense, it preserves its identity while it rotates.\nBut common sense is not shocked when the beam is broken up into a\nseries of events. A purely visual view of matter makes it much easier\nto regard all material things as series of events, like the rotating\nbeam.\u003c/p\u003e\n\n\u003cp\u003eOf course I am not suggesting that the other senses should be ignored\nas sources of knowledge concerning the physical world. What I am saying\nis that physics has tended, more and more, to interpret the information\nderived from the other senses by means of an imaginative picture\nderived from sight. Perhaps there are reasons for this; indeed, two\nsuggest themselves, one physical and one physiological. Anticipating\nlater discussions, we may say that fairly accurate perception is only\npossible when there is a causal chain, leading from the object to the\nsense-organ, which is to a considerable extent independent of what is\nto be found in the intermediate regions. Whether this is the case or\nnot is a question for physics. Touch is confined to bodies with which\nthe observer is in contact; smell and sound are not diffused very far.\nBut light-waves travel with extraordinarily little modification through\nempty space, and without very great modification through a clear\natmosphere. If we were to accept Professor Lewis\u0027s theory mentioned in\nChapter XIII., we could say that a light-quantum travels unchanged from\na star to a human eye. Even if this theory is not true, the mere fact\nthat it can be seriously proposed illustrates the causal \"purity\" (if I\nmay use such a word) of the passage of light from one body to another.\nThis is the physical merit of sight as a source of knowledge concerning\nthe external world.\u003c/p\u003e\n\n\u003cp\u003eThe other merit is physiological. One kind of physical\u003cspan class=\"pagenum\" id=\"Page_165\"\u003e[Pg 165]\u003c/span\u003e stimulus is\nbetter than another, as a source of information, if less energy is\nrequired to produce a noticeable sensation, and smaller physical\ndifferences are required to produce noticeable differences of\nsensation. In both these respects, light is peculiarly excellent. The\nenergy in the light from a just perceptible star is of the order of\none quantum per cubic metre.\u003ca id=\"FNanchor_37\" href=\"#Footnote_37\" class=\"fnanchor\"\u003e[37]\u003c/a\u003e Very small differences of wave-length\nproduce perceptible differences of colour, and stars are seen as\nseparate even when the angle between the rays from them to the eye\nis very minute. In these respects, sight is markedly the best of the\nsenses. It is therefore not surprising that physics has laid increasing\nstress upon visual data.\u003c/p\u003e\n\n\u003cp\u003eAt the level of common sense, the most important merit of sight is\nthat it makes us aware of objects at a distance. Sound and smell do\nthis to some extent—smell, however, is much more important to certain\nspecies of animals than to us. But neither sound nor smell carry over\ngreat distances, and they do not enable us to locate their source at\nall accurately. If we accept the usual causal theory of perception—as\nI think we should—the proximate physical cause of the physiological\noccurrences leading to a visual perception is not something happening\nin the object which we say we see, but something happening at the\nsurface of the eye. If this is to give us information about the distant\nobject, it must be, in the main, causally determined by the object,\nwithout regard to anything intervening between the object and the\neye. This is the physical merit of sight which we mentioned a moment\nago. It has, of course, very distinct limitations. The colour of the\nlight which reaches the eye will be different from that emitted by the\nobject if there is intervening mist or coloured glass. The direction\ncan be altered by a refracting medium. Mirrors deceive animals and\nyoung children. Then there are more subtle matters, such as the Doppler\neffect and aberration. But\u003cspan class=\"pagenum\" id=\"Page_166\"\u003e[Pg 166]\u003c/span\u003e after making all these allowances, sight\nremains supreme as a method of acquiring knowledge about distant\nobjects.\u003c/p\u003e\n\n\u003cp\u003eIn one respect, sight is defective—namely, in regard to distance. Some\npsychologists argue that depth can be, to a certain extent, perceived\nby sight alone, while others contend that it is wholly derived from\nother data. However that may be, it is certain that sight alone cannot\njudge any but very small distances. No one can distinguish between a\nhundred yards and a hundred miles by sight alone. Infants do not know\nat all, at first, which visual objects are within their grasp and which\nare not. For practical purposes, visual space has only two dimensions,\neven if this is not strictly correct in psychological theory. In\npractice, when we know the \"real\" size of a distant object, say a man\nor a cow, we can judge its distance by its apparent size.\u003ca id=\"FNanchor_38\" href=\"#Footnote_38\" class=\"fnanchor\"\u003e[38]\u003c/a\u003e But our\ninitial experience of distance is derived from the amount of bodily\nmovement required to establish contact. We may only have to stretch\nout an arm, we may have to lean the body, or we may have to walk for\nsome time. An hour\u0027s walk is a natural measure of distance—in fact,\nit is a league. We cannot arrive at the common-sense idea of space\nwithout bringing in movement. And measurement with a measuring rod\ninvolves movement, if the distance to be measured is longer than the\nrod. Of course there is space in our own body, which is known without\nmovement: we refer a headache to the head and a stomachache to the\nstomach. But this space is limited, and does not give spatial relations\nbetween our body and objects merely seen. To acquire a knowledge of\nthese relations, bodily movement is indispensable. And this would never\nhave been available for the purpose if there were not so many objects\nsurrounding us which are motionless relatively to the earth.\u003cspan class=\"pagenum\" id=\"Page_167\"\u003e[Pg 167]\u003c/span\u003e We can\ndiscover the distance of a house by walking to it, but not of a fox by\nthe distance we have to gallop before reaching him.\u003c/p\u003e\n\n\u003cp\u003eScience cannot dispense wholly with postulates, but as it advances\ntheir number decreases. I mean by a postulate something not very\ndifferent from a working hypothesis, except that it is more general: it\nis something which we assume without sufficient evidence, in the hope\nthat, by its help, we shall be able to construct a theory which the\nfacts will confirm. It is by no means essential to science to assume\nthat its postulates are true always or necessarily; it is enough if\nthey are often true. They ought to be so used that, when they are true,\nthey yield verifiable theories, but, when they are not true, \u003ci\u003eno\u003c/i\u003e\ntheory can be framed which will fit the facts—until we find a way of\nworking with different postulates.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_168\"\u003e[Pg 168]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe most important postulate of science is induction. This may be\nformulated in various ways, but, however formulated, it must yield\nthe result that a correlation which has been found true in a number\nof cases, and has never been found false, has at least a certain\nassignable degree of probability of being always true. I propose to\nassume the validity of induction, not because I know of any conclusive\ngrounds in its favour, but because it seems, in some form, essential\nto science and not deducible from anything very different from itself.\nI do not propose to discuss it, because the problem concerns empirical\nknowledge in general, not physics in particular; also because the\nsubject is so complicated that a discussion is useless unless it is\nvery lengthy. For the moment I must refer the reader to Mr Keynes and\nhis critics.\u003ca id=\"FNanchor_39\" href=\"#Footnote_39\" class=\"fnanchor\"\u003e[39]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe other postulates which were at one time thought necessary\nhave gradually been found to be superfluous. At one time, the\nindestructibility of matter would have been regarded as a postulate.\nNow, though electrons and protons are supposed to persist as a rule,\nit is seriously suggested that an electron and a proton may sometimes\ncombine so as to annihilate each other; Eddington has advanced this\nas an important possible source of stellar energy.\u003ca id=\"FNanchor_40\" href=\"#Footnote_40\" class=\"fnanchor\"\u003e[40]\u003c/a\u003e It is true\nthat, in this process, energy is supposed to be not destroyed; but the\nconservation of energy is no more than an empirical generalization, and\nis not thought to be strictly true.\u003c/p\u003e\n\n\u003cp\u003eSpatio-temporal continuity was, until lately, a postulate of science,\nbut the quantum theory has called it in question without intellectual\ndisaster. It \u003ci\u003emay\u003c/i\u003e be true, but we cannot say that it \u003ci\u003emust\u003c/i\u003e\nbe.\u003c/p\u003e\n\n\u003cp\u003eThe existence of causal laws perhaps deserves to rank as a postulate,\nor may perhaps be proved probable, on the existing evidence, if\ninduction is assumed. Here our proviso is relevant, that a postulate\nneed not be supposed to hold universally. We shall assume that there\nare causal laws, and try to discover them; but if none are found in\na given region, that merely means that science cannot conquer that\nregion. There are at present important regions of this kind. We do not\nknow why a radio-active atom disintegrates at one moment rather than\nanother, or why a planetary electron changes its orbit at one moment\nrather than another. We cannot be sure that these occurrences severally\nare governed by laws; but if they are not, science cannot deal with\nthem individually, and is confined to statistical averages. Whether\nthis will prove to be the case, we cannot yet say.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_37\" href=\"#FNanchor_37\" class=\"label\"\u003e[37]\u003c/a\u003e\nJeans, \u003ci\u003eop. cit.\u003c/i\u003e, p. 29.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_38\" href=\"#FNanchor_38\" class=\"label\"\u003e[38]\u003c/a\u003e\nTo show the depth of Dover cliff, Shakespeare says:\u003c/p\u003e\n\n\u003cdiv class=\"poetry-container\"\u003e\n\u003cdiv class=\"poetry\"\u003e\n \u003cdiv class=\"stanza\"\u003e\n \u003cdiv class=\"verse indent0\"\u003e\"The crows and choughs that wing the midway air\u003c/div\u003e\n \u003cdiv class=\"verse indent0\"\u003eShow scarce so gross as beetles.\"\u003c/div\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_39\" href=\"#FNanchor_39\" class=\"label\"\u003e[39]\u003c/a\u003e\n\u003ci\u003eA Treatise on Probability\u003c/i\u003e. By John Maynard Keynes.\nMacmillan, 1920.\u003c/p\u003e\n\n\u003cp\u003e\u003ci\u003eLe Problème Logique de l\u0027Induction\u003c/i\u003e. Par Jean Nicod. Paris,\nAlcan, 1924.\u003c/p\u003e\n\n\u003cp\u003eReview of the above by Braithwaite, \u003ci\u003eMind\u003c/i\u003e, 1925.\u003c/p\u003e\n\n\u003cp\u003e\u003ci\u003eThe Foundations of Probability\u003c/i\u003e. By R. H. Nisbet. Mind, January,\n1926.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_40\" href=\"#FNanchor_40\" class=\"label\"\u003e[40]\u003c/a\u003e\n\u003ci\u003eNature\u003c/i\u003e, May I, 1926, supplement.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_169\"\u003e[Pg 169]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XVII\"\u003eCHAPTER XVII\u003cbr\u003e\nWHAT IS AN EMPIRICAL SCIENCE?\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIT would be generally agreed that physics is an empirical science, as\ncontrasted with logic and pure mathematics. I want, in this chapter, to\ndefine in what this difference consists.\u003c/p\u003e\n\n\u003cp\u003eWe may observe, in the first place, that many philosophers in the past\nhave denied the distinction. Thorough-going rationalists have believed\nthat the facts which we regard as only discoverable by observation\ncould really be deduced from logical and metaphysical principles;\nthorough-going empiricists have believed that the premisses of pure\nmathematics are obtained by induction from experience. Both views\nseem to me false, and are, I think, rarely held in the present day;\nnevertheless, it will be as well to examine the reasons for thinking\nthat there is an epistemological distinction between pure mathematics\nand physics, before trying to discover its exact nature.\u003c/p\u003e\n\n\u003cp\u003eThere is a traditional distinction between necessary and contingent\npropositions, and another between analytic and synthetic propositions.\nIt was generally held before Kant that necessary propositions were\nthe same as analytic propositions, and contingent propositions were\nthe same as synthetic propositions. But even before Kant the two\ndistinctions were different, even if they effected the same division\nof propositions. It was held that every proposition is necessary,\nassertoric, or possible, and that these are ultimate notions, comprised\nunder the head of \"modality.\" I do not think much can be made of\nmodality, the plausibility of which seems to have come from confusing\npropositions with propositional functions.\u003c/p\u003e\n\n\u003cp\u003ePropositions may, it is true, be divided in a way corresponding\u003cspan class=\"pagenum\" id=\"Page_170\"\u003e[Pg 170]\u003c/span\u003e to\nwhat was meant by analytic and synthetic; this will be explained in a\nmoment. But propositions which are not analytic can only be true or\nfalse; a true synthetic proposition cannot have a further property of\nbeing necessary, and a false synthetic proposition cannot have the\nproperty of being possible. Propositional functions, on the contrary,\nare of three kinds: those which are true for all values of the argument\nor arguments, those which are false for all values, and those which are\ntrue for some arguments and false for others. The first may be called\nnecessary, the second impossible, the third possible. And these terms\nmay be transferred to propositions when they are not known to be true\non their own account, but what is known as to their truth or falsehood\nis deduced from knowledge of propositional functions. \u003ci\u003eE.g.\u003c/i\u003e \"it\nis possible that the next man I meet will be called John Smith\" is a\ndeduction from the fact that the propositional function \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a\nman and is called John Smith\" is possible—\u003ci\u003ei.e.\u003c/i\u003e true for some\nvalues of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and false for others. Where, as in this instance, it\nis worth while to say that a \u003ci\u003eproposition\u003c/i\u003e is possible, the fact\nrests upon our ignorance. With more knowledge, we should know who is\nthe next man I shall meet, and then it would be certain that he is John\nSmith or certain that he is not John Smith. Possibility in this sense\nthus becomes assimilated to probability, and may count as any degree of\nprobability other than 0 and 1. An \"assertoric\" proposition, similarly,\nwas, I think, a confused notion applicable to a proposition known to be\ntrue but also known to be a value of a propositional function which is\nsometimes false—\u003ci\u003ee.g.\u003c/i\u003e \"John Smith is bald.\"\u003c/p\u003e\n\n\u003cp\u003eThe distinction of analytic and synthetic is much more relevant to\nthe difference between pure mathematics and physics. Traditionally,\nan \"analytic\" proposition was one whose contradictory was\nself-contradictory, or, what came to the same thing in Aristotelian\nlogic, one which ascribed to a subject a predicate which was part of\nit—\u003ci\u003ee.g.\u003c/i\u003e \"white horses\u003cspan class=\"pagenum\" id=\"Page_171\"\u003e[Pg 171]\u003c/span\u003e are horses.\" In practice, however, an\nanalytic proposition was one whose truth could be known by means of\nlogic alone. This meaning survives, and is still important, although we\ncan no longer use the definition in terms of subject and predicate or\nthat in terms of the law of contradiction. When Kant argued that \"7 +\n5= 12\" is synthetic, he was using the subject-predicate definition, as\nhis argument shows. But when we define an analytic proposition as one\nwhich can be deduced from logic alone, then \"7 + 5 = 12\" is analytic.\nOn the other hand, the proposition that the sum of the angles of a\ntriangle is two right angles is synthetic. We must ask ourselves,\ntherefore: What is the common quality of the propositions which can be\ndeduced from the premisses of logic?\u003c/p\u003e\n\n\u003cp\u003eThe answer to this question given by Wittgenstein in his \u003ci\u003eTractatus\nLogico-Philosophicus\u003c/i\u003e seems to me the right one. Propositions\nwhich form part of logic, or can be proved by lope, are all\n\u003ci\u003etautologies\u003c/i\u003e—\u003ci\u003ei.e.\u003c/i\u003e they show that certain different\nsets of symbols are different ways of saying the same thing, or\nthat one set says part of what the other says. Suppose I say: \"If\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e, then not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e implies not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e.\" Wittgenstein\nasserts that \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e\" and \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e implies not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e\"\nare merely different symbols for one proposition: the fact which\nmakes one true (or false) is the same as the fact which makes the\nother true (or false). Such propositions, therefore, are really\nconcerned with symbols. We can know their truth or falsehood without\nstudying the outside world, because they are only concerned with\nsymbolic manipulations. I should add—though here Wittgenstein might\ndissent—that all pure mathematics consists of tautologies in the above\nsense. If this is true, then obviously empiricists such as J. S. Mill\nare wrong when they say that we believe 2 + 2 = 4 because we have found\nso many instances of its truth that we can make an induction by simple\nenumeration which has little chance of\u003cspan class=\"pagenum\" id=\"Page_172\"\u003e[Pg 172]\u003c/span\u003e being wrong. Every unprejudiced\nperson must agree that such a view \u003ci\u003efeels\u003c/i\u003e wrong: our certainty\nconcerning simple mathematical propositions does not seem analogous to\nour certainty that the sun will rise to-morrow. I do not mean that we\nfeel more sure of the one than of the other, though perhaps we ought to\ndo so; I mean that our assurance seems to have a different source.\u003c/p\u003e\n\n\u003cp\u003eI accept the view, therefore, that some propositions are tautologies\nand some are not, and I regard this as the distinction underlying\nthe old distinction of analytic and synthetic propositions. It is\nobvious that a proposition which is a tautology is so in virtue of its\nform, and that any constants which it may contain can be turned into\nvariables without impairing its tautological quality. We may take as\na stock example: \"If Socrates is a man and all men are mortal, then\nSocrates is mortal.\" This is a value of the general logical tautology:\u003c/p\u003e\n\n\u003cp\u003e\"For all values of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is an\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, and all \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e\u0027s are \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e\u0027s, then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e.\"\u003c/p\u003e\n\n\u003cp\u003eIn logic, it is a waste of time to deal with particular examples of\ngeneral tautologies; therefore constants ought never to occur, except\nsuch as are purely formal. The cardinal numbers turn out to be purely\nformal in this sense; therefore all the constants of pure mathematics\nare purely formal.\u003c/p\u003e\n\n\u003cp\u003eA proposition cannot be a tautology unless it is of a certain\ncomplexity, exceeding that of the simplest propositions. It is obvious\nthat there is more complexity in equating two ways of saying the same\nthing than there is in either way separately. It is obvious also that,\nwhenever it is actually useful to know that two sets of symbols say the\nsame thing, or that one says part of what the other says, that must be\nbecause we have some knowledge as to the truth or falsehood of what is\nexpressed by one of the sets. Consequently logical knowledge would be\nvery unimportant if it stood alone; its importance arises\u003cspan class=\"pagenum\" id=\"Page_173\"\u003e[Pg 173]\u003c/span\u003e through its\ncombination with knowledge of propositions which are not purely logical.\u003c/p\u003e\n\n\u003cp\u003eAll the propositions which are not tautologies we shall call\n\"synthetic.\" The simplest kinds of propositions must be synthetic,\nin virtue of the above argument. And if logic or pure mathematics\ncan ever be employed in a process leading to knowledge that is not\ntautological, there must be sources of knowledge other than logic and\npure mathematics.\u003c/p\u003e\n\n\u003cp\u003eThe distinctions hitherto considered in this chapter have been logical.\nIn the case of modality, it is true, we found a certain confusion from\nan admixture of epistemological notions; but modality was intended to\nbe logical, and in one form it was found to be so. We come now to a\ndistinction which is essentially epistemological, that, namely, between\n\u003ci\u003ea priori\u003c/i\u003e and empirical knowledge.\u003c/p\u003e\n\n\u003cp\u003eKnowledge is said to be \u003ci\u003ea priori\u003c/i\u003e when it can be acquired without\nrequiring any fact of experience as a premiss; in the contrary case,\nit is said to be empirical. A few words are necessary to make the\ndistinction clear. There is a process by which we acquire knowledge of\ndated events at times closely contiguous to them; this is the process\ncalled \"perception\" or \"introspection\"\u003ca id=\"FNanchor_41\" href=\"#Footnote_41\" class=\"fnanchor\"\u003e[41]\u003c/a\u003e according to the character\nof the events concerned. There is no doubt need of much discussion\nas to the nature of this process, and of still more as to the nature\nof the knowledge to be derived from it; but there can be no doubt of\nthe broad fact that we do acquire knowledge in this way. We wake up\nand find that it is daylight, or that it is still night; we hear a\nclock strike; we see a shooting star; we read the newspaper; and so\non. In all these cases we acquire knowledge of events, and the time\nat which we acquire the knowledge is the same, or nearly the same,\nas that at which the events take place. I shall call this process\n\"perception,\"\u003cspan class=\"pagenum\" id=\"Page_174\"\u003e[Pg 174]\u003c/span\u003e and shall, for convenience, include introspection—if\nthis is really different from what is commonly called \"perception.\" A\nfact of \"experience\" is one which we could not have known without the\nhelp of perception. But this is not quite clear until we have defined\nwhat we mean by \"could not\"; for clearly we may learn from experience\nthat 2 + 2 = 4, though we afterwards realize that the experience was\nnot logically indispensable. In such cases, we see afterwards that the\nexperience did not prove the proposition, but merely suggested it, and\nled to our finding the real proof. But, in view of the fact that the\ndistinction between empirical and \u003ci\u003ea priori\u003c/i\u003e is epistemological,\nnot logical, it is obviously possible for a proposition to change from\nthe one class to the other, since the classification involves reference\nto the organization of a particular person\u0027s knowledge at a particular\ntime. So regarded, the distinction might seem unimportant; but it\nsuggests some less subjective distinctions, which are what we really\nwish to consider.\u003c/p\u003e\n\n\u003cp\u003eKant\u0027s philosophy started from the question: How are synthetic \u003ci\u003ea\npriori\u003c/i\u003e judgments possible? Now we must first of all make a\ndistinction. Kant is concerned with \u003ci\u003eknowledge\u003c/i\u003e, not with mere\n\u003ci\u003ebelief\u003c/i\u003e. There is no philosophical problem in the fact that\na man can have a \u003ci\u003ebelief\u003c/i\u003e which is synthetic and not based on\nexperience—\u003ci\u003ee.g.\u003c/i\u003e that this time the horse on which he has put\nhis money will win. The philosophical problem arises only if there is a\nclass of synthetic \u003ci\u003ea priori\u003c/i\u003e beliefs which is always true. Kant\nconsidered the propositions of pure mathematics to be of this kind; but\nin this he was misled by the common opinion of his time, to the effect\nthat geometry, though a branch of pure mathematics, gave information\nabout actual space. Owing to non-Euclidean geometry, particularly as\napplied in the theory of relativity, we must now distinguish sharply\nbetween the geometry applicable to actual space, which is an empirical\nstudy forming part of physics, and the\u003cspan class=\"pagenum\" id=\"Page_175\"\u003e[Pg 175]\u003c/span\u003e geometry of pure mathematics,\nwhich gives no information as to actual space. Consequently this\ninstance of synthetic \u003ci\u003ea priori\u003c/i\u003e knowledge, upon which Kant\nrelied, is no longer available. Other kinds have been supposed to\nexist—for example, ethical knowledge, and the law of causality; but it\nis not necessary for our purposes to decide whether these kinds really\nexist or not. So far as physics is concerned, we may assume that all\nreal knowledge is either dependent (at least in part) upon perception,\nor analytic in the sense in which pure mathematics is analytic. The\nKantian synthetic \u003ci\u003ea priori\u003c/i\u003e knowledge, whether it exists or not,\nseems not to be found in physics—unless, indeed, the principle of\ninduction were to count as such.\u003c/p\u003e\n\n\u003cp\u003eBut the principle of induction, as we have already seen, has its\norigin in physiology, and this suggests a quite different treatment\nof a priori beliefs from that of Kant. Whether there is \u003ci\u003ea priori\nknowledge\u003c/i\u003e or not, there undoubtedly are, in a certain sense, \u003ci\u003ea\npriori beliefs\u003c/i\u003e. We have reflexes which we intellectualize into\nbeliefs; we blink, and this leads us to the belief that an object\ntouching the eye will hurt it. We may have this belief before we have\nexperience of its truth; if so, it is, in a sense, synthetic \u003ci\u003ea\npriori\u003c/i\u003e knowledge—\u003ci\u003ei.e.\u003c/i\u003e it is a belief, not based upon\nexperience, in a true synthetic proposition. Our belief in induction is\nessentially analogous. But such beliefs, even when true, hardly deserve\nto be called knowledge, since they are not all true, and therefore all\nrequire verification before they ought to be regarded as certain. These\nbeliefs have been useful in generating science, since they supplied\nhypotheses which were largely true; but they need not survive untested\nin modern science.\u003c/p\u003e\n\n\u003cp\u003eI shall therefore assume that, at any rate in every department relevant\nto physics, all knowledge is either analytic in the sense in which\nlogic and pure mathematics are analytic, or is, at least in part,\nderived from perception. And all knowledge\u003cspan class=\"pagenum\" id=\"Page_176\"\u003e[Pg 176]\u003c/span\u003e which is in any degree\nnecessarily dependent upon perception I shall call \"empirical.\" I shall\nregard a piece of knowledge as necessarily dependent upon perception\nwhen, after a careful analysis of our grounds for believing it, it is\nfound that among these grounds there is the cognition of an event in\ntime, arising at the same time as the event or very shortly after it,\nand fulfilling certain further criteria which are necessary in order to\ndistinguish perception from certain kinds of error. These criteria will\noccupy us in the next chapter.\u003c/p\u003e\n\n\u003cp\u003eIn a science, there are two kinds of empirical propositions. There are\nthose concerned with particular matters of fact, and those concerned\nwith laws induced from matters of fact. The appearances presented\nby the sun and moon and planets on certain occasions when they have\nbeen seen are particular matters of fact. The inference that the sun\nand moon and planets exist even when no one is observing them—in\nparticular, that the sun exists at night and the planets by day—is an\nempirical induction. Heraclitus thought the sun was new every day, and\nthere was no logical impossibility in this hypothesis. Thus empirical\nlaws not only depend upon particular matters of fact, but are inferred\nfrom these by a process which falls short of logical demonstration.\nThey differ from propositions of pure mathematics both through the\nnature of their premisses and through the method by which they are\ninferred from these premisses.\u003c/p\u003e\n\n\u003cp\u003eIn an advanced science such as physics, the part played by pure\nmathematics consists in connecting various empirical generalizations\nwith each other, so that the more general laws which replace them\nare based upon a larger number of matters of fact. The passage from\nKepler\u0027s laws to the law of gravitation is the stock instance. Each\nof the three laws was based upon a certain set of facts; all three\nsets of facts together formed the basis of the law of gravitation.\nAnd, as usually\u003cspan class=\"pagenum\" id=\"Page_177\"\u003e[Pg 177]\u003c/span\u003e happens in such cases, new facts, not belonging to\nany of the three previous sets, were found to support the new law—for\ninstance, the facts of tides, of lunar motion, and of perturbations.\nEpistemologically, in such cases, a fact is a premiss for a law;\nlogically, most of the relevant facts are consequences of the\nlaw—\u003ci\u003ei.e.\u003c/i\u003e all except those required to determine the constants\nof integration.\u003c/p\u003e\n\n\u003cp\u003eIn history and geography, the empirical facts are, at present, more\nimportant than any generalizations based upon them. In theoretical\nphysics, the opposite is the case: the fact that the sun and moon exist\nis chiefly interesting as affording evidence of the law of gravitation\nand the laws of the transmission of light. In a philosophic analysis\nof physics, we need not consider particular facts except when they\nform the evidence for a theory. It is of course part of the business\nof such an analysis to consider what all particular facts have in\ncommon, and how they come to be known; but such inquiries are general.\nWe are interested in the concept of topography, but not in the actual\ntopography of the universe; at least, we are not interested in it for\nits own sake, but only as affording the evidence for general laws.\u003c/p\u003e\n\n\u003cp\u003eWe have, in view of the above considerations, several different matters\nto consider, before we can return to actual physics. We have first\nto consider the nature and validity of the process we have called\n\"perception\"; next we have to investigate the general character of the\nfacts known by perception; and lastly we have to examine the inference\nfrom facts of perception to empirical laws. After disposing of these\ntopics, we shall resume contact with physics, asking ourselves now, not\nwhat physics asserts, but what justification it has for its assertions,\nand what inessential modifications will increase this justification.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_41\" href=\"#FNanchor_41\" class=\"label\"\u003e[41]\u003c/a\u003e\nI do not wish to prejudice the question whether there is\nsuch a process as \"introspection,\" but only to include it \u003ci\u003eif\u003c/i\u003e it\nexists.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_178\"\u003e[Pg 178]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XVIII\"\u003eCHAPTER XVIII\u003cbr\u003e\nOUR KNOWLEDGE OF PARTICULAR MATTERS OF FACT\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN this chapter, I wish to consider whatever would ordinarily pass\nfor knowledge of particular matters of fact, in so far as this is not\nobtained by a process of deliberate scientific inference. I want to\nconsider this as far as possible independently of the scientific laws\nbased upon it, though not completely without reference to the primitive\nbeliefs by which common sense draws inferences from perceptions. In\nparticular, I wish to abstain from introducing the causal theory of\nperception, unless, on investigation, this should prove impossible. It\nwill be understood that my purpose is epistemological: I am considering\nperception because it is involved in the premisses of empirical\nsciences, not because it is interesting as a mental process. It is of\ncourse necessary to consider its intrinsic character, but we do not do\nthis for its own sake, we do it for the sake of the light that it may\nthrow upon the character and extent of our knowledge.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_179\"\u003e[Pg 179]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eWe are met at the outset by a difficulty due to the fact that\nphilosophical terminology is inappropriate when the views to be\nexpressed are in any way unusual. \"Knowledge\" and \"belief\" both have\nconnotations which are inconvenient for the purpose I have in view.\nThey are both commonly applied in orthodox psychology to something\nconscious and explicit, such as is, or may be, already expressed in\nwords. For our purposes, it is desirable to include more primitive\noccurrences, such as may be supposed to exist in animals. Obviously a\nbird can see an approaching man, and fly away in consequence. I wish to\ninclude under \"perception\" what happens in the bird, and also to say\nthat the bird \"knows\" something when it sees a man, though I shall not\nventure to say what it knows.\u003c/p\u003e\n\n\u003cp\u003eBut at this point a good deal of caution is necessary. My knowledge of\nthe bird is part of my knowledge of the external world, and is partly,\nif not wholly, physical knowledge. Therefore when I am asking: how do\nI know about the physical world? I have no right to begin by comparing\nmy knowledge with that of a bird. I must start from myself and my own\ncognitions, and use the bird only to suggest hypotheses. This caution\napplies also to what was said in Chapter XV.\u003c/p\u003e\n\n\u003cp\u003eAgain, there is always a danger, in epistemology, of putting the\nless certain before the more certain. My knowledge of the process of\nperceiving is less certain, and less primitive, than my knowledge\nof percepts. When I say, \"I know that I have just heard a clap of\nthunder,\" I am saying something not so indubitable as when I say,\n\"There has just been a clap of thunder.\" It is facts of this latter\nkind that are required as premisses in physics. A man might be\ncompletely competent as a physicist if he knew such propositions as\n\"There has just been a clap of thunder\" even if he knew no propositions\nsuch as \"I know that there has just been a clap of thunder.\" The\nconsideration of our knowing, as opposed to what we know, is forced\non us by the fact that what we think we know sometimes turns out to\nbe false; if this were not the case, an analysis of matter need not\nconsider our knowing at all. As it is the case, we are compelled\nto examine our knowing, as well as what we know, with a view to\ndiscovering, if possible, how to minimize the risk involved in taking\nas knowledge what, on reflection, we still believe to be knowledge.\u003c/p\u003e\n\n\u003cp\u003eWe are often urged to adopt an artificial naivete in investigating\nproblems concerning what we know; if we do not do so, we are accused\nof the \"psychologist\u0027s fallacy.\" Now in certain problems this caution\nis quite proper, but in others it is not. My problem is: What do\nI, here and now, know about the external world, and how do I know\nit? It is obvious that my knowledge of the external world cannot be\ndependent\u003cspan class=\"pagenum\" id=\"Page_180\"\u003e[Pg 180]\u003c/span\u003e upon (say) how long it takes a fish to learn to recognize\nthe man who feeds it, since this supposes that I know all about the\nfish and the man and the feeding. Facts about the perceptions of\nbabies, such as we considered in Chapter XV., come under the same\nhead. Long before I can know that there are babies, I must know many\nother things about the external world. I want to start from what comes\nepistemologically first in \u003ci\u003emy\u003c/i\u003e existing knowledge \u003ci\u003enow\u003c/i\u003e; and\nin this problem, obviously, I cannot assume that I already know all\nabout the experiences of animals and babies. There must therefore be no\nartificial naivete, but a straightforward investigation of my knowledge\nas I find it.\u003c/p\u003e\n\n\u003cp\u003eThe position may be illustrated by Chuang-Tze\u0027s story of the two\nphilosophers on the bridge. The first says: \"See how the little fishes\nare darting about. Therein consists the pleasure of fishes.\" The second\nreplies: \"How do you, not being a fish, know wherein consists the\npleasure of fishes?\" To which the first retorts: \"How do you, not being\nI, know that I do not know wherein consists the pleasure of fishes?\" My\nposition is that of the second philosopher. If other philosophers know\n\"wherein consists the pleasure of fishes,\" I congratulate them; but I\nam not thus gifted.\u003c/p\u003e\n\n\u003cp\u003eWhen I try to disentangle the primitive from the inferred elements in\nwhat I take to be my knowledge, I find that the task is not really\nvery difficult, except in certain niceties. The primitive part seems\nsomething like this: There are coloured shapes which move, there\nare noises, smells, bodily sensations, the experiences which we\ndescribe as those of touch, and so on. There are relations among these\nitems: time-relations (earlier and later) among all of them, and\nspace-relations (up-and-down, right-and-left, and the relations by\nwhich localization in the body is effected) among many of them. There\nare recollections of some of these things; this seems indubitable,\nalthough it is not easy to say in what a\u003cspan class=\"pagenum\" id=\"Page_181\"\u003e[Pg 181]\u003c/span\u003e recollection consists, or\nhow it is related to what it recollects. There are also expectations;\nby this I mean something just as immediate as memory. Everyone knows\nthe story of the Orangeman who fell off a scaffolding and murmured\nas he fell: \"To Hell with the Pope, and now for the—bump.\" He was\nexperiencing expectation in the sense in which I mean it. Of thoughts\nother than memories and expectations, it is not necessary to take\naccount when our sole purpose is to reach the primitive basis of our\nknowledge of matter.\u003c/p\u003e\n\n\u003cp\u003eIn the above account, I have omitted many things which I formerly\n\"knew,\" and which, apparently, most other people \"know.\" I have omitted\n\"objects.\" In former days, my apparatus of non-inferential knowledge\nincluded tables and chairs and books and persons and the sun and moon\nand stars. I have come to regard these things as inferences. I do not\nmean that I inferred them formerly, or that other people do so now. I\nfully concede that I did not infer them. But now, as the result of an\nargument, I have become unable to accept the knowledge of them as valid\nknowledge, except in so far as it can be inferred from such knowledge\nas I still consider epistemologically primitive.\u003c/p\u003e\n\n\u003cp\u003eThe argument in question would naturally, but not validly, express\nitself in terms of the causal theory of perception. What I see—so\nit might be urged—is causally dependent upon the light waves that\nreach my eye, and these waves might be reflected or refracted in such\na way as to deceive me concerning their source. This way of stating\nthe argument is invalid because it assumes more knowledge of the\nphysical world than we have any right to assume at our present level.\nBut the facts upon which it relies can be easily made available,\nwithout any undue assumption of knowledge, for the purpose of proving\nour conclusion. In certain cases in which we seem to have immediate\nknowledge of objects, we find ourselves surprised by something\ntotally unexpected. The dog listening\u003cspan class=\"pagenum\" id=\"Page_182\"\u003e[Pg 182]\u003c/span\u003e to \"his master\u0027s voice\" on the\ngramophone may serve as an illustration. He thinks he perceives his\nmaster, but in fact he only perceives a noise. In restaurants which\nwish to look larger than they are, one whole wall sometimes consists\nof looking-glass, and it is easy to suppose that one perceives diners\nat tables, when in fact they are mere reflections. Perspective can\nbe made to deceive. When I say \"deceive,\" in this connection, I\nmean \"rouse expectations which are not fulfilled.\" It is useless to\nmultiply examples. The upshot is that what seems like perception of\nan object is really perception of certain sensible qualities together\nwith expectations of other sensible qualities—the commonest case\nbeing something visual which rouses tactual expectations. It is found\nthat the occasional deceptive experiences are not, in themselves,\ndistinguishable from those that are not deceptive. Hence we conclude\nthat we have to do with a correlation which is usual but not\ninvariable, and that, if we wish to construct an exact science, we must\nbe sceptical of the associations which experience has led us to form,\nconnecting sensible qualities with others with which they are often but\nnot always combined.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_183\"\u003e[Pg 183]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe above argument is based upon principles which common sense can be\nbrought to accept, and has a conclusion which physics has accepted,\nthough perhaps without fully realizing its scope. The argument is not\n\"philosophical,\" in the sense of coming from a region quite different\nfrom that of science and ordinary knowledge. It proceeds merely on the\nusual principle of trying to substitute something more accurate for a\nbelief which has been found to lead to error on occasion. It has as a\nconsequence that \"matter,\" in physics and in philosophy, if legitimate\nat all, cannot be altogether identified with the common-sense notion\nof a material object, though it will have a certain connection with\nthis notion, since the common-sense belief in material objects does not\n\u003ci\u003eusually\u003c/i\u003e lead to false expectations.\u003c/p\u003e\n\n\u003cp\u003eSome misunderstandings must be guarded against as regards expectation\nand error. Neither of these is primarily intellectual; I should be\ninclined to say that both are primarily muscular—or, we may say,\nnervous, in order not to seem paradoxical. Suppose you set to work to\nlift a watering-can: you may adjust your muscles in the way appropriate\nif the can is full, or in the way appropriate if it is empty. If\nthey are adjusted to a full can when the can is empty, you receive a\nshock of surprise on experiencing the lightness of the can. You would\ndescribe your experience by saying, \"I thought the can was full of\nwater.\" But as a rule, in such situations, there has not been anything\nthat could be called \"thought\"; there has been physiological adjustment\nas a result of a stimulus. Of course there may have been \"thought\";\nand whatever \"thought\" may be, it certainly can produce the kind of\nmuscular effects which we are considering. But these effects can be\nproduced more directly, and usually are. There is so little essential\ndifference between a process involving \"thought\" and one not involving\nit that it seems a mistake to confine the notions of truth and error\nto intellectual processes; they ought rather, it seems to me, to be\napplied to the complete reaction of a person to a situation, in which\n\"thought\" is only one element. But it will not do, at our present\nlevel, to introduce physiology, since we are considering how we know\nabout matter, and must not therefore assume that we already know about\nthe matter in our own body. However, the phenomena are easily described\nin the way which our problem demands. In the case of the watering-can,\nthe vivid part of the experience is the surprise. But by means of\nattention a number of other elements can be observed. We can observe\nthe feelings which are interpreted as meaning muscular adjustment to a\nheavy load; we can observe the visual appearance described as the can\ncoming up with a jerk; we can observe the sudden change in what, for\nshort, we may call muscular feelings. It is impossible\u003cspan class=\"pagenum\" id=\"Page_184\"\u003e[Pg 184]\u003c/span\u003e to describe all\nthis without circumlocution, since the natural words to use presuppose\nphysiology; but it is clear that there is a great deal that can be\ndirectly observed, without invoking any theory. In such a process, what\ncomes earlier may be described as \"error\" because of the emotion of\nsurprise which follows. Where the activity which has been begun runs\nits course without leading to this emotion, we shall say that there is\nnot error. I hesitate to ascribe \"truth\" to something pre-intellectual,\nbut at any rate we may say that there is \"correctness,\" or that what\nhas succeeded to the sensation (or perception) which came at the\nbeginning of the process has been \"correct.\" We may shorten this by\nsaying that the response to a stimulus may be \"correct\" or \"erroneous.\"\nBut the longer phrase has the merit of not assuming so much knowledge\nof causal relations.\u003c/p\u003e\n\n\u003cp\u003eIn the situations to which the above analysis applies, we have the\nadvantage of a perfectly definite criterion of correctness or error.\nThe feeling of surprise marks error, and the absence of this feeling\nmarks correctness. It must not be supposed that we have normally an\nexplicit prevision, still less an explicit inference; all that can be\nsaid is that we are in such a condition that one sort of event will\ncause surprise while another sort will not. Consider the experience\nwe have all had, of \"thinking\" we were at the bottom of a staircase\nwhen in fact there was another step to go down. In such a case, when\nwe \"think\" we are at the bottom, we do not think at all, for if we\ndid we should not make such a silly mistake. Indeed, we might say (or\nan Irishman might): \"I thought I was at the bottom because I wasn\u0027t\nthinking.\"\u003c/p\u003e\n\n\u003cp\u003eIt is fairly clear that all our elementary intellectual processes\nhave pre-intellectual analogues. The analogue of a general causal\nbelief is a reflex or a habit. A dog goes to the dining-room when he\nhears the dinner-bell, and so do we. In the case of the dog, it is\neasy to suppose that he has merely\u003cspan class=\"pagenum\" id=\"Page_185\"\u003e[Pg 185]\u003c/span\u003e acquired a habit, without having\nformulated the induction: \"Dinner-bells are a cause, or an effect,\nor an indispensable part of the cause, of dinner.\" We, however, can\nformulate this induction, and we shall then suppose that it is because\nwe have done so that we go into the dining-room when we hear the\nbell. In fact, however, we may be just as merely habitual as the dog.\nThe elementary inductions of common sense are first habits, and only\nsubsequently beliefs. We may say that if, in our experience, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is\naccompanied by \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e either often or in some emotionally important\nmanner, this fact causes first a habit which would be rational if \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nwere always accompanied by \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and then a belief that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is\nalways accompanied by \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e—the latter being a rationalization of the\npre-existing habit.\u003c/p\u003e\n\n\u003cp\u003eGeneral propositions may thus form part of our thinking from the start.\nSuch general propositions are merely the verbal expression of habits.\nThe hand-eye co-ordination becomes firmly fixed as a motor habit,\nand then, when we think, we conclude that what can be seen can often\nbe touched—in fact, that it can be touched in circumstances which\nwe know in practice, though we might have difficulty in formulating\nthem exactly. Such general propositions are synthetic, and are in a\ncertain sense \u003ci\u003ea priori\u003c/i\u003e; for, though experience has \u003ci\u003ecaused\u003c/i\u003e\nthem, they are not obtained by inference from other propositions, but\nby rationalizing and verbalizing our habits; that is to say, their\nantecedents are pre-intellectual. The trouble with them is that they\nare never quite right. Common sense, do what it will, cannot avoid\nbeing surprised occasionally. The object of science is to spare it\nthis emotion, and create mental habits which shall be in such close\naccord with the habits of the world as to secure that nothing shall be\nunexpected. Science has, of course, not yet achieved its ideal: the\nGreat War and the earthquake of Tokyo took people by surprise. But it\nis hoped that in time such events will no longer\u003cspan class=\"pagenum\" id=\"Page_186\"\u003e[Pg 186]\u003c/span\u003e disturb us, because\nwe shall have expected them. However, I do not wish at this stage\nto consider our knowledge of general propositions; it is particular\nmatters of fact that concern us at present.\u003c/p\u003e\n\n\u003cp\u003eAlthough, in our less intellectual moods, we act as the result of a\nsensation without stopping to think (\u003ci\u003ee.g.\u003c/i\u003e when we blink because\nwe see something approaching the eye), yet we can, when we choose,\nreact to a stimulus in the way which is called \"knowing\" it, and we\noften react involuntarily in this way. It is not necessary, in an\nanalysis of matter, to decide what \"knowing\" is; it is only necessary\nto decide what is known, in so far as this is relevant to our knowledge\nof physics. The list which I gave earlier in the present chapter was\ndesigned to be such as would exclude the risk of error, using \"error\"\nin the sense which I have been defining. Common sense is liable to\nerr—of this we have already given instances. We cannot therefore\ninclude the common-sense notion of an \"object\" or \"thing\" as part of\nwhat we know. But the sensible qualities which can be analyzed out of\nthe \"thing\" can be admitted without ever leading us into error. These,\ntherefore, are to be accepted as genuinely known.\u003c/p\u003e\n\n\u003cp\u003eIt is a remarkable fact that all such knowledge, when not inferential,\narises at about the same time as what is known, though it may survive\nfor an indefinite time in the form of memory. This is the essential\npeculiarity, which we mentioned earlier, that distinguishes the\nempirical premisses of empirical knowledge. These consist of facts\nwhich become known spontaneously at about the time when they occur, and\ncannot be known sooner except by elaborate and more or less doubtful\ninferences from other such facts. The process of getting to know such\nfacts without inference is called \"perception,\" and knowledge derived\nwholly or partly from perception is said to be based on experience.\nA Greek could know the multiplication table as well as we do, but he\ncould not know the biography of Napoleon.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_187\"\u003e[Pg 187]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XIX\"\u003eCHAPTER XIX\u003cbr\u003e\nDATA, INFERENCES, HYPOTHESES, AND THEORIES\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nWHEN a man of science speaks of his \"data,\" he knows very well in\npractice what he means. Certain experiments have been conducted, and\nhave yielded certain observed results, which have been recorded.\nBut when we try to define a \"datum\" theoretically, the task is\nnot altogether easy. A datum, obviously, must be a fact known by\nperception. But it is very difficult to arrive at a fact in which there\nis no element of inference, and yet it would seem improper to call\nsomething a \"datum\" if it involved inference as well as observation.\nThis constitutes a problem which must be briefly considered.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_188\"\u003e[Pg 188]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eWhat is recorded as the result of an experiment or observation is never\nthe bare fact perceived, but this fact as interpreted by the help of\na certain amount of theory. Take, say, the eclipse observations by\nwhich Einstein\u0027s theory of gravitation was confirmed. What in fact was\ngiven in perception was—apart from the previous arrangements—a visual\npattern of dots, interpreted as a photograph of stars near the sun; a\ntactual-visual experience called \"measuring,\" and finally coincidences\nof certain visual appearances with certain others called \"numbers on a\nscale.\" At least, whether this is actually a correct account or not,\nit represents the sort of thing that occurred. A considerable amount\nof theory was involved in merely measuring the photographs. And of\ncourse a vast structure was involved in interpreting the photographs\nas photographs of stars, and in inferring thence the course which the\nlight from the stars had pursued. It is the theoretical element in\nmeasuring the photographs that most needs to be stressed, since it is\neasily overlooked.\u003c/p\u003e\n\n\u003cp\u003eIt is sometimes maintained that there is something of the nature of\ninference at an even earlier stage. The effects of a given sensory\nstimulus upon two men with indistinguishable sense-organs but different\nexperiences may be very different. The most obvious illustration is\nthe effect of print upon a man who can read and upon a man who cannot.\nA child learning to read is aware of each letter in turn as a certain\nshape, and finally arrives, with pain and labour, at the word. A man\nwho learned to read as a child is quite unconscious of the letters,\nunless he is interested in typography or looking out for misprints;\nnormally, he passes straight to the words, and to the words as having\nmeaning, not as black marks on white paper. Nevertheless, he is very\nlikely to notice an oddity at once—say if someone omitted the \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e\nin \"Nietzsche.\" In writing to a philosopher to ask for a testimonial,\nit would be very unsafe to assume that he would not detect an error\nof this sort. But the detection of the error is due to the element of\nsurprise: the philosopher is expecting a \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e and has a shock when\nit is not there, like that of a man who has reached the bottom of a\nstaircase but thinks there is another step. The philosopher\u0027s body was\nexpecting a \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, though his mind was otherwise occupied.\u003c/p\u003e\n\n\u003cp\u003eA more orthodox illustration is the difference between the effect of\na visual stimulus upon an ordinary man and upon a man born blind but\nenabled to see as the result of an operation. The latter has not the\ntactual associations of the ordinary man, and cannot \"interpret\" what\nhe sees. Are we to include in perception this element of unconscious\ninterpretation, or are we to include only what we imagine that the\nsame stimulus would have produced if there had been no such previous\nexperience as would make interpretation possible? This is not an\naltogether easy question. On the one hand, the interpretation depends\nupon correlations which are frequent but probably not invariable, so\nthat, if it is included,\u003cspan class=\"pagenum\" id=\"Page_189\"\u003e[Pg 189]\u003c/span\u003e it might seem as though perception would\nsometimes contain an element of error. On the other hand, the element\nof interpretation can only be eliminated by an elaborate theory, so\nthat what remains—the hypothetical bare \"sensation\"—is hardly to be\ncalled a \"datum,\" since it is an inference from what actually occurs.\nThis last argument is, to my mind, conclusive. Perception must include\nthose elements which are irreducibly physiological, but it need not on\nthat account include those elements which come, or can be made to come,\nwithin the sphere of conscious inference. When we hear (say) a donkey\nbraying, we are quite conscious of inference from the noise of the\ndonkey, or at any rate we can easily become conscious of it. I should\nnot, therefore, in this case, include anything else of the donkey with\nthe perception, but only the noise. And if you see a donkey, though you\nmay have reactions connected with the sense of touch, these are never\nconfounded with what you feel when you actually touch him. I should\ntherefore say that a great deal of the interpretation that usually\naccompanies a perception can be made conscious by mere attention, and\nthat this part ought not to be included in the perception. But the\npart which can only be discovered by careful theory, and can never be\nmade introspectively obvious, ought to be included in the perception.\nPerhaps the line between the two is not so sharp as could be wished;\nbut I do not see how else to meet the conflicting considerations which\npresent themselves.\u003c/p\u003e\n\n\u003cp\u003eWe have still to ask ourselves whether perception, so defined, will\nsometimes contain an element of error. Here we must distinguish. It may\nbe, and often is, accompanied by expectations which are disappointed;\nand we agreed to take this as the mark of error. But the expectations\ncan be distinguished from the perception, although in practice this may\nnot always be easy. The tactual accompaniments of visual perceptions\nare of the nature of expectations. There are no\u003cspan class=\"pagenum\" id=\"Page_190\"\u003e[Pg 190]\u003c/span\u003e such accompaniments\nof perceptions of the heavenly bodies. I think that in all cases in\nwhich error occurs it is easy to distinguish the erroneous expectation\nfrom the perception. Whatever \"interpretation\" does not involve\nexpectations need not be regarded as erroneous. It is supposed that\nindistinguishable stimuli may fall upon indistinguishable sense-organs,\nand yet result in distinguishable perceptions because of differences in\nthe brains of the two percipients—these differences in their brains\nbeing the result of different experiences. But there is not on that\naccount anything erroneous in the perception of either. A different\nevent occurs in the one from that which occurs in the other; but\neach event really occurs. This topic,, however, cannot be adequately\ndiscussed until we come to the causal theory of perception and the\nrelation between perception and physical stimulus.\u003c/p\u003e\n\n\u003cp\u003eI come now to the question of inferences, which has already been\ntouched on. As we have seen, there is a purely physiological form of\ninference which belongs to an earlier stage than explicit inference,\nthough it persists in the habits of even the most sophisticated\nphilosopher, such as Hume. The next stage is where there is an\nactual passage from one belief to another, but the passage is a mere\noccurrence, not a transition motived by an argument. In this case, the\ntransition is usually caused by a physiological inference. Then there\nis inference based upon some belief; but even then the belief may be\nwholly irrational, or it may not logically warrant the inference, which\nis the case of fallacious reasoning. Lastly, there is valid inference\nby means of a true principle—but of this I cannot give an indubitable\ninstance.\u003c/p\u003e\n\n\u003cp\u003eIn historical fact, these types of inference emerge successively, but\na later type does not cause an earlier one to disappear. Moreover,\nthe later type tends to be adapted to the earlier. First we have\nphysiological inference: this is exemplified when a bird flies so\nas not to bump into solid\u003cspan class=\"pagenum\" id=\"Page_191\"\u003e[Pg 191]\u003c/span\u003e objects, and fails when it bumps into a\nwindow-pane. Then there is the transition from the belief expressing\nthe premiss of the physiological inference to that expressing\nits conclusion, without any consciousness of how the transition\nis effected. Then there is belief in a causal law which is the\nintellectualized expression of the habit embodied in the physiological\ninference. And last of all there is the search for criteria by which to\ndistinguish between true and false causal laws, these criteria being\nintellectual, not mere habits of the body. This last stage is only\nreached when we come to science.\u003c/p\u003e\n\n\u003cp\u003eOne of the main purposes of scientific inference is to justify beliefs\nwhich we entertain already; but as a rule they are justified with a\ndifference. Our pre-scientific general beliefs are hardly ever without\nexceptions; in science, a law with exceptions can only be tolerated\nas a makeshift. Scientific laws, when we have reason to think them\naccurate, are different in form from the common-sense rules which have\nexceptions: they are always, at least in physics, either differential\nequations, or statistical averages. It might be thought that a\nstatistical average is not very different from a rule with exceptions,\nbut this would be a mistake. Statistics, ideally, are accurate laws\nabout large groups; they differ from other laws only in being about\ngroups, not about individuals. Statistical laws are inferred by\ninduction from particular statistics, just as other laws are inferred\nfrom particular single occurrences. All this, however, is by the way;\nthe point is that inference as a practice has a long history before it\nbecomes scientific.\u003c/p\u003e\n\n\u003cp\u003eThe most important inference which science takes over from common\nsense is inference to unperceived entities. One form in which common\nsense makes this inference is that of a belief that objects which\nhave been perceived still exist when they are not perceived. If, at\na dinner-party, the electric light suddenly goes out, no one doubts\nthat his neighbours and the\u003cspan class=\"pagenum\" id=\"Page_192\"\u003e[Pg 192]\u003c/span\u003e dinner-table and the food and drink still\nexist, although at the moment they are unperceived. When the light\ngoes on again, this belief appears to be confirmed; if there are fewer\nspoons than before, we do not infer that they have ceased to exist,\nbut that someone present is a thief. This belief in the permanence\nof perceived objects has gone through all stages from physiological\ninference to advanced scientific or philosophical theory; the inquiry\ninto its justification is the central problem in the analysis of\nmatter, philosophically considered. No one, not even Berkeley, has\ntreated it with quite the seriousness that it deserves, because the\nphysiological inference is so irresistible that it is difficult to\nachieve a purely intellectual attitude towards the problem. This\ninference is the source of the philosophical notion of \"substance\" and\nthe physical notion of \"matter.\" For the present, I am only noting the\ninferences to be considered; I am not attempting to investigate their\nvalidity.\u003c/p\u003e\n\n\u003cp\u003eUnperceived entities are also inferred by common sense when it believes\nthat other people have \"minds.\" I wish to make it clear that even the\nmost rigid behaviourist makes this inference, although in a slightly\ndifferent form. Dr Watson, for example, would admit that his own\ntoothache can lead him to say, \"I have a toothache,\" whereas another\nperson\u0027s toothache will not lead him to say \"You have a toothache\"\nwithout some intermediate link. Whatever may be our analysis of\n\"knowledge,\" we certainly know things about our own bodies in ways\nwhich are not open to us where other people\u0027s bodies are concerned.\nThere is nothing mysterious about this: it is analogous to the fact\nthat some sounds are within earshot while others are not. The point is\nthat we infer, from the behaviour of others, the existence of things\n(such as toothaches) which we cannot perceive. Whether we say that\nthese things are \"mental\" or \"bodily\" makes no difference to the fact\nthat we make inferences.\u003cspan class=\"pagenum\" id=\"Page_193\"\u003e[Pg 193]\u003c/span\u003e These inferences, also, are at first purely\nphysiological.\u003c/p\u003e\n\n\u003cp\u003eFrom the point of view of physics, the inference to other people\u0027s\n\"minds\" has a twofold importance. The first, which is not specially\nphysical, is concerned with testimony. What is commonly accepted as the\nexperimental evidence on any topic of physics includes not only what a\ngiven physicist has himself observed, but whatever has been reliably\nrecorded. Everything that we learn from what other people say and write\ninvolves inference from something perceived (spoken or written words)\nto something unperceived—namely, the \"mental\" events of the speaker\nor writer. It may be that the primary inference is only to another\nperson\u0027s percepts, but it is none the less an inference to something\nwhich we do not perceive. The second point about the inference to other\npeople\u0027s percepts is specially physical; it concerns the fact that\ndifferent people live in a common world. The percepts of two different\npeople, if we accept testimony, are found to be often very similar,\nthough not exactly alike; this leads to the theory of a common external\ncause—\u003ci\u003ei.e.\u003c/i\u003e to the causal theory of perception, and to the\ndivision of the qualities of the perceived object into such as belong\nto the external cause and such as are supplied by the body or mind of\nthe percipient.\u003c/p\u003e\n\n\u003cp\u003eThe development of science out of common sense has not been by way of\na radically new start at any moment, but rather by way of successive\napproximations. That is to say, where some difficulty has arisen which\ncurrent common sense could not solve, a modification has been made\nat some point, while the rest of the common-sense view of the world\nhas been retained. Subsequently, using this modification, another\nmodification has been introduced elsewhere; and so on. Thus science has\nbeen an historical growth, and has assumed, at each moment, a more or\nless vague background of theory derived\u003cspan class=\"pagenum\" id=\"Page_194\"\u003e[Pg 194]\u003c/span\u003e from common sense. This is one\ndifference between science and philosophy: philosophy attempts, though\nnot always successfully, to set out its inferences in a form which\nassumes nothing on the mere ground that it has always been assumed\nhitherto. It may be doubted whether science can retain its vitality if\nit is severed from its root in our animal habits; when set forth quite\nabstractly, it loses plausibility. Induction, for example, is difficult\nto justify, and yet indispensable in science. In such cases, I shall\nallow myself to accept what seems necessary on pragmatic grounds, being\ncontent, as science is, if the results obtained are often verifiably\ntrue and never verifiably false. But wherever a principle is accepted\non such grounds as these, the fact should be noted, and we should\nrealize that there remains an intellectual problem, whether soluble or\nnot.\u003c/p\u003e\n\n\u003cp\u003eThe actual procedure of science consists of an alternation of\nobservation, hypothesis, experiment, and theory. The only difference\nbetween a hypothesis and a theory is subjective: the investigator\nbelieves the theory, whereas he only thinks the hypothesis sufficiently\nplausible to be worth testing. A hypothesis should accord with all\nknown relevant observations, and suggest experiments (or observations)\nwhich will have one result if the hypothesis is true, and another\nif it is false. This is an ideal: in actual fact, other hypotheses\nwill always exist which are compatible with what is meant to be an\n\u003ci\u003eexperimentum crucis\u003c/i\u003e. The crucial character can only be as\nbetween two hypotheses, not as between one hypothesis and all the\nrest. When a hypothesis has passed a sufficient number of experimental\ntests, it becomes a theory. The argument in favour of a theory is\nalways the formally invalid argument: \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e\nis true, therefore \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e is true.\" Here \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e is the theory, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e\nis the observed relevant facts. We are most impressed when \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e is\nvery improbable \u003ci\u003ea priori\u003c/i\u003e. For example,\u003ca id=\"FNanchor_42\" href=\"#Footnote_42\" class=\"fnanchor\"\u003e[42]\u003c/a\u003e\u003cspan class=\"pagenum\" id=\"Page_195\"\u003e[Pg 195]\u003c/span\u003e observation gives\nRydberg\u0027s constant as:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 22.863ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-356.png\" alt=\"\" data-tex=\"\\[\nR= 1·09678.10^{5} cm^{-1},\n\\]\"\u003e\u003c/span\u003e\nwhile Bohr\u0027s theory gives:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 19.469ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-357.png\" alt=\"\" data-tex=\"\\[\nR= 1·09.10^{5} cm^{-1},\n\\]\"\u003e\u003c/span\u003e\nwhich is within the degree of accuracy to be expected if the theory\nis right. Numerical confirmations of this kind are always the most\nstriking. Nevertheless, even they must be received with caution; Bohr\u0027s\ntheory of circular orbits required modification by the admission of\nelliptic orbits, and thus turned out to be not the only theory which\nwould give a correct value of Rydberg\u0027s constant.\u003c/p\u003e\n\n\u003cp\u003eWhen a theory fits a number of facts, but goes slightly astray in\nregard to certain others, it happens generally, though not always,\nthat it can be absorbed, by a slight modification, into a new theory\nwhich includes the hitherto discrepant facts. There are exceptions, of\nwhich the theory of relativity is perhaps the most notable: here an\nimmense theoretical reconstruction was required to account for very\nminute discrepancies. But in general a partially successful theory is\nan essential step towards its successor. And a result deduced from\na hitherto successful theory is more likely to be right than the\ntheory is: the theory is only right if \u003ci\u003eall\u003c/i\u003e its consequences\nare true (at least, so far as they can be tested), but a verifiable\nconsequence of the theory is likely to be true if \u003ci\u003emost\u003c/i\u003e of the\nverifiable consequences are true. That is why the practical value of\nscientific theories is so much greater than their philosophic value as\ncontributions to ultimate truth. To some extent, we can distinguish,\namong the consequences of a theory, which are the most reliable; they\nwill be those in the region of the facts which have given rise to the\ntheory. No one is surprised to find that an empirical law connecting\nspecific heat with temperature fails for temperatures much lower than\nthose for which it has been found to be correct; but if, in the middle\nof these latter, there was found to be a small range of temperatures\nwhere the law failed, we should be very much surprised. Thus there\nis a kind of common sense to be used in applying theories: some\napplications can be made with confidence, while others will be felt to\nbe questionable.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_42\" href=\"#FNanchor_42\" class=\"label\"\u003e[42]\u003c/a\u003e\nSommerfeld, \u003ci\u003eop. cit.\u003c/i\u003e, p. 217.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_197\"\u003e[Pg 197]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XX\"\u003eCHAPTER XX\u003cbr\u003e\nTHE CAUSAL THEORY OF PERCEPTION\u003ca id=\"FNanchor_43\" href=\"#Footnote_43\" class=\"fnanchor\"\u003e[43]\u003c/a\u003e\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nCOMMON sense holds—though not very explicitly—that perception reveals\nexternal objects to us directly: when we \"see the sun,\" it is the sun\nthat we see. Science has adopted a different view, though without\nalways realizing its implications. Science holds that, when we \"see\nthe sun,\" there is a process, starting from the sun, traversing the\nspace between the sun and the eye, changing its character when it\nreaches the eye, changing its character again in the optic nerve and\nthe brain, and finally producing the event which we call \"seeing the\nsun.\" Our knowledge of the sun thus becomes inferential; our direct\nknowledge is of an event which is, in some sense, \"in us.\" This\ntheory has two parts. First, there is the rejection of the view that\nperception gives direct knowledge of external objects; secondly, there\nis the assertion that it has external causes as to which something can\nbe inferred from it. The first of these tends towards scepticism; the\nsecond tends in the opposite direction. The first appears as certain\nas anything in science can hope to be; the second, on the contrary,\ndepends upon postulates which have little more than a pragmatic\njustification. It has, however, all the merits of a good scientific\ntheory—\u003ci\u003ei.e.\u003c/i\u003e its verifiable consequences are never found to\nbe false. Epistemologically, physics might be expected to collapse\nif perceptions have no external causes; therefore the matter must be\nexamined before we can go further.\u003c/p\u003e\n\n\u003cp\u003eWe must first give somewhat more precision to the common-sense\u003cspan class=\"pagenum\" id=\"Page_198\"\u003e[Pg 198]\u003c/span\u003e view\nwhich is rejected by the causal theory. We have to ask what is meant by\n\"external objects.\" One would naturally say \"spatially external.\" But\n\"space\" is very ambiguous: in visual space, the objects which we see\nare mutually external, and objects other than the visual appearances of\nparts of our own body are spatially external to those appearances. In\nthe space derived from the combination of touch and sight and bodily\nmovement, which is the ordinary space of common sense, there is the\nsame externality of visual appearances other than those of parts of\nour own body. Thus spatial externality, in the sense in which space\ncan be derived from the relations of our own percepts, is not what\nis meant. I think we shall come nearer to what is meant if we say\nthat two people can perceive the same object. In some sense, unless\nwe reject testimony, we must of course admit that this is true: we\ncan all see the sun unless we are blind. But this fact is differently\ninterpreted by common sense and by the causal theory: for common sense,\nthe percepts are identical when two people see the sun, whereas for\nthe causal theory they are only similar and related by a common causal\norigin.\u003c/p\u003e\n\n\u003cp\u003eIt would be a waste of time to recapitulate the arguments against\nthe common-sense view. They are numerous and obvious and generally\nadmitted. The laws of perspective may serve as an illustration: where\none man sees a circle, another sees an ellipse, and so on. These\ndifferences are not due to anything \"mental,\" since they appear equally\nin photographs from different points of view. Common sense thus becomes\ninvolved in contradictions. These do not exist for solipsism, but\nthat is a desperate remedy. The alternative is the causal theory of\nperception.\u003c/p\u003e\n\n\u003cp\u003eWe must not expect to find a \u003ci\u003edemonstration\u003c/i\u003e that perceptions have\nexternal causes, which may produce perceptions in a number of people\nat the same time. The most that we can\u003cspan class=\"pagenum\" id=\"Page_199\"\u003e[Pg 199]\u003c/span\u003e hope for is the usual ground\nfor accepting a scientific theory—namely, that it links together a\nnumber of known facts, that it does not have any demonstrably false\nconsequences, and that it sometimes enables us to make predictions\nwhich are subsequently verified. All these tests the causal theory\nfulfils; it must not be assumed, however, that no other theory could\nfulfil them. But let us examine the evidence.\u003c/p\u003e\n\n\u003cp\u003eFirst: there can be no question of logical proof. A certain\ncollection of facts is known to me by perception and recollection;\nwhat else I believe about the physical world is either the effect\nof unreasoning habit or the conclusion of an inference. Now there\ncannot be any logical impossibility in a world consisting of just\nthat medley of events which I perceive or remember, and nothing\nelse. Such a world would be fragmentary, absurd, and lawless, but\nnot self-contradictory.\u003ca id=\"FNanchor_44\" href=\"#Footnote_44\" class=\"fnanchor\"\u003e[44]\u003c/a\u003e I am aware that, according to many\nphilosophers, such a world would be self-contradictory. I am aware\nalso that, according to other philosophers, what we perceive is\nnot fragmentary, but really embraces the whole universe—what is\nfragmentary is only what we perceive that we perceive. The first of\nthese views is that of Hegel and his followers; the second is that of\nBergson and (perhaps) of Dr Whitehead. The Hegelian view rests upon\nan elaborate logic, which I have controverted on former occasions; at\npresent I am content to refer to what I have written before. The other\nview is traditionally associated with mysticism; my reasons for not\naccepting it are given in \u003ci\u003eMysticism and Logic\u003c/i\u003e. I say, therefore,\non grounds given in former writings, that the world of perception and\nmemory is fragmentary, but not self-contradictory. On grounds of logic,\nI hold that nothing existent can imply any other existent except a part\nof itself, if implication is taken in the sense of what Professor G. I.\nLewis calls \"strict implication,\" which is\u003cspan class=\"pagenum\" id=\"Page_200\"\u003e[Pg 200]\u003c/span\u003e the relevant sense for our\npresent discussion. If this is true, it follows that any selection of\nthe things in the world might be absent, so far as self-contradiction\nis concerned. Given a world consisting of particulars \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, … interrelated in various ways, the world which results from\nthe obliteration of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e must be logically possible. It follows that\nthe world consisting only of what we perceive and recollect cannot be\nself-contradictory; if, therefore, we are to believe in the existence\nof things which we neither perceive nor recollect, it must be either\non the ground that we have other non-inferential ways of knowing\nmatters of fact, or on the basis of an argument which has not the\ntype of cogency that we should demand in pure mathematics, in the\nsense that the conclusion is only probable. As for the fragmentary\ncharacter of the perceived world, those who deny it have to introduce\nminute perceptions, like Leibniz, or unconscious perceptions, or vague\nperceptions, or something of the kind. Now it seems to me unnecessary\nto inquire whether there are perceptions of such kinds; I certainly\nam not prepared to deny them dogmatically. But I do say that, even\nif they exist, they are useless as a basis for physics. Perceptions\nof which we are not sufficiently conscious to express them in words\nare scientifically negligible as data; our premisses must be facts\nwhich we have explicitly noted. Vagueness, no doubt, is omnipresent\nand unavoidable; but it is only in proportion as we overcome it that\nexact science becomes possible. And we overcome it most by analysis and\nconcentration, not by a diffused ecstatic mystical vision.\u003c/p\u003e\n\n\u003cp\u003eI return now to the question: What grounds have we for inferring\nthat our percepts and what we recollect do not constitute the entire\nuniverse? I believe that at bottom our main ground is the desire\nto believe in simple causal laws. But proximately there are other\narguments. When we speak to people, they behave more or less as we\nshould if we heard such words, not as we do when we speak them. When I\nsay\u003cspan class=\"pagenum\" id=\"Page_201\"\u003e[Pg 201]\u003c/span\u003e that they behave in a similar manner, I mean that our perceptions\nof their bodies change in the same sort of way as our perceptions of\nour own bodies would m correlative circumstances. When an officer who\nhas risen from the ranks gives the word of command, he sees his men\ndoing what he used to do when he heard the same sounds as a private;\nit is therefore natural to suppose that they have heard the word of\ncommand. One may see a crowd of jackdaws in a newly-ploughed field\nall fly away at the moment when one hears a shot; again it is natural\nto suppose that the jackdaws heard the shot. Again: reading a book\nis a very different experience from composing one; yet, if I were a\nsolipsist, I should have to suppose that I had composed the works of\nShakespeare and Newton and Einstein, since they have entered into my\nexperience. Seeing how much better they are than my own books, and how\nmuch less labour they have cost me, I have been foolish to spend so\nmuch time composing with the pen rather than with the eye. All this,\nhowever, would perhaps be the better for being set forth formally.\u003c/p\u003e\n\n\u003cp\u003eFirst, there is a preliminary labour of regularizing our own percepts.\nI spoke of seeing others do what we should do in similar circumstances;\nbut the similarity is obvious only as a result of interpretation. We\ncannot see our face (except the nose, by squinting) or our head or our\nback; but tactually they are continuous with what we can see, so that\nwe easily imagine what a movement of an invisible part of our body\nought to look like. When we see another person frowning, we can imitate\nhim; and I do not think the habit of seeing ourselves in the glass is\nindispensable for this. But probably this is explained by imitative\nimpulses—\u003ci\u003ei.e.\u003c/i\u003e when we see a bodily action, we tend to perform\nthe same action, in virtue of a physiological mechanism. This of course\nis most noticeable in children. Thus we first do what someone else has\ndone, and then realize that what we have done is what he did. However,\u003cspan class=\"pagenum\" id=\"Page_202\"\u003e[Pg 202]\u003c/span\u003e\nthis complication need not be pursued. What I am concerned with is the\npassage, by experience, from \"apparent\" shapes and motions to \"real\"\nshapes and motions. This process lies within the perceptual world: it\nis a process of becoming acquainted with congruent groups—\u003ci\u003ei.e.\u003c/i\u003e\nto speak crudely, with groups of visual sensations which correspond to\nsimilar tactual sensations. All this has to be done before the analogy\nbetween the acts of others and our own acts becomes obvious. But as it\nlies within the perceptual world, we may take it for granted. The whole\nof it belongs to early infancy. As soon as it is completed, there is\nno difficulty in interpreting the analogy between what we perceive of\nothers and what we perceive of ourselves.\u003c/p\u003e\n\n\u003cp\u003eThe analogy is of two kinds. The simpler kind is when others do\npractically the same thing as we are doing—for instance, applaud\nwhen the curtain goes down, or say \"Oh\" when a rocket bursts. In such\ncases, we have a sharp stimulus followed by a very definite act,\nand our perception of our own act is closely similar to a number of\nother perceptions which we have at the same time. These, moreover,\nare all associated with perceptions very like those which we call\nperceptions of our own bodies. We infer that all the other people have\nhad perceptions analogous to that of the stimulus to our own act. The\nanalogy is very good; the only question is: Why should not the very\nsame event which was the cause of our own act have been the cause of\nthe acts of the others? Why should we suppose that there had to be a\nseparate seeing of the fall of the curtain for each spectator, and not\nonly one seeing which caused all the appearances of bodies to appear to\napplaud? It may be said that this view is far-fetched. But I doubt if\nit would be unreasonable but for the second kind of analogy, which is\nincapable of a similar explanation.\u003c/p\u003e\n\n\u003cp\u003eIn the second kind of analogy, we see others acting as we should act\nin response to a certain kind of stimulus which,\u003cspan class=\"pagenum\" id=\"Page_203\"\u003e[Pg 203]\u003c/span\u003e however, we are not\nexperiencing at the moment. Suppose, for example, that you are a rather\nshort person in a crowd watching election returns being exhibited on\na screen. You hear a burst of cheering, but can see nothing. By great\nefforts, you manage to perceive a very notable result which you could\nnot perceive a few moments earlier. It is natural to suppose that\nthe others cheered because they saw this result. In this case, their\nperceptions, if they occurred, were certainly not \u003ci\u003eidentical\u003c/i\u003e with\nyours, since they occurred earlier; hence, if the stimulus to their\ncheering was a perception analogous to your subsequent perception, they\nhad perceptions which you could not perceive. I have chosen a rather\nextreme example, but the same kind of thing occurs constantly; someone\nsays \"There\u0027s Jones,\" and you look round and see Jones. It would seem\nodd to suppose that the words you heard were not caused by a perception\nanalogous to what you had when you looked round. Or your friend says\n\"Listen,\" and after he has said it you hear distant thunder. Such\nexperiences lead irresistibly to the conclusion that the percepts you\ncall other people are associated with percepts which you do not have,\nbut which are like those you would have if you were in their place. The\nsame principle is involved in the assumption that the words you hear\nexpress \"thoughts.\"\u003c/p\u003e\n\n\u003cp\u003eThe argument in favour of the view that there are percepts, connected\nwith other people, which are not among our own percepts, is presupposed\nin the acceptance of testimony, and comes first in logical order when\nwe are trying to establish the existence of things other than our own\npercepts, both because of its inherent strength, and because of the\nusefulness of testimony in the further stages. The argument for other\npeople\u0027s percepts seems to common sense so obvious and compelling\nthat it is difficult to make oneself examine it with the necessary\ndetachment. Nevertheless it is important to do so. As we have seen,\nthere are three stages. The first does\u003cspan class=\"pagenum\" id=\"Page_204\"\u003e[Pg 204]\u003c/span\u003e not take us outside our own\npercepts, but consists merely in the arrangement of them in groups.\nOne group consists of all the percepts which common sense believes to\nbe those of an identical object by different senses and from different\npoints of view. When we eliminate reference to an object, a group must\nbe constituted by correlations, partly between one percept and another\n(touch and sight when an object is held in the hand), partly between\none percept and the changes in another (bodily movement and changes\nof visual and tactual perceptions while we move). In assuming that\nthese correlations will hold in untested cases, we are of course using\ninduction; otherwise, the whole process is straightforward. The process\nenables us to speak of a \"physical object\" as a group of percepts, and\nto explain what we mean by saying that a near object and a distant\nobject are \"really\" of the same size and shape. Also we can explain\nwhat we mean by saying that a physical object does not \"really\" change\nas we walk away from it (\u003ci\u003ei.e.\u003c/i\u003e as we have the percepts which make\nus say we are walking). This is the first stage in the argument.\u003c/p\u003e\n\n\u003cp\u003eIn the second stage, we note the likeness of the physical objects\ncalled other people\u0027s bodies to each other and to our own body; we\nalso note the likeness of their behaviour to our behaviour. In the\ncase of our own behaviour, we can observe a number of correlations\nbetween stimulus and reaction (both being percepts). For example, we\nfeel hunger or thirst, and then we eat or drink; we hear a loud noise,\nand we jump; we see Jones, and we say \"Hullo, Jones.\" The behaviour\nof the percepts we call other people\u0027s bodies is similar to that\nof our own body in response to this or that stimulus; sometimes we\nexperience the stimulus, and behave just as others do, which is the\nsecond stage; sometimes we do not experience the stimulus, but suppose,\nfrom their behaviour, that other people have experienced it, which is\nthe third stage. This is a particularly plausible supposition if we\nourselves experience\u003cspan class=\"pagenum\" id=\"Page_205\"\u003e[Pg 205]\u003c/span\u003e the stimulus in question very shortly after we\nhave observed the behaviour which led us to infer it. The third stage\nis the more important, since in the second we \u003ci\u003emight\u003c/i\u003e attribute\nthe behaviour of others to the stimulus which we perceive, and thus\nescape inferring unperceived existents, while in the third stage this\nalternative is not open to us. It will be seen that, in the third\nstage, the argument is the usual causal-inductive type of argument\nupon which all empirical laws are based. We perceive \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e\nconjoined in a number of cases, and we then infer \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e in a\ncase in which we do not know by perception whether \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is present or\nnot. Moreover, the argument for other people\u0027s perceptions is the same\nin form and cogency as the argument for the future truth of laws of\ncorrelation among our own percepts. We have exactly as good reason for\nbelieving that others perceive what we do not as we have for believing\nthat we shall have a perception of touch if we stretch out our hand to\nan object which looks as if it were within reach.\u003c/p\u003e\n\n\u003cp\u003eThe argument is not demonstrative, either in the one case or in the\nother. A conjuror might make a waxwork man with a gramophone inside,\nand arrange a series of little mishaps of which the gramophone would\ngive the audience warning. In dreams, people give evidence of being\nalive which is similar in kind to that which they give when we are\nawake; yet the people we see in dreams are supposed to have no external\nexistence. Descartes\u0027 malicious demon is a logical possibility. For\nthese reasons, we may be mistaken in any given instance. But it\nseems highly improbable that we are \u003ci\u003ealways\u003c/i\u003e mistaken. From the\nobserved correlation of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e we may argue, as regards\ncases in which \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is observed but we do not know whether \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nexists or not, either: (1) \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is always present, or (2) \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is\ngenerally present, or (3) \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is sometimes present. Dreams suffice\nto show that we cannot assert (1). But dreams could be distinguished\nfrom waking life by a solipsist, unless\u003cspan class=\"pagenum\" id=\"Page_206\"\u003e[Pg 206]\u003c/span\u003e his dreams were unusually\nrational and coherent. We may therefore exclude them before beginning\nour induction. Even then, it would be very rash to assert (1). But (2)\nis more probable, and (3) seems extremely probable. Now (3) is enough\nto allow us to infer a proposition of great philosophic importance,\nnamely: there are existents which I do not perceive. This proposition,\ntherefore, if induction is valid at all, may be taken as reasonably\ncertain. And, if so, it increases the probability of other propositions\nwhich infer the existence of this or that unperceived existent.\nThe argument, though not demonstrative, is as good as any of the\nfundamental inductions of science.\u003c/p\u003e\n\n\u003cp\u003eWe have been considering hitherto, not the external world in general,\nbut the percepts of other people. We might say that we have been trying\nto prove that other people are alive, and not mere phantoms like the\npeople in dreams. The exact thing we have been trying to prove is\nthis: Given an observed correlation among our own percepts, in which\nthe second term is what one would naturally call a percept of our\nown bodily behaviour, and given a percept of similar behaviour in a\nphysical object not our own body but similar to it, we infer that this\nbehaviour was preceded by an event analogous to the earlier term in the\nobserved correlation among our percepts. This inference assumes nothing\nas to the distinction of mind and body or as to the nature of either.\u003c/p\u003e\n\n\u003cp\u003eIn virtue of the above argument, I shall now assume that we may\nenlarge our own experience by testimony—\u003ci\u003ei.e.\u003c/i\u003e that the noises\nwe hear when it seems to us that other people are talking do in\nfact express something analogous to what we should be expressing if\nwe made similar noises. This is a particular case of the principle\ncontained in the preceding paragraph. I think the evidence for other\npeople\u0027s percepts is the strongest we have for anything that we do\nnot perceive ourselves; therefore it seems right to establish this,\nso far as we\u003cspan class=\"pagenum\" id=\"Page_207\"\u003e[Pg 207]\u003c/span\u003e can, before proceeding to consider our evidence for\n\"matter\"—\u003ci\u003ei.e.\u003c/i\u003e for existents satisfying the equations of\nphysics. This must be our next task; but it will be well to begin with\ncommon-sense material \"things\" conceived as the causes of perceptions.\u003c/p\u003e\n\n\u003cp\u003eHaving now admitted the percepts of other people, we can greatly\nenlarge the group constituting one \"physical object.\" Within the\nsolipsistic world, we found means of collecting groups of percepts and\ncalling the group one physical object; but we can now enrich our group\nenormously. A number of people sitting near each other can all draw\nwhat they see, and can compare the resulting pictures; there will be\nsimilarities and differences. A number of stenographers listening to\na lecture can all take notes of it, and compare results. A number of\npeople can be brought successively into a room full of hidden roses,\nand asked \"What do you smell?\" In this way it appears that the world\nof each person is partly private and partly common. In the part which\nis common, there is found to be not identity, but only a greater or\nless degree of similarity, between the percepts of different people. It\nis the absence of identity which makes us reject the naive realism of\ncommon sense; it is the similarity which makes us accept the theory of\na common origin for similar simultaneous perceptions.\u003c/p\u003e\n\n\u003cp\u003eThe argument here is, I think, not so good as the argument for other\npeople\u0027s percepts. In that case, we were inferring something very\nsimilar to what we know in our own experience, whereas in this case we\nare inferring something which can never be experienced, and of whose\nnature we can know no more than the inference warrants. Nevertheless,\nthe common-sense arguments for an external cause of perception are\nstrong.\u003c/p\u003e\n\n\u003cp\u003eTo begin with, we can, without assuming anything that no one perceives,\nestablish a common space and time in which we all live. (Our discussion\nis necessarily confined to people on the surface of the earth, since\nother people, if they exist,\u003cspan class=\"pagenum\" id=\"Page_208\"\u003e[Pg 208]\u003c/span\u003e have not succeeded in communicating\nwith us; consequently the complications of relativity do not yet\narise.) The usual methods of determining latitude and longitude can\nbe applied, without assuming that the readings of clock and sextant\nhave the physical meaning usually assigned to them. Altitudes, also,\ncan be measured by the usual methods. By these means, observers can\nbe arranged in a three-dimensional order. Of course the resulting\nspace will not be a continuum, since it will contain only so many\n\"points\" as there are observers. But the motion of an observer can\nbe sensibly continuous, so that we can construct \"ideal\" points of\nview with defined mathematical properties, and thus build up, for\nmathematical purposes, a continuous space. We can thus arrive at the\nlaws of perspective, taken in a generalized sense; that is to say,\nwe can correlate the differences between correlated perceptions with\ndifferences in the situations of the percipients. And in the space\nderived from \"points of view\" we can place physical objects. For, let\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e be two observers, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e their correlated\nvisual percepts, which, being correlated, are described as percepts\nof one physical object \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e. If the angular dimensions of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e are\nlarger than those of \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, we shall say (as a definition) that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nis nearer to \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e than \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is. We can thus construct a number of routes\nconverging on \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e. We shall construct our geometry so that they\nintersect, and shall define their intersection as the place where\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e is. If \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e happens to be a human body, we shall find that the\nplace of \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e, so defined, is identical with the place of \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e as an\nobserver in the space of points of view.\u003ca id=\"FNanchor_45\" href=\"#Footnote_45\" class=\"fnanchor\"\u003e[45]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe correlation of the times of different percipients offers no\ndifficulty, since, as before observed, our percipients are all on the\nearth. The usual method of light-signals can be employed. But here we\ncome upon one of the arguments for the causal theory of perception,\nas against both common sense\u003cspan class=\"pagenum\" id=\"Page_209\"\u003e[Pg 209]\u003c/span\u003e and phenomenalism. (We may define\nphenomenalism, at least for the moment, as the view that there are only\npercepts.) Suppose a gun on a hilltop is fired every day at twelve\no\u0027clock: many people both see and hear it fired, but the further they\nare from it the longer is the interval between seeing and hearing.\nThis makes it very difficult to accept a naively realistic view as to\nthe hearing, since, if that view were correct, there would have to be\na fixed interval of time (presumably zero) between the sight and the\nsound. It also makes it natural to adopt a causal view of sound, since\nthe retardation of the sound depends upon the distance, not upon the\nnumber of intermediate percipients. But hitherto our space was purely\n\"ideal\" except where there were percipients; it seems odd, therefore,\nthat it should have an actual influence. It is much more natural to\nsuppose that the sound travels over the intervening space, in which\ncase something must be happening even in places where there is no one\nwith ears to hear. The argument is perhaps not very strong, but we\ncannot deny that it has \u003ci\u003esome\u003c/i\u003e force.\u003c/p\u003e\n\n\u003cp\u003eMuch stronger arguments, however, are derivable from other sources.\nSuppose a room arranged with a man concealed behind a curtain, and\nalso a camera and a dictaphone. Suppose two men came into the room,\nconverse, dine, and smoke. If the record of the dictaphone and the\ncamera agrees with that of the man behind the curtain, it is impossible\nto resist the conclusion that something happened where they were which\nbore an intimate relation to what the hidden man perceived. For that\nmatter, one might have two cameras and two dictaphones, and compare\ntheir records. Such correspondences, which are only more extreme\nforms of those with which primitive common sense is familiar, make\nit inconceivably complicated and implausible to suppose that nothing\nhappens where there is no percipient. If the dictaphone and the hidden\nman give the same report of the conversation, one\u003cspan class=\"pagenum\" id=\"Page_210\"\u003e[Pg 210]\u003c/span\u003e must suppose some\ncausal connection, since otherwise the coincidence is in the highest\ndegree improbable. But the causal connection is found to depend upon\nthe position of the dictaphone at the time of the conversation, not\nupon the person who hears its record. This seems very strange, if its\nrecord does not exist until it is heard, as we shall have to suppose if\nwe confine the world to percepts. I will not emphasize the more obvious\noddities of such a world, as, \u003ci\u003ee.g.\u003c/i\u003e, the one once brought forward\nby Dr G. E. Moore, that a railway train would only have wheels when it\nis not going, since, while it is going, the passengers cannot see them.\u003c/p\u003e\n\n\u003cp\u003eBefore accepting such arguments, however, we must see what could be\nsaid against them by a phenomenalism. Let us, therefore, proceed to\nstate the case for phenomenalism.\u003c/p\u003e\n\n\u003cp\u003eIt may be suggested that our argument is, after all, not so strong as\nit looks, since all the facts can be interpreted by means of \"ideal\"\npercipients. The doubt I have in mind is suggested by a certain kind\nof construction, of which a good example is the introduction of\n\"ideal\" points, lines, and planes in descriptive geometry.\u003ca id=\"FNanchor_46\" href=\"#Footnote_46\" class=\"fnanchor\"\u003e[46]\u003c/a\u003e For\nour purposes, \"ideal\" points will suffice. The process by which they\nare constructed is as follows. Take all the straight lines which pass\nthrough a given point; these form a group of lines having other notable\nproperties besides that of all possessing a common point. These other\nproperties belong also to certain groups of lines which have no point\nin common—\u003ci\u003ee.g.\u003c/i\u003e in Euclidean geometry, to the group consisting\nof all lines parallel to a given line. We then define a group of lines\npossessing these properties as an \"ideal\" point.\u003ca id=\"FNanchor_47\" href=\"#Footnote_47\" class=\"fnanchor\"\u003e[47]\u003c/a\u003e Thus some \"ideal\"\npoints correspond to\u003cspan class=\"pagenum\" id=\"Page_211\"\u003e[Pg 211]\u003c/span\u003e real points, while others do not. In this way,\nby proceeding to \"ideal\" lines and planes, we arrive at last at a\nprojective geometry, in which any two planes have a common line, and\nany two lines in a plane a common point, which immensely simplifies the\nstatement of our propositions.\u003c/p\u003e\n\n\u003cp\u003eThe analogy with our problem is perhaps closer than might be thought.\nWe have, in the first place, real percepts, collected into groups each\nof which is defined by the characteristic that common sense would\ncall all its members percepts of one physical object. These real\npercepts, as we saw, vary from one percipient to another in such a\nway as to allow us to construct a space of percipients, and to locate\nphysical objects in this space. Let us, for the moment, adopt the view\nthat nothing exists except percepts, our own and other people\u0027s. We\nshall then observe that the percepts forming a given group can always\nbe arranged about a centre in the space of percipients, and we can\nfill out the group by interpolating \"ideal\" percepts, continuous in\nquality with actual percepts, in regions where there are no actual\npercipients. (A region of space which is \"ideal\" at one moment may\nbe actual at another owing to motion of a percipient. The successive\npositions of an observer watching Cleopatra\u0027s Needle from a passing\ntram form a sensibly continuous series.) If a number of people hear a\ngun fired, there are differences in the loudness and the time of their\npercepts; we can fill out the actual percepts by \"ideal\" noises varying\ncontinuously from one actual one to another. The same can be done with\ncorrelated visual percepts; also with smells. We will call a group\nthus extended by interpolation and extrapolation a \"full\" group: its\nmembers are partly real, partly ideal. Each group has a centre in the\nspace of percipients; this centre is real if occupied by a percipient,\nwhile otherwise it is ideal. (Our space is not assumed to be a smooth\ngeometrical space, and the centre may be a finite volume.) As a rule,\neven when the\u003cspan class=\"pagenum\" id=\"Page_212\"\u003e[Pg 212]\u003c/span\u003e centre is occupied by a percipient, it nevertheless\ncontains no member of the group, not even an ideal member: \"the eye\nsees not itself.\" A group, that is to say, is hollow: when we get\nsufficiently near to its centre it ceases to have members. This is a\npurely empirical observation.\u003c/p\u003e\n\n\u003cp\u003eA full group which contains any real members will be called a \"real\"\ngroup; a group whose members are all ideal will be called \"ideal.\" It\nremains to show how we are to define an ideal group.\u003c/p\u003e\n\n\u003cp\u003eIn addition to the laws correlating percepts forming one group—which\nmay be called, in an extended sense, laws of I perspective—there are\nalso laws as to the manner in which percepts succeed one another. These\nare causal laws in the ordinary sense; they are included in the usual\nlaws of physics. When we know a certain number of members of a full\ngroup, we can infer the others by the laws of perspective; it is found\nthat some exist and some do not, but all that do exist are members\nof the calculated full group. In like manner, when we are given a\nsufficient number of full groups, we can calculate other full groups at\nother times. It is found that some of the calculated full groups are\nreal, some ideal, but that all real groups are included among those\ncalculated. (I am assuming an impossible perfection of physics.) Two\ngroups belonging to different times may, in virtue of causal relations\nwhich we shall explain when we come to discuss substance, be connected\nin the way which makes us regard them as successive states of one\n\"thing\" or \"body.\" (The time of a full group, by the way, is not\nexactly the time at which its members occur, but slightly earlier than\nthe earliest real member—or much earlier, in the case of a star. The\ntime of a full group is the time at which physics places the occurrence\nsupposed to be perceived.) The whole series of groups belonging to a\ngiven \"thing\" is called a \"biography.\" The causal laws are such as\nto allow us sometimes to infer \"things.\" A thing is \"real\"\u003cspan class=\"pagenum\" id=\"Page_213\"\u003e[Pg 213]\u003c/span\u003e when its\nbiography contains at least one group which is \"real,\" \u003ci\u003ei.e.\u003c/i\u003e\ncontains at least one percept; otherwise a thing is \"ideal.\" This\nconstruction is closely analogous to that of \"ideal\" points, lines, and\nplanes in descriptive geometry. We have to ask ourselves whether there\nare any reasons for or against it.\u003c/p\u003e\n\n\u003cp\u003eThe above construction preserves the whole of physics, at least\nformally; and it gives an interpretation, in terms of percepts and\ntheir laws, to every proposition of physics which there is any\nempirical reason to believe. \"Ideal\" percepts, groups, and things, in\nthis theory, are really a shorthand for stating the laws of actual\npercepts, and all empirical evidence has to do with actual percepts.\nThe above account, therefore, preserves the truth of physics with the\nbare minimum of hypothesis. Of course there should be also rules for\ndetermining when a calculated percept is real and when it is ideal; but\nthis is difficult, since such rules would have to contain a science\nof human actions. It may be known that you will see certain things\nif you look through a telescope, but it is difficult to know whether\nyou will look through it. This completion of our science is therefore\nnot possible at present; but that is no argument against the truth of\nour science so far as it goes. It is obvious that the method might be\nextended so as to make all perceptions except one\u0027s own \"ideal\"; we\nshould then have a completely solipsistic interpretation of physics. I\nshall, however, ignore this extension, and consider only that form of\nthe theory in which all percepts are admitted.\u003c/p\u003e\n\n\u003cp\u003eThe metaphysic which we have been developing is essentially Berkeley\u0027s:\nwhatever is, is perceived. But our reasons are somewhat different\nfrom his. We do not suggest that there is any impossibility about\nunperceived existents, but only that no strong ground exists for\nbelieving in them. Berkeley believed that the grounds against them\nwere conclusive; we only suggest that the grounds in their favour are\u003cspan class=\"pagenum\" id=\"Page_214\"\u003e[Pg 214]\u003c/span\u003e\ninconclusive. I am not asserting this: I am proposing it as a view to\nbe considered.\u003c/p\u003e\n\n\u003cp\u003eThe great difficulty in the above theory of \"ideal\" elements is that\nit is hard to see how anything merely imaginary can be essential to\nthe statement of a causal law. We have to explain the dictaphone which\nrepeats the conversation. We will suppose that it was seen in place\nbefore and after the conversation, but not during it. Consequently,\non the view we are examining, it did not exist at all during the\nconversation. Causal laws, stated without fictitious elements, will\nthus involve action at a distance in time and space. Moreover, our\npercepts are not sufficient to determine the course of nature: we\nderive causal laws from close observation, and preserve them in other\ncases by inventing \"ideal\" things. This would not be necessary if\npercepts sufficed for the causal determination of future percepts. Thus\nthe view we are examining is incompatible with physical determinism, in\nfact though not in form. We could multiply difficulties of this kind\nindefinitely. No one of them is conclusive, but in the aggregate they\nsuffice to account for the fact that it is almost impossible to compel\noneself to believe such a theory. Perhaps continuity (not in a strict\nmathematical sense) is one of the strongest objections. We experience\nsensible continuity when we move our own body, and when we fixedly\nobserve some object which does not explode. But if we repeatedly open\nand shut our eyes we experience visual discontinuity, which we find it\nimpossible to attribute to the physical objects which we alternately\nsee and do not see, the more so as, to another spectator, they remain\nunchanged all the time. Causation at a distance in time, though not\nlogically impossible, is also repugnant to our notions of the physical\nworld. Therefore, although it is logically possible to interpret\nthe physical world in terms of ideal elements, I conclude that this\ninterpretation is implausible, and that it has no positive grounds in\nits favour.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_215\"\u003e[Pg 215]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eNevertheless the above construction remains valid and important, as a\nmethod of separating perceptual and non-perceptual elements of physics,\nand of showing how much can be achieved by the former alone. As such, I\nshall continue to utilize it in the sequel. The only thing rejected is\nthe view that \"ideal\" elements are unreal.\u003ca id=\"FNanchor_48\" href=\"#Footnote_48\" class=\"fnanchor\"\u003e[48]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eThe matter would, of course, be otherwise in this last respect if we\ncould accept the argument for idealism, whether of the Berkeleyan\nor the German variety. These arguments profess to prove that what\nexists \u003ci\u003emust\u003c/i\u003e have a mental character, and therefore compel\nus to interpret physics accordingly. I reject such \u003ci\u003ea priori\u003c/i\u003e\nargumentation, whatever conclusion it may be designed to prove. There\nis no difficulty in interpreting physics idealistically, but there is\nalso, I should say, no necessity for such an interpretation. \"Matter,\"\nI shall contend, is known only as regards certain very abstract\ncharacteristics, which might quite well belong to a manifold of mental\nevents, but might also belong to a different manifold. In fact, the\nonly manifolds known for certain to possess the mathematical properties\nof the physical world are built up out of numbers, and belong to\npure mathematics. Our reason for not regarding \"matter\" as actually\nbeing an arithmetical structure derived from the finite integers is\nthe connection of \"matter\" with perception; that is why our present\ndiscussion is necessary. But this connection, as I shall try to show,\ntells us extremely little about the character of the unperceived events\nin the physical world. Unlike idealists and materialists, I do not\nbelieve that there is any other source of knowledge from which this\nmeagre result can be supplemented. Like other people, I allow myself\nto speculate; but that is an exercise of imagination, not a process of\ndemonstrative reasoning.\u003c/p\u003e\n\n\u003cp\u003eI shall assume henceforth not only that there are percepts\u003cspan class=\"pagenum\" id=\"Page_216\"\u003e[Pg 216]\u003c/span\u003e which I do\nnot perceive, connected with other people\u0027s bodies, but also that there\nare events causally connected with percepts, as to which we do not know\nwhether they are perceived or not. I shall assume, \u003ci\u003ee.g.\u003c/i\u003e that\nif I am alone in a room and I shut my eyes, the objects in it which I\nno longer see (\u003ci\u003ei.e.\u003c/i\u003e the causes of my visual percepts) continue\nto exist, and do not suddenly become resurrected when I re-open my\neyes. This must be taken in conjunction with what was said earlier\nabout perspective in a generalized sense, and about the common space\nin which we locate the physical objects which, for common sense, are\nperceived by several people at once. We collect correlated percepts\ninto a group, and we suppose that there are other members of the group,\ncorresponding to places where there is no percipient—or, to speak\nmore guardedly, where there is not known to be a percipient. But we\nno longer assume, as when we were constructing \"ideal\" elements, that\nwhat at such places is what we should perceive if we went to them. We\nthink, \u003ci\u003ee.g.\u003c/i\u003e, that light consists of waves of a certain kind,\nbut becomes transformed, on contact with the eye, into a different\nphysical process. Therefore what occurs before the light reaches an\neye is presumably different from what occurs afterwards, and therefore\ndifferent from a visual percept. But it is supposed to be causally\ncontinuous with the visual percept; and it is largely for the sake of\nthis causal continuity that a certain re-interpretation of the physical\nworld seems desirable.\u003c/p\u003e\n\n\u003cp\u003eIn some ways, the language of causation is perhaps not the best for\nexpressing what is intended. What is intended may be expressed as\nfollows. Confining ourselves, to begin with, to the percepts of various\nobservers, we can form groups of percepts connected approximately,\nthough not exactly, by laws which may be called laws of \"perspective.\"\nBy means of these laws, together with the changes in our other percepts\nwhich are connected with the perception of bodily movement,\u003cspan class=\"pagenum\" id=\"Page_217\"\u003e[Pg 217]\u003c/span\u003e we can\nform the conception of a space in which percipients are situated, and\nwe find that in this space all the percepts belonging to one group\n(\u003ci\u003ei.e.\u003c/i\u003e of the same physical object, from the standpoint of\ncommon sense) can be ordered about a centre, which we take to be the\nplace where the physical object in question is. (For us, this is a\n\u003ci\u003edefinition\u003c/i\u003e of the place of a physical object.) The centre is not\nto be conceived as a point, but as a volume, which may be as small as\nan electron or as large as a star. The essential assumption for what is\ncommonly called the causal theory is, that the group of percepts can\nbe enlarged by the addition of other events, ranged in the same space\nabout the same centre, and connected both with each other and with\nthe group of percepts by laws which include the laws of perspective.\nThe essential points are (1) the arrangement about a centre, (2) the\ncontinuity between percepts and correlated events in other parts of the\nspace derived from percepts and locomotion. The first is a matter of\nobservation; the second is a hypothesis designed to secure simplicity\nand continuity in the laws of correlation suggested by the grouping of\npercepts. It cannot be demonstrated, but its merits are of the same\nkind as those of any other scientific theory, and I shall therefore\nhenceforth assume it.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_43\" href=\"#FNanchor_43\" class=\"label\"\u003e[43]\u003c/a\u003e\nOn this subject, \u003ci\u003ecf.\u003c/i\u003e chap. \u003cspan class=\"allsmcap\"\u003eIV\u003c/span\u003e. of Dr\nBroad\u0027s \u003ci\u003ePerception, Physics, and Reality\u003c/i\u003e, Cambridge, 1914.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_44\" href=\"#FNanchor_44\" class=\"label\"\u003e[44]\u003c/a\u003e\nPerhaps it would not really be lawless; I shall discuss\nthis at a later stage.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_45\" href=\"#FNanchor_45\" class=\"label\"\u003e[45]\u003c/a\u003e\nOn this subject, \u003ci\u003ecf.\u003c/i\u003e my \u003ci\u003eKnowledge of the\nExternal World\u003c/i\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_46\" href=\"#FNanchor_46\" class=\"label\"\u003e[46]\u003c/a\u003e\nSee Dr Whitehead\u0027s tract on this subject (Cambridge\nUniversity Press). Also Pasch, \u003ci\u003eNeuere Geometrie\u003c/i\u003e, Leipzig, 1882.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_47\" href=\"#FNanchor_47\" class=\"label\"\u003e[47]\u003c/a\u003e\nThe definition of an \"ideal\" point is as follows. Let\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-358.png\" alt=\"\" data-tex=\"\\(l\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e be any two lines in one plane, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e any point not in this\nplane. Then the planes \u003cimg style=\"vertical-align: -0.025ex; width: 2.371ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-359.png\" alt=\"\" data-tex=\"\\(Al\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 3.683ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-360.png\" alt=\"\" data-tex=\"\\(Am\\)\"\u003e have a line in common, say \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e.\nThe class of all such lines as \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e, when \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is varied while \u003cimg style=\"vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-358.png\" alt=\"\" data-tex=\"\\(l\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e remain fixed, is the \"ideal\" point determined by the two\nlines \u003cimg style=\"vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-358.png\" alt=\"\" data-tex=\"\\(l\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_48\" href=\"#FNanchor_48\" class=\"label\"\u003e[48]\u003c/a\u003e\nThe character of the \"ideal\" elements, also, will be less\nsimilar to that of percepts than in the above construction, or at least\ncannot be known to be so similar.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_218\"\u003e[Pg 218]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXI\"\u003eCHAPTER XXI\u003cbr\u003e\nPERCEPTION AND OBJECTIVITY\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nWHEN a number of people are, from the standpoint of common sense,\nobserving the same object, there are both likenesses and differences\namong their percepts. For common sense, with its naive realism, the\ndifferences constitute a difficulty, since they render the percepts\nmutually inconsistent if taken to be each wholly a revelation of one\nand the same physical object. But to the causal theory of perception\nthis difficulty is non-existent. We have now, however, an opposite\ndifficulty—namely, that of deciding what elements in a percept I can\nbe used for inference as to the existence of something other I than\nitself, and as to the nature of the inferences when they can be drawn.\nFor the moment, I am not thinking of inferences involving motion, but\nonly of inferences as to the present state of the physical object which\nis being observed.\u003c/p\u003e\n\n\u003cp\u003eWe must be on our guard against a confusion which is difficult to avoid\nin such inquiries. Perception, as an event in our own history, is a\nrecognizable occurrence; its psychological meaning is fairly definite.\nBut it has also an epistemological meaning, and this is hardly capable\nof being made as definite as could be wished. Perception is interesting\nto us, in our present discussion, because it is a source of knowledge,\nnot because it is an occurrence which a psychologist can recognize. So\nlong as naive realism remained tenable, perception was knowledge of a\nphysical object, obtained through the senses, not by inference. But in\naccepting the causal theory of perception we have committed ourselves\nto the view that perception gives no immediate knowledge of a physical\nobject, but at best a datum for inference. A perception does,\u003cspan class=\"pagenum\" id=\"Page_219\"\u003e[Pg 219]\u003c/span\u003e however,\nstill give knowledge of something: if I perceive a round red patch, I\nknow that there is a round red patch in the world now, and no account\nof the causes of my perception can destroy this knowledge. It may be\nconceded that, in saying this, I am using \"perception\" more narrowly\nthan it might be used in psychology: I am confining it to cases where\nwe notice explicitly what we are perceiving. For epistemological\npurposes, this restriction is essential. I am deliberately refraining\nfrom all analysis of \"knowing\" since that would take us too far from\nour subject.\u003c/p\u003e\n\n\u003cp\u003eThe inferences to be primarily drawn from a perception are, as to\nother members of the group to which the percept concerned belongs.\nThis is done, in a confused way, by common sense, when it infers the\n\"real\" size or shape of an object from its \"apparent\" size or shape,\n\u003ci\u003ei.e.\u003c/i\u003e from the real size or shape of the percept. The \"real\" size\nor shape is a norm, from which the percept of a spectator in a given\nrelative situation can be inferred. Ordinarily, there is no conscious\ninference involved; but conscious inference can be used without\ninvoking any fresh knowledge. For example, an architect can show the\nview of a proposed house from any angle when he knows its measurements,\nand for this purpose he uses only systematized common sense; and he\ncan infer the measurements approximately when he has viewed an actual\nhouse from several angles. The \"real\" object, as opposed to its\n\"appearances,\" is thus something of the nature of a formula by means\nof which all sufficiently near \"appearances\" can be determined. Given\nthe measurements of a house, we can infer its apparent shape at a given\ndistance in a given direction. If perception were perfectly accurate\nand regular, a few percepts belonging to a given group would enable us\nto determine all percepts, actual and possible, belonging to that group.\u003c/p\u003e\n\n\u003cp\u003eThis is found to be not in fact the case. From seeing a drop of water\nwith the naked eye, we cannot know that under the\u003cspan class=\"pagenum\" id=\"Page_220\"\u003e[Pg 220]\u003c/span\u003e microscope it will\nbe found to be full of bacilli. When we see a man a hundred yards away,\nwe cannot tell whether he is handsome or plain. When we can only just\ndistinguish a person\u0027s voice, we cannot tell what is being said. These\nare all cases of \"vagueness,\" in a certain perfectly precise sense. In\nany group of percepts, those nearer the centre have a many one relation\nto those farther off—\u003ci\u003ei.e.\u003c/i\u003e two things which look alike from a\ndistance look different when seen close to. In this sense, the more\ndistant percepts are vaguer than the nearer ones: the former can be\ninferred from the latter, but not the latter from the former.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, a converse fact—namely, that what may be called\nthe \"regular\" law for inferring distant from near appearances may be\ninterfered with by intervening things. The sun may be visible from a\ngreat altitude when clouds make it invisible from the earth\u0027s surface.\nSounds may be stopped by obstacles, and die away completely at a\nsufficient distance from their source. Smells die away still more\nquickly, and are even more dependent upon the wind. This set of facts\ninterferes with the inference from near to distant appearances, just\nas the former set interfered with the inference from distant to near\nappearances.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, an important difference between the two sets of\nfacts. The increasing vagueness of distant appearances is an intrinsic\nlaw of groups of percepts, whereas the uncertainty as to distant\nappearances when near appearances are given depends always upon outside\ninterference. This distinction is of a kind which we shall find to be\nvery important in various ways. Let us try to state it clearly in the\ncase in question.\u003c/p\u003e\n\n\u003cp\u003eSuppose two persons to be both observing a given object which is\nstationary on the earth\u0027s surface, and suppose that one of the persons\nremains at rest while the other moves about. We will suppose that to\nthe person who remains at rest there\u003cspan class=\"pagenum\" id=\"Page_221\"\u003e[Pg 221]\u003c/span\u003e is no perceptible change in the\nobject throughout the time concerned. To the other person there will be\nchanges which, in general, are approximately according to the laws of\nperspective, especially for small changes in the observer\u0027s position.\nBut sometimes, to take the most obvious example, the object in question\nbecomes invisible when the observer takes up certain positions—those,\nnamely, from which some opaque object is between the observer and the\nobject which he had been seeing. As a rule, this happens gradually:\nat first both objects are visible, gradually their angular distance\nbecomes less, and at last only the nearer object remains visible. The\nnearer object has thus had an effect upon the appearance of the farther\nobject. Fog, smoke, glass, blue spectacles, etc., similarly modify\nthe appearances of distant objects. That is to say, in calculating\nthe appearance which a body will present in such and such a place, we\nhave to take account, not only of the body\u0027s appearances elsewhere,\nbut also of the bodies between it and the place in question. These\nintervening bodies are sometimes sensible, sometimes not; when they are\nnot, they are inferred as being necessary in order to preserve the laws\nwhich have been found to hold when they were sensible. The principle\nis the following: If we compare neighbouring members of a group of\npercepts, we find, in a great many cases, that their first-order\ndifferences are in accordance with the laws of perspective, while their\nsecond-order differences are functions of groups with other centres;\nor rather, since the above statement is too precise for the facts, we\nmay say simply that the differences between neighbouring positions\nare compounded of the laws of perspective together with functions of\ngroups with other centres. Suppose, \u003ci\u003ee.g.\u003c/i\u003e, that you are seeing\nan object through glass which is slightly distorting. The glass is a\ntactual group between you and the object; as you move, the distortions\ndue to the glass change, and have to be compounded with the laws of\nperspective in order to\u003cspan class=\"pagenum\" id=\"Page_222\"\u003e[Pg 222]\u003c/span\u003e calculate one member of a group from another.\nIn other cases, by carefully comparing a number of members of a group,\nwe can discover that their departure from perspective laws proceeds\naccording to a law which is a function of a position not perceptibly\noccupied. The previous illustration will apply to this case also, if\nwe have not touched the distorting glass. Human beings are superior to\nbirds and insects in the fact that they can infer glass in such cases,\nwithout any scientific apparatus, whereas birds and insects repeatedly\nbump into it.\u003c/p\u003e\n\n\u003cp\u003eLike much of what has to be said in the transition from perception to\nscience, the above statement is not capable of being made in an exact\nform. The methods by which we collect a number of percepts into one\ngroup are rough and ready, and become impossible if there is very great\ndistortion by the intervening medium. But these methods are successful\nin a sufficient number of cases to give rise to the notion of events\ngrouped about a centre, changing partly in accordance with the laws\nof perspective and partly in ways which are functions of groups with\nother centres. Having arrived at this notion, it is not very difficult\nto modify it in such a way that it shall become capable of scientific\nprecision.\u003c/p\u003e\n\n\u003cp\u003eI come now to the question of \"objectivity\" in a perception. This is a\nmatter of degree: the more correct are the inferences we can draw from\na percept as to other events (whether percepts or not) belonging to the\nsame group, the more \"objective\" is the perception. (I propose this as\na definition.) A percept may not belong to a group at all; in that case\nit has no objectivity. Hallucinations and dreams come under this head.\nOr we may be mistaken as to the position of the centre of the group;\nthis is the case with a mirage, or with a reflection not recognized as\nsuch. Or we may perceive a colour or shape which is erratic, say owing\nto intervening smoke, and thus misleads us as to the colour or shape\nwhich others will see. I should not regard a perception as failing in\nobjectivity through\u003cspan class=\"pagenum\" id=\"Page_223\"\u003e[Pg 223]\u003c/span\u003e mere vagueness. Vagueness diminishes the number of\ninferences that we can draw, but not their correctness. From a distance\nwe perceive correctly that what is approaching is a man; when he gets\nnear we perceive that he is Jones. But our previous perception did not\nfail in objectivity through failing to show that it was Jones. It would\nhave failed of objectivity if, owing to intervening lenses, it had\nshown us a man standing on his head.\u003c/p\u003e\n\n\u003cp\u003eWhen two people simultaneously have percepts which they regard as\nbelonging to one group, if the inferences of the one differ from those\nof the other, one of them at least must be drawing false inferences,\nand must therefore have an element of subjectivity in his perception.\nIt is only where the inferences of the two observers agree that both\nperceptions may be objective. It will be seen that, according to this\nview, the objectivity of a perception does not depend only upon what\nit is in itself, but also upon the experience of the percipient. A man\naccustomed to being short-sighted can judge objects much more correctly\nthan a man whose vision suddenly acquires the same defect. Fatigue as\nwell as alcohol may make us see double, but fatigue will not deceive us\nwhen it does so.\u003c/p\u003e\n\n\u003cp\u003eSubjectivity in perceptions may be traced to three sources, physical,\nphysiological, and psychological; or, better perhaps, physical,\nsensory, and cerebral. In all cases in which a percept is really a\nmember of a group constituting a physical object, any element of\nsubjectivity that it may possess is due to the distortions connected\nwith intervening physical objects—that, at least, is the theory which\nhas been found successful. When these objects are between the body\nof the percipient and the centre of the group to which the percept\nbelongs, the subjectivity is physical; when they are in the body of the\npercipient but not in his brain, they are sensory when they are in his\nbrain, they are cerebral. The last of these, however, is usually purely\nhypothetical; the \u003ci\u003ediscoverable\u003c/i\u003e causes of the\u003cspan class=\"pagenum\" id=\"Page_224\"\u003e[Pg 224]\u003c/span\u003e subjectivity which\nwe are calling cerebral are as a rule psychological.\u003c/p\u003e\n\n\u003cp\u003ePhysical subjectivity exists equally in a photograph or gramophone\nrecord; it is present already in the events, external to the\npercipient\u0027s body, which belong to the group in question and are\nvery near to the sense-organ concerned in the perception. The stick\nthat looks bent when it is half in water is an obvious example of\nphysical subjectivity. So are many effects of reflexion, refraction,\netc. The theory of relativity has brought to light a new kind of\nphysical subjectivity, dependent upon relative motion. The prevention\nof mistaken inferences owing to physical subjectivity is part of the\nbusiness of physics, and does not involve physiology or psychology.\u003c/p\u003e\n\n\u003cp\u003ePhysiological (or sensory) subjectivity arises through defects of the\nsense-organs or afferent nerves; it may also be produced by drugs.\nWe can discover such defects by the comparison of different people\u0027s\nperceptions in a given situation. It should be observed that the\nintrinsic quality of a percept is unimportant in this respect: if one\nperson sees red where another sees green, and green where another sees\nred, the fact will be undiscoverable and harmless. But if, where one\nperson sees two colours, red and green, another only sees one, we have\na discoverable difference, which is correctly described as a defect\nin the vision of the person who only sees one. It is always assumed\nthat if two stimuli produce noticeably different effects in a given\npercipient at a given time, there must be differences in the stimuli\ncorrelated with the differences in their effects; while if the effects\nare not noticeably different, there may nevertheless be differences in\nthe stimuli. Consequently \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\u0027s senses are better than \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e\u0027s if\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e perceives differences when \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e does not. For the same reason,\nthe microscope and the telescope are better than the naked eye. But\nthis has, as a rule, more to do with vagueness than with\u003cspan class=\"pagenum\" id=\"Page_225\"\u003e[Pg 225]\u003c/span\u003e subjectivity.\nSubjectivity only enters in when we are led to make false inferences,\nnot when we are merely unable to make inferences which another can\nmake. A mere deficiency, such as blindness or deafness, does not amount\nto subjectivity, but seeing double does if it deceives us. It deceives\nus when it leads to false inferences—\u003ci\u003ee.g.\u003c/i\u003e that there are two\ntactual objects, or that a person near us will see two objects.\u003c/p\u003e\n\n\u003cp\u003eCerebral (or psychological) subjectivity arises as a result of past\nexperience. An obvious example is a sensation which appears to be in\na leg which has been amputated. We are liable to this kind of error\nwhenever two things usually associated are for some reason dissociated.\nCertain sensations have, in the past, been generally associated with\na stimulus in the leg; but they have had as intermediaries conditions\nof the nerves between the leg and the brain. If these previously\nintermediate conditions arise in a person who has lost his leg, he\nwill interpret them as sensations in his leg, if he has momentarily\nforgotten that he has lost his leg—\u003ci\u003ee.g.\u003c/i\u003e on waking from sleep.\nIn all perception (except perhaps during the first weeks of life) there\nis a large element of interpretation due to past experience, and this\nelement is subjective when the present situation does not contain the\ncorrelations whose past occurrence has caused the interpretation.\u003c/p\u003e\n\n\u003cp\u003eAll these sources of error have to be guarded against if perception\nis not to mislead us. The ways of guarding against them are those\nsuggested by common sense and perfected by science; they are all\nsuch as to substitute laws with few or no exceptions for laws with a\ncomparatively large number of exceptions.\u003c/p\u003e\n\n\u003cp\u003eIt will be seen that very little can be inferred with confidence from\na single percept; we need observation from different points of view,\nand throughout a certain period of time. It is true that we shall\n\u003ci\u003eusually\u003c/i\u003e be right in what we infer from a single percept, but\nthat is because the objects\u003cspan class=\"pagenum\" id=\"Page_226\"\u003e[Pg 226]\u003c/span\u003e that surround us mostly belong to familiar\nkinds—men, horses, motor-cars, etc. But it would not be difficult\nto construct situations which would deceive at the first glance,\nespecially if we could be suddenly transported into a quite unfamiliar\nworld, like Wells\u0027s Martians. Water, for example, would completely\npuzzle a person who had never seen a liquid, if such a person could\nexist. In this matter, as elsewhere, we proceed step by step from the\neasy but precarious inferences of common sense to the difficult but\nmore reliable inferences of science.\u003c/p\u003e\n\n\u003cp\u003eWhere the intervening medium is relevant in inferring other members\nof a group from a percept, it is obvious that the single percept\nis theoretically inadequate as a basis for inference, since, by a\nchange in the medium, the same percept might be associated with a\ndifferent group. In this case, the distorting element in the medium\nmay be directly discovered by other percepts—\u003ci\u003ee.g.\u003c/i\u003e glass may be\ntouched—or it may be merely inferred by examining the way in which\npercepts belonging to one group change from place to place—\u003ci\u003ee.g.\u003c/i\u003e\nrefraction in air. When it has been inferred, the inference needs to be\ntested by examining whether it has further consequences which can be\nverified. All this is a commonplace.\u003c/p\u003e\n\n\u003cp\u003eIt remains to say something about the inference from percepts to\nevents which no one perceives. It is not its validity that I wish to\nexamine now, but its scope—\u003ci\u003ei.e.\u003c/i\u003e how much we can know about\nunperceived events, assuming the causal theory of perception. It is\nsometimes urged that an unperceived cause of a perception must be a\nmere \u003ci\u003eDing-an-sich\u003c/i\u003e or Spencerian Unknowable. This seems to me\nonly very partially true, if we accept the usual canons of scientific\ninference. We assume that differences in percepts imply differences in\nstimuli—\u003ci\u003ei.e.\u003c/i\u003e if a person hears two sounds at once, or sees two\ncolours at once, two physically different stimuli have reached his ear\nor his eye. This principle,\u003cspan class=\"pagenum\" id=\"Page_227\"\u003e[Pg 227]\u003c/span\u003e together with spatio-temporal continuity,\nsuffices to give a great deal of knowledge as to the \u003ci\u003estructure\u003c/i\u003e\nof stimuli. Their intrinsic characters, it is true, must remain\nunknown; but we may assume that the stimuli causing us to hear notes\nof different pitches form a series in respect of some character which\ncorresponds causally with pitch, and we may make similar assumptions in\nregard to colour or any other character of sensations which is capable\nof serial arrangement. And we can without difficulty extend geometry\nto the world outside our perceptions, although the space of that world\nwill only correspond to the space of perception in certain respects,\nand will be by no means identical with the space of perception.\u003c/p\u003e\n\n\u003cp\u003eWhat we assume is, formally, something like this: there is a roughly\none-one relation between stimulus and percept—\u003ci\u003ei.e.\u003c/i\u003e between\nthe events just outside the sense-organ and the event which we call a\nperception. This enables us to infer certain mathematical properties\nof the stimulus when we know the percept, and conversely enables us\nto infer the percept when we know these mathematical properties of\nthe stimulus Consequently, except when we are studying physiology or\npsychology, we may suppose that what is happening in a place is what\na person would perceive in that place, provided we use, in inference,\nonly those properties of the percept which it shares with the stimulus.\n\u003ci\u003eE.g.\u003c/i\u003e we must not use the blueness of blue, but we may use its\ndifference from red or yellow. We cannot argue that because a picture\nlooks beautiful, therefore there is beauty in the system of stimuli,\nbecause beauty may depend upon the actual qualities.\u003ca id=\"FNanchor_49\" href=\"#Footnote_49\" class=\"fnanchor\"\u003e[49]\u003c/a\u003e But nothing\nin physical science ever depends upon the actual qualities. Hence\nfor practical purposes in physics the difference between percept\nand stimulus only compels us to confine ourselves to the structural\nproperties of percepts; so long as we do this,\u003cspan class=\"pagenum\" id=\"Page_228\"\u003e[Pg 228]\u003c/span\u003e we need hardly trouble\nto remember that percept and stimulus are different. In physiology and\npsychology this does not hold, since we are concerned with the process\nintervening between stimulus and perception, or with perception itself.\u003c/p\u003e\n\n\u003cp\u003eEven in physics, it does not hold strictly, because the relation\nof stimulus and perception is not strictly one-one. It is only\napproximately so, even when we confine ourselves to stimuli to a given\nsense of a given person at a given time—\u003ci\u003e e.g.\u003c/i\u003e two colours\nwhich I perceive side by side. Even here, vagueness comes in, so that\nslightly different stimuli may give indistinguishable perceptions. This\nconstitutes an essential limitation to our knowledge, enshrined in the\nnotion of \"probable error.\" It can, however, be reduced to a minimum\nby the usual methods and constitutes, therefore, rather a practical\ndifficulty than a theoretical problem.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_49\" href=\"#FNanchor_49\" class=\"label\"\u003e[49]\u003c/a\u003e\nIf we accepted the theory that beauty depends only upon\n\"significant form,\" we should have to say that a musical score is as\nbeautiful as the music which it represents.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_229\"\u003e[Pg 229]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXII\"\u003eCHAPTER XXII\u003cbr\u003e\nTHE BELIEF IN GENERAL LAWS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHROUGHOUT out discussion of perception and the physical object, we\nhave assumed the validity of general laws. This is always assumed in\nscientific practice, but the reasons for assuming it are not very dear.\nAlthough the subject is not one on which it is easy to say anything\ndefinite, yet it seems necessary to examine it.\u003c/p\u003e\n\n\u003cp\u003eLike other scientific postulates, the belief in general laws is rooted\nin the properties of nervous tissue—the same properties which make\nus believe in induction and enable us to learn from experience. This\norigin, of course, affords no warrant for the truth of the belief,\nbut equally gives no reason against it. Indeed, so far as it goes, it\naffords a slight presumption in favour of the view that a great many\nevents are in accordance with general laws, since it shows that animals\nwhich act in a way which the truth of this belief would render rational\ncan survive. I should not wish, however, to lay stress upon such an\nargument.\u003c/p\u003e\n\n\u003cp\u003eWhen we first begin to think, we find ourselves acting in certain\nways which seem to succeed, and we set to work to rationalize our\nbehaviour. The natural way to do this is to say: Things \u003ci\u003ealways\u003c/i\u003e\nhappen that way. This so often succeeds that we acquire the habit of\nalways supposing that there is some general law according to which\nany particular event has occurred. This belief has two practical\nconsequences. First, when a set of events are all in accordance with\nsome law, we expect other similar events to be in accordance with it.\nSecondly, when a set of events appears irregular, we invent hypotheses\nto regularize it. Both procedures are important.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_230\"\u003e[Pg 230]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe first of these procedures is simply induction. As such, it is\nfundamental, in some form or other, and I propose to say no more about\nit.\u003c/p\u003e\n\n\u003cp\u003eThe second is more interesting for our purposes. When an induction\nfails in a surprising way—\u003ci\u003ee.g.\u003c/i\u003e when there is an eclipse—there\nare two things which a primitive man may do. He may regard the failure\nas a \"portent,\" in no way invalidating the general validity of the\ninduction, but showing that there is something strange, and probably\nterrifying, in the special circumstances connected with the astonishing\nevent. Or he may look for some general law different from that which\nhas hitherto proved adequate, in the hope that the new law may account\nfor the exceptional occurrence as well. The latter course will seldom\nbe adopted until a high degree of intellectual culture has been\nattained. If the odd event is on a large scale, it will be considered\nsuperstitiously, and if not, it will be simply ignored. Sometimes,\nhowever, a general law is found by accident, as a result of the careful\nrecords inspired by superstition. This evidently happened with the\nEgyptian priesthood, who learnt to predict eclipses, and probably only\nthen ceased to regard them with awe. Gradually, the view that there\nmust be \u003ci\u003esome\u003c/i\u003e law according to which strange things happened\nbecame more widespread. Dr Whitehead, in his \u003ci\u003eScience and the Modern\nWorld\u003c/i\u003e\u003ca id=\"FNanchor_50\" href=\"#Footnote_50\" class=\"fnanchor\"\u003e[50]\u003c/a\u003e traces the belief in natural laws to various sources,\nsuch as: Fate in Greek tragedy, the supremacy of Roman law, and the\nrationality of God in mediæval theology. In effect, however, he regards\nthe belief as having only acquired a firm hold of the scientific mind\nat the renaissance. Everything that he says on this subject is so\nexcellent that it is unnecessary to cover the ground again.\u003c/p\u003e\n\n\u003cp\u003eAlthough the belief in the universality of natural law was, at the time\nof the renaissance, a bold faith going far in advance of the evidence,\nit has since been so successful that it is now\u003cspan class=\"pagenum\" id=\"Page_231\"\u003e[Pg 231]\u003c/span\u003e possible to defend it\non inductive grounds. But there is some difficulty in deciding what we\nare to mean by it. I have dealt with this subject before,\u003ca id=\"FNanchor_51\" href=\"#Footnote_51\" class=\"fnanchor\"\u003e[51]\u003c/a\u003e and shall\nnow consider it only briefly.\u003c/p\u003e\n\n\u003cp\u003eThe regularities which we first observe, and in which we first believe,\nare of the simple form: \"\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is always accompanied (or preceded or\nsucceeded) by \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e.\" But all such regularities are capable of having\nexceptions, and science soon seeks laws of a different kind. We arrive\nin the end (possibly not at the very end) at differential equations. I\nthink that these are of two kinds, those expressing persistence, and\nthose expressing accelerations (in a generalized sense). The former are\nconcealed, more or less, by the assumption of permanent substance; but\nthis is a topic which I shall consider in the next chapter. The latter\nare the ordinary differential equations of the second order which occur\nthroughout mathematical physics. But in addition to these, in order to\nproduce observed macroscopic results, there must be statistical laws\ngoverning quantum changes and radio-active disruptions of atoms. I want\nto inquire whether we are saying anything significant in assuming that\nthere are laws governing the course of the physical world, or whether\n\u003ci\u003eany\u003c/i\u003e set of percepts must be amenable to law by a sufficiently\nliberal use of hypothesis.\u003c/p\u003e\n\n\u003cp\u003eIt is by no means clear that the accepted laws of physics make certain\nimaginable series of percepts impossible; still less that the mere\nexistence of laws would have this effect. Take, \u003ci\u003ee.g.\u003c/i\u003e continuity.\nChanges which appear sudden (\u003ci\u003ee.g.\u003c/i\u003e explosions) can be resolved\ninto a number of continuous though rapid changes: \u003ci\u003eper contra\u003c/i\u003e,\nsituations in which there appears to be no change (\u003ci\u003ee.g.\u003c/i\u003e a\nsteadily glowing gas) are resolved into a number of discontinuous\nchanges. Thus we can neither infer the absence of physical continuity\nfrom the absence of\u003cspan class=\"pagenum\" id=\"Page_232\"\u003e[Pg 232]\u003c/span\u003e continuity in percepts, nor the presence of\nphysical continuity from the presence of continuity in percepts.\nAgain: if percepts change in unexpected ways, we infer unperceived\nmatter; and by a sufficient amount of unperceived matter almost any\nseries of percepts could be explained. Of course a particular law is\nstrengthened when it enables us to predict percepts, but this belongs\nto the arguments in favour of such-and-such laws, not to the arguments\nin favour of laws in general. We can have evidence in favour of\nsuch-and-such a law without having evidence for laws in general. But\nhere we must make some distinctions. Evidence in favour of a particular\nlaw is evidence that a certain class of phenomena are subject to a rule\nwhich we have succeeded in discovering. If so, they are sure to be also\nsubject to other rules sensibly indistinguishable from the one for\nwhich we have evidence; but these will in general be more complicated\nthan the rule which we adopt. Complication may be of two kinds: it may\nbe in the formula, or in the amount of hypothetical matter needed to\nmake the rule work. The great merit of Newtonian gravitation was that\nit was simple in both respects. But clearly any set of observations\non planetary motions could have been fitted into the Newtonian\nformula by postulating a sufficient number of invisible bodies or a\nsufficient complication in the law of attraction. For any given set\nof observations, there would have been many such possible methods of\nbringing harmony between observation and theory; most of these would\nnot have been compatible with a fresh set of observations, but some of\nthem would have been, given sufficient mathematical ingenuity. What\nis remarkable, therefore, is not the reign of law, but the reign of\n\u003ci\u003esimple\u003c/i\u003e laws. If the transfer of energy were subject to laws as\ncomplicated as those governing the transfer of English land, we should\nnever succeed in discovering them: there would always remain a number\nof possible codes, all of which would fit all known relevant facts.\u003cspan class=\"pagenum\" id=\"Page_233\"\u003e[Pg 233]\u003c/span\u003e\nThe principle of induction, as practically employed, is the principle\nthat the \u003ci\u003esimplest\u003c/i\u003e law which fits the known facts will also fit\nthe facts to be discovered hereafter. This principle, in all its naked\nsimplicity, has come to the fore in Einstein\u0027s theory of gravitation,\nwhich consists in taking the simplest available tensor equation in\npreference to the others that are mathematically possible.\u003c/p\u003e\n\n\u003cp\u003eIt may be said that the principle of simple laws is purely heuristic,\nand of course this is true to a considerable extent. No sensible\nmathematician would test a complicated formula before testing a simple\none. But the remarkable thing is that the simple formula so often turns\nout right. From the trend of physics, it seems as though complication\nwere geographical rather than legal. Organic compounds have an\nimmensely complicated structure, but there is no reason to suppose that\ntheir fundamental laws are other than those which govern the hydrogen\natom. Professor J. B. Haldane, it is true, thinks otherwise, and so\ndo all varieties of vitalists. But, to a layman, their arguments seem\ninconclusive, and they are rejected by many competent authorities. It\nis therefore at least a tenable hypothesis that all matter is governed\nby very simple laws. This is so remarkable that it almost suggests\nsome relation to Mr Keynes\u0027s \"principle of limitation of variety,\"\nand seems to confirm his hint that Nature may be really like the urn\ncontaining white and black balls which plays such a prominent part in\nthe theory of probability. Some Mendelians would make us think of human\nbeings in this way. Suppose there were a hundred pairs of characters,\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.825ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-361.png\" alt=\"\" data-tex=\"\\(a\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.598ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-293.png\" alt=\"\" data-tex=\"\\(b\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.607ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-294.png\" alt=\"\" data-tex=\"\\(c\u0027\\)\"\u003e, etc., such that every\nhuman being possessed by inheritance one but not both of the characters\nin each pair. This would make the number of differing human embryos\n\u003cimg style=\"vertical-align: 0; width: 3.719ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-362.png\" alt=\"\" data-tex=\"\\(2^{100}\\)\"\u003e—\u003ci\u003ei.e.\u003c/i\u003e about \u003cimg style=\"vertical-align: -0.05ex; width: 4.05ex; height: 2.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-363.png\" alt=\"\" data-tex=\"\\(10^{20}\\)\"\u003e. If this is thought too few,\nwe can take more pairs of characters. Views of this sort cannot be\nrejected out of hand, and they are strongly suggested by the success\nof\u003cspan class=\"pagenum\" id=\"Page_234\"\u003e[Pg 234]\u003c/span\u003e induction and the prevalence of simple laws. Let us, therefore, ask\nonce more: What evidence is there that simple laws prevail, and how\nmuch reason have we to be surprised by the degree of their prevalence?\u003c/p\u003e\n\n\u003cp\u003eAs I have pointed out on a former occasion, it would be fallacious to\nargue inductively from the simplicity of the laws we have discovered\nto the probable simplicity of undiscovered laws. For, if some laws are\nsimple and some complicated, we are likely to discover the simple laws\nfirst. We have toi proceed more cautiously. First, is it surprising\nthat there are \u003ci\u003eany\u003c/i\u003e simple laws? Secondly, have we any ground for\nbelieving, as was suggested just now, that \u003ci\u003eall\u003c/i\u003e phenomena are\ngoverned by simple laws?\u003c/p\u003e\n\n\u003cp\u003eSimplicity is best established at the two opposite extremes of size:\nastronomy and the atom. The latter, however, is much more significant\nfor our inquiry, since the simplicity of astronomy may result from\naveraging. As we saw in Part I., the theory of the atom amounts,\nbroadly, to this: An atom is composed of electrons and protons, the\nlatter being all in the nucleus, the former partly in the nucleus\n(except in hydrogen), partly planetary. The number of protons in the\nnucleus gives the atomic weight; the excess of the number of protons\nover that of electrons in the nucleus gives the atomic number. When\nthe atom is unelectrified, the number of planetary electrons is equal\nto the atomic number. If the quantum theory is correct, an atom has a\ncertain number of characters, each measured by integers called quantum\nnumbers, which are always small. It has also a property called energy,\nwhich is a function of the quantum numbers; and in connection with each\nof the quantum numbers there is a periodic process which is subject to\nquantum rules. Each quantum number is capable of changing suddenly from\none integer to another. When the atom is left to itself, these changes\nwill only be such as to diminish the energy, but when it is receiving\nenergy from\u003cspan class=\"pagenum\" id=\"Page_235\"\u003e[Pg 235]\u003c/span\u003e elsewhere the changes may increase the energy. All this,\nhowever, is more or less hypothetical. What we really know about is the\ninterchange of energy between the atom and the surrounding space; here\nthere are simple laws as to the form the radiant energy will take. But\nthere are at present no laws determining \u003ci\u003ewhen\u003c/i\u003e quantum changes\nwill take place in the atom, though the changes that are possible are a\ndefinite known set.\u003c/p\u003e\n\n\u003cp\u003eAs we are only considering how far simple laws \u003ci\u003ecan\u003c/i\u003e account for\nthe phenomena, we may accept the view of the atom as a miniature solar\nsystem, governed, except as to quantum changes, by attractions and\nrepulsions among its electrons. Nevertheless it remains a fact that\nthe atom only indicates its presence when it suffers a quantum change,\nand that we know of no laws determining why, at a given moment, such\na change takes place in some atoms rather than in others. The laws\ngoverning the intensity of the light emitted by a gas are statistical\nlaws. This suggests a world in which the number of possibilities is\nfinite, but the choice among possibilities is left purely to chance. We\nmight suppose, as Poincaré once suggested, and as Pythagoras apparently\nbelieved, that space and time are granular, not continuous—\u003ci\u003ei.e.\u003c/i\u003e\nthe distance between two electrons may be always an integral multiple\nof some unit, and so may the time between two events in the history of\none electron. This, together with the fact that the number of electrons\nis finite, would give a finite number of possible situations for each\nelectron. And it may be that the choice among possible situations is\nwholly a matter of chance. In that case, the apparent regularity of the\nworld will be due to the \u003ci\u003eabsence\u003c/i\u003e of laws. I think it improbable\nthat such a view could be developed satisfactorily, but at least we\nmust take account of it before we attach undue importance to the\nappearance of law in the world.\u003c/p\u003e\n\n\u003cp\u003eThe real objection to a philosophy founded upon such a\u003cspan class=\"pagenum\" id=\"Page_236\"\u003e[Pg 236]\u003c/span\u003e theory of the\nuniverse as we have been considering is that, after all, we still\nneed statistical laws, which will involve a \"random distribution,\" or\nsomething of the kind. Such laws are still laws, though they differ\nfrom others by seeming \u003ci\u003ea priori\u003c/i\u003e probable instead of improbable.\nTo this extent, it is a gain if we can base science upon them; but it\nwould not be correct to say that, in that case, science would have\nsucceeded in doing without laws. We could no longer say, however, that\nthe laws of science were surprising; on the contrary, we should be\nsurprised by their failure.\u003c/p\u003e\n\n\u003cp\u003eThere is another question to be considered, and that is as to the scope\nof simple laws. It cannot be pretended that we \u003ci\u003eknow\u003c/i\u003e the laws\ngoverning the hydrogen atom to be sufficient to account for all that\nhappens to matter, especially to organic matter. This is at present\nmerely a hypothesis. All science uses laws based upon observation,\nwhich may or may not be deducible by a celestial mathematician from\nthe laws governing electrons, but are not likely ever to be deducible\nby mathematicians on this planet. And when we come to such matters as\nphysiology, the laws are no longer such as to enable us to say, with\nany confidence, just what is going to happen; they give tendencies\nrather than precise mathematical rules. It would be rash to maintain\nthat such rules must exist; we may do well to look for them, but not\nwell to feel quite certain that they are to be found.\u003c/p\u003e\n\n\u003cp\u003eOn the whole, the tendency of the foregoing discussion has been to\nsuggest that it is easy to exaggerate the evidence for simple laws\nin the physical world. Where we know most—\u003ci\u003e i.e.\u003c/i\u003e in regard to\nthe structure of the atom—there is, so far as we know, a complete\nabsence of law in certain very important respects. Where we know less,\nthe laws may be purely statistical. The amount of law known to exist\nin the physical world is, therefore, less surprising than it seems\nat first sight, and there is no conclusive reason for believing that\nall natural\u003cspan class=\"pagenum\" id=\"Page_237\"\u003e[Pg 237]\u003c/span\u003e occurrences happen in accordance with laws which suffice\nto determine them given a sufficient knowledge of their antecedents.\nScience must continue to postulate laws, since it is coextensive with\nthe domain of natural law. But it need not assume that there are\nlaws everywhere; it need only assume, what is evident since it is a\ntautology, that there are laws wherever there is science.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_50\" href=\"#FNanchor_50\" class=\"label\"\u003e[50]\u003c/a\u003e\nChap, \u003cspan class=\"allsmcap\"\u003eI\u003c/span\u003e., especially p. 5 ff.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_51\" href=\"#FNanchor_51\" class=\"label\"\u003e[51]\u003c/a\u003e\nCf. \"On the Notion of Cause,\" in \u003ci\u003eMysticism and\nLogic\u003c/i\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_238\"\u003e[Pg 238]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXIII\"\u003eCHAPTER XXIII\u003cbr\u003e\nSUBSTANCE\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE question of substance in the philosophy of physics has three\nbranches: logical, physical, and epistemological. The first is a\nproblem in pure philosophy: is the notion of \"substance\" in any sense a\n\"category,\" \u003ci\u003ei.e.\u003c/i\u003e forced upon us by the general nature either of\nfacts or of knowledge? The second is a question of the interpretation\nof mathematical physics: is it (\u003ci\u003ea\u003c/i\u003e) necessary or (\u003ci\u003eb\u003c/i\u003e)\nconvenient to interpret our formulæ in terms of permanent entities\nwith changing states and relations? The third concerns the special\ntopics with which we are concerned in Part II.—namely, the relation\nof perception to the physical world. The first and second problems\nreally belong to other portions of the philosophy of matter, but I\nshall discuss them here in order to obtain a unified discussion of the\nproblem of substance.\u003c/p\u003e\n\n\u003cp\u003eLogically, \"substance\" has played a very important part in the past,\nand is still perhaps less obsolete than might be supposed. A substance\nmay be defined in purely logical terms as \"that which can only enter\ninto a proposition as subject, never as predicate or relation.\" This\ndefinition is practically that of Leibniz, except that he does not\nmention relations, since he held them to be unreal. We shall do well,\nhowever, to include them, because the logical position of substance is\nnot much affected thereby, and it may, I hope, be now taken for granted\nthat relations areas \"real\" as predicates.\u003c/p\u003e\n\n\u003cp\u003eMetaphysically, substances have generally been held to be\nindestructible. But this opinion is not justified by the logical\ndefinition, though many philosophers have supposed that it\u003cspan class=\"pagenum\" id=\"Page_239\"\u003e[Pg 239]\u003c/span\u003e was. When\nI wish to discuss a substance having this further attribute, I shall\nspeak of it as a \"permanent substance\"; when I use the word \"substance\"\nwithout qualification, I shall mean only substance in the logical\nsense, leaving the question of duration open.\u003c/p\u003e\n\n\u003cp\u003eIt is extraordinarily difficult, in considering substance from the\npoint of view of logic, to avoid being unduly influenced by the\nstructure of language. All languages commonly known to civilized people\nconsist of sentences which can be analyzed into subject and predicate,\ntwo subjects and a dyadic relation, three subjects and a triadic\nrelation, etc., together with relations between such units, expressed\nby \"or\" or \"if\" or some analogous word. I do not know whether the same\ncan be said of African, Australian, or other uncivilized languages.\nBut certainly it can be said of all the languages that philosophers\nhave known. Logic, as ordinarily conceived, takes over this linguistic\nscheme, and is inclined to attribute metaphysical importance to it.\nWe can hardly resist the belief that the structure of the sentence\nreproduces the structure of the fact which it asserts, or, in the case\nof false sentences, of the fact which would exist if the assertion\nwere true. This belief, natural as it is, seems very implausible when\nexplicitly stated. Nevertheless, I believe that it has some element of\ntruth, though it is very hard to disentangle this element. An attempt\nwas made by Wittgenstein,\u003ca id=\"FNanchor_52\" href=\"#Footnote_52\" class=\"fnanchor\"\u003e[52]\u003c/a\u003e and I have been much influenced by his\npoint of view.\u003c/p\u003e\n\n\u003cp\u003eIf we admit, as it seems natural to do, that some sentences, taken in\ntheir usual meaning, correspond to facts, while others do not, we must\nsuppose that the structure of sentences is related, in some way, to\nthe structure of facts, since otherwise such correspondence would be\nimpossible. Moreover, a sentence is a physical fact, and may therefore\nbe expected to be capable of correspondence with other physical facts.\nThese\u003cspan class=\"pagenum\" id=\"Page_240\"\u003e[Pg 240]\u003c/span\u003e two arguments come from quite different intellectual regions,\nthe one being logical, the other physical. If we were discussing\nanything other than physics, they would work in opposite directions,\nand tend to show that we cannot understand (at least verbally)\nanything having a structure radically different from that of events in\nspace-time. For our purposes, however, the two arguments are concurrent.\u003c/p\u003e\n\n\u003cp\u003eLet us, for a moment, consider a sentence as a physical occurrence.\nWe must distinguish between spoken and written sentences, since the\nformer are evanescent events while the latter are pieces of matter. We\nmust also distinguish between a sentence in the sense in which it is\nunique on each occasion when it is uttered or written, and a sentence\nin the sense in which the same sentence occurs at a given place in\neach copy of the same book. \u003ci\u003eE.g.\u003c/i\u003e Jeremiah xvii. 9 is a sentence\nin the latter sense; in the former sense, the particular series of\nshapes at that point in my Bible constitute a sentence, while those in\nyours constitute another (similar) sentence. The former sense comes\nfirst when we are considering a sentence as a physical occurrence; the\nlatter, when we are considering it as having \"meaning.\"\u003c/p\u003e\n\n\u003cp\u003eA spoken sentence, considered physically, is a series of noises from\nthe point of view of the hearer, and a series of movements in the mouth\nand throat from the point of view of the speaker. The \"meaning\" of the\nsentence depends upon the causes of the spoken words and the effects of\nthe heard words.\u003ca id=\"FNanchor_53\" href=\"#Footnote_53\" class=\"fnanchor\"\u003e[53]\u003c/a\u003e But for the moment let us ignore \"meaning.\" Then\nwe find that the sentence consists essentially of noises in order: the\norder is as essential as the character of the noises. (In a language\nlike Latin, this is not so true of the separate words as in a modern\nlanguage, but it is just as true of the parts of words: \"Roma\" is a\ndifferent word from \"amor.\") Considered as physical occurrences, the\nwords expressing different\u003cspan class=\"pagenum\" id=\"Page_241\"\u003e[Pg 241]\u003c/span\u003e parts of speech are indistinguishable;\nnevertheless there are relations which are symbolized by relations\namong words, not by words. Consider \"Brutus killed Cæsar\" and \"Cæsar\nkilled Brutus.\" The difference between these two statements is\nindicated, in an uninflected language, not by a word, but by a relation\namong words. Thus a spoken sentence consists of certain noises in a\ncertain temporal order. In the sentence, we can distinguish terms and\nrelations: the terms are the words (or, more strictly, the elementary\nnoises which, in a phonetic system, would each be represented by a\nseparate letter), and the relations are temporal relations among\nevents. According to our definition, the elementary noises composing\nthe sentence may count as \"substances,\" in spite of the fact that they\nare evanescent.\u003c/p\u003e\n\n\u003cp\u003eIn the case of written words, the sentence is no longer a temporal\nseries of events, but a spatial series of material structures. It\nis not essential to a written sentence that its parts should stand\nfor sounds: in some languages (\u003ci\u003ee.g.\u003c/i\u003e Chinese) this is not the\ncase, and there is some reason to think that writing developed from\npictures, not from the attempt to symbolize speech. We may therefore\ntreat the written language as an independent method of conveying\nmeaning. It is obvious that its efficacy in this respect depends upon\nits capacity for causing visual perceptions (or tactual perceptions in\nthe case of \"Braille\"). Written words, even Chinese ideograms, consist\nessentially of parts with a structure, and the structure is essential\nto the meaning. This is equally the case with a sentence, even in\nLatin. Take \"Cæsar amat Brutum\" and \"Cæsarem amat Brutus.\" Here the\ncase-endings may be regarded as separate words (which they probably\nwere originally), whose position relative to the stem \"Brut\" or \"Cæsar\"\nindicates the \"sense\" of the relation asserted.\u003c/p\u003e\n\n\u003cp\u003eThe written language depends upon the causal theory of\u003cspan class=\"pagenum\" id=\"Page_242\"\u003e[Pg 242]\u003c/span\u003e perception\nand the existence of physical objects; the spoken language involves\nthe former, but not the latter. Thus in the written language the\n\"substantial\" elements have a permanence (throughout some finite time)\nwhich they do not have in the spoken language. Their permanence,\nhowever, is not metaphysical or absolute; it is only like that of\nhouses or trees. It depends upon the fact that matter arranged in\ncertain patterns will often retain those patterns for a long time,\nthough not for ever. And the essential thing about writing is its\ncapacity for causing visual events.\u003c/p\u003e\n\n\u003cp\u003eSo far, we have seen no reason to suppose that the suggestions of\nlanguage are misleading where the physical world is concerned,\nsince language is a physical phenomenon, and must share whatever\nstructure all such phenomena have in common. But the philosophy which\nhas been based on language—or, perhaps, has moulded language—has\nfurther elements which are more dubious. These are derived from the\ndistinctions between parts of speech. Philosophers have, as a rule,\nfailed to notice more than two types of sentence, exemplified by\nthe two statements \"this is yellow\" and \"buttercups are yellow.\"\nThey mistakenly supposed that these two were one and the same type,\nand also that all propositions were of this type. The former error\nwas exposed by Frege and Peano; the latter was found to make the\nexplanation of order impossible. Consequently the traditional view\nthat all propositions ascribe a predicate to a subject collapsed, and\nwith it the metaphysical systems which were based upon it, consciously\nor unconsciously. This did away with the objections to pluralism as a\nmetaphysic.\u003c/p\u003e\n\n\u003cp\u003eBut there remain certain linguistic distinctions which \u003ci\u003emay\u003c/i\u003e have\nmetaphysical importance. There are proper names, adjectives, verbs,\nprepositions, and conjunctions. It is natural to hold that, in an ideal\nlanguage, proper names would indicate substances, adjectives would\nindicate the properties\u003cspan class=\"pagenum\" id=\"Page_243\"\u003e[Pg 243]\u003c/span\u003e by means of which substances are collected\ninto classes, verbs and prepositions would indicate relations, and\nconjunctions would indicate the relations between propositions by means\nof which we build up what are called \"truth-functions.\"\u003ca id=\"FNanchor_54\" href=\"#Footnote_54\" class=\"fnanchor\"\u003e[54]\u003c/a\u003e If there\nreally are these categories in the world, it is desirable that language\nshould symbolize them, and metaphysical errors are likely to result if\nlanguage performs this task inaccurately. For my part, I believe that\nthere are such categories, except, perhaps, conjunctions. But I will\nnot argue the question at this point, since I wish, as far as possible,\nto avoid metaphysic.\u003c/p\u003e\n\n\u003cp\u003eOne point in which language tends to mislead is that the words which\nsymbolize relations are themselves just as substantial as other words.\nIf we say \"Cæsar loves Brutus,\" the word \"loves,\" considered as a\nphysical event, is of exactly the same kind as the words \"Cæsar\" and\n\"Brutus,\" but is supposed to mean something of a totally different\nkind. It follows that the relation of a word to its meaning must be\ndifferent according to the category to which the meaning belongs. There\nis in the above sentence a relation which is symbolized by a relation,\nnot by a word; this is the three-term relation of love to Cæsar and\nBrutus. This is symbolized by the order of the words—\u003ci\u003ei.e.\u003c/i\u003e by\na three-term relation. But in order to mention this relation, it is\nnecessary to treat \"love\" grammatically as a substantive, which tends\nto confuse the distinction between a substance and a relation. However,\nit is not very difficult to avoid the false suggestions due to this\npeculiarity of language, when once the danger of them has been pointed\nout.\u003c/p\u003e\n\n\u003cp\u003eI come now to the second part of our inquiry concerning substance.\nAssuming that the physical world consists of substances with qualities\nand relations, are these substances to be taken as permanent bits of\nmatter, or as brief events?\u003cspan class=\"pagenum\" id=\"Page_244\"\u003e[Pg 244]\u003c/span\u003e Common sense holds the former view, though\nits \"things\" are only quasi-permanent. But science has found means of\nresolving \"things\" into groups of electrons and protons, each of which\n\u003ci\u003emay\u003c/i\u003e be quite permanent. As we saw in Part I., there are some who\nthink that an electron and a proton can annihilate each other, so that\neven they are not quite permanent. But the question of permanence is\nnot the one which most concerns us. The question is: Are electrons and\nprotons part of the ultimate stuff of the world, or are they groups of\nevents, or causal laws of events?\u003c/p\u003e\n\n\u003cp\u003eWe have already seen that the physical object, as inferred from\nperception, is a group of events arranged about a centre. There\n\u003ci\u003emay\u003c/i\u003e be a substance in the centre, but there can be no reason\nto think so, since the group of events will produce exactly the same\npercepts; therefore the substance at the centre, if there is one,\nis irrelevant to science, and belongs to the realm of mere abstract\npossibility. If we can reach the same conclusion as regards matter in\nphysics, we have diminished the difficulty involved in building our\nbridge from perception to physics.\u003c/p\u003e\n\n\u003cp\u003eThe substitution of space-time for space and time has made it much\nmore natural than formerly to conceive a piece of matter as a group of\nevents. Physics starts, nowadays, from a four-dimensional manifold of\nevents, not, as formerly, from a temporal series of three-dimensional\nmanifolds, connected with each other by the conception of matter\nin motion. Instead of a permanent piece of matter, we have now the\nconception of a \"world-line,\" which is a series of events connected\nwith each other in a certain way. The parts of one light-ray are\nconnected with each other in a manner which enables us to consider them\nas forming, together, one light-ray; but we do not conceive a light-ray\nas a substance moving with the velocity of light. Just the same kind\nof connection may be held to constitute the unity of an electron. We\nhave\u003cspan class=\"pagenum\" id=\"Page_245\"\u003e[Pg 245]\u003c/span\u003e a series of events connected together by causal laws; these may\nbe taken to \u003ci\u003ebe\u003c/i\u003e the electron, since anything further is a rash\ninference which is theoretically useless.\u003c/p\u003e\n\n\u003cp\u003eWhat is peculiar about a string of events which physics takes as\nbelonging to one electron is a character which is present approximately\nin the common-sense \"thing,\" a character which I should define as the\nexistence of a first-order differential law connecting successive\nevents along a linear route. That is to say, given an event belonging\nto an electron at one place in space-time, there will be other events\nat certain neighbouring regions of space-time, separated from the\nfirst and from each other by small time-like intervals, such that,\nwhen the intervals are taken small enough, if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e are\nthree such events, and the interval between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e is equal\nto that between \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, then the difference between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e tends towards equality with the difference between \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, in certain measurable respects. This is a way of saying that\naccelerations are always finite—or, where they are not (as perhaps in\nquantum phenomena), there are other characteristics involved which are\nsubject to a condition analogous to finite acceleration. Let us take\nfirst the common-sense \"thing.\" If I watch a moving object, I have a\nseries of percepts which change gradually, both as regards position\nand as regards qualities—colour, shape, etc. The gradualness of the\nchange is the criterion by which I am led to regard the percepts as\nall belonging to one \"thing.\" But on a common-sense basis there are\nexceptions, such as explosions. Science deals with these as rapid, but\nnot instantaneous, changes, and so removes the exceptions. We thus\narrive at the conclusion that, given an event \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e at a time \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e,\nthere will be closely analogous events at neighbouring times. We may\nsymbolize this by saying that, if there is an event \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e at time\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-84.png\" alt=\"\" data-tex=\"\\(t\\)\"\u003e, there will be, at any neighbouring time \u003cimg style=\"vertical-align: -0.186ex; width: 5.576ex; height: 1.756ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-364.png\" alt=\"\" data-tex=\"\\(t + dt\\)\"\u003e an event:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 22.73ex; height: 2.565ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-365.png\" alt=\"\" data-tex=\"\\[\nx + f_1(x)dt + f_2(x)dt^{2},\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_246\"\u003e[Pg 246]\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.566ex; width: 5.151ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-366.png\" alt=\"\" data-tex=\"\\(f_1(x)\\)\"\u003e is a continuous function of the time, while \u003cimg style=\"vertical-align: -0.566ex; width: 5.151ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-367.png\" alt=\"\" data-tex=\"\\(f_2(x)\\)\"\u003e\nis determined by the second-order differential equations of physics.\nThe string of events so connected is called one piece of matter. In the\ncase of the sudden changes contemplated by the quantum theory, there is\nstill continuity in everything except spatial position, and the spatial\nposition undergoes a change which is one of a small number of possible\nchanges. Thus in this case also the new occurrences can be causally\nconnected with the old, though the laws of the connection are somewhat\ndifferent from what they are in the usual case.\u003c/p\u003e\n\n\u003cp\u003eThus the string of events constituting one material unit is\ndistinguished from others by the existence of an intrinsic causal law,\nthough this law is only differential. A light-wave, in this respect, is\nanalogous to a material unit; it differs in the fact that it spreads\nspherically instead of travelling along a linear route.\u003ca id=\"FNanchor_55\" href=\"#Footnote_55\" class=\"fnanchor\"\u003e[55]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eIt will be seen that, if a piece of matter is a string of events, the\ndistinction between motion and other continuous changes is not so\nsimple as it seemed. We could form continuous series of events which\nwould not all belong to one piece of matter; therefore the change\nfrom one to another would not be a \"motion.\" A \"motion\" is a string\nof events connected| with each other according to the laws of motion.\nThis might seem like a vicious circle, but in fact it is not. What we\nassert is: Strings of events exist which are connected with each other\naccording to the laws of motion; one such string is called (me piece of\nmatter, and the transition from one event in the string to another is\ncalled a motion. This contains as much as can be verifiable in physics,\nsince every percept is an event. There is no mathematical advantage\nin asserting more, and to assert more is to go beyond the evidence.\nTherefore it is prudent, in physics, to regard an electron as a group\nof events connected\u003cspan class=\"pagenum\" id=\"Page_247\"\u003e[Pg 247]\u003c/span\u003e together in a certain way. An electron may be a\n\"thing,\" but it is absolutely impossible to obtain any evidence for or\nagainst this possibility, which is scientifically unimportant, because\nthe group of events has all the requisite properties.\u003c/p\u003e\n\n\u003cp\u003eThe light thrown on the notion of substance by the connection between\nphysics and perception, which was the third branch of our problem, has\nalready been touched upon. We saw in former chapters that the physical\nobject to be inferred from perception is a group of events, rather\nthan a single \"thing.\" Percepts are always events, and common sense is\nrash when it refers them to \"things\" with changing states. There is\ntherefore every reason, from the standpoint of perception, to desire\nan interpretation of physics which dispenses with permanent substance.\nAs we have seen that such an interpretation is possible, we shall\nhenceforth adopt it.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, a view not uncommon in philosophy, and perhaps\nnearer to common sense than the view which I have adopted. This view\nis, I think, that of Dr Whitehead. It holds that the different events\nwhich constitute a group—whether those which make up a physical\nobject at one time or those which make up the history of a physical\nobject—are not \u003ci\u003elogically\u003c/i\u003e self-subsistent, but are mere\n\"aspects,\" implying other aspects in some sense which is not merely\ncausal or inductively derived from observed correlations. I consider\nthis view impossible on purely logical grounds, and have so argued\nelsewhere. But at the moment I prefer to argue that it is empirically\nuseless. Given a group of events, the evidence that they are \"aspects\"\nof one \"thing\" must be inductive evidence derived from perception,\nand must be exactly the same as the evidence upon which we have\nrelied in collecting them into causal groups. The supposed logical\nimplications, if they exist, cannot be discovered by logic, but only\nby observation; no one, by mere reasoning, could avoid being deceived\nby the three-card trick. Moreover, in calling two events \"aspects\" of\none \"thing,\" we imply that their likeness is more important than their\ndifference; but for science both are facts, and of exactly the same\nimportance. One may say that the theory of relativity has grown up by\npaying attention to small differences between \"aspects.\" I conclude,\ntherefore, that the \"thing\" with \"aspects\" is as useless as permanent\nsubstance, and represents an inference which is as unwarrantable as it\nis unnecessary.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_52\" href=\"#FNanchor_52\" class=\"label\"\u003e[52]\u003c/a\u003e\n\u003ci\u003eTractatus Logico-Philosophicus.\u003c/i\u003e\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_53\" href=\"#FNanchor_53\" class=\"label\"\u003e[53]\u003c/a\u003e\nCf. \u003ci\u003eAnalysis of Mind\u003c/i\u003e, chap. \u003cspan class=\"allsmcap\"\u003eX\u003c/span\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_54\" href=\"#FNanchor_54\" class=\"label\"\u003e[54]\u003c/a\u003e\nSee \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. \u003cspan class=\"allsmcap\"\u003eI\u003c/span\u003e.,\nIntroduction to second edition.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_55\" href=\"#FNanchor_55\" class=\"label\"\u003e[55]\u003c/a\u003e\nThe non-substantial character of the election emerges\neven more forcibly from the Heisenberg theory mentioned in Chapter IV.\nthan from the older theory.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_249\"\u003e[Pg 249]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXIV\"\u003eCHAPTER XXIV\u003cbr\u003e\nIMPORTANCE OF STRUCTURE IN SCIENTIFIC INFERENCE\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE inference from perception to physics, which we have been\nconsidering, is one which depends upon certain postulates, the chief of\nwhich, apart from induction, is the assumption of a certain similarity\nof structure between cause and effect where both are complex. I want,\nin this chapter, to inquire more closely into this postulate, not with\na view to establishing its validity, which I shall take for granted,\nbut with a view to discovering what it asserts and what are its\nconsequences.\u003c/p\u003e\n\n\u003cp\u003eThe first point is to be clear as to what we mean by structure. The\nnotion is not applicable to classes, but only to relations or systems\nof relations. It is fully defined, and made the basis of a general kind\nof arithmetic, in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e.\u003ca id=\"FNanchor_56\" href=\"#Footnote_56\" class=\"fnanchor\"\u003e[56]\u003c/a\u003e But as the later\nparts of that book are not read, I may be excused for repeating, in\noutline, what is needed for our present purposes.\u003c/p\u003e\n\n\u003cp\u003eTwo relations \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e are said to be \"similar\" if there is a\none-one relation between the terms of their fields, which is such\nthat, whenever two terms have the relation \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, their correlates\nhave the relation \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e, and vice versa. The most familiar example\nis that of series: two series are similar when their terms can be\ncorrelated without change of order. But it would be a great mistake to\nsuppose that series are the only important application of the notion\nof similarity between relations. A map, for example, if accurate, is\nsimilar to the region which it maps. A book spelt phonetically is\nsimilar to the sounds produced when it is read aloud. A gramophone\nrecord is similar to the music which it produces. And so on.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_250\"\u003e[Pg 250]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIt should be observed that similarity applies not only to two-term\nrelations, but to relations with any number of terms. Suppose we have\ntwo relations \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e, \u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e each \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e-adic; suppose there is a one-one\nrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e which relates all the terms in the field of \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e to all\nthe terms in the field of \u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e; let \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e, … \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e be\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e terms which have the relation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e and let \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-368.png\" alt=\"\" data-tex=\"\\(x_1\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-369.png\" alt=\"\" data-tex=\"\\(x_2\u0027\\)\"\u003e,\n… \u003cimg style=\"vertical-align: -0.576ex; width: 2.442ex; height: 2.294ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-370.png\" alt=\"\" data-tex=\"\\(x_n\u0027\\)\"\u003e be the terms correlated with them by the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e.\nThen \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e and \u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e are similar if there is a one-one relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e\nsuch that, when the above conditions are fulfilled, \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-368.png\" alt=\"\" data-tex=\"\\(x_1\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.583ex; width: 2.282ex; height: 2.3ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-369.png\" alt=\"\" data-tex=\"\\(x_2\u0027\\)\"\u003e,\n… \u003cimg style=\"vertical-align: -0.576ex; width: 2.442ex; height: 2.294ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-370.png\" alt=\"\" data-tex=\"\\(x_n\u0027\\)\"\u003e have the relation \u003cimg style=\"vertical-align: -0.048ex; width: 2.345ex; height: 1.765ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-351.png\" alt=\"\" data-tex=\"\\(R\u0027\\)\"\u003e, and conversely.\u003c/p\u003e\n\n\u003cp\u003eTwo relations which are similar have the same \"structure\" or\n\"relation-number.\" The \"relation-number\" of a relation is the same as\nits \"structure,\" and is defined as the class of all relations similar\nto the given relation. Relation-numbers satisfy all the formal laws\nof arithmetic which are satisfied by transfinite ordinal numbers;\nordinal numbers, both finite and transfinite, are a particular kind\nof relation-numbers—namely, the relation-numbers of relations which\ngenerate well-ordered series.\u003c/p\u003e\n\n\u003cp\u003eThe formal laws satisfied by relation-numbers are:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -6.813ex; width: 31.488ex; height: 14.758ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-371.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\n(\\alpha+\\beta)+\\gamma \u0026 =\\alpha+(\\beta+\\gamma) \\\\\n(\\alpha \\times \\beta) \\times \\gamma \u0026 =\\alpha \\times(\\beta \\times \\gamma) \\\\\n(\\beta+\\gamma) \\times \\alpha \u0026 =(\\beta \\times \\alpha)+(\\gamma \\times \\alpha) \\\\\n\\alpha^{\\beta} \\times \\alpha \\gamma \u0026 =\\alpha^{\\beta+\\gamma} \\\\\n(\\alpha \\beta) \\gamma \u0026 =\\alpha^{\\beta \\times \\gamma}.\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nThey do not in general satisfy the commutative law, nor the\nother form of the distributive law, viz.:\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 56.81ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-372.png\" alt=\"\" data-tex=\"\\[\n\\alpha \\times(\\beta+\\gamma)=(\\alpha \\times \\beta)+(\\alpha \\times \\gamma), \\text { nor } \\alpha^{\\gamma} \\times \\beta \\gamma=(\\alpha \\times \\beta) \\gamma.\n\\]\"\u003e\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eRelation-numbers are important for the following reason. In addition\nto the propositions which can be \u003ci\u003eproved\u003c/i\u003e by logic (considered\nin Chapter XVII.), there are other propositions which can be\n\u003ci\u003eenunciated\u003c/i\u003e by logic, though they cannot be proved or disproved\nexcept by empirical evidence. Such, for example, is the proposition:\n\"There are classes which are not\u003cspan class=\"pagenum\" id=\"Page_251\"\u003e[Pg 251]\u003c/span\u003e finite.\" This is a proposition which\nis purely logical in content, but there is no \u003ci\u003ea priori\u003c/i\u003e way of\nknowing whether it is true or false. (Many such have been proposed,\nbut they are all fallacious.) Then, again, there are propositions\nwhich contain some particular constituent, but would be capable of\nenunciation in logical terms if that constituent were turned into a\nvariable. Take, \u003ci\u003ee.g.\u003c/i\u003e: \"\u003ci\u003eBefore\u003c/i\u003e is a transitive relation.\"\nThis is not a statement which pure logic can enunciate, because\n\u003ci\u003ebefore\u003c/i\u003e is an empirical relation. But \"\u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e is a transitive\nrelation,\" where \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e is variable, can be enunciated by pure logic.\nWe will say that a proposition containing a certain constituent \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\nattributes a \"logical property\" to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e if, when \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is replaced\nby a variable \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, the result is a propositional function which can\nbe expressed by logic. The test of a logical property is very simple:\napart from the constant \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, there must be no constants involved—except\nsuch purely formal constants as \"incompatibility\" and \"for all\nvalues of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\" which are not constituents of the propositions in\nwhose verbal or symbolic expression they occur. It will be seen that\ntransitiveness, \u003ci\u003ee.g.\u003c/i\u003e, is a logical property of a relation; so is\nasymmetry or symmetry; so is having \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e terms in its field; so is,\nin the case of a three-term relation (\u003ci\u003ebetween\u003c/i\u003e), the property\nof generating a Euclidean space; so is, in the case of a four-term\nrelation (\u003ci\u003eseparation of couples\u003c/i\u003e), the property of generating\na projective space; and so on. We can now state the proposition on\naccount of which structure is important.\u003c/p\u003e\n\n\u003cp\u003e\u003ci\u003eWhen two relations have the same structure\u003c/i\u003e (\u003ci\u003eor\nrelation-number\u003c/i\u003e), \u003ci\u003eall their logical properties are identical.\u003c/i\u003e\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_252\"\u003e[Pg 252]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eLogical properties include all those which can be expressed in\nmathematical terms. Moreover, the inferences from perceptions to their\ncauses, assuming such inferences to be valid, are concerned mainly, if\nnot exclusively, with logical properties. This latter proposition is\none which we must now examine.\u003c/p\u003e\n\n\u003cp\u003eTake first the relation between the space of physics and the space of\nperception. Within the private space of one percipient, there is a\ndistinction between perceived space-relations and inferred ones. There\nis a space into which all the percepts of one person fit, but this\nis a constructed space, the construction being achieved during the\nfirst months of life. But there are also perceived space-relations,\nmost obviously among visual percepts. These space-relations are not\nidentical with those which physics assumes among the corresponding\nphysical objects, but they have a certain kind of correspondence\nwith those relations. If we represent the position, for physics,\nof visible objects by polar co-ordinates, taking the percipient as\norigin, the two angular co-ordinates correspond to perceived relations\namong visual percepts, while the radius vector (except possibly for\nvery small distances) is inferred by means of causal laws. Let us\nconfine ourselves to the angular co-ordinates. My point is that the\nrelations which physics assumes in assigning angular co-ordinates are\nnot identical with those which we perceive in the visual field, but\nmerely correspond with them in a manner which preserves their logical\n(mathematical) properties. This follows from the assumption that any\ndifference between two simultaneous percepts implies a correlative\ndifference in their stimuli. Consequently, assuming that light travels\nin straight lines, two objects which produce percepts which differ in\nperceived direction must differ in some respect which corresponds with\nperceived direction. But we need not assume that physical direction has\nanything in common with visual direction except the logical properties\nimplied by the above assumption. I shall, in Part III., attempt a\nconstruction of physical space which will supply some of the detail of\nthe correspondence; for the present, I am concerned to point out that\nwe can only infer the logical (or mathematical) properties of physical\nspace, and must not suppose that it is identical with the space of our\u003cspan class=\"pagenum\" id=\"Page_253\"\u003e[Pg 253]\u003c/span\u003e\nperceptions. Indeed, as I shall try to prove later, the whole of a\nman\u0027s visual space is, for physics, inside his head; this will follow\nfrom causal considerations.\u003c/p\u003e\n\n\u003cp\u003eThe same sort of considerations apply to colours and sounds. Colours\nand sounds can be arranged in an order with respect to several\ncharacteristics; we have a right to assume that their stimuli can be\narranged in an order with respect to corresponding characteristics,\nbut this, by itself, determines only certain logical properties of\nthe stimuli. This applies to all varieties of percepts, and accounts\nfor the fact that our knowledge of physics is mathematical: it is\nmathematical because no non-mathematical properties of the physical\nworld can be inferred from perception.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, one exception to this limitation, at least\napparently. The exception I mean is \u003ci\u003etime\u003c/i\u003e. We always assume that\nthe time between percepts is the same as the time in the physical\nworld. I do not know whether this view is correct or not; but I will\ntry to set forth the arguments on either side.\u003c/p\u003e\n\n\u003cp\u003eIn the first place, we must adapt our language to the theory of\nrelativity. I shall assume (what I shall argue in Part III.) that,\nwhen we are speaking of physical space, all our percepts are in our\nhead. Consequently psychological time is the same as time measured by\nour watches, assuming that we carry them on our person. Our head moves\nalong a world-line, and our psychological time-intervals are measured\nphysically by integrating ds along this world-line. Thus there is no\ndifficulty in adapting the statement that psychological and physical\ntime are identical to the requirements of the theory of relativity.\nIn this respect, time differs from space, because physically all our\nsimultaneous percepts are in one place.\u003c/p\u003e\n\n\u003cp\u003eI think, however, that the time-intervals between percepts are only to\nbe obtained by means of inferences of the same sort as those which lead\nus to the physical world. \u003ci\u003ePerceived\u003c/i\u003e\u003cspan class=\"pagenum\" id=\"Page_254\"\u003e[Pg 254]\u003c/span\u003e relations are not between\nevents at different times, but between a percept and a recollection,\nboth of which occur at the same time; or again, where very short\ntimes are concerned, between a sensation of maximum vividness and a\nfading (akoluthic) sensation. Sensations do not decay suddenly, but\nfade gradually, though very quickly. That is why a quick movement can\nbe apprehended as a whole: the sensations belonging to earlier parts\nare still present, though less vivid, when the sensations belonging\nto later parts arise. Thus our knowledge of time seems to be inferred\nfrom perceived relations which are not strictly temporal. These\nrelations are, I think, of three sorts. Two sorts have been mentioned:\nthe relation of a vivid to a fading sensation, and the relation of a\npercept to a recollection. But in addition to these there is an order\nwithin recollections: we can recollect a process in the right order.\nHere, also, however, all that we perceive is in the present, and the\ntime-order of the original events is inferred from relations among\nthe simultaneous events which constitute our present recollection.\nThus the conclusion seems to be: Psychological time may be identified\nwith physical time, because neither is a datum, but each is derived\nfrom data by inferences of the sort we have found elsewhere, namely,\ninferences which allow us to know only the logical or mathematical\nproperties of what we infer.\u003c/p\u003e\n\n\u003cp\u003eThus it would seem that, wherever we infer from perceptions, it is\nonly structure that we can validly infer; and structure is what can be\nexpressed by mathematical logic, which includes mathematics.\u003c/p\u003e\n\n\u003cp\u003eBefore concluding this discussion, we must consider an extension of the\nnotion of similarity which has considerable importance in relation to\nthe inferences leading to the physical world. In defining similarity,\nwe used a one-one relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e. But we may substitute a many-one\nrelation, and still obtain something useful. The importance of this\nis that, as we have seen, if we take a group of events constituting\na physical\u003cspan class=\"pagenum\" id=\"Page_255\"\u003e[Pg 255]\u003c/span\u003e object, the relation of the events which are nearer the\nobject to those which are further from it is many-one, not one-one. If\nwe are observing a man half a mile away, his appearance is not changed\nif he frowns, whereas it is changed for a man observing him from a\ndistance of three feet. Considerable events may happen in the sun\nwithout being perceptible to us even with the best telescopes; but near\nthe sun they may have effects which would be important to a percipient\nsituated where these effects occur. It is obvious as a matter of logic\nthat, if our correlating relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e is many-one, not one-one,\nlogical inference in the sense in which \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e goes is just as feasible\nas before, but logical inference in the opposite sense is more\ndifficult. That is why we assume that differing percepts have differing\nstimuli, but indistinguishable percepts need not have exactly similar\nstimuli. If we have \u003cimg style=\"vertical-align: -0.05ex; width: 4.675ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-373.png\" alt=\"\" data-tex=\"\\(xSx\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 4.304ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-374.png\" alt=\"\" data-tex=\"\\(ySy\u0027\\)\"\u003e, where \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e is many-one, and\nif \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.736ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-102.png\" alt=\"\" data-tex=\"\\(y\u0027\\)\"\u003e differ, we can infer that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.922ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-101.png\" alt=\"\" data-tex=\"\\(x\u0027\\)\"\u003e differ;\nbut if \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.736ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-102.png\" alt=\"\" data-tex=\"\\(y\u0027\\)\"\u003e do not differ, we cannot infer that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.922ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-101.png\" alt=\"\" data-tex=\"\\(x\u0027\\)\"\u003e do not differ. We find often that indistinguishable percepts are\nfollowed by different effects—\u003ci\u003ee.g.\u003c/i\u003e one glass of water causes\ntyphoid and another does not. In such cases we assume imperceptible\ndifferences—which the microscope may render perceptible. But where\nthere is no discoverable difference in the effects, we can still not be\nsure there is not a difference in the stimuli which may become relevant\nat some later stage.\u003c/p\u003e\n\n\u003cp\u003eWhen the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e is many-one, we shall say that the two systems\nwhich it correlates are \"semi-similar.\"\u003c/p\u003e\n\n\u003cp\u003eThis consideration makes all physical inference more or less\nprecarious. We can construct theories which fit the known facts, but\nwe can never be sure that other theories would not fit them equally\nwell. This is an essential limitation on scientific inference, which\nis generally recognized by men of science: no prudent man of science\nwould maintain that such-and-such a theory is so firmly established\nthat it will never call\u003cspan class=\"pagenum\" id=\"Page_256\"\u003e[Pg 256]\u003c/span\u003e for modification. Newtonian gravitation came\nnearer to this certainty than any other theory has ever done; yet\nNewtonian gravitation has had to be modified. The fundamental reason\nfor this uncertainty, which remains even when we assume all the canons\nof scientific inference, is the fact that our relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e, which\nconnects the physical object with the percept, is many-one and not\none-one.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_56\" href=\"#FNanchor_56\" class=\"label\"\u003e[56]\u003c/a\u003e\nVol. \u003cspan class=\"allsmcap\"\u003eII.\u003c/span\u003e, part \u003cspan class=\"allsmcap\"\u003eIV.\u003c/span\u003e, *150 ff.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_257\"\u003e[Pg 257]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXV\"\u003eCHAPTER XXV\u003cbr\u003e\nPERCEPTION FROM THE STANDPOINT OF PHYSICS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nHITHERTO we have been taking perception as our starting-point, and\nconsidering how physics could be obtained as an inference from\nperception. In the present chapter, I want to pursue the opposite\ncourse, and consider how, assuming physics, percepts can find their\nplace in the physical world.\u003c/p\u003e\n\n\u003cp\u003eLet us first of all exclude certain problems which are not relevant to\nthis inquiry. A \"percept,\" considered as the epistemological basis of\nphysics, must be a \"datum\"—it must be something noticed. Obviously,\ntherefore, whatever may be true of percepts in general, those which\nafford empirical premisses for physics have to be \"known.\" But it\nis unnecessary for us to define \"knowing\": for physics, only the\npercepts are important, and our relation to them may be taken for\ngranted. Similarly we need not consider whether, when we perceive,\nthe occurrence is relational, involving a percept and a percipient,\nor whether the occurrence of the percept is all that happens at the\nmoment, and its \"mental\" character is conferred by memory (in its most\ngeneral sense). Such psychological questions need not concern us. What\nI wish to discuss is the physical status of percepts, \u003ci\u003ei.e.\u003c/i\u003e\nof patches of colour, noises, smells, hardnesses, etc., as well as\nperceived spatial relations. And in this discussion I am now assuming\nordinary physics, subject to the latitude of interpretation explained\nin Chapter I.\u003c/p\u003e\n\n\u003cp\u003eDr Whitehead\u0027s books are a protest against the \"bifurcation of nature\"\nwhich has resulted from the causal theory of perception. With this\nprotest I am in complete agreement. Locke\u0027s belief, that the primary\nqualities belong to the object\u003cspan class=\"pagenum\" id=\"Page_258\"\u003e[Pg 258]\u003c/span\u003e and the secondary to the percipient,\nhas been that of science in practice, whatever individual scientific\nmen may have thought in their philosophic moments. The view which I\nwish to advocate is quite different. I hold that the world is very full\nof events, that often a group of these events, or some characteristic\nwhich the members of the group possess in varying degrees, is such as\nto suggest arrangement in an order, generally a symmetrical order about\na centre—\u003ci\u003ee.g.\u003c/i\u003e the percepts of different people when they look\nat a penny may be ordered by their size and by their shape. The orders\nderived from different sources are roughly identical: \u003ci\u003ee.g.\u003c/i\u003e if\nwe move so as to make the big drum look larger, we also move so as to\nmake it sound louder. In this way we construct a space containing both\npercipients and physical objects; but percepts have a twofold location\nin this space, namely that of the percipient and that of the physical\nobject. Keeping one half of this location fixed, we obtain the view of\nthe world from a given place; keeping the other half fixed, we obtain\nthe views of a given physical object from different places. The first\nof these is a percipient, the second is a physical object. But the\nfirst half of this statement is to be taken with a grain of salt.\u003c/p\u003e\n\n\u003cp\u003eThe physical world, I suggest, considered as perceptible, consists of\noccurrences having this twofold location. For the moment I am concerned\nto assign the place of perception in such a scheme.\u003c/p\u003e\n\n\u003cp\u003eConsider a spherical light-wave proceeding from a momentary flash.\n\u003ci\u003eIn vacuo\u003c/i\u003e, it advances in accordance with Maxwell\u0027s equations,\nbut when it encounters matter it becomes transformed in one way or\nanother according to circumstances. What do I mean by saying that it\n\"encounters matter\"? The answer is quite straightforward. Connected\nwith each electron or proton there is a gravitational field and an\nelectromagnetic field; these are displayed by laws modifying the\n\"undisturbed\" distribution about other centres of such\u003cspan class=\"pagenum\" id=\"Page_259\"\u003e[Pg 259]\u003c/span\u003e things as\nlight-waves. In fact, the fields may be said actually to consist\nof the formulæ of such modification. Therefore when I say that a\nlight-wave \"encounters matter,\" I mean that it is near the centre of\nsome such systematic modification. The eye is a collection of such\ncentres, and after traversing it the process which was a light-wave\nobeys a different set of laws. The percept is a term of this process,\ncharacterized by the fact that it occurs after traversing a region of\na certain sort—to wit, an eye, an optic nerve, and part of a brain.\nOwing to its causal continuity with other parts of the process, it has,\nas its twofold location, on the one hand the source of light, on the\nother hand the brain. If it is said that a percept is \"obviously\" not\nin the brain, that is because we are thinking of its location in the\nphysical object, and comparing this with the location of the brain as a\nphysical object.\u003c/p\u003e\n\n\u003cp\u003eCertain explanations are called for, chiefly in virtue of Dr Broad\u0027s\ncriticisms.\u003ca id=\"FNanchor_57\" href=\"#Footnote_57\" class=\"fnanchor\"\u003e[57]\u003c/a\u003e In the first place, it is suggested that the above\ntheory takes a common-sense view of the percipient\u0027s body, and derives\nfrom this an undue plausibility for the view which it suggests as to\nexternal objects. This is not the case, but in order to dispel the\nappearance of such an error it is necessary to explain the twofold\ncharacter of a physical object. On the one hand, it is a group of\n\"appearances\"—\u003ci\u003ei.e.\u003c/i\u003e of connected events—differing, from next to\nnext, approximately according to the laws of perspective. On the other\nhand, a physical object has an influence upon the appearances of other\nobjects, especially appearances in its neighbourhood, causing these to\ndepart, in a greater or less degree, from what they would be if they\nfollowed the laws of perspective strictly. The sense organs have only\nthis second function to perform in the theory of perception, while\nthe object perceived has the first function. It is this difference\nof function, in the theory of perception, which makes it seem as\nif we were treating the\u003cspan class=\"pagenum\" id=\"Page_260\"\u003e[Pg 260]\u003c/span\u003e percipient\u0027s body more realistically than\nexternal objects. But this is only a matter of degree. The appearance\nof an external object is modified also by other external objects—\u003ci\u003e\ne.g.\u003c/i\u003e by blue spectacles or by a microscope. I conceive the part\nplayed by the eye as essentially analogous to that played by a\nmicroscope; and I take the same view as to the part played by the optic\nnerve.\u003c/p\u003e\n\n\u003cp\u003eAnother objection urged by Dr Broad is that the above theory is at\nbest only suitable to visual objects, not to objects known by other\nsenses. Now I certainly hold that vision is much the most important\nand least misleading of the senses, when considered as a source of the\nfundamental notions of physics. But I do not admit that the view which\nI have suggested is in any way inapplicable to the other senses. This\nsubject, however, demands some discussion.\u003c/p\u003e\n\n\u003cp\u003eLet us take first the sense of touch. This sense is complicated by the\nfact that it has no special organ, such as the eye, but is diffused\nthroughout the surface of the body. In order to avoid complications,\nlet us assume that only the tip of the forefinger of the right hand\nis being used. I do not know what, exactly, is supposed to be the\nphysical process in touch, but we may suppose that it is somewhat as\nfollows: the electrons and protons of a certain part of the skin come\ninto such close proximity to those of an external body that electrical\ndisturbances are set up, which travel along the afferent nerves to\nthe proper part of the brain, and produce corresponding disturbances\nthere. It does not matter for our purposes if this view is not quite\nright, since the exact nature of the process is irrelevant. But there\nis one point of some importance, and that is, that the change or lack\nof change in a sensation of touch has more importance than in the case\nof sight. A printed letter, and even a printed word, can be seen at a\nglance; but to read \"Braille\" it is necessary to let the finger travel\nround the contours of the letters. Thus shape,\u003cspan class=\"pagenum\" id=\"Page_261\"\u003e[Pg 261]\u003c/span\u003e in the case of touch,\nis, in the main, inferred by means of movement; the momentary datum is\nmuch simpler than many visual data. The inference to shape depends, of\ncourse, upon the assumption that the object touched has not changed\nits shape meanwhile; it would be difficult for a blind man to acquire\ncorrect views as to the shape of an eel. But when there is doubt the\nfinger can be allowed to travel repeatedly round the contours of the\nobject; if the result is similar on each occasion, it may be assumed\nthat the object has kept an approximately unchanging shape.\u003c/p\u003e\n\n\u003cp\u003eThere is another respect in which touch is inferior to sight, and\nthat is, that the spatial relation of the physical object to the\npercipient\u0027s body is much more restricted. The physical object must be\nvery close to the part of the percipient\u0027s body which is said to be\ntouching it. This means that its location is confined within a certain\nsmall region. Within that region touch can locate it rather well,\nprovided a sensitive part of the skin is used; we know the position of\nour hand by means of feelings connected with the muscles, and thence we\nknow the position of anything in contact with the hand. The intervening\nmedium, in the case of touch, is always a part of the percipient\u0027s\nbody; but its influence is shown in the difference between the touch\nsensations when a physical object touches one part of the body and when\nit touches another. Thus our theory applies to touch just as well as to\nsight.\u003c/p\u003e\n\n\u003cp\u003eSound is, in many ways, very analogous to light. It is a disturbance\nhaving a centre, and is greatest near the centre. What we hear is\nloudest when we are near the centre. The direction of the sound can\nbe gauged roughly, though not with anything approaching the precision\nwith which we can gauge the direction of a visual object. Here, also,\nwe have a certain physical process, which obeys certain laws in air,\nbut obeys somewhat different laws in the ear and nerves and brain.\nThese differences, however, may be conceived to be of the\u003cspan class=\"pagenum\" id=\"Page_262\"\u003e[Pg 262]\u003c/span\u003e same kind,\nessentially, as those normally produced in physical processes by the\npresence of matter. I cannot see, therefore, that sound offers any\ndifficulty.\u003c/p\u003e\n\n\u003cp\u003eThe other senses are much less important as sources of physical\nknowledge, and it seems unnecessary to discuss them in detail.\nPhysiology, however, tends to show that any abnormal condition of the\nsense organs or of the afferent nerves tends to modify percepts in such\na way as requires, for its explanation, some such theory as ours. It is\na fallacy to argue, as is sometimes done, that, if we cannot trust our\nsenses, we cannot know that we have sense organs, or that there is any\ntruth in physiology. If We find that several people, looking at Jones,\nsee him just as usual, while one person sees him looking queer; if the\nseveral see nothing queer in each other\u0027s eyes, while they all see\nsomething queer in the eyes of the one; in such circumstances, I say,\nit is natural and proper to correlate the two queernesses. The man who\nsees Jones differently from usual sees him through a medium which has\nan unusual effect; there is no more ground for scepticism than is to\nbe derived from the effect of opera glasses. The sceptical argument is\nonly valid as against naive realism, and derives its rhetorical force\nfrom our tendency to relapse into naive realism whenever we are not on\nour guard.\u003c/p\u003e\n\n\u003cp\u003eThe cognitive efficacy of perception depends upon two factors, one\nphysical and one psychological (and physiological). The psychological\nfactor is memory and the whole effect of experience upon mind and body.\nThis is a large subject, which I mention only to dismiss. The physical\nfactor, however, may be pointed out once more. It is, the fact that\nphysical occurrences tend to be grouped about centres, the members\nof one group being approximately related according to laws which we\nhave called the laws of perspective. This enables us to infer from\na percept other percepts which we should have if we moved, or which\nother percipients have now.\u003cspan class=\"pagenum\" id=\"Page_263\"\u003e[Pg 263]\u003c/span\u003e When one astronomer sees an eclipse of\nthe moon, he can be pretty sure that others see it too if they are\nlooking in the right direction. When one man sees the Derby, he can be\npretty sure that the other spectators are also seeing it—\u003ci\u003ei.e.\u003c/i\u003e\nthat they have percepts which can be inferred approximately from his\nby the laws of perspective. As to what is happening where there is no\npercipient, we can, on certain assumptions, infer a good deal as to its\nmathematical structure, but nothing as to its intrinsic quality. In a\nword, the inferential power of perception depends upon the fact that\nphysical events occur in connected groups, and is limited by the fact\nthat this is only true to a certain degree of approximation.\u003c/p\u003e\n\n\u003cp\u003eThere remains one matter of considerable importance to be discussed\nin this connection—I mean, the \u003ci\u003eprima facie\u003c/i\u003e difference between\na percept and a physical process. At first sight, a light-wave seems\nvery different from a visual percept, and a sound-wave from an auditory\npercept. But this apparent gulf is due to comparison of events of\ndifferent orders. A physical disturbance, such as a light-wave, must be\nregarded as much more complex in reality than in mathematics. Events\nin the physical world are correlated according to certain laws, and\nwe can, for mathematical purposes, treat a whole group of correlated\nevents as if it were one event. There is no theoretical reason why a\nlight-wave should not consist of groups of occurrences, each containing\na member more or less analogous to a minute part of a visual percept.\nWe cannot perceive a light-wave, since the interposition of an eye and\nbrain stops it. We know, therefore, only its abstract mathematical\nproperties. Such properties may belong to groups composed of any kind\nof material. To assert that the material \u003ci\u003emust\u003c/i\u003e be very different\nfrom percepts is to assume that we know a great deal more than we do\nin fact know of the intrinsic character of physical events. If there\nis any advantage in supposing that the light-wave, the process in the\neye, and the process in the optic nerve, contain\u003cspan class=\"pagenum\" id=\"Page_264\"\u003e[Pg 264]\u003c/span\u003e events qualitatively\ncontinuous with the final visual percept, nothing that we know of the\nphysical world can be used to disprove the supposition.\u003c/p\u003e\n\n\u003cp\u003eThe gulf between percepts and physics is not a gulf as regards\nintrinsic quality, for we know nothing of the intrinsic quality of\nthe physical world, and therefore do not know whether it is, or is\nnot, very different from that of percepts. The gulf is as to what we\nknow about the two realms. We know the quality of percepts, but we do\nnot know their laws so well as we could wish. We know the laws of the\nphysical world, in so far as these are mathematical, pretty well, but\nwe know nothing else about it. If there is any intellectual difficulty\nin supposing that the physical world is intrinsically quite unlike that\nof percepts, this is a reason for supposing that there is not this\ncomplete unlikeness. And there is a certain ground for such a view,\nin the fact that percepts are part of the physical world, and are the\nonly part that we can know without the help of rather elaborate and\ndifficult inferences.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_57\" href=\"#FNanchor_57\" class=\"label\"\u003e[57]\u003c/a\u003e\n\u003ci\u003eScientific Thought\u003c/i\u003e, Kegan Paul, 1923, pp. 531 fl.,\nesp. p. 533.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_265\"\u003e[Pg 265]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXVI\"\u003eCHAPTER XXVI\u003cbr\u003e\nNON-MENTAL ANALOGUES TO PERCEPTION\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nAS we saw in Chapter XXV., the cognitive value of\nperception—\u003ci\u003ei.e.\u003c/i\u003e its capacity for giving rise to inferences\nwhich are often valid—is a product of two factors, one depending\nupon the human mind and body, the other purely physical. The factor\nwhich depends upon the human mind and body is that which is concerned\nwith \"mnemic\" phenomena. These occur wherever there is life, and to\nsome slight extent in \"dead\" matter; but the higher the type of life\nthe more notable they become. It is, however, the physical factor in\nperception that I wish to consider in this chapter, as it appears when\nseparated from the mnemic factor. That is to say, I want to emphasize\nthe fact that a percept is one of a system of correlated events, all\nstructurally similar or semi-similar, and that the physical world, so\nfar as known, consists of such events. My main purpose in dwelling upon\nthis topic is to make it clear that percepts fit easily and naturally\ninto their place in the physical world, and are not to be regarded as\nsomething quite different from the processes with which physics is\nconcerned.\u003c/p\u003e\n\n\u003cp\u003eLet us revert to our earlier illustration of the dictaphone and camera\nwhich record a conversation with its accompanying action, and are found\nto agree with the recollections of eyewitnesses. When we considered\nthis coincidence in a previous chapter, we were concerned with\nfundamental doubts; now we will assume the four-dimensional manifold\nof physics and the justification (in principle) of the inference\nfrom perceived to unperceived events. Assuming this, what can we\ninfer as to the relation between (\u003ci\u003ea\u003c/i\u003e) the sounds heard by the\nlistener, (\u003ci\u003eb\u003c/i\u003e) the events just outside his ear when he hears,\n(\u003ci\u003ec\u003c/i\u003e) the events\u003cspan class=\"pagenum\" id=\"Page_266\"\u003e[Pg 266]\u003c/span\u003e at the dictaphone at the same time, (\u003ci\u003ed\u003c/i\u003e)\nthe dictaphone record, (\u003ci\u003ee\u003c/i\u003e) the sounds heard by the man when he\nlistens to the dictaphone?\u003c/p\u003e\n\n\u003cp\u003eThe similarity between (\u003ci\u003ea\u003c/i\u003e) and (\u003ci\u003ee\u003c/i\u003e) is fundamental, and\nis known by a comparison of a percept with a memory. Thus the problem\nof the relation between perception and memory is involved; but as\nthis problem is psychological, I will only say that the inference\nfrom a recollection (which occurs now) to what is recollected (which\noccurred at a former time) appears to me to be essentially similar to\nthe inferences in physics, and to warrant only a belief in identity\n(or close similarity) of structure between the recollection and the\nevent recollected. The grounds for the trustworthiness of memory seem\nto be of the same kind as those for the trustworthiness of perception.\nBut I shall take all this for granted, since our theme is physics, not\npsychology. I shall therefore assume that (\u003ci\u003ea\u003c/i\u003e) and (\u003ci\u003ee\u003c/i\u003e) can\nbe known to be similar in structure, in the sense explained in Chapter\nXXIV.\u003c/p\u003e\n\n\u003cp\u003eWe have thus a chain of processes, (\u003ci\u003ea\u003c/i\u003e) at one end and (\u003ci\u003ee\u003c/i\u003e)\nat the other; the end-processes are similar in the technical sense, and\nwe assume that the intermediate processes are also similar, both to\neach other and to the end-processes. Let us consider this in somewhat\nmore detail. The relation of (\u003ci\u003ea\u003c/i\u003e) and (\u003ci\u003eb\u003c/i\u003e) is that of\npercept and stimulus—\u003ci\u003ei.e.\u003c/i\u003e a relation of effect to cause. The\neffect is a complex process; we assume that recognizably different\npercepts must have different stimuli; therefore the cause must be a\ncomplex process, at least semi-similar to the effect. We may take it\nas similar, not merely semi-similar, by ignoring those respects, if\nany, in which the structure of the cause is more complex than the\nstructure of the effect. A similar argument will enable us to treat\n(\u003ci\u003ed\u003c/i\u003e) and (\u003ci\u003ee\u003c/i\u003e) as similar. Since (\u003ci\u003ea\u003c/i\u003e) and (\u003ci\u003ee\u003c/i\u003e)\nare similar, it follows that (\u003ci\u003eb\u003c/i\u003e) and (\u003ci\u003ed\u003c/i\u003e) are similar. We\ncannot attribute this similarity to chance, since it is found to exist\nwhenever the necessary conditions have\u003cspan class=\"pagenum\" id=\"Page_267\"\u003e[Pg 267]\u003c/span\u003e been fulfilled. Hence we infer\nthat (\u003ci\u003ec\u003c/i\u003e) must also be similar to the other processes. Since the\ndictaphone may be placed anywhere in the neighbourhood of the speakers,\nwe infer that throughout a region surrounding them there are physical\nevents similar in structure to the aural percepts of the listener. For\nlight, the same thing follows from photographs. Consequently a percept,\nconsidered physically, is not very different from other physical\nevents. We may suppose, if we choose, that it differs from them in\nintrinsic quality, and we know that it differs causally, since it\ngives rise to memories and inferences. Even these, however, are not so\ndifferent from certain physical processes as they seem at first sight.\u003c/p\u003e\n\n\u003cp\u003eMemory is shown by the capacity for producing events similar in\nstructure to certain previous events, when the right stimulus is\napplied. We are not always remembering everything that we can remember;\nwe remember things when we are asked about them, or when something\noccurs which recalls them by association. The dictaphone \"remembers\"\nin this sense. It is true that it cannot \"infer\": it will not answer\na question which it has never heard answered. But physiological\ninference, which is causally the basis of all other inference, is not\nvery unlike other physical processes, and may quite possibly proceed\naccording to the laws of physics. However, I do not wish to pursue\nthese psychological topics; it is only perception and its non-mental\nanalogues that I wish to consider.\u003c/p\u003e\n\n\u003cp\u003eWe have to suppose that a great many events are taking place\neverywhere, since both light and sound can be recorded by instruments\nand observed by percipients. Our visual field is very complex, and the\nphysical stimulus must have at least equal complexity: if this were\nnot the case, we could not see a number of objects at once, nor could\na photographic plate photograph them. Physics, however, simplifies all\nthis by taking the stimulus to a sensation to be a periodic process,\u003cspan class=\"pagenum\" id=\"Page_268\"\u003e[Pg 268]\u003c/span\u003e\nnot a static event. Our perception of colour, for example, does not\nseem to be a periodic process analogous to a light-wave; in this\nrespect, the apparent structure of a visual percept differs from that\nwhich physics assumes in the external cause. A few words must be said\ncm this topic, in order to make clear its relation to our general\ntheory of similarity of structure.\u003c/p\u003e\n\n\u003cp\u003eFirst: in a transaction such as the passage from stimulus to percept,\nwe cannot expect \u003ci\u003ecomplete\u003c/i\u003e similarity of structure: at most we\ncan expect as much as we find in purely physical transactions. There is\na great deal of difference between a light-wave and a quantum change\nin an atom, yet they are related as effect to cause. What we know\nabout the atom we know in virtue of the light-waves which make us see\nthings; unless differences in light-waves corresponded to differences\nin atoms, light-waves would not be vehicles of information about atoms.\nNow when light-waves reach the eye, they have effects upon the matter\nof the eye, which reverse the previous process from quantum changes to\nlight-waves. It is possible, in view of such theories as we considered\nin Chapter XIII., that the relation between what happens in the atom\nand what happens in the eye is more direct than the above account would\nsuggest, but it would not be prudent to assume that this is the case\nuntil the theory of light quanta has become more adequate. We cannot,\ntherefore, assume any very close relation between the physical process\nin the eye and the physical process in the atom from which the light\ncomes. And \u003ci\u003ea fortiori\u003c/i\u003e we cannot assume a very close relation\nbetween the percept and the process in the radiating atom. Yet it is\nonly in so far as such a relation exists that vision can be accepted as\na source of physical knowledge; in so far as the correspondence fails,\nvision ceases to be trustworthy.\u003c/p\u003e\n\n\u003cp\u003eSecondly: there is no reason why the degree of correspondence between\nstimulus and percept which is required\u003cspan class=\"pagenum\" id=\"Page_269\"\u003e[Pg 269]\u003c/span\u003e should not exist between\na periodic process and a static occurrence. So long as different\nprocesses give rise to different percepts, the requisites in the way\nof correspondence are satisfied. There is therefore no theoretic\ndifficulty in the view that the stimulus to a sensation of red is a\nvibration, while the sensation of red itself has not this character,\nbut is a steady state capable of continuing for a short finite time.\u003c/p\u003e\n\n\u003cp\u003eThirdly: we do not really know that our percept of a colour does not\nhave the rhythmic character of the stimulus. We know something about\npercepts, but not all about them. We all know that if an object is\nmade to rotate rapidly, for instance on a top, we can see it rotating\nif it does not go too fast, but when it passes a certain speed we see\nonly a continuous band. This is to be expected in view of the existence\nof akoluthic sensations. But it by no means follows that there is\nnot a flicker in the percept, although we cannot perceive a flicker.\nExactly the same thing applies to light and sound generally, and to the\napparent continuity of motion in the cinema. We cannot know, unless in\nvirtue of some elaborate argument, whether our percepts are static or\nrhythmical, nor yet whether their physical stimuli are continuous or\ndiscrete. Such knowledge is rendered impossible by the fact that we can\nonly assume semi-similarity, not full similarity, between percept and\nstimulus.\u003c/p\u003e\n\n\u003cp\u003eThere is therefore no difficulty in the accepted theory that the\nstimuli to our most important percepts are rapid periodic processes.\nOn the other hand, there is a great advantage in this theory, in that\nit simplifies the physical world which has to be assumed as the cause\nof our perceptions. A physical system, conceived merely as a set of\nmaterial units in space-time, is capable of an indefinite variety of\nrhythmic movements. Some physical structures are resonant for one\nperiod, some for another. Thus our sense-organs can select one sort of\nmovement as the stimulus to which they will respond, and\u003cspan class=\"pagenum\" id=\"Page_270\"\u003e[Pg 270]\u003c/span\u003e reject all\nthe rest. In fact, it may be said that the essential characteristic\nof a sense-organ is sensitiveness to one sort of stimulus, which, in\nthe case of the eye or the ear, must be a periodic movement. In this\nthe sense-organs do not differ from lifeless instruments, such as\nphotographic plates and gramophones. Such instruments have something\nclosely analogous to perception, when we leave out of account the\nmental consequences which we observe in ourselves as a result of\nperception. And in a certain extended sense we may say that every body\nwhich behaves in a characteristic manner when a certain stimulus is\npresent, and only then, has a \"perception\" of that stimulus. We can\ninfer the stimulus from the behaviour of such a body just as well as\nfrom our own percepts—sometimes better, as in the case of a very\nsensitive photographic plate.\u003c/p\u003e\n\n\u003cp\u003eThe outcome of the discussion we have been conducting in Part II. has\nbeen to justify the ordinary scientific attitude, and to minimize\nthe gulf which seems at first sight to exist between perception and\nphysics. We have seen that the inference from percepts to unperceived\nphysical events, though it cannot be made mathematically cogent,\nis quite as good as any inductive inference can hope to be. And we\nhave found that there is no ground in philosophy for supposing the\nphysical world to be very different from what physics asserts it to\nbe. But we have found it necessary to emphasize the extremely abstract\ncharacter of physical knowledge, and the fact that physics leaves\nopen all kinds of possibilities as to the intrinsic character of the\nworld to which its equations apply. There is nothing in physics to\nprove that the physical world is radically different in character\nfrom the mental world. I do not myself believe that the philosophical\narguments for the view that all reality must be mental are valid. But\nI also do not believe that any valid arguments against this view are\nto be derived from physics. The only legitimate attitude about the\nphysical\u003cspan class=\"pagenum\" id=\"Page_271\"\u003e[Pg 271]\u003c/span\u003e world seems to be one of complete agnosticism as regards\nall but its mathematical properties. However, something can be done\nin the way of constructing possible physical worlds which fulfil the\nequations of physics and yet resemble rather more closely the world of\nperception than does the world ordinarily presented in physics. Such\nconstructions have the merit of making the inference from perception\nto physics seem more reliable, since they save us from the necessity\nof assuming anything radically different from what we know. From this\npoint of view, they have a certain interest, and I shall partially\ndevelop them, at least as regards space-time, in Part III. But they\nmust not be confounded with scientific knowledge: they are hypotheses\nwhich may hereafter prove fruitful, and which have already a certain\nimaginative value. But they are not to be regarded as necessitated by\nany recognized principle of scientific inference.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_273\"\u003e[Pg 273]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"PART_III\"\u003ePART III\u003cbr\u003e\n\u003cbr\u003e\nTHE STRUCTURE OF THE PHYSICAL WORLD\u003c/h2\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_275\"\u003e[Pg 275]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXVII\"\u003eCHAPTER XXVII\u003cbr\u003e\nPARTICULARS AND EVENTS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nWE shall be concerned, in what follows, with the construction of a map\nof the physical world, in part more or less conjectural, but never\nin contradiction to the physical or epistemological results hitherto\nconsidered. We shall seek to construct a metaphysic of matter which\nshall make the gulf between physics and perception as small, and the\ninferences involved in the causal theory of perception as little\ndubious, as possible. We do not want the percept to appear mysteriously\nat the end of a causal chain composed of events of a totally different\nnature; if we can construct a theory of the physical world which\nmakes its events continuous with perception, we have improved the\nmetaphysical status of physics, even if we cannot prove more than that\nour theory is possible. In what follows, some portions will be more\nconjectural than others, but I shall try to indicate, at each stage,\nwhether I am advancing what I believe to be a well-grounded inference\nby induction and analogy, or whether I am concerned only with an\nillustrative hypothesis designed to exhibit the possibilities that are\ncompatible with the abstract scientific knowledge to be derived from\nphysics.\u003c/p\u003e\n\n\u003cp\u003eWe have found, hitherto, that what we know of the physical world falls\ninto two parts: on the one hand, the concrete but disjointed knowledge\nof percepts; on the other hand, the abstract but systematic knowledge\nof the physical world as a whole. Certain questions as to structure\nare answered by physics, while others are left open. The questions\nwhich are left open are of a sort of which some must always remain\nopen—namely. Is any further analysis of the terms which are\u003cspan class=\"pagenum\" id=\"Page_276\"\u003e[Pg 276]\u003c/span\u003e ultimate\nfor physics possible, and, if so, what means exist of conjecturing its\nnature? In science, we have evidence of structure down to a certain\npoint, while beyond that point we have no evidence. There can never\nbe evidence that the point we have reached is one beyond which there\nis no structure—\u003ci\u003ei.e.\u003c/i\u003e that we have arrived at simple units\ntotally devoid of parts; therefore analysis is essentially incapable\nof reaching a term \u003ci\u003eknown\u003c/i\u003e to be final, even if it has in fact\nreached a final term. I think that, in the case of physics, there is\nreason to think that its terms are not final, and that it is possible\nto suggest a further analysis which is at least likely to be true.\u003c/p\u003e\n\n\u003cp\u003eWhen we wish to describe a structure, we have to do so by means of\nterms and relations. It may turn out that the terms themselves have a\nstructure, as, \u003ci\u003ee.g.\u003c/i\u003e, in arithmetic, when cardinal integers are\ndefined as classes of similar classes. In the technique of mathematical\nphysics, there is a considerable apparatus which belongs to the formal\nmethod, and would not be regarded by most physicists as having any\nphysical reality. Such is the manifold of space-time points. Space-time\nis held to represent a system of physical facts, but its mathematical\npoints are generally conceded to be fictions. Such a state of affairs\nis unsatisfactory until we can say just what non-fictional assertion\nis implicit in a true proposition of physics which technically uses\n\"points.\" I propose to deal with this problem in the next chapter.\u003c/p\u003e\n\n\u003cp\u003eBut what shall we say of electrons? Are they physical realities, or\nare they mathematical conveniences, like points? Or are they something\nintermediate between these two extremes? We think of a light-ray as a\nseries of events; is an electron perhaps something similar? But the\nlight-ray also raises problems: it has a certain assigned mathematical\nstructure, but it is difficult to say what we axe to think of the\nmathematical terms of this structure. Formerly, the conception of\na transverse wave in the æther seemed fairly clear:\u003cspan class=\"pagenum\" id=\"Page_277\"\u003e[Pg 277]\u003c/span\u003e the æther was\ncomposed of particles, each of which could move in the required manner.\nBut nowadays the æther is grown insubstantial and incapable of \"motion\"\nin any straightforward sense; certainly few people would venture to\nregard it as composed of point-particles, like the homogeneous fluid of\na hydrodynamical text-book. Thus the light-wave has become a structure\nin the air, like a genealogical tree whose members are all imaginary.\nThis illustrates a necessity in describing a structure: the terms are\nas important as the relations, and we cannot rest content with terms\nwhich we believe to be fictitious. It is the terms of the physical\nstructure that will concern us in the present chapter.\u003c/p\u003e\n\n\u003cp\u003eI shall give the name \"particulars\" to the ultimate terms of the\nphysical structure—ultimate, I mean, in relation to the whole of our\npresent knowledge. A \"particular,\" that is to say, will be something\nwhich is concerned in the physical world merely through its qualities\nor its relations to other things, never through its own structure, if\nany. The difference between a transverse wave and a longitudinal wave\nis a difference of structure; therefore neither can be a \"particular\"\nin the technical sense in which I mean it. An atom is a structure of\nelectrons and protons; therefore an atom is not a \"particular.\" But\nwhen I call something a \"particular,\" I do not mean to assert that\nit certainly has no structure; I assert only that nothing in the\nknown laws of its behaviour and relations gives us reason to infer\na structure. From the standpoint of logic, a particular fulfils the\ndefinition of \"substance\" which we gave in Chapter XXIII. But it\nfulfils this definition only in the existing state of knowledge;\nfurther discoveries may require us to recognize structure within it,\nand it will then cease to fulfil the definition of substance. This does\nnot falsify former statements as to the structure of the world, in\nwhich the particular in question was taken as unanalyzable; it merely\nadds new propositions, in which it is no\u003cspan class=\"pagenum\" id=\"Page_278\"\u003e[Pg 278]\u003c/span\u003e longer so treated. Atoms\nwere formerly particulars; now they have ceased to be so. But that\nhas not falsified the chemical propositions which can be enunciated\nwithout taking account of their structure. The word \"particular,\" as\nabove defined, is, therefore, a word relative to our knowledge, not an\nabsolute metaphysical term.\u003c/p\u003e\n\n\u003cp\u003eLet us begin with a few general considerations as to our knowledge\nof structure. Part of this knowledge is obtainable by analysis of\npercepts, part depends upon inferences involving unperceived entities.\nI shall call a relation \"perceived\" or \"perceptual\" if the fact that\nthis relation holds between certain terms can be discovered by mere\nanalysis of percepts. Thus before-and-after is a perceptual relation,\nwhen it occurs between terms both of which belong to the specious\npresent. Spatial relations within the visual field are perceptual; so\nare those between simultaneous tactual sensations in different parts\nof the body. Tactual sensations in the same part of the body, say\na finger-tip, may have perceived relations, if both are within the\nspecious present; these must be important in the recognition of shape\nby blind people. There are perceived relations between a percept and\na recollection, which lead us to refer the latter to the past. There\nare perceived relations of comparison, which may sometimes be rather\ncomplicated—\u003ci\u003ee.g.\u003c/i\u003e \"The resemblance of blue and green is greater\nthan the resemblance of blue and yellow.\" (Here the blue and green and\nyellow are supposed to be particular given patches of colour.) There\nis also, I should say, a perceived relation of simultaneity. I do not\nsuggest that the above list is complete, but it indicates the kinds of\ncases in which relations can be perceived.\u003c/p\u003e\n\n\u003cp\u003eThere is a well-advertised type of difficulty in such cases as the\nanalysis of a perceived motion. If I move my hand before my eyes from\nleft to right, and attend to the visual percept, it seems qualitatively\ndifferent from the successive\u003cspan class=\"pagenum\" id=\"Page_279\"\u003e[Pg 279]\u003c/span\u003e perceptions of my hand in a number of\ndifferent positions. On a watch, we can \"see\" the motion of the second\nhand, but not of the minute hand. There is no doubt that there is an\noccurrence which we naturally describe as the perception of a motion.\nWe are aware of perceiving a process: if I move my hand from left to\nright, the impression is different from what it is if I move my hand\nfrom right to left, and it is obvious to everyone that the difference\nis in the \"sense\" of the motion. We can, in fact, distinguish earlier\nand later parts of the motion, so that the motion does not appear to\nbe without structure. But the parts of it seem to be other motions,\nwhich, presumably, must each have its own structure. This leads to the\nnotion of infinite divisibility, not based upon a definable structure\nof indivisibles, but upon a process in which the parts are always\ncomposed of parts similar in structure to themselves, and simple parts\nare nowhere attainable. The paradoxes of motion, the antinomies,\nBergson\u0027s objection to analysis, and the philosophers\u0027 insistence\nthat the Cantorian continuum does not resolve their difficulties, are\nall derived from this one puzzle, that a motion seems to consist of\nmotions—or, as Kant says, that a space consists of spaces.\u003c/p\u003e\n\n\u003cp\u003eIt is important to clear up this problem of the analysis of the percept\nof motion, since it applies to all perception of change, and has\nbeen thought to constitute a difficulty in the attempt to harmonize\npsychology and physics. To begin with, continuity in the percept is no\nevidence of continuity in the physical process; it is easy to produce a\nstaccato process which causes a continuous (or apparently continuous)\npercept—\u003ci\u003ee.g.\u003c/i\u003e in the cinema. Next, it is noteworthy that, if a\nstaccato physical process is gradually accelerated, the percept will\nretain its staccato character longer if we are wide awake and have\nacute senses than if we are sleepy or have feeble senses. Everybody\nknows the experience of being awakened from a doze by a striking clock:\nat first, the noise of the strike seems\u003cspan class=\"pagenum\" id=\"Page_280\"\u003e[Pg 280]\u003c/span\u003e continuous. It is therefore a\ntenable hypothesis, if desirable on other grounds, to maintain that all\nphysical processes are staccato, and continuity in percepts is merely a\ncase of vagueness, in the sense of a many-one relation between stimulus\nand percept. I am not asserting such a view; I am only saying that it\nfits in with what we know of the relation between stimulus and percept\nin the case of swift processes. \u003ci\u003eA fortiori\u003c/i\u003e, the mathematical\ncontinuum, if it existed in the stimulus process, would produce\nthe percepts we call continuous. There is therefore nothing in our\nperception of process to make us feel that the mathematical analysis\nof continuity must be inadequate to physics, nor yet to show that a\nquantized time and space could not produce the sort of percepts which\nwe call \"seeing a motion.\" All physical possibilities are left open, so\nfar as the immediate character of the percept is concerned.\u003c/p\u003e\n\n\u003cp\u003eThe argument advanced by those who lay stress upon the perceived\ncharacter of perceptual continuity is, however, not as to the nature\nof the physical stimulus, but as to the nature of the percept. The\ncontinuity of the percept, they maintain, is quite obviously not that\nof the mathematical continuing nor yet the deceptive appearance of\ncontinuity which would exist if the percept were a rapid staccato\nprocess. In saying this, they seem to me to go beyond what the evidence\nwarrants. Consider a case which is analogous in some respects, but not\nin others—namely, the case of slightly different shades of colour.\nSuppose we have a series of colours, \u003ci\u003eA\u003c/i\u003e, \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.873ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-260.png\" alt=\"\" data-tex=\"\\(D\\)\"\u003e, …\nsuch that each is sensibly indistinguishable from its neighbour, but\nnot from the rest. That is to say, we can see no difference between\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e or between \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, but we can see a difference\nbetween \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e. We are then compelled to infer a difference\nbetween \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and between \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e, although we cannot\nperceive any difference. There is no theoretical difficulty in such an\ninference, for, although \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e\u003cspan class=\"pagenum\" id=\"Page_281\"\u003e[Pg 281]\u003c/span\u003e are percepts, and\nthe difference between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e is a percept, there is no reason\nwhy the differences between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and between \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e should be percepts: the relations between percepts are sometimes\npercepts and sometimes not. Now, instead of different static shades\nof colour, let us suppose that we are watching a chameleon gradually\nchanging. We may be quite unable to \"see\" a process of change, and yet\nable to know that, after a time, a change has taken place. This will\noccur if, supposing \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e to be the shades at the beginning\nand end of a specious present, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e are indistinguishable,\nwhile A recollected is distinguishable from \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e when \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e occurs.\nThe supposition we have to make about a perceived motion is not quite\nanalogous to this, but has certain points in common with it. Suppose\nthat we are perceiving a motion in a case where we know the physical\nstimulus to consist of a discrete series, as in the cinema. Let us\nsuppose that \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e of these stimuli can be comprised within one specious\npresent, and that each produces an element in the percept. Then the\npercept at one instant consists of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e elements \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e,\n… \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, which are arranged in an order by the degree of fading.\nLet us suppose that we cannot distinguish \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e from \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e nor\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e from \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e, but that we can distinguish \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e from \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-3.png\" alt=\"\" data-tex=\"\\(x_3\\)\"\u003e.\nIn that case our present percept will be indistinguishable from the\npercept of a continuous motion. The percept will in fact contain parts\nthat are not processes, but these parts will be imperceptible. The\nanalogy with the case of the colours arises through the existence, in\neach case, of a series in which differences of neighbouring terms are\nimperceptible while those of distant terms are perceptible. And it\nelicits the important principle that a percept may have parts which\nare not percepts, so that the structure of a percept may be only\ndiscoverable by inference. It follows also that we need not assume\nanything mysterious about the kind of complexity belonging to a percept\nof motion, but may regard its complexity\u003cspan class=\"pagenum\" id=\"Page_282\"\u003e[Pg 282]\u003c/span\u003e as of the same kind as that\nbelonging to the stimulus according to mathematical physics.\u003c/p\u003e\n\n\u003cp\u003eI wish now to consider the general question: how can we infer structure\nwhen it is not perceived? The above discussion of motion involved a\nparticular case of such inference, but now I wish to consider the\nproblem more generally.\u003c/p\u003e\n\n\u003cp\u003eFor reasons analogous to those which arise in analyzing motion, we are\nled to the view that all our percepts are composed of imperceptible\nparts. We can, for instance, perceive a heap of fine powder, and\nremove the whole heap grain by grain, where at each stage there is no\nperceptible difference. Our original percept may have had perceptible\nparts, but these were apparently always complex. It is not strictly\nnecessary to suppose the percepts complex; they might form a series\nof gradually varying quality. But we may say, in a sense, that the\ndifference of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e (supposed perceptible) is compounded of\nthe differences between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e (supposed\nimperceptible). Thus we arrive at virtually the same result in\nregard to qualitative differences as we have otherwise in regard to\nsubstantial parts. All such arguments rest ultimately upon the logical\npremiss that exact similarity is transitive, and the empirical premiss\nthat indistinguishability is not transitive. These two together are the\nsource of much of our inference as regards structure.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, another source, derived from causal arguments.\nTwo indistinguishable percepts are found to be followed by different\nresults. Inverting the maxim \"same cause, same effect,\" we argue:\n\"Different effects, different causes.\" Often the difference in the\ncauses becomes perceptible under the microscope; but we assume it\nin any case. It is this, more than anything else, that has led to\nthe minuteness of the processes inferred by physics. There are\nnoticeable differences in the effects in cases where we know that the\ndifference in the causes, if any, must be very small; we are\u003cspan class=\"pagenum\" id=\"Page_283\"\u003e[Pg 283]\u003c/span\u003e therefore\ncompelled to attribute to the physical world a structure which is very\nfine-grained relatively to perception.\u003c/p\u003e\n\n\u003cp\u003eIt is necessary to consider the very usual form of analysis into\ndiversity of \"substance,\" because, for reasons already given, we\ncannot regard this form of analysis as ultimate. Let us take the\nmost elementary of scientific examples: the analysis of water into\nhydrogen and oxygen. We recognize water by a group of characteristic\npercepts and processes; by another group we recognize hydrogen,\nand by yet another oxygen. We find that we can—\u003ci\u003ee.g.\u003c/i\u003e by\nelectrolysis—produce hydrogen and oxygen where formerly there was\nwater; we find that the masses of the two bear a fixed proportion to\neach other, and add up to the mass of the previous water; we find\nfurther that, if we let them come together, water reappears, equal in\namount to what was lost by electrolysis. Such facts are interpreted\nin science by means of the postulate that matter is indestructible.\nIf we accept this postulate, the facts prove that water consists of\nhydrogen and oxygen. Exactly similar arguments lead us on from atoms\nto electrons and protons, where, for the present, the process of\nsubstantial analysis ceases.\u003c/p\u003e\n\n\u003cp\u003eWithout questioning the \u003ci\u003econvenience\u003c/i\u003e of substantial analysis, it\nmay be asked whether it is metaphysically accurate, and even whether,\nat the stage we have reached, it is adequate to all the needs of\nphysics. We must now examine the arguments on this question.\u003c/p\u003e\n\n\u003cp\u003eAs regards adequacy for physics: we have already (in Chapter IV.) given\na brief account of Heisenberg\u0027s theory, which, in effect, resolves the\nelectron into a series of radiations. We have also seen that electrons\nand protons are not now supposed to be strictly indestructible, but\nare thought by many to be capable of annihilating each other. Thus the\nindestructibility of matter is no longer accepted as a universal law\nof the physical world. With this goes the fact that proper\u003cspan class=\"pagenum\" id=\"Page_284\"\u003e[Pg 284]\u003c/span\u003e mass is\nnot supposed to be exactly conserved, and that relative mass has been\nabsorbed into energy. Mass was supposed to be \"quantity of matter.\"\nThis certainly could not be said of relative mass, which depends upon\nthe choice of axes and belongs also to light-waves. And if it be said\nof proper mass, we must conclude that the \"quantity of matter\" is not\nquite constant. On all these grounds, persistent units of matter,\nthough still convenient, have no longer the metaphysical status that\nthey were formerly supposed to have.\u003c/p\u003e\n\n\u003cp\u003eThis conclusion is reinforced by arguments of economy. We perceive\nevents, not substances; that is to say, what we perceive occupies\na volume of space-time which is small in all four dimensions, not\nindefinitely extended in one dimension (time). And what we can\nprimarily infer from percepts, assuming the validity of physics,\nare groups of events, again not substances. It is a mere linguistic\nconvenience to regard a group of events as states of a \"thing,\" or\n\"substance,\" or \"piece of matter.\" This inference was originally made\non the ground of the logic which philosophers inherited from common\nsense. But the logic was faulty, and the inference is unnecessary.\nBy defining a \"thing\" as the group of what) would formerly have been\nits \"states,\" we alter nothing in the detail of physics, and avoid an\ninference as precarious as it is useless.\u003c/p\u003e\n\n\u003cp\u003eWhat, then, shall we say about the analysis of water into hydrogen and\noxygen? We shall say something of this sort: Water has, for common\nsense, a certain amount of permanence: although puddles dry up, the\nsea is always there. This permanence, interpreted without the use of\n\"substance,\" means certain intrinsic causal laws: the behaviour of\nthe sea can, to a considerable extent, be discovered by observing\nonly the sea, without taking account of other things. Similarity on\ndifferent occasions is the most obvious of these approximate causal\nlaws. But water can change into ice or snow or steam:\u003cspan class=\"pagenum\" id=\"Page_285\"\u003e[Pg 285]\u003c/span\u003e here we can\nobserve the gradual transformation, and continuity takes the place of\nlikeness for common sense. In all changes, we find, on examination,\nthat there is some continuity like that between water and ice; we\nthus trace a causal chain, more or less separable from other causal\nchains, and having enough intrinsic unity to be regarded as successive\nstates of one \"substance.\" When we throw over \"substance,\" we preserve\nthe causal chain, substituting the unity of a causal process for\nmaterial identity. Thus the persistence of substance is replaced by\nthe persistence of causal laws, which was, in fact, the criterion by\nwhich the supposed material identity was recognized. We thus preserve\neverything that there was reason to suppose true, and reject only a\npiece of unfruitful metaphysic.\u003c/p\u003e\n\n\u003cp\u003eThe analysis of water into hydrogen and oxygen represents, therefore,\nthe analysis of one approximate causal law into two more nearly\naccurate causal laws. If you infer that where there was water yesterday\nthere is water to-day, you are employing a causal law which is not\nalways correct. If you infer that where there was hydrogen and oxygen\nthere is hydrogen and oxygen (or at least that there is hydrogen and\noxygen in places connected by a continuous route with where they were\nyesterday), you are very unlikely to be wrong, unless the place is in\nthe neighbourhood of Sir Ernest Rutherford. It is assumed (what is only\npartially true at present) that the properties of water can be inferred\nfrom those of oxygen and hydrogen together with the manner in which\nthey are combined in the molecules of water. Thus by means of analysis\nyou have obtained causal laws which are at once more true and more\npowerful than those which common sense could obtain by supposing that\nall the parts of water were water.\u003c/p\u003e\n\n\u003cp\u003eWe may say that this is the characteristic merit of analysis as\npractised in science: it enables us to arrive at a structure\u003cspan class=\"pagenum\" id=\"Page_286\"\u003e[Pg 286]\u003c/span\u003e such\nthat the properties of the complex can be inferred from those of the\nparts.\u003ca id=\"FNanchor_58\" href=\"#Footnote_58\" class=\"fnanchor\"\u003e[58]\u003c/a\u003e And it enables us to arrive at laws which are permanent,\nnot merely temporary and approximate. This is an ideal, only partially\nverified as yet; but the degree of verification is abundantly\nsufficient to justify science in constructing the world out of minute\nunits.\u003c/p\u003e\n\n\u003cp\u003eFrom what has been said about substance, I draw the conclusion that\nscience is concerned with groups of \"events,\" rather than with \"things\"\nthat have changing \"states.\" This is also the natural conclusion to\ndraw from the substitution of space-time for space and time. The old\nnotion of substance had a certain appropriateness so long as we could\nbelieve in one cosmic time and one cosmic space; but it does not fit\nin so easily when we adopt the four-dimensional space-time framework.\nI shall therefore assume henceforth that the physical world is to be\nconstructed out of \"events,\" by; which I mean practically, as already\nexplained, entities or structures occupying a region of space-time\nwhich is small in all four dimensions. \"Events\" may have a structure,\nbut it is convenient to use the word \"event,\" in the strict sense,\nto mean something which, if it has a structure, has no space-time\nstructure, \u003ci\u003ei.e.\u003c/i\u003e it does not have parts which are external to\neach other in space-time. I do not assume that an event can ever\noccupy only a point of space-time; the construction of \"points\" out of\nfinitely extended events will form the subject of the next chapter.\nNor do I assign a maximum to the duration of an event, though I hold\nthat any event, in the broad sense, which lasts for more\u003cspan class=\"pagenum\" id=\"Page_287\"\u003e[Pg 287]\u003c/span\u003e than about a\nsecond can, if it is a percept, be analyzed into a structure of events.\nBut this is a merely empirical fact.\u003c/p\u003e\n\n\u003cp\u003eThere are certain purely logical principles which are useful in regard\nto structure. When we are dealing with inferred entities, as to which,\nas explained in Part II., we know nothing beyond structure, we may be\nsaid to know the equations, but not what they mean: so long as they\nlead to the same results as regards percepts, all interpretations are\nequally legitimate. Let us take an example. Suppose we have a set of\npropositions about an electron which we will call \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e. According to\nthe subject-predicate logic, and according to the view that matter is\na substance, there is a certain entity \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e which is mentioned in all\nstatements about this electron. According to the view which resolves\nan electron into a series of events, the propositions in question will\nbe differently analyzed. Assuming a certain schematic simplicity, we\nmight set the matter out as follows: there is a certain relation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e\nwhich sometimes holds between events, and when it holds between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e are said to be events in the biography\nof the same electron. If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e belongs to the field of \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e, \"the\nelectron to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e belongs\" will mean the relation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e with its\nfield limited to terms belonging to the \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e-family of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e; and the\n\u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e-family of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e consists of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e together with the terms which\nhave the relation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and the terms to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e has the\nrelation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e. \"This electron\" will mean \"the electron to which this\nbelongs.\" \"An electron\" will mean \"a series such that there is an \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\nsuch that the series is the electron to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e belongs.\" In order\nto mention some particular electron, we must be able to mention some\nevent connected with it, \u003ci\u003ee.g.\u003c/i\u003e the scintillation when it hits\na certain screen. Thus, instead of saying \"the event \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e happened\nto the electron \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e\" we shall say \"the event \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e happened to the\nelectron to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e happened,\" or, more simply, \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e belongs to\nthe \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e-family of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e.\" The formal properties of the propositional\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e belongs to the \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e-family\u003cspan class=\"pagenum\" id=\"Page_288\"\u003e[Pg 288]\u003c/span\u003e of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\" (\u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e being\nconstant) are the same as those of \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e belongs to the electron\n\u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e.\" If we want any two electrons to be mutually exclusive, in the\nsense that no event can happen to both, we can insure it by assuming\nthat if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e has the relation \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e (or the converse relation) to both\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e belongs to the \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e-family of \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e. If\nwe do not want this, we do not make this assumption about \u003cimg style=\"vertical-align: -0.048ex; width: 1.717ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-8.png\" alt=\"\" data-tex=\"\\(R\\)\"\u003e. It is\nbecause of the identity in formal properties that the one propositional\nfunction can be substituted for the other. Whenever we suggest a new\nview as to structure, we have to make sure that it does not falsify any\nof the old formulæ, though it may give them a new interpretation.\u003c/p\u003e\n\n\u003cp\u003eAnother illustration, more purely logical, may be useful. It\nseems natural to say that any given shade of colour is a quality,\n\u003ci\u003ei.e.\u003c/i\u003e that when we say \"this is red,\" we are saying that \"this\"\nhas a characteristic which we cannot express otherwise than by a\npredicate—assuming, for the moment, that \"red\" stands for just one\nshade of colour. But although this \u003ci\u003emay\u003c/i\u003e be the right view, there\nis no logical necessity for supposing that it is. We might define\none shade of colour as \"all the coloured surfaces which have exact\ncolour-similarity to a given surface.\" Thus \"this has the colour \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e\"\nis replaced by \"this is one of the class of entities that have exact\ncolour-similarity with \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\"; and \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e is a colour\" will be replaced\nby \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e is the class of all entities having exact colour-similarity\nwith a given entity.\" In this case, no facts can be conceived which\nwould give reason for preferring one form of statement to the other,\nsince any ascertainable fact can be interpreted equally well on either\ntheory.\u003c/p\u003e\n\n\u003cp\u003eWe have, in fact, something more or less analogous to the arbitrariness\nof co-ordinates in the general theory of relativity. Provided our\nsymbols have the same interpretation when they apply to percepts, their\ninterpretation elsewhere is arbitrary, since, so long as the formulæ\nremain the same, the \u003ci\u003estructure\u003c/i\u003e\u003cspan class=\"pagenum\" id=\"Page_289\"\u003e[Pg 289]\u003c/span\u003e asserted is the same whatever\ninterpretation we give. Structure, and nothing else, is just what is\nasserted by formulæ in which the meaning of the terms is unknown,\nbut the purely, logical symbols have definite meanings (see Chapter\nXVII.). Even the purely logical symbols are arbitrary to a certain\nlimited extent, as we saw in the above example of colours. But often,\nwhen facts from different regions have to be brought into connection,\none interpretation is much simpler than another. Often, also, one\ninterpretation involves less inference than another, and is therefore\nless likely to be wrong. These are the main motives governing any\nsuggested interpretation of the symbols which occur in mathematical\nphysics.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_58\" href=\"#FNanchor_58\" class=\"label\"\u003e[58]\u003c/a\u003e\nDr C. D. Broad, in \u003ci\u003eThe Mind and its Place in\nNature\u003c/i\u003e, lays stress upon what he calls \"emergent\" properties of\ncomplexes—\u003ci\u003ei.e.\u003c/i\u003e such as cannot be inferred from the properties\nand relations of the parts. I believe that \"emergent\" properties\nrepresent merely scientific incompleteness, which would not exist in\nthe ideal physics. It is difficult to advance any conclusive argument\non either side as to the ultimate character of apparently \"emergent\"\nproperties, but I think my view is supported by such examples as\nthe explanation of chemistry in terms of physics by means of the\nRutherford-Bohr theory of atomic structure.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_290\"\u003e[Pg 290]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXVIII\"\u003eCHAPTER XXVIII\u003cbr\u003e\nTHE CONSTRUCTION OF POINTS\u003ca id=\"FNanchor_59\" href=\"#Footnote_59\" class=\"fnanchor\"\u003e[59]\u003c/a\u003e\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE subject of this chapter is one which has been treated with\nwonderful ingenuity by Dr Whitehead, to whom is due the whole\nconception of a method which arrives at \"points\" as systems of\nfinitely-extended events. In advocating this method, it is not\nnecessary to maintain that mathematical points are \u003ci\u003eimpossible\u003c/i\u003e\nas simple entities (or \"particulars\"); all that it is necessary to\nmaintain is that we have no good ground for regarding them as such.\nWhat we know about points is that they are useful technically—so\nuseful that we must seek an interpretation of the propositions in\nwhich, symbolically, they occur. But there is no ground for denying\nstructure to a point; on the contrary, there are two grounds for\nassigning structure to a point. One is the familiar argument of Occam\u0027s\nrazor: we can make structures having the mathematical properties of\npoints, and to suppose that there are points in any other sense is\nan inference which is useless to science and not warranted by any\nprinciple, logical or scientific. The other argument is much more\ndifficult to state, but the more one studies logical construction\nthe more weight one feels inclined to attach to it. It rests upon a\nmaxim which might be enunciated as a supplement to Occam\u0027s razor:\n\"What is logically convenient is likely to be artificial.\" To me\npersonally, the first example of this maxim was the definition of\nreal numbers. Mathematicians found it convenient to suppose that all\nseries of rationals have limits, while nevertheless\u003cspan class=\"pagenum\" id=\"Page_291\"\u003e[Pg 291]\u003c/span\u003e some do not have\n\u003ci\u003erational\u003c/i\u003e limits. They therefore postulated irrational limits,\nsupposed to be homogeneous with the rationals. Although the method of\nDedekind cuts was familiar, nobody thought of saying: An irrational is\na Dedekind cut, or at least its inferior portion. Yet this definition\nsolves all difficulties. We have now first ratios (which cannot be\nirrational), then segments of the series of ratios. Segments which have\na limit are rational, segments which have no limit are irrational.\nThe square root of 2 is the class of ratios whose square is less than\n2. Segments of the series of ratios are real numbers the series of\nreal numbers has both Dedekindian and Cantorian continuity. Thus it is\nmathematically convenient; but its logical structure is more complex\nthan that of the series of ratios. The logical analysis of mathematics\naffords many examples of this procedure, such as the construction of\n\"ideal\" points, lines, and planes alluded to in Chapter XX.\u003c/p\u003e\n\n\u003cp\u003eIt will be seen that the phrase \"what is logically convenient is\nartificial\" does not express what is meant with as much precision\nas is to be desired. What we mean is this: Given a set of terms\nhaving properties which \u003ci\u003esuggest\u003c/i\u003e certain general mathematical\n(or logical) properties, but are subject to exceptions in regard\nto these properties, it is a mistake to postulate other terms,\nlogically homogeneous with the original set, and such as to remove the\nexceptions; the proper procedure is to look for logical structures\ncomposed of the original terms, and such that these structures always\nhave the mathematical properties in question. It will be found that,\nwhere the assumption of such properties has proved fruitful, this\nprocedure is usually possible.\u003c/p\u003e\n\n\u003cp\u003eStarting from events, there are many ways of reaching points.\nOne is the method adopted by Dr Whitehead, in which we consider\n\"enclosure-series.\" Speaking roughly, we may say that this method\ndefines a point as all the volumes\u003cspan class=\"pagenum\" id=\"Page_292\"\u003e[Pg 292]\u003c/span\u003e which contain the point. (The\nniceties of the method are required to prevent this definition from\nbeing circular; also to distinguish a set of volumes having only a\npoint in common from such as have a line or surface in common.) As a\npiece of logic, this method is faultless. But as a method which aims at\nstarting with the actual constituents of the world it seems to me to\nhave certain defects. Dr Whitehead assumes that every event encloses\nand is enclosed by other events. There is, therefore, for him, no lower\nlimit or minimum, and no upper limit or maximum, to the size of events.\nEach of these assumptions demands consideration.\u003c/p\u003e\n\n\u003cp\u003eLet us begin with the absence of a lower limit or minimum. Here we\nare confronted with a question of fact, which might conceivably be\ndecided against Dr Whitehead, but could not conceivably be decided\nin his favour. The events which we can perceive all have a certain\nduration, \u003ci\u003ei.e.\u003c/i\u003e they are simultaneous with events which are not\nsimultaneous with each other. Not only are they all, in this sense,\nfinite, but they are all above an assignable limit. I do not know what\nis the shortest perceptible event, but this is the sort of question\nwhich a psychological laboratory could answer. We have not, therefore,\ndirect empirical evidence that there is no minimum to events. Nor can\nwe have indirect empirical evidence, since a process which proceeds\nby very small finite differences is sensibly indistinguishable from\na continuous process, as the cinema shows. \u003ci\u003ePer contra\u003c/i\u003e, there\nmight be empirical evidence, as in the quantum theory, that events\ncould not have less than a certain minimum spatio-temporal extent. Dr\nWhitehead\u0027s assumption, therefore, seems rash. At the same time, there\nis a confusion to be avoided: space-time may be continuous even if\nthere is a lower limit to events. Suppose every elementary event filled\na four-dimensional cube, \u003ci\u003ee.g.\u003c/i\u003e a cubic centimetre lasting for the\ntime that light takes to travel a centimetre; and suppose, conversely,\nthat every such four-dimensional\u003cspan class=\"pagenum\" id=\"Page_293\"\u003e[Pg 293]\u003c/span\u003e cube was occupied by an event. The\nspace-time of such a world would be continuous, given suitable axioms,\nalthough events had a minimum. And, conversely, the absence of a\nminimum to events does not insure spatio-temporal continuity. The two\nquestions are thus wholly distinct.\u003c/p\u003e\n\n\u003cp\u003eI conclude that there is at present no means of knowing whether events\nhave a minimum or not; that there never can be conclusive evidence\nagainst their having a minimum; but that conceivably evidence may\nhereafter be found in favour of a minimum. It remains to consider the\nquestion of a maximum.\u003c/p\u003e\n\n\u003cp\u003eOn the question of a maximum to events, the arguments are rather\nlogical than empirical. In a certain sense, any series of events may\nbe called one event; the Battle of Waterloo, for instance, may count\nas a single occurrence. But in a complex event of this sort, there are\nparts which have spatio-temporal and causal relations to each other;\nno single entity devoid of physical structure persists throughout the\nwhole period. I mean by this that anything simultaneous with everything\nthat happened during the Battle of Waterloo is a complex of parts not\nall simultaneous with each other. Whether we are to call such a complex\nan \"event\" or not is merely a question of words. But if our object is\nto exhibit the structure of the physical world, it is clear that we\nmust distinguish objects having physical structure from such as are\nonly component parts of such structures. It is therefore convenient\nto have a word for the latter. The word I shall use is \"event.\" But\nI shall not go so far as to say that an \"event\" must have \u003ci\u003eno\u003c/i\u003e\nstructure. I shall assume only that any structure which it may have is\nirrelevant both to physics and to psychology; in other words, that its\nparts, if any, do not have scientifically distinguishable relations to\nother objects. When the word \"event\" is used in this sense, it is plain\nthat,\u003cspan class=\"pagenum\" id=\"Page_294\"\u003e[Pg 294]\u003c/span\u003e so far as our experience goes, no event lasts for more than a\nfew seconds at most. There is no a priori reason why this should be the\ncase; it is merely an empirical fact. But I think a phraseology which\nobscures it can only lead to confusion.\u003c/p\u003e\n\n\u003cp\u003eFor the above reasons, I am unable to accept Dr Whitehead\u0027s\nconstruction of points by means of enclosure-series as an adequate\nsolution of the problem which it is designed to solve. This problem\nis: to discover structures having certain geometrical properties, and\ncomposed of the raw material of the physical world.\u003c/p\u003e\n\n\u003cp\u003eThere is another method, which may be called that of \"partial\noverlapping.\" In my \u003ci\u003eKnowledge of the External World\u003c/i\u003e, I\napplied this method to the definition of instants. It is easy to\nsee that it is adequate for this purpose in psychology, where we\nhave a one-dimensional time-order which remains definite in spite of\nrelativity. But in physics it is the \"point-instant\" that has to be\ndefined, \u003ci\u003ei.e.\u003c/i\u003e a completely definite position in space-time, not\nmerely in space or merely in time. Here the method is only applicable\nwith suitable modifications. However, the method must first be\nexplained as applied to the one-dimensional psychological time-series.\u003c/p\u003e\n\n\u003cp\u003eWe assume that two events may have a relation which I will call\n\"compresence,\" which means, practically, that they overlap in\nspace-time. Take, for instance, notes played by different instruments\nin orchestral music: if one is heard beginning before the other\nhas ceased to be heard, the auditory percepts of the hearer have\n\"compresence.\" If a group of events in one biography are all compresent\nwith each other, there will be some place in space-time which is\noccupied by all of them. This place will be a \"point\" if there is\nno event outside the group which is compresent with all of them. We\nmay therefore define a \"point-instant,\" or simply a \"point,\" in one\nbiography, as a group of events having the following two properties:\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_295\"\u003e[Pg 295]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003e(1) Any two members of the group are compresent;\u003c/p\u003e\n\n\u003cp\u003e(2) No event outside the group is compresent with every member of the\ngroup.\u003c/p\u003e\n\n\u003cfigure class=\"figright width500\" id=\"i_295a\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-295a.jpg\" width=\"200\" height=\"208\" alt=\"A Venn\ndiagram showing three overlapping circles arranged in a triangular\nformation, creating a central area where all three intersect.\nSimple black outline on white background, illustrating set theory\nrelationships.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eWhen we pass beyond one dimension, this method is no longer applicable.\nTake, for example, the three circles in the accompanying figure: each\noverlaps with the other two, but there is no region common to all\nthree. If we try to remedy this (as I believe we can) by starting,\nin two dimensions, with a relation of \u003ci\u003ethree\u003c/i\u003e events, which is\nto hold when all three have a region in common, we are still met by\ndifficulties. The three circles \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e have a region in\ncommon, and the shaded area \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e has a region in common with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, also with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, and also with \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, yet\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e have no region in common. Therefore if\nevents may have queer shapes such as \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, our new three-term relation\nwill still not enable us to define a \"point.\"\u003c/p\u003e\n\n\u003cfigure class=\"figleft width500\" id=\"i_295b\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-295b.jpg\" width=\"200\" height=\"187\" alt=\" Venn\ndiagram with three overlapping circles labeled a, b, and c. The central\nintersection where all three circles meet is shaded with diagonal\nlines, highlighting the common area shared by all three sets.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_296\"\u003e[Pg 296]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eSince the problem with which we are concerned belongs to \u003ci\u003eanalysis\nsitus\u003c/i\u003e, in which we are occupied only with such properties of\nfigures as are unaffected by continuous deformation, we cannot simply\ndeclare in advance that no events are to have odd shapes. But before\nattempting to deal with this difficulty, it will be as well to\nconsider certain points in \u003ci\u003eanalysis situs\u003c/i\u003e, which will show us\nwhat are the requisites of a solution of our problem. In \u003ci\u003eanalysis\nsitus\u003c/i\u003e we start with two conceptions, that of a point, and that of\n\"neighbourhoods of a given point\"—the latter being collections of\npoints. Certain definitions obtained in this way will be useful.\u003c/p\u003e\n\n\u003cp\u003eThe following definitions are due to Leopold Vietoris.\u003ca id=\"FNanchor_60\" href=\"#Footnote_60\" class=\"fnanchor\"\u003e[60]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eIf \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e is a set of points, a point \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e is called a \"Häufungspunkt\"\nof \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e if in every neighbourhood of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e there is a point other than\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eTwo collections of points \"touch\" each other in a point \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e\nbelongs to one collection and is a \"Häufungspunkt\" of the other.\u003c/p\u003e\n\n\u003cp\u003eA set of points \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e is \"continuous from \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e\" if it\ncontains \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, and any two parts of it whose sum is \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e,\nof which one contains \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and the other \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, touch each other (in\nat least one point).\u003c/p\u003e\n\n\u003cp\u003eA set of points \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e is a \"Linienstück\" from \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e if it, but\nnone of its proper parts, is continuous from \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eHausdorff\u003ca id=\"FNanchor_61\" href=\"#Footnote_61\" class=\"fnanchor\"\u003e[61]\u003c/a\u003e has defined a \"metrical\" space and a \"topological\" space\nin the following terms.\u003c/p\u003e\n\n\u003cp\u003eA \"metrical\" space is a manifold such that with any two points \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e is associated a real not-negative number \u003cimg style=\"vertical-align: -0.464ex; width: 2.403ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-297.png\" alt=\"\" data-tex=\"\\(xy\\)\"\u003e having the\nfollowing three properties: (\u003ci\u003ea\u003c/i\u003e) \u003cimg style=\"vertical-align: -0.464ex; width: 7.823ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-375.png\" alt=\"\" data-tex=\"\\(yx = xy\\)\"\u003e; (\u003ci\u003eb\u003c/i\u003e) \u003cimg style=\"vertical-align: -0.464ex; width: 2.403ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-297.png\" alt=\"\" data-tex=\"\\(xy\\)\"\u003e\nis only zero when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e are identical; (\u003ci\u003ec\u003c/i\u003e) \u003cimg style=\"vertical-align: -0.464ex; width: 7.329ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-376.png\" alt=\"\" data-tex=\"\\(xy + yz\\)\"\u003e\nis greater than or equal to \u003cimg style=\"vertical-align: -0.025ex; width: 2.346ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-377.png\" alt=\"\" data-tex=\"\\(xz\\)\"\u003e.\u003ca id=\"FNanchor_62\" href=\"#Footnote_62\" class=\"fnanchor\"\u003e[62]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eA \"topological\" space is a manifold whose elements \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e are associated\nwith sub-classes \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e of the manifold such that:\u003c/p\u003e\n\n\u003cp\u003e(A) To every \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e corresponds at least one \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e, and every \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e\ncontains \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e;\u003c/p\u003e\n\n\u003cp\u003e(B) If \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.357ex; width: 2.422ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-379.png\" alt=\"\" data-tex=\"\\(V_x\\)\"\u003e are both neighbourhoods of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e there is a\nneighbourhood of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, say \u003cimg style=\"vertical-align: -0.357ex; width: 3.239ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-380.png\" alt=\"\" data-tex=\"\\(W_x\\)\"\u003e which is contained in the common\npart of \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.357ex; width: 2.422ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-379.png\" alt=\"\" data-tex=\"\\(V_x\\)\"\u003e;\u003c/p\u003e\n\n\u003cp\u003e(C) If y is a member of \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e, there is a neighbourhood of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\nwhich is contained in \u003cimg style=\"vertical-align: -0.357ex; width: 2.648ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-378.png\" alt=\"\" data-tex=\"\\(U_x\\)\"\u003e;\u003c/p\u003e\n\n\u003cp\u003e(D) Given any two distinct points, there is a neighbourhood of the one\nand there is a neighbourhood of the other such that the two have no\ncommon point.\u003ca id=\"FNanchor_63\" href=\"#Footnote_63\" class=\"fnanchor\"\u003e[63]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_297\"\u003e[Pg 297]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIn order to be able to apply the usual methods of limits to a\ntopological space, Hausdorff has need of an \"Abzählbarkeitsaxiom,\"\nor \"denumerative axiom.\" He gives two such axioms (p. 263), of which\nthe first is the weaker, and is for some purposes insufficient. The\nfirst states that the number of neighbourhoods of a given point is\nnever greater than \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-135.png\" alt=\"\" data-tex=\"\\(\\aleph_{0}\\)\"\u003e; the second states that the total\nnumber of neighbourhoods of all points is together \u003cimg style=\"vertical-align: -0.05ex; width: 2.514ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-381.png\" alt=\"\" data-tex=\"\\(\\aleph{0}\\)\"\u003e. This\nsecond axiom suffices for all the usual kinds of argument, without the\nintroduction of any metrical ideas.\u003c/p\u003e\n\n\u003cp\u003eP. Urysohn\u003ca id=\"FNanchor_64\" href=\"#Footnote_64\" class=\"fnanchor\"\u003e[64]\u003c/a\u003e has shown that every topological space which satisfies\nHausdorff\u0027s second denumerative axiom and has one further property\n(which he calls \"normality\"\u003ca id=\"FNanchor_65\" href=\"#Footnote_65\" class=\"fnanchor\"\u003e[65]\u003c/a\u003e) is metricizable.\u003c/p\u003e\n\n\u003cp\u003eThese are the main points from \u003ci\u003eanalysis situs\u003c/i\u003e that are relevant\nto the solution of our problem.\u003c/p\u003e\n\n\u003cp\u003eFor the present, we are not concerned with metrical properties,\nbut only with such as belong to \"topological\" spaces. In virtue of\nUrysohn\u0027s theorem, it will be possible to introduce a metric if we can\nconstruct the right sort of topological space. But when one metric is\npossible, an infinite number are possible. The metric which is actually\nintroduced in theory of relativity is introduced for empirical reasons;\nit uses a quantitative relation which might be called degree of causal\nproximity. The existence of this relation is not implied by anything\nwith which we are at present concerned. Moreover, the metrical manifold\nwhich we require in physics is not a \"metrical space\" according to\nHausdorff\u0027s definition given above, since interval in relativity does\nnot possess the properties (\u003ci\u003eb\u003c/i\u003e) and (\u003ci\u003ec\u003c/i\u003e)\u003cspan class=\"pagenum\" id=\"Page_298\"\u003e[Pg 298]\u003c/span\u003e which distance\npossesses in Hausdorff\u0027s definition. However, so far as topological\nconsiderations are concerned, we may, without appreciable inaccuracy,\nassign to small regions the topological properties which belong to a\nsmall region of Euclidean space lasting for a short time, \u003ci\u003ei.e.\u003c/i\u003e\nto a continuous series of small regions of Euclidean space all\ngeometrically indistinguishable.\u003c/p\u003e\n\n\u003cp\u003eIn \u003ci\u003eanalysis situs\u003c/i\u003e, both points and neighbourhoods are given.\nWe, on the other hand, wish to define our points in terms of \"events\"\nwhere \"events\" will have a one-one correspondence with certain\nneighbourhoods. We want our \"events\" to correspond with neighbourhoods\nwhich are above a certain minimum and below a certain maximum when, at\na later stage, the empirical metric is introduced. We have to assign\nto our events such properties as will enable us to define the points\nof a topological space as classes of events, and the neighbourhoods of\nthe points as classes of points. But we have to remember that we do not\nwant to construct merely a topological space: what we want to construct\nis the four-dimensional space-time of the general theory of relativity.\u003c/p\u003e\n\n\u003cp\u003eThe following illustration will serve to introduce the problem.\nConsider a three-dimensional Euclidean numerical space, \u003ci\u003ei.e.\u003c/i\u003e the\nmanifold of all ordered triads of real numbers (\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e),\nwith the usual definition of distance. Consider, in this space, all the\nspheres having a given radius and having centres whose co-ordinates are\nrational. The number of such spheres is \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e. Let us define\na group of these spheres as \"co-punctual\" if it is such that every\nfour chosen out of the group have a common region; and let us define\na co-punctual group as \"punctual\" if it cannot be enlarged without\nceasing to be co-punctual. Then there is a one-one correspondence\nbetween the original points of our space and the punctual groups of\nspheres. Consequently the punctual groups of spheres form a Euclidean\nspace. If the spheres are all distorted in any continuous way,\u003cspan class=\"pagenum\" id=\"Page_299\"\u003e[Pg 299]\u003c/span\u003e they\nwill still enable us to construct punctual groups in the same way, and\nthe manifold of punctual groups will still have all the topological\nproperties which are possessed by a three-dimensional Euclidean space.\nTherefore if we are to use this method of constructing points out\nof \"events,\" we shall have to assume that, in the resulting space,\nthere is a possible metric according to which the points of which a\ngiven event is a member always form a spherical volume. Although this\nis expressed in metrical language, it is in reality a topological\nproperty, since it is unaffected by continuous deformation. It must be\npossible to express it in non-metrical language, though I must confess\nthat I lack the necessary skill.\u003c/p\u003e\n\n\u003cp\u003eI propose, therefore, to regard events as occupying regions of\nspace-time which, in some possible metric, are spheres so far as their\nspace-dimensions are concerned, and between a certain maximum and a\ncertain minimum so far as their time-dimension is concerned. The region\n\"occupied\" by an event is the class of points of which it is a member.\u003c/p\u003e\n\n\u003cp\u003eAs the fundamental relation in the construction of points, we take a\nfive-term relation of \"co-punctuality,\" which holds between five events\nwhen there is a region common to all of them. A group of five or more\nevents is called \"co-punctual\" when every quintet chosen out of the\ngroup has the relation of co-punctuality.\u003c/p\u003e\n\n\u003cp\u003eA \"point\" is a co-punctual group which cannot be enlarged without\nceasing to be co-punctual.\u003c/p\u003e\n\n\u003cp\u003eIn order to demonstrate the existence of points so defined, it\nis sufficient to assume that all events (or at least all events\nco-punctual with a given co-punctual quintet) can be well ordered. If\nZermelo\u0027s axiom is true, this must be the case; if not, it may involve\nsome limitation as to the number of events. I have been led by the\narguments, first of Dr H. M. Sheffer, and then of Mr F. P. Ramsey, to\nthe view that Zermelo\u0027s axiom is true; I am therefore less reluctant\nthan I\u003cspan class=\"pagenum\" id=\"Page_300\"\u003e[Pg 300]\u003c/span\u003e should have been formerly to assume that events can be well\nordered.\u003c/p\u003e\n\n\u003cp\u003eTo prove that every event is a member of at least one point, we proceed\nas follows—assuming that there are co-punctual quintets.\u003c/p\u003e\n\n\u003cp\u003eLet \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e be a well-ordered series whose field consists of all events;\nput\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 38.907ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-382.png\" alt=\"\" data-tex=\"\\[\nP = (x_1, x_2, … x_n, … x_{\\omega}, x_{\\omega + 1} … x_{\\nu} …)\n\\]\"\u003e\u003c/span\u003e\nLet \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e be a co-punctual quintet. If\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e is the only event co-punctual \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, then\nthe class whose only members are \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e\nis a point according to the definition. If, on the other hand, there\nare \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\u0027s other than \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e which are co-punctual with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e, let \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e be the first of them. If\nno \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e other than \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e is co-punctual with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e form a point. Otherwise, let \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e be\nthe first \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e other than \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e and co-punctual with\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-383.png\" alt=\"\" data-tex=\"\\(y_3\\)\"\u003e must be\nlater in the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-series than \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e. If this process comes to an end\nwith \u003cimg style=\"vertical-align: -0.464ex; width: 2.256ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-329.png\" alt=\"\" data-tex=\"\\(y_n\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-328.png\" alt=\"\" data-tex=\"\\(y_2\\)\"\u003e, …\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.256ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-329.png\" alt=\"\" data-tex=\"\\(y_n\\)\"\u003e together form a point. If it does not come to an end with any\nfinite \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e, it may happen that no \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e outside the series (\u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e, … \u003cimg style=\"vertical-align: -0.464ex; width: 2.256ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-329.png\" alt=\"\" data-tex=\"\\(y_n\\)\"\u003e,…) is co-punctual with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e and all the \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s; in that case, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e and\nthese \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s form a point. But if there are \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\u0027s other than the\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s and co-punctual with all of them, let \u003cimg style=\"vertical-align: -0.464ex; width: 2.291ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-384.png\" alt=\"\" data-tex=\"\\(y_{\\omega}\\)\"\u003e be the\nfirst of them. Then \u003cimg style=\"vertical-align: -0.464ex; width: 2.291ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-384.png\" alt=\"\" data-tex=\"\\(y_{\\omega}\\)\"\u003e is later in the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-series than\nany of the finite \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s. We proceed in this way as long as possible,\nusing two principles: (1) given a series of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s ending with\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.144ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-385.png\" alt=\"\" data-tex=\"\\(y_{\\nu}\\)\"\u003e, let \u003cimg style=\"vertical-align: -0.471ex; width: 4.189ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-386.png\" alt=\"\" data-tex=\"\\(y_{\\nu + 1}\\)\"\u003e be the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e in the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-series\nafter \u003cimg style=\"vertical-align: -0.464ex; width: 2.144ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-385.png\" alt=\"\" data-tex=\"\\(y_{\\nu}\\)\"\u003e and co-punctual with the group of all the previous\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s; (2) given a series of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s having no last term, take as\nthe next \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e in the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-series which is after all\nthe \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s hitherto selected and co-punctual with all of them. If, at\nany stage, there is no such \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, the \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s already selected form\na point. Now this process must end sooner or later; for the \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s\n(other than \u003cimg style=\"vertical-align: -0.464ex; width: 2.096ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-327.png\" alt=\"\" data-tex=\"\\(y_1\\)\"\u003e) form an ascending series\u003cspan class=\"pagenum\" id=\"Page_301\"\u003e[Pg 301]\u003c/span\u003e selected from \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, and\ntherefore, sooner or later, there will be no \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e\u0027s later than all\nthe \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s previously selected. At this stage, if not before, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e and the \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e\u0027s already selected will form a point.\nHence if all events can be well ordered, every event is a member of\nat least one point, provided every event is a member of a co-punctual\nquintet. The proof still holds if we only assume that all events\nco-punctual with a given quintet can be well ordered.\u003c/p\u003e\n\n\u003cp\u003eGiven any class of events \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, let \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e be the class of those\nevents which are co-punctual with a. Then by definition a is a point\nif \u003cimg style=\"vertical-align: -0.566ex; width: 9.39ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-388.png\" alt=\"\" data-tex=\"\\(\\alpha = R(\\alpha)\\)\"\u003e. The necessary and sufficient condition that all\nthe members of a should have a point in common is that a should\nbe contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e. This condition is necessary, for, if\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e is a point and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e, it\nfollows that \u003cimg style=\"vertical-align: -0.566ex; width: 4.482ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-390.png\" alt=\"\" data-tex=\"\\(R(\\delta)\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e, and that\n\u003cimg style=\"vertical-align: -0.566ex; width: 8.504ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-391.png\" alt=\"\" data-tex=\"\\(\\delta=R(\\delta)\\)\"\u003e, so that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e. The\nproof that the condition is sufficient is longer; it is as follows.\u003c/p\u003e\n\n\u003cp\u003eIf \u003cimg style=\"vertical-align: -0.566ex; width: 9.39ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-388.png\" alt=\"\" data-tex=\"\\(\\alpha = R(\\alpha)\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is a point. If not, let \u003cimg style=\"vertical-align: -0.566ex; width: 4.667ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-392.png\" alt=\"\" data-tex=\"\\(S(\\alpha)\\)\"\u003e denote the part\nof \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e which is outside \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e. Using again the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-series of all\nevents, put\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -7.985ex; width: 47.333ex; height: 17.102ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-393.png\" alt=\"\" data-tex=\"\\[\n\\begin{aligned}\nz_1 \u0026= \\text{the first member of}\\,\\, S(\\alpha)\\,\\, \\text{in the}\\, P-\\text{order}.\\\\\n\\zeta_{1} \u0026= \\alpha\\,\\, \\text{together with}\\,\\, z_1.\\\\\nz_1 \u0026= \\text{the first member of}\\,\\, S(\\zeta_1)\\,\\, \\text{in the}\\, P-\\text{order}.\\\\\n\\zeta_{2} \u0026= \\zeta_1\\,\\, \\text{together with}\\,\\, z_2.\\\\\n\\zeta_{\\omega} \u0026= \\alpha\\,\\, \\text{together with all the finite}\\,\\, z\u0027s.\\\\\nz_{\\omega} \u0026= \\text{the first member of}\\,\\, S\\left(\\zeta_{\\omega}\\right)\\,\\, \\text{in the}\\, P-\\text{order},\n\\end{aligned}\n\\]\"\u003e\u003c/span\u003e\nand so on, as long as possible. If \u003cimg style=\"vertical-align: -0.489ex; width: 5.58ex; height: 1.71ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-394.png\" alt=\"\" data-tex=\"\\(\\mu \u003c \\nu\\)\"\u003e, \u003cimg style=\"vertical-align: -0.685ex; width: 2.204ex; height: 1.685ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-395.png\" alt=\"\" data-tex=\"\\(z_{\\mu}\\)\"\u003e precedes\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.088ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-396.png\" alt=\"\" data-tex=\"\\(z_{\\nu}\\)\"\u003e in the \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e-order. Hence, as before, there must come a\nstage when no fresh \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e\u0027s can be constructed. If \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e is the\nclass consisting of a together with all the \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e\u0027s yielded by the\nmethod, \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e is a point. For (1) all the quintets in \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e\nare co-punctual, by the construction; (2) a term co-punctual with all\nthe quartets of \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e cannot be later than all the \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e\u0027s,\nbecause if there were such a term we could construct more \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e\u0027s; (3)\nsuch a term cannot be\u003cspan class=\"pagenum\" id=\"Page_302\"\u003e[Pg 302]\u003c/span\u003e earlier than some member of \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e because,\nif it were, it would have been chosen as the \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e of that stage in\nthe construction; hence no event outside \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e is co-punctual with\nevery quartet of \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e. Hence \u003cimg style=\"vertical-align: -0.462ex; width: 1.066ex; height: 2.054ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-397.png\" alt=\"\" data-tex=\"\\(\\zeta\\)\"\u003e is a point.\u003c/p\u003e\n\n\u003cp\u003eTo say that a collection of events have a point in common is to say\nthat the collection is part (or the whole) of the class which is the\npoint. Conversely, a collection of events may contain a sub-class which\nis a point; the necessary and sufficient condition for this is that\n\u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e should be contained in \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, where \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is\nthe collection in question. The proof proceeds exactly as before, if\nwe now make \u003cimg style=\"vertical-align: -0.566ex; width: 4.667ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-392.png\" alt=\"\" data-tex=\"\\(S(\\alpha)\\)\"\u003e mean the part of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e which is not contained in\n\u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eA group of events a is \"co-punctual\" if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained in\n\u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e, and a \"point\" is a co-punctual group which cannot be\nenlarged without ceasing to be co-punctual.\u003c/p\u003e\n\n\u003cp\u003eA few purely logical properties of points may be noted. Given any\ntwo classes \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained in\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.566ex; width: 4.758ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-398.png\" alt=\"\" data-tex=\"\\(R(\\beta)\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e. Hence\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are points and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained\nin \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are identical; for in that\ncase \u003cimg style=\"vertical-align: -0.566ex; width: 4.758ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-398.png\" alt=\"\" data-tex=\"\\(R(\\beta)\\)\"\u003e and \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e are respectively identical with\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, and therefore if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e is contained in\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, so that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are identical.\u003c/p\u003e\n\n\u003cp\u003eEvery co-punctual group of events contains at least one point. This has\nalready been proved, since to say that a is a co-punctual group is to\nsay that a is contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.925ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-387.png\" alt=\"\" data-tex=\"\\(R(\\alpha)\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIt may be taken that, in general, there are a number of points of which\nany given event is a member. Such a set of points will fill a \"region,\"\nbut not every region will be the set of points to which some one event\nbelongs. This topic, however, cannot be dealt with until we have\ndiscussed space-time order.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_59\" href=\"#FNanchor_59\" class=\"label\"\u003e[59]\u003c/a\u003e\nIn this chapter and the next, I owe much to the criticism\nand suggestions of Mr M. H. A. Newman of St. John\u0027s College, Cambridge,\nwho must not, however, be held responsible for their contents; on me\ncontrary, I am convinced that he could construct a much better theory\nthan that which follows.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_60\" href=\"#FNanchor_60\" class=\"label\"\u003e[60]\u003c/a\u003e\n\u003ci\u003eStetige Mengen\u003c/i\u003e, Monatshafte für Mathematik u.\nPhysik., \u003cspan class=\"allsmcap\"\u003eXXXI.\u003c/span\u003e, 1921, pp. 173-204.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_61\" href=\"#FNanchor_61\" class=\"label\"\u003e[61]\u003c/a\u003e\nGrundzüge der Mengenlehre, Leipzig, 1914.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_62\" href=\"#FNanchor_62\" class=\"label\"\u003e[62]\u003c/a\u003e\n\u003ci\u003eIb.\u003c/i\u003e, p. 211.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_63\" href=\"#FNanchor_63\" class=\"label\"\u003e[63]\u003c/a\u003e\n\u003ci\u003eIb.\u003c/i\u003e, p. 213.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_64\" href=\"#FNanchor_64\" class=\"label\"\u003e[64]\u003c/a\u003e\n\u003ci\u003eZum Metrisationsproblem\u003c/i\u003e, Math. Annalen 94 (1925),\npp. 309-315.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_65\" href=\"#FNanchor_65\" class=\"label\"\u003e[65]\u003c/a\u003e\nHe defines a topological space as \"normal\" when any two\nnon-overlapping closed manifolds \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e can be separated by\ntwo non-overlapping regions \u003cimg style=\"vertical-align: -0.345ex; width: 3.166ex; height: 1.94ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-399.png\" alt=\"\" data-tex=\"\\(G_A\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 3.18ex; height: 1.934ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-400.png\" alt=\"\" data-tex=\"\\(G_B\\)\"\u003e which respectively contain\nthem and have no boundary-points. \u003ci\u003eIb.\u003c/i\u003e, p. 310, and Hausdorff,\n\u003ci\u003eop. cit.\u003c/i\u003e, p. 215. A \"boundary-point\" of a collection is one\nwhich has a neighbourhood that is not a sub-class of the collection.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_303\"\u003e[Pg 303]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXIX\"\u003eCHAPTER XXIX\u003cbr\u003e\nSPACE-TIME ORDER\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN the present chapter I shall show how to develop spatio-temporal\norder, in the sense in which it is assumed by the general theory\nof relativity, without any apparatus beyond that of the preceding\nchapter, except a few hypotheses of the sort to be expected in founding\n\u003ci\u003eanalysis situs\u003c/i\u003e.\u003c/p\u003e\n\n\u003cp\u003eThe transformations of co-ordinates which are admissible in tensor\nanalysis are not unlimited; they are such, only, as leave relations\nof \u003ci\u003eneighbourhood\u003c/i\u003e unchanged.\u003ca id=\"FNanchor_66\" href=\"#Footnote_66\" class=\"fnanchor\"\u003e[66]\u003c/a\u003e That is to say, a small\ndisplacement in one system of co-ordinates must correspond to a small\ndisplacement in any other. This requires that, independently of\nmetrical considerations, the events of the space-time manifold should\nhave certain relations of order. It must be possible, in certain\ncircumstances, to say that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is nearer to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e than to \u003cimg style=\"vertical-align: -0.05ex; width: 1.719ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-208.png\" alt=\"\" data-tex=\"\\(C\\)\"\u003e,\nwithout presupposing any quantitative measure of distance. It must be\npossible to construct lines along which there is a definite order, but\nit must be impossible to distinguish certain lines as \"straight.\" A\nclosed curve will be distinguishable from an open curve, but two open\ncurves will not be distinguishable from each other, provided they have\nno singularities. Generally, we shall be able to make propositions\nbelonging to \u003ci\u003eanalysis situs\u003c/i\u003e, at any rate in a sufficiently\nsmall region. But propositions about a configuration must, in the\ngeometry we are to construct, be only such as would remain true if the\nconfiguration were subjected to any kind of deformation which does not\nviolate continuity. It is this pre-co-ordinate geometry that concerns\nus in the present chapter.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_304\"\u003e[Pg 304]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe order to be introduced is of two sorts, macroscopic and\nmicroscopic. We will treat first of the former.\u003c/p\u003e\n\n\u003cp\u003eLet us observe, to begin with, that events may be divided into\nzones with respect to a given event. There are first those that are\ncompresent with a given event, then those not compresent with it, but\ncompresent with an event compresent with it, and so on. The \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003eth\nzone will consist of events that can be reached in \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e steps, but not\nin \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-401.png\" alt=\"\" data-tex=\"\\(n -1\\)\"\u003e, \"step\" being taken as the passage from an event to another\nwhich is compresent with it. We will call two points \"connected\" when\nthere is an event which is a member of both. The passage from event to\nevent by the relation of compresence may be replaced by the passage\nfrom point to point by the relation of connection. Thus points also can\nbe collected into zones. If there is a minimum to the size of events,\nwe may assume that it is always possible to pass from one event to\nanother by a finite number of \"steps.\" If so, there must be a smallest\nnumber of steps in which the passage can be made; thus every event\nwill belong to some definite zone with respect to a given event. This\nis useful in the introduction of order, because we can agree that the\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003eth zone is to be nearer the origin than the \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003eth if \u003cimg style=\"vertical-align: -0.09ex; width: 6.361ex; height: 1.312ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-402.png\" alt=\"\" data-tex=\"\\(m \u003c\nn\\)\"\u003e so that it only remains to introduce order among the members of\na given zone. And even here we only want such order as is involved\nin \u003ci\u003eanalysis situs\u003c/i\u003e, not such more rigid order as is involved,\n\u003ci\u003ee.g.\u003c/i\u003e, in projective geometry.\u003c/p\u003e\n\n\u003cp\u003eWhen an event can be reached from another in \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e steps but not in\n\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-403.png\" alt=\"\" data-tex=\"\\(n – 1\\)\"\u003e, we may regard the intermediate events as forming a sort of\nquantized geodesic route between the two events.\u003c/p\u003e\n\n\u003cp\u003eIn virtue of the above division into zones, which can be effected with\nrespect to any point as origin, we can define a rather small region of\nspace-time by means of four integers, representing the number of steps\nin which any point in the region can be reached from four given points.\nIt is only within a small region of this sort, therefore, that we need\nthe\u003cspan class=\"pagenum\" id=\"Page_305\"\u003e[Pg 305]\u003c/span\u003e more delicate methods of microscopic order, to which we shall now\nproceed.\u003c/p\u003e\n\n\u003cp\u003eGiven two points \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, let us denote by \"\u003cimg style=\"vertical-align: -0.027ex; width: 2.613ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-404.png\" alt=\"\" data-tex=\"\\(x\\lambda\\)\"\u003e\"\ntheir logical product, \u003ci\u003ei.e.\u003c/i\u003e the events which are members of\nboth, or, in geometrical language, the events which contain both. It\nis obvious that, taking the view of events explained at the beginning\nof the preceding chapter, \u003cimg style=\"vertical-align: -0.027ex; width: 2.613ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-404.png\" alt=\"\" data-tex=\"\\(x\\lambda\\)\"\u003e will be null unless \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e are fairly near together. As already stated, we say that\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e are \"connected\" when \u003cimg style=\"vertical-align: -0.027ex; width: 2.613ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-404.png\" alt=\"\" data-tex=\"\\(x\\lambda\\)\"\u003e is not null.\nMicroscopic order is confined to connected points, at any rate to begin\nwith.\u003c/p\u003e\n\n\u003cfigure class=\"figright width500\" id=\"i_305\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-305.jpg\" width=\"200\" height=\"105\" alt=\"A\nmathematical diagram showing two nested, overlapping curved loops\nresembling a torus or linked rings. The inner overlapping region is\nshaded with diagonal lines, and point x marks the intersection area.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eWe now define \"\u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e\" as meaning:\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e are points such that \u003cimg style=\"vertical-align: -0.489ex; width: 2.658ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-405.png\" alt=\"\" data-tex=\"\\(x\\mu\\)\"\u003e is not null\nand is a proper part of \u003cimg style=\"vertical-align: -0.027ex; width: 2.613ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-404.png\" alt=\"\" data-tex=\"\\(x\\lambda\\)\"\u003e.\" An equivalent definition is:\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e are points such that \u003cimg style=\"vertical-align: -0.489ex; width: 2.658ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-405.png\" alt=\"\" data-tex=\"\\(x\\mu\\)\"\u003e is not null,\nand is contained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, but \u003cimg style=\"vertical-align: -0.027ex; width: 2.613ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-404.png\" alt=\"\" data-tex=\"\\(x\\lambda\\)\"\u003e is not contained in\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e.\" By the help of suitable axioms, \"between,\" so defined, can be\nmade to give rise to the spatio-temporal order presupposed in assigning\nco-ordinates in the general theory of relativity. What the definition\nsays, in geometrical language, is that every event which contains both\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e contains \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, but not every event which\ncontains both \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e contains \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eWe must not imagine that all the points between two others lie on one\nline; each lies on \u003ci\u003esome\u003c/i\u003e short route joining the end-points,\na \"short\" route being one composed wholly of points between the\nend-points; but none lies on \u003ci\u003eall\u003c/i\u003e short routes.\u003c/p\u003e\n\n\u003cp\u003eBefore developing the formal consequences of this definition, it may be\nas well to consider its geometrical import. In the accompanying figure,\n\u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e will be between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e if there are events which\ncontain all three, but there are none which contain \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-192.png\" alt=\"\" data-tex=\"\\(\\mu\\)\"\u003e\nwithout containing \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e. (I represent events by areas.) Now if\nevents can often be of irregular shapes\u003cspan class=\"pagenum\" id=\"Page_306\"\u003e[Pg 306]\u003c/span\u003e such as that of the shaded\narea in the figure, it would seem that one event is not likely ever to\nbe between two others according to the definition. I shall therefore\nassume that we may picture events as free from re-entrant angles and\nsimilar oddities. I imagine them as all oval; but formally it would\ndo just as well if they were all four-dimensional cubes, and it would\nnot matter whether they were large or small, provided they did not\ndiffer too much, and were all above a certain minimum. These pictorial\nrequisites are rather for the \u003ci\u003eimportance\u003c/i\u003e of the theory to be\ndeveloped than for its truth. In the preceding chapter, we assumed that\nevents are such as to be all spheres according to one possible metric.\nFormally, we might equally well have assumed that there is a metric\nin which they are all cubes. Some assumption of this kind, as we saw,\nis necessary for the success of our definition of points. The other\nassumptions needed for its truth will be explicitly stated as they are\nintroduced. The assumptions introduced so far in this chapter and its\npredecessor are:\u003c/p\u003e\n\n\u003cp\u003e(1) Compresence is symmetrical.\u003c/p\u003e\n\n\u003cp\u003e(2) Defining \"events\" as the field of compresence, every event is\ncompresent with itself.\u003c/p\u003e\n\n\u003cp\u003e(3) Events can be well ordered; or at least those compresent with a\ngiven event can be.\u003c/p\u003e\n\n\u003cp\u003e(4) Any two events have a relation which is a finite power of\ncompresence. (This is required for mapping space-time into zones.)\nIn other words, the ancestral relation derived from compresence is\nconnected.\u003c/p\u003e\n\n\u003cp\u003eWe will now define a set of points as \"collinear\" if every pair of the\nset are connected, and every triad \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e\nare such that either \u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e, or\n\u003cimg style=\"vertical-align: -0.489ex; width: 2.676ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-407.png\" alt=\"\" data-tex=\"\\(\\alpha\\gamma\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e. We will define a set of\npoints as a \"line\" if (1) it is collinear, (2) it is not contained in\nany larger collinear group with the same extremities. It will be seen\nthat this definition is analogous to that of points. We may define\na set of events\u003cspan class=\"pagenum\" id=\"Page_307\"\u003e[Pg 307]\u003c/span\u003e as \"co-punctual\" when every quintet of the set are\nco-punctual; and we can then define a set of events as a \"point\"\nwhen (1) it is co-punctual, (2) it is not contained in any larger\nco-punctual group. This way of stating our previous definition of\n\"points\" brings out the analogy.\u003c/p\u003e\n\n\u003cp\u003eThe \"lines\" that we are defining are not to be supposed \"straight\";\nstraightness is a notion wholly foreign to the geometry we are\ndeveloping. Perhaps it might be better to call them \"routes\"; but there\nis no harm in calling them \"lines\" provided we remember that they are\nnot supposed to be straight. For the present, we shall not be concerned\nwith lines, but only with collinear groups of points.\u003c/p\u003e\n\n\u003cp\u003eLet us define a set of points as \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear\" if (1) every\npair of the set is connected; (2) given any two, \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e,\neither \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e or \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, or \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e is\nbetween \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e. We shall want such axioms as will\nenable us to show that such a set of points is collinear, not\nmerely \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear, and that their order is independent of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e. It is obvious that, if we put \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e before \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e\nwhenever \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, we obtain a serial\norder of any set of points which is \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear. But to insure\nthat the order shall be independent of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e we require the\nfollowing three axioms:\u003c/p\u003e\n\n\u003cp\u003e(1) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e are points, and\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.489ex; width: 2.115ex; height: 2.081ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-410.png\" alt=\"\" data-tex=\"\\(\\xi\\eta\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.489ex; width: 2.572ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-411.png\" alt=\"\" data-tex=\"\\(\\alpha\\eta\\)\"\u003e is\ncontained in \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e are distinct, then\n\u003cimg style=\"vertical-align: -0.489ex; width: 2.405ex; height: 2.084ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-412.png\" alt=\"\" data-tex=\"\\(\\beta\\eta\\)\"\u003e is not contained in \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003e(2) If \u003cimg style=\"vertical-align: -0.489ex; width: 2.572ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-411.png\" alt=\"\" data-tex=\"\\(\\alpha\\eta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.464ex; width: 2.271ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-413.png\" alt=\"\" data-tex=\"\\(\\beta\\xi\\)\"\u003e is\ncontained in \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, than \u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is contained in the sum\nof \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e. (It follows at once that \u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is\ncontained in \u003cimg style=\"vertical-align: -0.489ex; width: 2.115ex; height: 2.081ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-410.png\" alt=\"\" data-tex=\"\\(\\xi\\eta\\)\"\u003e.)\u003c/p\u003e\n\n\u003cp\u003e(3) If \u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.489ex; width: 2.572ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-411.png\" alt=\"\" data-tex=\"\\(\\alpha\\eta\\)\"\u003e\nis contained in \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.464ex; width: 2.271ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-413.png\" alt=\"\" data-tex=\"\\(\\beta\\xi\\)\"\u003e is contained in the sum\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e. (It follows at once that \u003cimg style=\"vertical-align: -0.464ex; width: 2.271ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-413.png\" alt=\"\" data-tex=\"\\(\\beta\\xi\\)\"\u003e is\ncontained in \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e.)\u003c/p\u003e\n\n\u003cp\u003eThe practical effects of these three axioms are:\u003c/p\u003e\n\n\u003cfigure class=\"figcenter width500\" id=\"i_307\" style=\"width: 300px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-307.jpg\" width=\"300\" height=\"77\" alt=\"A simple line\nsegment with four points marked along it, labeled from left to right as\n(alpha), (xi), (eta), and (beta), representing points on a continuous\nline in mathematical notation.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_308\"\u003e[Pg 308]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003e(1) If \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e are between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, and\n\u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is not between\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003e(2) If \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e is\nbetween \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e are between\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003e(3) If \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e is\nbetween \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e.\u003c/p\u003e\n\n\u003cfigure class=\"figleft width500\" id=\"i_308\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-308.jpg\" width=\"200\" height=\"232\" alt=\"A simple\ncircle diagram with four points marked around its circumference,\nlabeled with Greek letters: (alpha) at bottom left, (beta) at bottom,\n(xi) at top left, and (eta) at top right.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eFrom these axioms we can deduce that a set of points which is\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear is collinear. Also that, given a set of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear points, if \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e is one of them, the points\nof the set which are beyond \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e from a are \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e-collinear,\nand retain the same order when arranged with reference to \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e\nas they had when arranged with reference to \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e. Also that, if\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e is one of a set of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e-collinear points, those of the\nset which are between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e-collinear,\nand have, when arranged with reference to \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, the converse order\nto that which they had when arranged with reference to \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e.\nThese propositions show that we have a satisfactory definition of order\namong the points of a collinear set.\u003c/p\u003e\n\n\u003cp\u003eThe above axioms are logically adequate, but regarded as asserting\nphysical truths about events they may perhaps be regarded as more or\nless doubtful. We have to remember that our lines are not straight,\nand may therefore return into themselves. Routes with very great\ncurvature are, however, excluded by our definition of collinearity.\nConsider, \u003ci\u003ee.g.\u003c/i\u003e, such a route as that in the accompanying figure.\nWe may suppose that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e, \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e are all\nconnected, but \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e will not be between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e according to the definition, because obviously an event\nmay contain \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e without containing \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e and\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e. Thus if we wish to regard the above route from \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e\nto \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e as, in \u003ci\u003esome\u003c/i\u003e sense, a line, it will have to be in\nan extended sense, namely, that it can be divided into a number of\nsmall finite parts, each of which is a line. And a set of points may\nbe regarded as collinear in an extended sense if it is capable\u003cspan class=\"pagenum\" id=\"Page_309\"\u003e[Pg 309]\u003c/span\u003e of a\nserial order such that any sufficiently small consecutive stretch of\nthe series is a collinear set—provided that such stretch must contain\nnot less than four points.\u003c/p\u003e\n\n\u003cp\u003eWe can now prove, by the help of one further axiom, that any\nprogression of collinear points all lying between two points \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e\nand \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e must have a limit.\u003c/p\u003e\n\n\u003cp\u003eLet our set of points be \u003cimg style=\"vertical-align: -0.566ex; width: 23.221ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-414.png\" alt=\"\" data-tex=\"\\(x = (\\xi_1, \\xi_2, \\xi_3,\\ldots \\xi_n\\ldots)\\)\"\u003e\nall lying on a line between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, in an\norder from \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e towards \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e. Let \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e be the sum of\nall the points in \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e (\u003ci\u003ei.e.\u003c/i\u003e the class of members of members\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e), and \u003cimg style=\"vertical-align: -0.023ex; width: 1.873ex; height: 0.998ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-415.png\" alt=\"\" data-tex=\"\\(\\varpi\\)\"\u003e their product, \u003ci\u003ei.e.\u003c/i\u003e the events which\nbelong to every member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. Then \u003cimg style=\"vertical-align: -0.023ex; width: 1.873ex; height: 0.998ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-415.png\" alt=\"\" data-tex=\"\\(\\varpi\\)\"\u003e is not null, because\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.729ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-406.png\" alt=\"\" data-tex=\"\\(\\alpha\\beta\\)\"\u003e is contained in it, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e are\nconnected (in virtue of the definition of collinearity).\u003c/p\u003e\n\n\u003cp\u003eLet \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e consist of all the \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e\u0027s except \u003cimg style=\"vertical-align: -0.464ex; width: 1.979ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-416.png\" alt=\"\" data-tex=\"\\(\\xi_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-2.png\" alt=\"\" data-tex=\"\\(x_2\\)\"\u003e of\nall the \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e except \u003cimg style=\"vertical-align: -0.464ex; width: 1.979ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-417.png\" alt=\"\" data-tex=\"\\(\\xi_2\\)\"\u003e, etc. Let be the events belonging to\nall members of \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-1.png\" alt=\"\" data-tex=\"\\(x_1\\)\"\u003e and generally let \u003cimg style=\"vertical-align: -0.357ex; width: 3.021ex; height: 1.332ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-418.png\" alt=\"\" data-tex=\"\\(\\varpi_n\\)\"\u003e be the events\nbelonging to all members of \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e; and let \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e be the sum of\nall the \u003cimg style=\"vertical-align: -0.023ex; width: 1.873ex; height: 0.998ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-415.png\" alt=\"\" data-tex=\"\\(\\varpi\\)\"\u003e\u0027s. Then \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e consists of all those events\nwhich belong to all sufficiently late \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e\u0027s; \u003ci\u003ei.e.\u003c/i\u003e to say\nthat an event is a member of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is to say that there is an\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e such that the event is a member of \u003cimg style=\"vertical-align: -0.471ex; width: 4.788ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-419.png\" alt=\"\" data-tex=\"\\(\\xi_{n+m}\\)\"\u003e for all values of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIt will be observed that \u003cimg style=\"vertical-align: -0.471ex; width: 5.091ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-420.png\" alt=\"\" data-tex=\"\\(x_{n+m}\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-4.png\" alt=\"\" data-tex=\"\\(x_n\\)\"\u003e, therefore\n\u003cimg style=\"vertical-align: -0.357ex; width: 3.021ex; height: 1.332ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-418.png\" alt=\"\" data-tex=\"\\(\\varpi_n\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.471ex; width: 5.67ex; height: 1.446ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-421.png\" alt=\"\" data-tex=\"\\(\\varpi_{n+m}\\)\"\u003e. It follows that, if\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.68ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-103.png\" alt=\"\" data-tex=\"\\(z\u0027\\)\"\u003e are two members of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, there is an \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e such\nthat \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.68ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-103.png\" alt=\"\" data-tex=\"\\(z\u0027\\)\"\u003e are both members of \u003cimg style=\"vertical-align: -0.357ex; width: 3.021ex; height: 1.332ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-418.png\" alt=\"\" data-tex=\"\\(\\varpi_n\\)\"\u003e. Hence they are\nboth members of \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-422.png\" alt=\"\" data-tex=\"\\(\\xi_{n+1}\\)\"\u003e. Hence any five members of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e\nare co-punctual, and therefore there is at least one point which\ncontains the whole of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, since \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is contained in\n\u003cimg style=\"vertical-align: -0.566ex; width: 4.796ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-423.png\" alt=\"\" data-tex=\"\\(R(\\lambda)\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIf there is a limit, say \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e, to the series of \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e\u0027s, we\nrequire:\u003c/p\u003e\n\n\u003cp\u003e(1) That \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e should be beyond all the \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e\u0027s, \u003ci\u003ei.e.\u003c/i\u003e\nthat for every \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e we should have \u003cimg style=\"vertical-align: -0.464ex; width: 3.143ex; height: 2.086ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-424.png\" alt=\"\" data-tex=\"\\(\\delta\\xi_n\\)\"\u003e contained\nin \u003cimg style=\"vertical-align: -0.471ex; width: 4.788ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-419.png\" alt=\"\" data-tex=\"\\(\\xi_{n+m}\\)\"\u003e \u003ci\u003ei.e.\u003c/i\u003e that we should have \u003cimg style=\"vertical-align: -0.025ex; width: 2.296ex; height: 1.647ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-425.png\" alt=\"\" data-tex=\"\\(\\delta\\sigma\\)\"\u003e\ncontained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e;\u003c/p\u003e\n\n\u003cp\u003e(2) That there should be no point beyond all the \u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-408.png\" alt=\"\" data-tex=\"\\(\\xi\\)\"\u003e\u0027s but between\nthem and \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e, \u003ci\u003ei.e.\u003c/i\u003e that, if \u003cimg style=\"vertical-align: -0.489ex; width: 1.124ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-409.png\" alt=\"\" data-tex=\"\\(\\eta\\)\"\u003e is any point such\nthat \u003cimg style=\"vertical-align: -0.489ex; width: 2.416ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-426.png\" alt=\"\" data-tex=\"\\(\\eta\\sigma\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.489ex; width: 2.416ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-426.png\" alt=\"\" data-tex=\"\\(\\eta\\sigma\\)\"\u003e is\ncontained in \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_310\"\u003e[Pg 310]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eA sufficient condition is, therefore, \u003cimg style=\"vertical-align: -0.186ex; width: 6.632ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-427.png\" alt=\"\" data-tex=\"\\(\\delta\\sigma = \\lambda\\)\"\u003e.\nIf there is a point \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e fulfilling this condition, it is the\nrequired limit.\u003c/p\u003e\n\n\u003cp\u003eIf there is an event \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e such that every quartet of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e\nis co-punctual with \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e and every quartet of \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e which is\nco-punctual with \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e is a part of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, then there is a point\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e which contains \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e and has \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e for a member, and\nthis point will be such that \u003cimg style=\"vertical-align: -0.186ex; width: 6.632ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-427.png\" alt=\"\" data-tex=\"\\(\\delta\\sigma = \\lambda\\)\"\u003e, so that it\nwill be the required limit. But if there is no such event as \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e, we\nmust proceed differently.\u003c/p\u003e\n\n\u003cp\u003eIn this case we need a new axiom, namely:\u003c/p\u003e\n\n\u003cp\u003eIf \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a\nmember of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e but not of \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, then there is a quartet\nwhich is contained in \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and \u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-79.png\" alt=\"\" data-tex=\"\\(\\gamma\\)\"\u003e but is not co-punctual\nwith \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIn the figure, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e represents a member of such a quartet.\u003c/p\u003e\n\n\u003cp\u003eGiven this axiom, we proceed as follows.\u003c/p\u003e\n\n\u003cfigure class=\"figleft width500\" id=\"i_310\" style=\"width: 200px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-310.jpg\" width=\"200\" height=\"133\" alt=\"A geometric\ndiagram showing two circles of different sizes positioned side by side.\nThe larger circle (right) is labeled with x and y, while the smaller\ncircle (left) has labels (alpha), (beta), and (gamma), with arrows\nindicating direction.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eSince \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-422.png\" alt=\"\" data-tex=\"\\(\\xi_{n+1}\\)\"\u003e is between \u003cimg style=\"vertical-align: -0.464ex; width: 2.139ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-428.png\" alt=\"\" data-tex=\"\\(\\xi_n\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a\nmember of \u003cimg style=\"vertical-align: -0.464ex; width: 2.139ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-428.png\" alt=\"\" data-tex=\"\\(\\xi_n\\)\"\u003e but not of \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-422.png\" alt=\"\" data-tex=\"\\(\\xi_{n+1}\\)\"\u003e, there is a quartet which\nis contained in \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e and \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-422.png\" alt=\"\" data-tex=\"\\(\\xi_{n+1}\\)\"\u003e, but is not co-punctual\nwith \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. Now \u003cimg style=\"vertical-align: -0.471ex; width: 5.464ex; height: 2.066ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-429.png\" alt=\"\" data-tex=\"\\(\\beta\\xi_{n+1}\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e;\ntherefore there is a quartet which is a part of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e but is not\nco-punctual with \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. It follows by transposition that if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is\na member of \u003cimg style=\"vertical-align: -0.464ex; width: 2.139ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-428.png\" alt=\"\" data-tex=\"\\(\\xi_n\\)\"\u003e and every quartet of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is co-punctual\nwith \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a member of \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-422.png\" alt=\"\" data-tex=\"\\(\\xi_{n+1}\\)\"\u003e. It follows that\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a member of \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-430.png\" alt=\"\" data-tex=\"\\(\\xi_{n+2}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.471ex; width: 4.183ex; height: 2.063ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-431.png\" alt=\"\" data-tex=\"\\(\\xi_{n+3}\\)\"\u003e, … so that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is\na member of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e. Hence, since \u003cimg style=\"vertical-align: -0.464ex; width: 2.139ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-428.png\" alt=\"\" data-tex=\"\\(\\xi_n\\)\"\u003e may be any member of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, it follows that any member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e which is co-punctual\nwith the whole of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is a member of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e. Now the\nterms co-punctual with the whole of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e constitute the class\n\u003cimg style=\"vertical-align: -0.566ex; width: 4.796ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-423.png\" alt=\"\" data-tex=\"\\(R(\\lambda)\\)\"\u003e. Hence the common part of \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e and \u003cimg style=\"vertical-align: -0.566ex; width: 4.796ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-423.png\" alt=\"\" data-tex=\"\\(R(\\lambda)\\)\"\u003e\nis contained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, and is therefore equal to \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e,\nsince \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e and in \u003cimg style=\"vertical-align: -0.566ex; width: 4.796ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-423.png\" alt=\"\" data-tex=\"\\(R(\\lambda)\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_311\"\u003e[Pg 311]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eNow if \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e is a point which contains \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, it follows\nthat \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.566ex; width: 4.796ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-423.png\" alt=\"\" data-tex=\"\\(R(\\lambda)\\)\"\u003e; hence \u003cimg style=\"vertical-align: -0.025ex; width: 2.296ex; height: 1.647ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-425.png\" alt=\"\" data-tex=\"\\(\\delta\\sigma\\)\"\u003e\nis contained in \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e, and is therefore equal to \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e,\nsince \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-246.png\" alt=\"\" data-tex=\"\\(\\lambda\\)\"\u003e is contained in \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e and in \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-196.png\" alt=\"\" data-tex=\"\\(\\sigma\\)\"\u003e. Hence\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-389.png\" alt=\"\" data-tex=\"\\(\\delta\\)\"\u003e is the required limit.\u003c/p\u003e\n\n\u003cp\u003eIt follows from this that a compact series of points contained within\na stretch of collinear points is continuous. It does not follow that\nthere are compact series of points; this would require existence-axioms\nwhich there is no object in introducing, since we do not know whether\nspace-time is continuous or not. It is, however, interesting to observe\nthat an initial apparatus of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e events suffices to generate\na continuous space-time of points, by means of the relations of\nco-punctuality and logical inclusion.\u003c/p\u003e\n\n\u003cp\u003eThe further development of our geometry, so as to include surfaces,\nvolumes, and four-dimensional regions, obviously presents no difficulty\nin principle, and I do not propose to enlarge upon it. I will merely\nobserve that it is possible to extend the method by which we have\ndefined points and lines so as to obtain something which we may call\nsurfaces and regions, though not quite in the usual sense. Probably\nvarious ways of doing this are possible; the one that I suggest is the\nfollowing.\u003c/p\u003e\n\n\u003cp\u003eA class of lines will be called \"co-superficial\" when any two\nintersect, but there is no point common to all the lines of the class.\u003c/p\u003e\n\n\u003cp\u003eA \"surface\" is a co-superficial class of lines which cannot be\naugmented without ceasing to be co-superficial.\u003c/p\u003e\n\n\u003cp\u003eA class of surfaces is \"co-regional\" when any two have a line in\ncommon, but no line is common to all the surfaces of the class.\u003c/p\u003e\n\n\u003cp\u003eA \"region\" is a co-regional class of surfaces which cannot be augmented\nwithout ceasing to be co-regional.\u003c/p\u003e\n\n\u003cp\u003eIt is obvious that this method could be extended to any number of\ndimensions; also that it requires limitations and extensions. But it\nseems unnecessary to pursue the matter further, since it is plain that\nwe have what is needed for the pre-co-ordinate geometry of space-time.\u003c/p\u003e\n\n\u003cp\u003eLet us now compare our constructed space-time with the spatial\nmanifolds of \u003ci\u003eanalysis situs\u003c/i\u003e. In the preceding chapter\u003cspan class=\"pagenum\" id=\"Page_312\"\u003e[Pg 312]\u003c/span\u003e we quoted\nHausdorff\u0027s definition of a \"topological\" space, and we saw that, in\norder to prove the usual propositions about limits, it is necessary\nthat the total number of neighbourhoods should be \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e. Let\nus now define as a \"neighbourhood\" of a point \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e any set of points\neach of which contains as a sub-class a certain finite co-punctual\nclass of events which is a sub-class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. That is to say, if a is\na co-punctual class of events each of which is a member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, the\nset of all the points of which a is a sub-class will be a neighbourhood\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e. With this definition of a \"neighbourhood,\" it is obvious that\nour space has the four characteristics by which Hausdorff (\u003ci\u003eloc.\ncit.\u003c/i\u003e, p. 213) defines a topological space. In order to insure that\nour space shall also satisfy his second denumerative axiom (\u003ci\u003eloc.\ncit.\u003c/i\u003e, p. 263), it is necessary and sufficient to assume that the\ntotal number of events is \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e. With this assumption, the\ntheorems of \u003ci\u003eanalysis situs\u003c/i\u003e become applicable to our space-time\nmanifold of points.\u003c/p\u003e\n\n\u003cp\u003eIt remains to say a word on the subject of dimensions. We have not\nso far said anything explicit on this subject, though our original\nintroduction of co-punctuality as a five-term relation could only\nprove satisfactory in a four-dimensional manifold. The most suitable\ndefinition of dimensions from our point of view is that of Poincaré,\nwhich is inductive. He defines a space \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e as one-dimensional if,\ngiven any two points \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e, there is an isolated set of points\n\u003cimg style=\"vertical-align: 0; width: 1.928ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-432.png\" alt=\"\" data-tex=\"\\(X\\)\"\u003e such that no connected part of \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e-not-\u003cimg style=\"vertical-align: 0; width: 1.928ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-432.png\" alt=\"\" data-tex=\"\\(X\\)\"\u003e contains both\n\u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e. And he defines a space \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e as \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e-dimensional if,\ngiven any two points \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e, there is an (\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-401.png\" alt=\"\" data-tex=\"\\(n -1\\)\"\u003e)-dimensional\nset of points \u003cimg style=\"vertical-align: 0; width: 1.928ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-432.png\" alt=\"\" data-tex=\"\\(X\\)\"\u003e such that no connected part of \u003cimg style=\"vertical-align: 0; width: 2.378ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-307.png\" alt=\"\" data-tex=\"\\(M\\)\"\u003e-not-\u003cimg style=\"vertical-align: 0; width: 1.928ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-432.png\" alt=\"\" data-tex=\"\\(X\\)\"\u003e\ncontains both \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e. Using this definition, or any other\nwhich is purely topological, we set up the axiom that our topological\nspace-time is to be four-dimensional.\u003ca id=\"FNanchor_67\" href=\"#Footnote_67\" class=\"fnanchor\"\u003e[67]\u003c/a\u003e This completes the material\nrequired for the topological treatment of space-time.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_66\" href=\"#FNanchor_66\" class=\"label\"\u003e[66]\u003c/a\u003e\nFor a geometry based on \"neighbourhood,\" see Hausdorff,\n\u003ci\u003eGrundzüge der Mengenlehre\u003c/i\u003e (Leipzig, 1914), chaps, \u003cspan class=\"allsmcap\"\u003eVII\u003c/span\u003e.\nand \u003cspan class=\"allsmcap\"\u003eVIII\u003c/span\u003e.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_67\" href=\"#FNanchor_67\" class=\"label\"\u003e[67]\u003c/a\u003e\nFor an account of the modern theory of dimensions, see\nKarl Menger, \u003ci\u003eBericht über die Dimensionstheorie\u003c/i\u003e, Jahresbericht\nder deutschen Mathematiker-Vereinigung, 35, pp. 113-150 (1926).\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_313\"\u003e[Pg 313]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXX\"\u003eCHAPTER XXX\u003cbr\u003e\nCAUSAL LINES\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE notion of causality has been greatly modified by the substitution\nof space-time for space and time. We may define causality in its\nbroadest sense as embracing all laws which connect events at different\ntimes, or, to adapt our phraseology to modern needs, events the\nintervals between which are time-like. Now owing to the fact that\nthe formula for \u003cimg style=\"vertical-align: -0.023ex; width: 3.225ex; height: 1.909ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-168.png\" alt=\"\" data-tex=\"\\(ds^{2}\\)\"\u003e is formally the same for time-like and for\nspace-like intervals, there is no longer the difference that formerly\nexisted between causal and geometrical relations. Geodesics are\ngeometrical, but they are also the paths of material particles. It is\nhardly correct to say that a particle \u003ci\u003emoves\u003c/i\u003e in a geodesic; it\nis more correct to say that a particle is a geodesic (though not all\ngeodesics are particles). To say that a particle moves in a geodesic is\nto use language appropriate to the conception of a space which persists\nthrough time, involving the notion of a position which may be occupied\neither at one time or at another. We think, for example, that it is\npossible to move from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e or from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e; but such a\nview is incompatible with the theory of space-time. According to that\ntheory, every position of a body has a date, and it is impossible to\noccupy the same position at another date, since the date is one of the\nco-ordinates of the position. When we travel from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, the\ndate is continually advancing; the return journey, having different\ndates, does not cover the same route. Thus geometry and causation\nbecome inextricably intertwined.\u003c/p\u003e\n\n\u003cp\u003eDr A. A. Robb has laid stress upon the fact that, when two events\nhave a space-like interval, there can be no direct causal relation\nbetween them. This means that, given two such events \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, if any inference is possible from the one to\u003cspan class=\"pagenum\" id=\"Page_314\"\u003e[Pg 314]\u003c/span\u003e the other, it\nmust be by way of a common causal ancestor. Two men may see the sun\nat the same moment, so that the interval between their percepts is\nspace-like; the inference that so-and-so is seeing the sun now arises\nfrom our knowledge of radiation, and requires that we should trace his\npercept and our own to a common ancestry in the sun. We may therefore\ndistinguish time-like and space-like intervals by saying that the\nformer occur where there is some direct causal relation, while the\nlatter occur where both events are related to a common ancestor or a\ncommon descendant. And possibly the magnitude of the interval may be\nderivable from the magnitude of the causal relation. But if this is to\nbe possible, it will be necessary to achieve considerable precision as\nto what we mean by causal relations.\u003c/p\u003e\n\n\u003cp\u003eAs we saw in Part II., perception as a source of knowledge concerning\nphysical objects would be impossible if there were not, in the physical\nworld, semi-independent causal chains, or causal lines as we may call\nthem. The light which comes to us from a printed page retains the\nstructure of the page; if it did not, reading would be impossible. The\nretention is only approximate; it ceases at a distance from the book.\nAnd it ceases within the eye if we have defective vision. But where\nthere is such failure, perception ceases—or rather, it fades away as\nthe failure to preserve structure increases. Thus it is essential to\nperception as a source of knowledge that there should be in the world\ncausal series which are, within limits, independent of the rest of the\nworld.\u003c/p\u003e\n\n\u003cp\u003eAnother point concerning causation emerges from the consideration\nof perception. A number of simultaneous percepts—\u003ci\u003ee.g.\u003c/i\u003e the\nletters of a word which we read at a glance—are to be regarded\nas \"co-punctual\" in the sense of our two preceding chapters.\nEach of these percepts has its own causal antecedents, different\nfrom those of the other percepts. It is true that there may be\nmutual modification—\u003ci\u003ee.g.\u003c/i\u003e a\u003cspan class=\"pagenum\" id=\"Page_315\"\u003e[Pg 315]\u003c/span\u003e colour looks different in the\nneighbourhood of another colour from what it looks against a dark\nbackground. But this is recognized as \"modification,\" \u003ci\u003ei.e.\u003c/i\u003e\nas effecting a change from a norm, which must remain within limits\nif perception is to be successful. Thus the percipient is the\nmeeting-place of a number of more or less independent causal series—as\nmany, at least, as there are distinguishable elements in his total\nmomentary perceptual field. But although these lines have converged\nupon him more or less independently, the totality of his percepts\nnow becomes a causal unit, as is seen in mnemic phenomena. Given a\nnumber of simultaneous percepts, a percept very similar to one of\nthem, occurring on a future occasion, recalls something similar to the\nothers, or at least may do so; here the co-punctuality of the percepts\nis essential to the character of their total effect.\u003c/p\u003e\n\n\u003cp\u003eIn the physical world, the same sort of thing must be supposed to\noccur, though to a less striking degree. According to the theory of\nChapter XXVIII., any event in the physical world occupies a finite\nregion of space-time, whose finiteness consists in the fact that the\nsaid event is compresent with events which are not compresent with\neach other. On the analogy of mnemic phenomena, a group of co-punctual\nevents may have effects which would have been impossible if the events\nhad not been co-punctual. That is the reason why physics is compelled\nto resort to points in stating its causal laws. Until we have a\ncomplete group of co-punctual events, \u003ci\u003ei.e.\u003c/i\u003e a point, we cannot\nbe quite sure as to the effect which will follow from any one of the\nevents; such knowledge as we can have will be more or less approximate.\u003c/p\u003e\n\n\u003cp\u003eIt is these two opposite laws, of approximately separable causal lines\non the one hand, and interactions of co-punctual events on the other,\nwhich make the warp and woof of the world, both physical and mental.\nIn this chapter, I want to attain more precision as to the separable\ncausal lines.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_316\"\u003e[Pg 316]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe possibility of perception, as we have seen already, depends upon\nthe occurrence in the physical world of processes which may be called\n\"radiations,\" provided the word is used somewhat more widely than is\ncustomary. The processes commonly called radiations are, naturally,\nthe most perfect examples. In these, when they are undisturbed, we\nhave a condition of some kind which spreads outward from a centre,\nchanging in an apparently continuous manner as it travels. Something\nmay be met with on the way which alters the law of change, or even\nstops the radiation in some direction altogether; but in the absence of\nobstacles the process proceeds according to its own intrinsic laws. The\npublic senses—sight, hearing, and smell—depend upon radiations, in a\ngeneralized sense in the case of smell. Bodily senses, including touch,\nare more analogous to electric currents in their manner of propagation:\nthey travel along nerves, but not through air or empty space. The\npublic senses, also, travel along nerves, but the disturbance in the\nnerves is a prolongation, with alterations, of a process in the world\noutside the percipient\u0027s body, which is not the case with the bodily\nsenses. It is owing to the existence of radiations that we live in\na common world, since this depends upon the fact that neighbouring\npercipients receive similar stimuli at about the same time. The\nphysical account of radiations is, however, very different in different\ncases. In the case of smell, the emission theory is universally\naccepted: we smell a body because portions of it travel from it to\nthe nose. In the case of sound, only a process, not actual matter, is\ntransmitted, but the process is in matter. In the case of light, if\nwe accept the undulatory theory, the process consists of a transverse\nvibration, which may be said to be in the æther if that brings comfort\nto the speaker, but is certainly not in ordinary matter. If we could\naccept the light-quantum theory, we should still suppose that there is\nsome periodic process, such that the action during one period\u003cspan class=\"pagenum\" id=\"Page_317\"\u003e[Pg 317]\u003c/span\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e\n(Planck\u0027s constant); the light consists of (so to speak) atoms, each\nof which is such a process. There is a great difference of physical\nimportance between these three cases of smell, sound, and light; the\nfirst is quite unimportant physically, the second a somewhat late\ndevelopment from more fundamental principles, the third a corner-stone\nof physical theory.\u003c/p\u003e\n\n\u003cp\u003eIn the ideal case of a radiation, a few observations should suffice to\ndetermine its centre, and then, its laws being known, we could infer\nthe whole connected system of events which constitutes it, in so far\nas the events enter into physical laws. The case of light from a fixed\nstar very nearly realizes the ideal. The places in the universe where\nthe light encounters obstacles are very few, though unfortunately they\ninclude the places where we live. It is because this example of light\n\u003ci\u003ein vacuo\u003c/i\u003e is so nearly perfect that we know as much as we do\nabout astronomy.\u003c/p\u003e\n\n\u003cp\u003eRadiation independent of matter, however, is only one form of causal\nprocess in the physical world. Apart from quantum changes, there are at\nleast two others which are of great importance: one is the motion of\nmatter, and the other is the transmission of a process by matter. The\ndifference involved is essentially one as to causal laws: one sort of\ncausal connection between events makes us regard them as part of the\nhistory of one piece of matter, while another does not, but there is no\nmore intimate connection between an electron at one time and the same\nelectron at another time than between two parts of one light-ray. Let\nus consider for a moment the nature of the causal laws which define one\npiece of matter.\u003c/p\u003e\n\n\u003cp\u003eOne \u003ci\u003eprima facie\u003c/i\u003e difference is that the propagation of light\nis spherical (or conical, in the case of a directed beam), whereas\nthe motion of matter is linear. The history of a piece of matter is a\n\"world-line\"; the history of a light-wave is not. This difference may\nno longer exist if some adaptation of the\u003cspan class=\"pagenum\" id=\"Page_318\"\u003e[Pg 318]\u003c/span\u003e light-quantum theory can\nbe made satisfactory; but, if so, we shall feel that the difference\nbetween light and matter has been much diminished. Another difference\nis the relative indestructibility of matter. One form of energy changes\ninto another, but the energy represented by the proper mass of an\nelectron or proton is not \u003ci\u003eknown\u003c/i\u003e to change into other forms, and\napparently never does so under terrestrial conditions: it does not\nradiate at all in any circumstances that we can produce or observe.\nThen there is the fact that the velocity of a body relative to any\nobserver is always less than that of light. But in spite of the doubt\nas to light-quanta, the main feature of the causal laws that constitute\nmatter seems to be their linear rather than spherical character. It is\nthis that enables us to locate a given piece of matter at a given time.\nThe light emitted by a flash is, at a given moment, diffused over the\nsurface of a sphere, but an electron is as concentrated at one time as\nat another, and does not tend to spread itself out. A unit of matter\nmay, therefore, be appropriately defined as a \"causal line.\"\u003c/p\u003e\n\n\u003cp\u003eBefore pursuing this subject, however, it will be well to dispose of\nthe other kind of causal process which we mentioned just now, namely\nthe transmission of a process by matter. This is itself of two sorts,\none illustrated by sound, the other by the conduction of an electric\ncurrent. In the case of sound we have a radiation; in the other case\nwe have a more or less linear process. In each case, however, actual\npieces of matter move, and cause others to move. The former belongs to\nthe notion of a \"causal line,\" to which we shall return in a moment.\nThe latter belongs to the causal laws as to the interactions of\ndifferent pieces of matter, which I do not wish to consider until I\nhave elicited the intrinsic causal laws which constitute the definition\nof one piece of matter. These, as we saw, have been somewhat obscured\nby the notion of substance, which made it plausible to take for granted\ncertain connections\u003cspan class=\"pagenum\" id=\"Page_319\"\u003e[Pg 319]\u003c/span\u003e between events at different times, which, for us,\nare causal, and demand explicit recognition. It is these intrinsic\nlaws which replace substance that I wish to consider now, leaving the\ninteractions between different pieces of matter for a later stage.\u003c/p\u003e\n\n\u003cp\u003eWhat, then, constitutes a \"causal line\"? In other words, what\nconstitutes one electron? Before asking ourselves what makes us call an\nelectron at one time the same as an electron at another time, it may be\nwell to ask ourselves: What constitutes an electron at one time?\u003c/p\u003e\n\n\u003cp\u003eWe must find some reality for the electron, or else the physical world\nwill run through our fingers like a jelly-fish. There is the same\nsort of reason, however, for not regarding an electron as an ultimate\nparticular as there was for refusing this status to a space-time point.\nThe electron has very convenient properties, and is therefore probably\na logical structure upon which we concentrate attention just because\nof these properties. A rather haphazard set of particulars may be\ncapable of being collected into groups each of which has very agreeable\nsmooth mathematical properties; but we have no right to suppose Nature\nso kind to the mathematician as to have created particulars with just\nsuch properties as he would wish to find. We have, therefore, to ask\nourselves: Can we construct an electron out of events, in the same\nsort of way in which we constructed space-time points? To this inquiry\nwe must now address ourselves, confining ourselves, at first, to the\nelectron at one time.\u003c/p\u003e\n\n\u003cp\u003eWhen I speak of \"electrons\" in this discussion, I shall include\n\"protons,\" since everything that is to be said about the one is to be\nsaid about the other also.\u003c/p\u003e\n\n\u003cp\u003eWe do not know much about the contents of any part of the world except\nour own heads; our knowledge of other regions, as we have seen, is\nwholly abstract. But we know our percepts, thoughts, and feelings in a\nmore intimate fashion.\u003cspan class=\"pagenum\" id=\"Page_320\"\u003e[Pg 320]\u003c/span\u003e Whoever accepts the causal theory of perception\nis compelled to conclude that percepts are in our heads, for they come\nat the end of a causal chain of physical events leading, spatially,\nfrom the object to the brain of the percipient. We cannot suppose that,\nat the end of this process, the last effect suddenly jumps back to\nthe starting-point, like a stretched rope when it snaps. And with the\ntheory of space-time as a structure of events, which we developed in\nthe last two chapters, there is no sort of reason for not regarding\na percept as being in the head of the percipient. I shall therefore\nassume that this is the case, when we are speaking of physical, not\nsensible, location.\u003c/p\u003e\n\n\u003cp\u003eIt follows from this that what the physiologist sees when he examines a\nbrain is in the physiologist, not in the brain he is examining. What is\nin the brain by the time the physiologist examines it if it is dead, I\ndo not profess to know; but while its owner was alive, part, at least,\nof the contents of his brain consisted of his percepts, thoughts, and\nfeelings. Since his brain also consisted of electrons, we are compelled\nto conclude that an electron is a grouping of events, and that, if\nthe electron is in a human brain, some of the events composing it are\nlikely to be some of the \"mental states\" of the man to whom the brain\nbelongs. Or, at any rate, they are likely to be parts of such \"mental\nstates\"—for it must not be assumed that part of a mental state must\nbe a mental state. I do not wish to discuss what is meant by a \"mental\nstate\"; the main point for us is that the term must include percepts.\nThus a percept is an event or a group of events, each of which belongs\nto one or more of the groups constituting the electrons in the brain.\nThis, I think, is the most concrete statement that can be made about\nelectrons; everything else that can be said is more or less abstract\nand mathematical.\u003c/p\u003e\n\n\u003cp\u003eWe have arrived at the conclusion that an electron at an instant is\na grouping of events; the question is: what sort of group is it?\nObviously it includes all the events that\u003cspan class=\"pagenum\" id=\"Page_321\"\u003e[Pg 321]\u003c/span\u003e happen where the electron\nis. If we may regard the electron as a material point, the events\nconstituting an electron will have the two characteristic properties\nof points, viz. any five are co-punctual, and not all sub-classes of\nfour events are co-punctual with any event outside the group. I do not\nknow whether there is any valid ground for supposing that an electron\nis of finite size; none of the usual arguments seem at all conclusive,\nsince they only show the forces developed in the neighbourhood of an\nelectron. However, it is usual to assume a finite size, and for us the\nmatter is one of indifference. If we assume a finite size, the events\nbelonging to the electron can be grouped into many points, not only\ninto one; in this case, the electron is a group of points, \u003ci\u003ei.e.\u003c/i\u003e\na class of classes of events. It will save circumlocution to speak\nof the electron as a point, and leave it to the reader to make the\nnecessary verbal alterations for adaptation to the hypothesis of finite\nsize. But it should be remembered that in Heisenberg\u0027s theory the\nelectron is neither a point nor of finite size, since ordinary spatial\nconceptions are inapplicable to it. For the moment, we will, however,\nconfine ourselves to the older theory of the electron.\u003c/p\u003e\n\n\u003cp\u003eIf the electron is a point, it is a \u003ci\u003ematerial\u003c/i\u003e point, and thus\ndiffers from points in empty space. This difference, I believe, does\nnot consist in anything characteristic of the electron at an instant,\nbut in its causal laws. What distinguishes a material point from a\npoint of empty space-time is that we can recognize a series of earlier\nand later material points as all parts of the history of one electron.\nIn the Newtonian theory, one could say the same of a point of absolute\nspace; but with the abandonment of absolute space we have become\nunable to regard a point at one time as in any sense the same as a\npoint at another time, except in the case of a \u003ci\u003ematerial\u003c/i\u003e point.\nThe existence of this connection may be taken as the definition of\n\"matter\"; and obviously the connection is causal.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_322\"\u003e[Pg 322]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIn order to develop this further, we must return to the view suggested\nin connection with perception, that events occur, usually, in groups\narranged about centres. These centres may be taken to be places where\nthere is matter. It is found that, given events arranged about a\ncentre at one time, there are generally similar events arranged about\nneighbouring centres at slightly earlier or later times. By taking\nthe centre very small, and by continually diminishing the time-like\ninterval concerned, this statement can be made more and more nearly\ntrue; in the limit, when stated in the language of differentials, it\nmay be exactly true, except where quantum phenomena are concerned. In\ntheir case, continuity is not the criterion, at least not continuity in\nall respects. There is continuity in some respects, and in others there\nis a jump of a definite amount connected with the quantum theory. This\ncase shows, however, that continuity is not the essence of material\nidentity; the essence is inferribility of a group of phenomena at one\ntime from a group at another, when both groups are arranged about\ncentres.\u003ca id=\"FNanchor_68\" href=\"#Footnote_68\" class=\"fnanchor\"\u003e[68]\u003c/a\u003e The time must be very short, and the inference is only\napproximate, except in the limit, as the time tends towards zero.\nMoreover, the time of the group is not any of the times at which the\nseveral members of the group occur, but the calculated time at which\nthe group began to be propagated from the centre. The centre is \"where\nthe piece of matter is,\" and the route of the piece of matter is\ndetermined by the differential equations which result from the above\nprinciple. But as to what are the actual events at the centre, we know\nnothing except what follows from the fact that our percepts and \"mental\nstates\" are among the events which constitute the matter of our brains.\u003c/p\u003e\n\n\u003cp\u003eThus each material unit is a causal line whose neighbouring\u003cspan class=\"pagenum\" id=\"Page_323\"\u003e[Pg 323]\u003c/span\u003e points\nare connected by an intrinsic differential law. The simplest form of\nsuch a law is the first law of motion, from which it follows that\nif a body covers a given distance in a very short time, it will\ncover a very nearly equal distance in the next very short time. I\nconceive—though this is conjectural—that, given any event anywhere in\nspace-time, there is usually some qualitatively very similar event in a\nneighbouring place in space-time, and that, if there is any measurable\nrelation between the two events, the \"velocity\" of the change varies\ncontinuously, so that at a third neighbouring point there will be an\nevent differing from the second by very nearly the same amount as that\nby which the second differed from the first, provided the interval\nbetween the second and third points is equal to that between the first\nand second. This, together with the fact that events can be grouped\nabout centres by the sort of laws which we have called \"perspective,\"\nseems to explain the utility of matter in stating the causal laws of\nthe physical world. But there is need of caution owing to quantum\nphenomena, as explained in the preceding paragraph. Continuity is the\nrule, but it may have exceptions. So long as the exceptions are subject\nto ascertainable laws, they do not make the whole system impossible.\u003c/p\u003e\n\n\u003cp\u003eSo far, I have said nothing about extrinsic causal laws, \u003ci\u003ei.e.\u003c/i\u003e\nthose which we naturally regard as exemplifying the influence of one\npiece of matter upon another. Einstein\u0027s theory of gravitation has\nthrown a new light upon these; but this is matter for a new chapter.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_68\" href=\"#FNanchor_68\" class=\"label\"\u003e[68]\u003c/a\u003e\nIn this case, however, if Heisenberg is right, we cannot\nidentify an electron at one time with an electron at another. This\nwould be a difficulty if an electron were conceived as a substance, but\nfor us it is merely an empirical limitation of the empirical conception\nof a causal line.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_324\"\u003e[Pg 324]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXI\"\u003eCHAPTER XXXI\u003cbr\u003e\nEXTRINSIC CAUSAL LAWS\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nI MEAN by an \"extrinsic\" causal law any formula in which one piece of\nmatter is mentioned as concerned in the behaviour of another. Newtonian\ngravitation afforded a perfect example of an extrinsic causal law, but\nEinsteinian gravitation, \u003ci\u003eprima facie\u003c/i\u003e, does not. The question\nI want to consider is: Can we, in the last analysis, dispense with\nsuch laws altogether, and regard each piece of matter as completely\nself-determined? Or must we admit them, and, if so, in what form? And\nwhat are we to say of such matters as the emission and absorption of\nlight?\u003c/p\u003e\n\n\u003cp\u003eLet us first consider Einsteinian gravitation. The theory consists in\nascribing to every region of space-time a metrical structure which is\nobtained (roughly speaking) by superposing a number of structures which\nare symmetrical about centres, the centres being portions of matter;\nand, given the structure, each piece of matter moves in a geodesic, or\nrather is a geodesic. It is not very easy to see what this means when\nit is translated from the technical language of theoretical physics\ninto the language of groups of events. Nevertheless, we must make the\nattempt.\u003c/p\u003e\n\n\u003cp\u003eTo begin with: Can we make \"matter\" into a mere law according to which\nevents occur in the places where there is no matter? This question is\nanalogous to that of phenomenalism as discussed in Chapter XX. We there\nconsidered the possibility of explaining unperceived \"things\" as laws\nconcerning the behaviour of perceived \"things.\" Similarly we might take\nevents which occur in empty space, and find that they were subject to\nlaws symmetrical about centres, and\u003cspan class=\"pagenum\" id=\"Page_325\"\u003e[Pg 325]\u003c/span\u003e define each such law as a piece of\nmatter situated at the centre. Conversely, we might regard the supposed\nevents in empty space as mere laws connecting events in different\npieces of matter; this becomes phenomenalism if we confine the pieces\nof matter to human brains. There are many possible ways of turning\nsome things hitherto regarded as \"real\" into mere laws concerning the\nother things. Obviously there must be a limit to this process, or else\nall the things in the world will merely be each other\u0027s washing. But\nthe only obvious final limit is that set by phenomenalism—perhaps one\nought to say, rather, that set by solipsism. If we have once admitted\nunperceived events, there is no very obvious reason for picking and\nchoosing among the events which physics leads us to infer.\u003c/p\u003e\n\n\u003cp\u003eThis argument, however, hardly warrants us in assuming events inside\nan electron. If we assume an electron of the Rutherford type, we shall\nhave to say that, if anything does take place inside the electron,\nwe can know nothing about it. No physical process passes through the\nelectron, so that the inside, if it exists, is a prison from which\nnothing can escape. No event inside an electron can be compresent with\nan event outside it; consequently, according to the theory of Chapter\nXXIX., no line can cross the boundary of an electron. What goes on\ninside, if anything does, is irrelevant to the rest of the universe,\nand is not really in the same space-time as what-goes on outside.\nNow the world of physics is intended to be a causally interconnected\nworld, and must be such if it is not to be a groundless fairy tale,\nsince our inferences depend upon causal laws. Therefore if anything\noccurs which is causally isolated, we cannot include it in physics.\nWe have no ground whatever for saying that nothing is causally\nisolated, but we can never have ground for saying: Such-and-such a\ncausally isolated event exists. The physical world is the world which\nis causally continuous with percepts, and what is not so continuous\nlies outside physics. Thus if\u003cspan class=\"pagenum\" id=\"Page_326\"\u003e[Pg 326]\u003c/span\u003e anything occurs inside an electron,\nsuch an occurrence does not belong to the world of physics. It would\nseem to follow that, if the electron is to have a definite position in\nspace-time, it must be either a point or a hole. The former, however,\nis physically unsatisfactory, and the latter seems scarcely capable of\nan intelligible interpretation. Thus the Rutherford type of electron\nraises problems, however we may interpret it.\u003c/p\u003e\n\n\u003cp\u003eThe Heisenberg electron offers a way out of these difficulties. This\nelectron is not in a definite place, and nothing happens inside it.\nIt is essentially a collection of radiations observable in other\nplaces than that in which the electron would formerly have been said\nto be. Thus the electron is reduced to a law as to occurrences in a\ncertain region. We cannot say, on this view, that the electron is a\npoint, or that it is a certain finite region, or that it is a hole;\nit is, so to speak, something of a different logical type, connected\nwith a region through the fact that the radiations concerned have\ndiminishing intensity as we pass away from this region, but not capable\nof \u003ci\u003eaccurate\u003c/i\u003e correlation with either a region or a point. Thus on\nthis view matter consists merely of laws as to occurrences in \"empty\"\nspace.\u003c/p\u003e\n\n\u003cp\u003eOwing to the fact that an electron at one time cannot be identified\nwith an electron at another time where quantum changes have intervened,\nthe conception of motion loses its definiteness where electrons\nare concerned. This, however, only raises difficulties when we are\nconcerned with very minute phenomena, such as those which occur within\nan atom. For large-scale phenomena, such as those with which astronomy\nis concerned, we may still regard the electron as persisting and as\nmoving in space-time.\u003c/p\u003e\n\n\u003cp\u003eWe can now return to the Einsteinian theory of gravitation, which\nnecessitated this long digression. According to this theory, each\nelectron is associated with a crinkle, which grows\u003cspan class=\"pagenum\" id=\"Page_327\"\u003e[Pg 327]\u003c/span\u003e less marked as\nwe get away from the electron, but extends theoretically throughout\nspace. The actual metrical structure of space-time in any region is\nobtained (roughly speaking) by superposing these crinkles. Now the\nmetrical properties of space-time are nothing but a method of stating\ncausal laws. In the case of gravitation, these laws have to do with\nthe way in which the movement of one electron is connected with the\npositions of the others. We must suppose that the formula for interval\nrepresents something in the state of affairs at each place, and that\nbodies left to themselves move in geodesics, and that, so long as\nelectromagnetic phenomena are left out of account, the formula for\ninterval at any place is found approximately by superposing a number\nof spherically symmetrical formulæ, each of which corresponds to an\nelectron in its central region. It is natural to ask, at this point,\nwhether interval has any more physical reality than force. But I do not\nwish to raise this question yet, as I propose to consider it in later\nchapters. For the present we may say (\u003ci\u003ea\u003c/i\u003e) that we can recognize\npeculiar regions in space-time, which are those that would naturally be\nregarded as in the immediate neighbourhood of matter; (\u003ci\u003eb\u003c/i\u003e) that\nthe formula for interval at any place is a function of the geodesic\ndistances from that place to neighbouring pieces of matter; (\u003ci\u003ec\u003c/i\u003e)\nthat pieces of matter travel along geodesics.\u003c/p\u003e\n\n\u003cp\u003eThe question whether, in such a theory, there is \"action at a distance\"\nis really one of words. The formula by which we determine what will\nhappen in a given region will contain references to distant regions,\nand it may be said that this is all we can mean by \"action at a\ndistance.\" To mean more, it may be said, is to regard causality as\nsomething more than correlation, which there can be no reason for\ndoing. If what happens in one place is correlated with what happens\nin another, we may be told, nothing more could be imagined in the way\nof action at a distance. But this is not quite what\u003cspan class=\"pagenum\" id=\"Page_328\"\u003e[Pg 328]\u003c/span\u003e in fact occurs.\nWhat happens in one place is not correlated with \u003ci\u003ewhat happens\nin\u003c/i\u003e another place, but with another place, which is a different\nthing. Different neighbourhoods have different characters, and the\ndifferences can be represented by a combination of formulæ which are\nspherically symmetrical. This is not action at a distance, but action\n\u003ci\u003eaccording\u003c/i\u003e to a distance; there is nothing that cam properly be\ncalled an effect of one thing upon another at a distance from it. Thus\nso far, pending the discussion of interval, we have found nothing that\ncam properly be described as am extrinsic causal law.\u003c/p\u003e\n\n\u003cp\u003eElectromagnetic phenomena, if we accept Weyl\u0027s theory, will not\ndiffer importantly, so far as our present question is concerned,\nfrom gravitation. An electromagnetic field will be represented by\ngauge-relations between points in a neighbourhood, and there will be\nno ground for supposing that one piece of matter influences another;\nall that we can say is that a piece of matter corresponds to a metrical\nstate of affairs which makes the geodesics different from what they\nwould otherwise be. The motion of an electron or proton is then due to\nthe peculiarities of the metrical state of affairs where it is, not\nto something even so near as the hydrogen nucleus is to its planetary\nelectron.\u003c/p\u003e\n\n\u003cp\u003eBut what are we to say of the emission and absorption of light? It is\nclear that whenever we perceive light we absorb it, that is to say,\nthe energy in the waves of light (or light-quanta?) that hit the eye\nis transformed into a different kind of energy, though I should not\nventure to say what kind. Therefore all visual percepts involve this\nprocess of absorbing light. And if perception can ever be a source of\nknowledge as to things outside the percipient\u0027s body, there must be\ncausal laws connecting what happens to the percipient with what goes on\noutside. It is, of course, obvious that there are such laws; we cannot\nrevive Leibniz\u0027s windowless monads. The process of absorption and\nemission of light will serve as a\u003cspan class=\"pagenum\" id=\"Page_329\"\u003e[Pg 329]\u003c/span\u003e special case, about which we have\nconsiderable knowledge, in which we can hope to analyze exactly what\noccurs.\u003c/p\u003e\n\n\u003cp\u003eLet us take, for simplicity, two hydrogen atoms, of which one emits\nenergy which the other absorbs. But for the theory of quanta, and\nsuch phenomena as the photo-electric effect, a supposition of this\nsort would be impossible. If the energy radiated from a hydrogen atom\nin the form of light really has the shape of a spherical wave, it is\nimpossible that the whole of it should be absorbed by one other atom,\nany more than the whole of the light radiated from the sun can fall\non the earth. But if the light emitted by a single atom travels in a\nstraight line (approximately), like a material particle, then it may\nhappen to hit one atom and be absorbed whole, just as Jonah might\nhave been swallowed by another whale. We shall have to suppose, in\nthis case, that the spherical distribution of light round a radiating\nbody is a statistical phenomenon, like bullets fired from a fort in\nall directions. This suggests the hypothesis which we have already\nconsidered in Chapter XIII., according to which nothing at all happens\nbetween the emission of light by one body and its absorption by\nanother. In that case, empty space collapses just as the electron did,\nand only the surface of the electron remains. This, however, seems\nhardly a tenable view. The intervening space might be described as\nnon-existent from a metrical point of view, since the interval between\nthe emission and the absorption of a light-ray is zero; but from an\nordinal point of view this is not the case, since, if \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e\nare two points on a light-ray, we can distinguish the case in which the\nray goes from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e from that in which it goes from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e to\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. This difference can be stated in metrical terms. For example:\nLet us take as our time co-ordinate the proper time of no matter what\nbody; whatever body we choose, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e will be earlier than \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, or\nelse, whatever body we choose, \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e will be earlier than \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. Again:\nSuppose that at \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e there are mirrors, which reflect part\nof the\u003cspan class=\"pagenum\" id=\"Page_330\"\u003e[Pg 330]\u003c/span\u003e ray in such a way that an observer \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e sees both reflected\nrays. Then either every such observer will see the reflection from\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e before that from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, or else every such observer will see the\nreflection from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e before that from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. We can free this from\ndependence on an observer by the following method of statement: Let\n\u003cimg style=\"vertical-align: 0; width: 2.325ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-433.png\" alt=\"\" data-tex=\"\\(A\u0027\\)\"\u003e be a point on the ray reflected from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, and \u003cimg style=\"vertical-align: 0; width: 2.345ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-434.png\" alt=\"\" data-tex=\"\\(B\u0027\\)\"\u003e a point\non the ray reflected from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, so chosen that the interval between\n\u003cimg style=\"vertical-align: 0; width: 2.325ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-433.png\" alt=\"\" data-tex=\"\\(A\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.345ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-434.png\" alt=\"\" data-tex=\"\\(B\u0027\\)\"\u003e is time-like. Then, however \u003cimg style=\"vertical-align: 0; width: 2.325ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-433.png\" alt=\"\" data-tex=\"\\(A\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.345ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-434.png\" alt=\"\" data-tex=\"\\(B\u0027\\)\"\u003e may\nbe chosen, either \u003cimg style=\"vertical-align: 0; width: 2.325ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-433.png\" alt=\"\" data-tex=\"\\(A\u0027\\)\"\u003e is always before \u003cimg style=\"vertical-align: 0; width: 2.345ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-434.png\" alt=\"\" data-tex=\"\\(B\u0027\\)\"\u003e, or \u003cimg style=\"vertical-align: 0; width: 2.345ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-434.png\" alt=\"\" data-tex=\"\\(B\u0027\\)\"\u003e is always\nbefore \u003cimg style=\"vertical-align: 0; width: 2.325ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-433.png\" alt=\"\" data-tex=\"\\(A\u0027\\)\"\u003e. This is stated in the language of the special theory,\nbut it is still valid, \u003ci\u003emutatis mutandis\u003c/i\u003e, in the general theory.\nThus when we say that the interval between two points on a light-ray is\nzero we are not denying that there is an important sense in which one\nis earlier than the other, and in which one can be regarded as cause\nand the other as effect. This suggests that the zero interval is not\nquite so significant as it might seem to be, and I cannot therefore\naccept the view that there are no events along the path of a light-ray\nin empty space.\u003c/p\u003e\n\n\u003cp\u003eLet us now return to the emission of light, ignoring absorption for the\npresent; and let us still consider a single hydrogen atom. We are told\nto suppose that the electron revolves about the proton for a certain\ntime, say in a circular orbit four times as large as the minimum orbit;\nthen, suddenly, it decides to revolve in the minimum orbit. When this\nchange occurs, the atom loses a certain amount of energy, which is\ntransformed into light whose frequency is obtained by dividing the\nloss of energy by \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e (Planck\u0027s constant). Whether the light travels\nonly in one direction, or in a spherical wave, we are compelled, in\nthe present state of physical knowledge, to leave an open question.\nBut we do assume that something travels away from the electron, and\nthat, if light is absorbed by another atom, that light has traversed\na route from its place or places of origin. We assume also that the\nlight has a frequency, \u003ci\u003ei.e.\u003c/i\u003e that what travels is a periodic\nprocess. When\u003cspan class=\"pagenum\" id=\"Page_331\"\u003e[Pg 331]\u003c/span\u003e the light is absorbed, it ceases to exist as light,\nalthough it may reappear (in fluorescence). But often its energy exists\nin discoverable forms—chemical forms in chlorophyl, for example.\nWhen, however, the energy exists in the form of a steady motion of the\nelectron in its orbit, it is not discoverable until there is a change\nof orbit. If we had sufficiently powerful microscopes, we could see\na glowing gas dissolving into a comparatively small number of spots\nof light, while the atoms in steady motion would be invisible. Thus\nwe seem to reach the conclusion that the causal laws which genuinely\nconnect one piece of matter with another are quantum laws, in which\nthere are various stages: first, a periodic process having no outside\neffect; secondly, a sudden disruption of the energy of this process\ninto two parts, one being a new periodic process in the original body,\nthe other a periodic process travelling in empty space; thirdly, the\narrival of the travelling process at another body; fourthly, a quantum\nchange in this other body, involving absorption of the radiant energy\nin the production of a new steady state in the absorbing body. All\ngenuine causal relations between different bodies, we may suppose,\ninvolve this process of sudden loss of energy by one body and its\nsudden acquisition, later, by another body. The older physical laws, as\nreinterpreted by relativity, can apparently be so stated as to leave\nbodies independent of each other; but I cannot see how the quantum laws\ncan be so stated.\u003c/p\u003e\n\n\u003cp\u003eIf one could adopt what may be called the \"parcels-post\" theory of\nradiation, according to which, when energy leaves an atom, it does so\nwith a definite destination in view, we could simplify our account\nof the matter. In that case, atoms would, at most times, live a\nself-contained life, \"the world forgetting, by the world forgot.\"\nBut sometimes they would give a parcel of energy to the postman, and\nsometimes they would receive one from him. The postman (who is perhaps\nnot a teetotaller) sways from side to side as he travels, and the\u003cspan class=\"pagenum\" id=\"Page_332\"\u003e[Pg 332]\u003c/span\u003e\nbigger the parcel the faster he sways. But he travels at the same rate\nwhether his parcel is big or small; and he is the only link between the\natom and the rest of the world.\u003c/p\u003e\n\n\u003cp\u003eFor the present, we dare not assume that the question is as simple\nas in the parcels-post illustration. Energy may (as the orthodox\ntheory supposes) be lost by radiation into the void—lost, I mean, not\nmathematically, but practically. The difficulty is that we cannot put\nan instrument into the void to see what happens there; the attempt\nis just like trying to go and see what things look like from a place\nwhere there is no eye. All our actual knowledge is concerned with the\nboundary surfaces between matter and empty space: what is inside and\noutside these surfaces is conjectural. I cannot help believing that\nsome far simpler logical scheme of physics is possible than any yet\nevolved, and that the simplification is most likely to come through\ngiving up the attempt to make physical space resemble the space of\npercepts, of which a beginning has been made by the Heisenberg quantum\nmechanics. The theory of space-time developed in Chapters XXVIII. and\nXXIX. was, perhaps, unduly orthodox and unimaginative. Perhaps a great\ndeal of apparatus could be cut away if we could free ourselves from the\nbelief that we must preserve, in physics, characteristics which we find\nin psychological space and time. To this topic I shall devote the next\nchapter.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_333\"\u003e[Pg 333]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXII\"\u003eCHAPTER XXXII\u003cbr\u003e\nPHYSICAL AND PERCEPTUAL SPACE-TIME\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN Part II., when we were considering the transition from perception\nto physics, we took over from common sense certain rough-and-ready\napproximations which, at our present stage, we must seek to replace\nby something more exact. We want now to make a second approximation:\nhaving inferred a certain kind of physical world from our percepts,\nwe can use the properties of this inferred world to reinterpret the\nrelation of percepts to the outer world, and we can consider more\ncarefully whether any of the properties we assigned to the outer\nworld were accepted without sufficient reason, merely because they\nwere such as we think we find in the perceptual world. The subject is\nimaginatively difficult, and it is not easy to disentangle different\nlevels of inference, but it is important to do so.\u003c/p\u003e\n\n\u003cp\u003eStarting from percepts, we observe that different people have similar\npercepts, whose differences proceed approximately according to the\nlaws of perspective. The first picture of the physical world to be\nderived from a comparison of percepts (when we start with a developed\nlogic, not with common sense) is, that there are groups of more or less\nsimilar events arranged about centres; that the first-order laws as to\nthe differences between events in one group are spherically symmetrical\nwith respect to the centre of the group; and that the second-order\nlaws are obtained by combining a number of laws of \"distortion,\" each\nof which has its own centre. In this picture of the world, we use a\nphysical space which is derived from, and also correlated with, the\nspace of percepts, in the manner explained in discussing phenomenalism\nin Chapter XX. I\u003cspan class=\"pagenum\" id=\"Page_334\"\u003e[Pg 334]\u003c/span\u003e shall here repeat and amplify this construction, with\na view to suggesting modifications of it derived from physics.\u003c/p\u003e\n\n\u003cp\u003eWe cannot wholly eliminate the subjective factor in our knowledge of\nthe world, since we cannot discover experimentally what the world looks\nlike from a place where there is no one to see it. But we can make\nthe subjective factor approximately constant, and thus be reasonably\nconvinced that the differences which remain are due to causes that are\nnot subjective. I shall therefore suppose that, at a given moment,\na number of photographs are taken of some object, say a chair or a\ntable, from different places, with cameras and plates as similar as\npossible. I shall suppose that the photographs are compared by a person\nsitting motionless, who places them successively on a fixed stand in\nfront of him. It is then reasonable to assume that the differences\nbetween his percepts of the photographs are due to physical causes;\nalso, within limits, that the likenesses between them are due to\nlikenesses in the stimuli to the photographic plates. We find that the\ndifferences between the photographs proceed according to certain laws,\nwhich we call the laws of perspective; these laws are correlated with\nthe differences between the appearances of the different cameras to\nan observer who sees them all at the moment when the photographs are\ntaken, and so on. In fact, they can all be expressed as functions of\nthe \"co-ordinates\" of the cameras and the parts of the table, where\n\"co-ordinates\" may be defined by relation to the single observer.\n\u003ci\u003eE.g.\u003c/i\u003e he may get another man to go with one end of a stretched\ntape-measure to each camera in turn, while he holds the other end; he\ncan read the length \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e of the tape-measure, and observe, on scales,\nthe angular co-ordinates \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e of the tape-measure.\nThese facts lead us to attribute a measure of objectivity to our\nco-ordinates, since, although they are all observed by us from our\npoint of view, they determine the sort of photograph that a camera will\ntake. Further, they lead us to think that, all\u003cspan class=\"pagenum\" id=\"Page_335\"\u003e[Pg 335]\u003c/span\u003e round the table or\nchair which is being photographed, there are events which are connected\nwith each other according to the laws of perspective as stated with\nreference to a certain centre as defined by our polar co-ordinates. Our\nobserver\u0027s \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e are facts concerning his own\npercepts, yet they suffice mathematically to determine the \"percepts\"\nof the cameras; they must therefore have some significance which is not\npurely private to him.\u003c/p\u003e\n\n\u003cp\u003eThis argument, elaborated and extended in obvious ways, gives the\nground for supposing that our perceptual space has some objective\ncounterpart, \u003ci\u003ei.e.\u003c/i\u003e that there is some relation between the camera\nand the table corresponding to the relation between the co-ordinates of\nour percepts of them. (I am throughout assuming the causal theory of\nperception.) If we now use one camera to make one photograph containing\nvarious objects, we shall again find that the spatial relations of\nthe representations of the objects in the photograph are such as can\nbe calculated from the co-ordinates of the objects and the camera.\nWe cannot know the intrinsic quality of the events at the camera\nwhich cause the photograph, but we can infer a certain similarity of\nstructure between these events and our percept of the photograph. All\nthis leads us to the notion of groups of events arranged about centres,\nthe centres having to each other relations whose causal properties can\nbe inferred from relations between certain of our percepts. That is\nto say, given a group \u003cimg style=\"vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-156.png\" alt=\"\" data-tex=\"\\(G\\)\"\u003e, of which one member is a percept \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e,\nand another group \u003cimg style=\"vertical-align: -0.05ex; width: 2.406ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-435.png\" alt=\"\" data-tex=\"\\(G\u0027\\)\"\u003e, of which one member is a percept \u003cimg style=\"vertical-align: -0.439ex; width: 1.766ex; height: 2.156ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-436.png\" alt=\"\" data-tex=\"\\(p\u0027\\)\"\u003e,\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e are the co-ordinates of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e, and\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.648ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-121.png\" alt=\"\" data-tex=\"\\(r\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.689ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-437.png\" alt=\"\" data-tex=\"\\(\\theta\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 2.107ex; height: 2.21ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-438.png\" alt=\"\" data-tex=\"\\(\\varphi\u0027\\)\"\u003e are the co-ordinates of \u003cimg style=\"vertical-align: -0.439ex; width: 1.766ex; height: 2.156ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-436.png\" alt=\"\" data-tex=\"\\(p\u0027\\)\"\u003e, there\nis a relation between \u003cimg style=\"vertical-align: -0.05ex; width: 1.778ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-156.png\" alt=\"\" data-tex=\"\\(G\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.406ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-435.png\" alt=\"\" data-tex=\"\\(G\u0027\\)\"\u003e which can be inferred from\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.648ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-121.png\" alt=\"\" data-tex=\"\\(r\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.689ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-437.png\" alt=\"\" data-tex=\"\\(\\theta\u0027\\)\"\u003e, \u003cimg style=\"vertical-align: -0.493ex; width: 2.107ex; height: 2.21ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-438.png\" alt=\"\" data-tex=\"\\(\\varphi\u0027\\)\"\u003e.\nThese facts give the grounds for regarding space as objective, though,\neven on the basis of these facts, the space which is objective will not\nbe identical with the space of perception, but only correlated with it.\u003c/p\u003e\n\n\u003cp\u003eThe events which cause a photograph obviously take place\u003cspan class=\"pagenum\" id=\"Page_336\"\u003e[Pg 336]\u003c/span\u003e at the\nsurface of the photographic plate; what happens between this and\nthe object photographed consists of causal antecedents, not of the\nimmediate cause. And the resulting photograph is in the plate, not\nin the object. Similarly the events which are the immediate causal\nantecedents of our percept are in the eye and optic nerve, and the\npercept is in us, not in the outer world, when we are speaking of\nphysical space. The whole of our perceptual world is, for physics,\nin our heads, since otherwise there would be a spatio-temporal jump\nbetween stimulus and percept which would be quite unintelligible. Any\ntwo events which we experience together—\u003ci\u003ee.g.\u003c/i\u003e a noise and a\ncolour which we perceive to be simultaneous—are \"compresent.\" I should\nnot say, however, that two percepts which are not both \"conscious\"\n\u003ci\u003emust\u003c/i\u003e be compresent. Two events are compresent when they form\ntogether one causal unit or part of one—this is a sufficient, but\nperhaps not a necessary, condition. When two percepts are experienced\ntogether, they are thus causally conjoined; but when either is\n\"unconscious\" they may not be, and therefore we cannot be sure that\nthey are compresent. It is not necessary, consequently, to suppose that\nthe mind occupies a mere point in physical space.\u003c/p\u003e\n\n\u003cp\u003eIt is now necessary to point out the limitations to the accuracy of\nthe above account. In the first place, there are departures from the\nlaws of perspective which can be easily fitted in—opaque bodies,\nprisms, looking-glasses, echoes, etc. These cases are easy because\nthe departure from regularity as regards one sense is accompanied by\nevidence, from another sense, of the existence of a physical object\nat the centre of the disturbance, or at the apex if the disturbance\nis conical, like a shadow. Then there are the cases where a physical\nobject is inferred from the disturbance, although there is no direct\nevidence of its existence. But none of these are really important. The\ntwo important matters are: (1) The difficulties\u003cspan class=\"pagenum\" id=\"Page_337\"\u003e[Pg 337]\u003c/span\u003e about measurement; (2)\nthe difference between a percept as it seems and a stimulus as it is\ninferred.\u003c/p\u003e\n\n\u003cp\u003e(1) The difficulties about measurement have already been discussed,\nbut we must now endeavour to reach conclusions about them. As already\npointed out, every measurement, however inaccurate, records a fact,\nthough not always the fact which it is intended to record. We saw a\nmoment ago that, if we measure the co-ordinates \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-74.png\" alt=\"\" data-tex=\"\\(\\theta\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.493ex; width: 1.48ex; height: 1.493ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-187.png\" alt=\"\" data-tex=\"\\(\\varphi\\)\"\u003e of an object to be photographed and of a number of cameras,\nwe can make inferences as to the pictures which the various cameras\nwill make of the object. We inferred that the co-ordinates represented\nrelations to our body which have certain peculiar properties of\nthe sort called geometrical, in the sense that when we know the\nco-ordinates of two bodies relatively to ourselves, we can infer\ntheir co-ordinates relatively to each other. All this is only roughly\ntrue if our measurements are careless: in that case, when we mean to\ndiscover intrinsic relations, we are only discovering very complicated\nrelations involving our sense-organs and perhaps even our desires. We\nseek a technique for eliminating all circumstances except those with\nwhich we wish to be concerned, and to a great extent we are successful.\nBut relativity informs us that there is a residue of variability in\nmeasures which cannot be eliminated, because, in fact, the relations we\ntry to measure are partially non-existent. Or, more correctly, they are\nrelations involving more terms than we thought they did. We supposed\nthat co-ordinates represented relations to the axes. But if we had two\nsets of axes momentarily coinciding, while one was moving relatively\nto the other, the co-ordinates of an event would not in general be the\nsame with respect to both. And we cannot even, in any strict sense,\ndiscover any exact relation between distant points such as could give\nphysical significance to co-ordinates. The appearance to the contrary\nis only an approximate truth, which cannot be made precise.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_338\"\u003e[Pg 338]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eAll this represents a failure of correspondence between physical\nspace-time and perpetual space and time. If we assume that the human\nbody moves in a geodesic, perceptual time may be identified with the\nintegral of ds taken along that geodesic, while perceptual space\nconsists of certain relations between simultaneous percepts (the word\n\"simultaneous\" raises no difficulties, since all percepts are in our\nheads), partly themselves perceptual, partly inferred, but all just\nwhat they are, whatever physics may say. There are certain respects\nin which we can modify perceptual space to suit physics, and certain\nothers in which we cannot. We can, for example, infer that percepts\nconsist of imperceptible parts, if physics gives us ground for thinking\nso. But where we perceive some relation between percepts, we cannot\ndeny that there is such a relation, however little physics may allow\nit to subsist between the objects said to be perceived. The rule is:\nWe can infer extra complexity of structure in percepts if physics\nrequires it, but, however much physics may require it, we cannot infer\na \u003ci\u003esmaller\u003c/i\u003e complexity than is demanded by the study of percepts\non their own account. In the world of percepts, the distinction between\nspace and time does really exist, and space does really have certain\nproperties which relativity denies to physical space. Thus to this\nextent the correspondence between perceptual and physical space breaks\ndown, and measurement, which has to do primarily with percepts, fails\nto give us quite such good data as we hoped to obtain for inferences as\nto the physical world.\u003c/p\u003e\n\n\u003cp\u003e(2) I come now to the difference between a percept as it seems and a\nstimulus as it is inferred. But this is not the whole scope of the\nproblem to be discussed. The word \"perception\" implies relation to a\nphysical object; we are supposed to \"perceive\" a chair or a table or a\nperson. If physics is correct, the relation of a percept to a physical\nobject is very remote and curious. In ordinary cases, we see objects\nby\u003cspan class=\"pagenum\" id=\"Page_339\"\u003e[Pg 339]\u003c/span\u003e means of light which is reflected or scattered, which increases the\ncomplication. To take the simplest possible case, let us suppose that\nwe are seeing a glowing gas. The percept seems to be a patch of bright\ncolour of a certain shape, sensibly continuous in perceptual space, and\napproximately constant in perceptual time. Perception gives knowledge\nonly in so far as this percept corresponds to what is really taking\nplace in the gas. Now if physics is true, there are great differences\nbetween the apparent structure of the percept and the real structure\nof what is taking place in the gas. (Differences other than structural\nmay be ignored.) Instead of something steady and continuous, such as\nthe percept seems to be, the process in the gas is supposed to be a\nlarge number of separated sudden discrete upheavals. It is true that\nthere are important similarities between the percept and the physical\nevent. The shape of the percept corresponds to the shape of the region\nin which the upheavals are taking place, with the limitations mentioned\njust now in connection with measurement. The colour of the percept\ncorresponds to the amount of energy lost by each atom in an upheaval.\nThe constancy of the percept corresponds to the statistical constancy\nin the rate at which upheavals occur in any not too small portion of\nthe gas. Thus everything in the percept represents a statistical fact\nabout the gas, with the exception of the colour, which is supposed to\nrepresent a fact about each atom. This, by the way, is an odd reversal\nof Locke\u0027s dictum about secondary qualities: the colour is the most\nnearly objective of all the elements in the percept.\u003c/p\u003e\n\n\u003cp\u003eThese differences are all of one kind in a certain respect: they\nattribute \u003ci\u003emore\u003c/i\u003e structure to the physical occurrence than to the\npercept. This is in line with the general principle that the relation\nof distant to near appearances is one-many, so that differences in the\npercept imply differences in the object, but not vice versa. The finer\nstructure of the object is all, in\u003cspan class=\"pagenum\" id=\"Page_340\"\u003e[Pg 340]\u003c/span\u003e the last analysis, inferred from\nthe grosser structure of percepts, but it involves the comparison of\nmany percepts and the search for invariable causal laws, in the manner\nwhich we considered in Part II. There is therefore no inconsistency in\nthe view that the physical event differs from the percept in the way\nsuggested by physics, since the difference consists in attributing more\nstructure to the physical event, not in denying to it those elements of\nstructure which are possessed by the percept.\u003c/p\u003e\n\n\u003cp\u003eIt is possible, if we choose, to attribute to the percept the same\nstructure as that possessed by the physical occurrence, or rather the\nsame structure as that possessed by the immediate external stimulus. It\ncannot be proved that this hypothesis is untrue, but it is less useful\nthan it might be supposed to be, because only what is \u003ci\u003eknown\u003c/i\u003e\nabout percepts is epistemologically important, and such structure,\nif it exists, is certainly unperceived. What we only discover about\npercepts by means of inference does not belong to the part which\naffords premisses for science, but is, from the standpoint of theory\nof knowledge, in the same position as events in the external world.\nTherefore, although percepts may have an unperceived structure, this\ndoes not diminish the significance of the fact that the structure we\n\u003ci\u003eperceive\u003c/i\u003e in percepts has only a one-many relation to that of\ntheir stimuli.\u003c/p\u003e\n\n\u003cp\u003eThe question must be faced: Is physical space-time perhaps much more\nunlike the space and time of perception than we have supposed? Have\nwe been victims of imaginative laziness in our merely piecemeal\nmodifications of common-sense prejudices? Dr Whitehead, most\nemphatically, is not open to such a charge; his \"fallacy of simple\nlocation,\" when avoided, leads to a world-structure quite different\nfrom that of common sense and early science. But his structure depends\nupon a logic which I am unable to accept, namely the logic which\nsupposes that \"aspects\" may be not quite alike, and\u003cspan class=\"pagenum\" id=\"Page_341\"\u003e[Pg 341]\u003c/span\u003e yet may be in some\nsense numerically one. To my mind, such a view, if taken seriously, is\nincompatible with science, and involves a mystic pantheism. But I shall\nnot pursue this topic here, having treated it on former occasions. The\nquestion I wish to ask is: Without adopting heroic measures, what could\nwe suppose about physical space-time, if we were anxious to preserve\nwhat is probably true in physics, but not anxious to keep as near as\npossible to common sense? In particular, can space-time itself be\natomic, as the existence of the unit of action \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e seems to suggest?\nAnd first, how are we to conceive \"action\"?\u003c/p\u003e\n\n\u003cp\u003eAction is usually defined as the time-integral of energy; since\nenergy can be identified with mass, \"action\" may also be defined as\nmass multiplied by time. Gravitational mass is a length; \u003ci\u003ee.g.\u003c/i\u003e\nthe mass of the sun is 1·47 kilometres.\u003ca id=\"FNanchor_69\" href=\"#Footnote_69\" class=\"fnanchor\"\u003e[69]\u003c/a\u003e Since gravitational and\ninertial mass are equal, we might regard action as length multiplied by\ntime. Dr Jeans (\u003ci\u003eAtomicity and Quanta\u003c/i\u003e, p. 8) says:\u003c/p\u003e\n\n\u003cdiv class=\"blockquot\"\u003e\n\n\u003cp\u003e\"There can hardly be an atomicity of the continuum itself, for, if\nthere were, a universal constant of the physical dimensions of space\nmultiplied by time ought to pervade the whole of physical science.\nNothing of the kind is even suspected, nor, so far as I know, has ever\nbeen so much as surmised. Thus science can to-day proclaim with high\nconfidence that both space and time are continuous.\"\u003c/p\u003e\n\u003c/div\u003e\n\n\u003cp\u003eIn this passage, the \"high confidence\" seems to me to go beyond what\nis warranted. If there were a scientific gain in conceiving the\nspace-time structure atomically, I do not believe that any theoretical\narguments to the contrary could interpose a veto. Arguments from\ndimensions, such as Dr Jeans employs, have no longer the definiteness\nthat they had before the introduction of relativity. As we have just\nseen, we \u003ci\u003ecould\u003c/i\u003e define \"action\" so that its dimensions would\nbe length multiplied by time. Now there is a universal constant of\naction, namely \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. Perhaps, if we were to take action as one of\nthe basic conceptions of physics, we might be able to construct a\nphysics which would be atomistic all through, and yet would contain\nall that is verifiable. I do not \"proclaim with high confidence\" that\nthis is possible; I only invite attention to the hypothesis, as worth\ninvestigating on the chance of its affording a simplification of the\nconceptual apparatus of physics. In the following chapters, this\nhypothesis is to be borne in mind.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_69\" href=\"#FNanchor_69\" class=\"label\"\u003e[69]\u003c/a\u003e\nEddington, \u003ci\u003eop. cit\u003c/i\u003e. 87.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_343\"\u003e[Pg 343]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXIII\"\u003eCHAPTER XXXIII\u003cbr\u003e\nPERIODICITY AND QUALITATIVE SERIES\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE periodic character of many physical occurrences has been obvious\never since men observed their own respiration and the alternation of\nnight and day, but it has acquired a quite new importance with the\ndiscovery of the quantum. The quantum characterizes a whole period\nof a rapid periodic process, not any one moment of the period; it\nthus requires us to consider the period as a whole, and in some sense\nreverses what has hitherto been the trend of physical laws, namely\nto proceed away from integrals towards differentials. It will be\nremembered that the quantum principle, as enunciated by Wilson and\nSommerfeld, states: Given a periodic or quasi-periodic process, the\nkinetic energy of which has been expressed by means of \"separated\"\nco-ordinates, if \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e is any one of these co-ordinates and\n\u003cimg style=\"vertical-align: -0.357ex; width: 4.203ex; height: 1.895ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-439.png\" alt=\"\" data-tex=\"\\(E_{kin}\\)\"\u003e is the kinetic energy, then\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.084ex; width: 18.897ex; height: 5.231ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-440.png\" alt=\"\" data-tex=\"\\[\n\\int \\frac{\\partial E_{k i n}}{\\partial \\dot{q}_{k}} d q_{k}=n_{k} h\n\\]\"\u003e\u003c/span\u003e\nwhere the integration is to extend over one complete period of \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e,\nand \u003cimg style=\"vertical-align: -0.357ex; width: 2.379ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-441.png\" alt=\"\" data-tex=\"\\(n_k\\)\"\u003e is a small integer which is the quantum number associated\nwith \u003cimg style=\"vertical-align: -0.439ex; width: 2.03ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-53.png\" alt=\"\" data-tex=\"\\(q_k\\)\"\u003e. This law is essentially concerned with a whole period,\nand thus makes periodicity fundamental in physics in quite a new way.\u003c/p\u003e\n\n\u003cp\u003eBefore going further, it will be well to consider how far periodicity\nretains this importance in the newer quantum mechanics inaugurated by\nHeisenberg. For this purpose, we may concentrate attention upon the one\nfundamental equation involving h in the new system. This equation takes\nthe form:\u003ca id=\"FNanchor_70\" href=\"#Footnote_70\" class=\"fnanchor\"\u003e[70]\u003c/a\u003e\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.577ex; width: 16.958ex; height: 4.676ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-442.png\" alt=\"\" data-tex=\"\\[\np q-q p=\\frac{h}{2 \\pi i} \\cdot \\mathbf{I}\n\\]\"\u003e\u003c/span\u003e\u003cspan class=\"pagenum\" id=\"Page_344\"\u003e[Pg 344]\u003c/span\u003e\nwhere \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e are matrices, \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e being a Hamiltonian\nco-ordinate in the new sense, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e the corresponding \"impulse,\"\nalso in the new sense; while \u003cimg style=\"vertical-align: 0; width: 0.817ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-443.png\" alt=\"\" data-tex=\"\\(\\mathrm{I}\\)\"\u003e is the unit matrix. This\nequation is asserted to hold for \u003ci\u003eall\u003c/i\u003e motions, not only for such\nas are periodic. But in the case of motions which are not periodic,\nit gives a result which approximates to that of classical mechanics.\nThus it remains the case that the new mechanics is only necessitated by\nperiodic motions, although it is technically possible to find a quantum\nprinciple which is also applicable to non-periodic motions. Hence the\nimportance of periodicity remains intact from an empirical point of\nview, though somewhat diminished from the point of view of a statement\nof fundamental laws. In any case, it remains sufficiently important to\ndemand a separate discussion.\u003c/p\u003e\n\n\u003cp\u003eTraditionally, periodicity in physics was a question of motion: a\nbody described the same path in space over and over again. With the\ncoming of relativity, it has become necessary to modify this account\nsomewhat. In space-time, every point has a date, and cannot be occupied\ntwice; neither the earth nor an electron can describe again the orbit\nit described on a former occasion. And periodicity will be relative\nto a given system of co-ordinates: if, in one system, a co-ordinate\nruns through a given range of values repeatedly, and always in equal\ntimes, it may happen that, in another system, even if there is an\noscillating co-ordinate, its periods are not all equal. A change\nof axes may even take away all trace of periodic character from a\nprocess. Since, however, the quantum principle compels us to treat\nperiodicity as physically important, it would seem that we must regard\nit as a character belonging to a process when referred to axes which\nmove with it, since this would overcome the difficulties connected\nwith relativity. If, in certain cases, this method is not open to us,\nsome other must be found which equally avoids these difficulties. But\nwhere processes connected with\u003cspan class=\"pagenum\" id=\"Page_345\"\u003e[Pg 345]\u003c/span\u003e matter (as opposed to electromagnetic\nprocesses) are concerned we shall, I think, find no other possibility\nexcept to take axes which move with the matter concerned. But this\nmakes it impossible to treat periodicity as fundamentally a character\nexhibited in a motion, since we have reduced to rest the body in which\nthe periodic process is taking place. The suggestion I have to make is\nthat, fundamentally, periodicity is constituted by the recurrence of\n\u003ci\u003equalities\u003c/i\u003e.\u003c/p\u003e\n\n\u003cp\u003eIn the present chapter, I wish to consider what can be meant by the\n\"quality\" of an event; I wish also to investigate the connection of\nquality with causality and motion and periodicity.\u003c/p\u003e\n\n\u003cp\u003ePhysics traditionally ignores quality, and reduces the physical\nworld to matter in motion. This view is no longer adequate. Energy\nturns out to be more important than matter, and light possesses many\nproperties—\u003ci\u003ee.g.\u003c/i\u003e gravitation—which were formerly regarded as\ncharacteristic of matter. The substitution of space-time for space\nand time has made it natural to regard events, rather than persistent\nsubstances, as the raw material of physics. Quantum phenomena have\nthrown doubt on continuity of motion. For these and other reasons, the\nold simplicities have disappeared.\u003c/p\u003e\n\n\u003cp\u003eWhen we start from perception instead of from mathematical physics,\nwe find that the events with which we are best acquainted have\n\"qualities,\" by means of which they can be arranged in classes\nand series. All colours have something in common which is not\npossessed by sounds. Two colours may be so similar as to be almost\nor quite indistinguishable, but they may also be very dissimilar. As\nGestalt-psychologie has emphasized, shapes are perceived qualitatively,\nnot analytically as a system of interrelated parts. But this whole\nconception of quality, which plays such a large part in our perceptual\nlife, has been wholly absent from traditional physics. Colours, sounds,\ntemperatures, etc., have all been\u003cspan class=\"pagenum\" id=\"Page_346\"\u003e[Pg 346]\u003c/span\u003e regarded as caused by various kinds\nof motions. There was no objection to this so far as it succeeded, but,\nif and where it proves insufficient, there can also be no objection to\nre-introducing qualitative differences into the physical world.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, one essential limitation. We may find reasons for\nsupposing qualitative differences, in order to be able to build up\nthe kind of structure which we have inferred; but we cannot have any\nmeans of knowing what are the qualities which differ. This point was\ndiscussed in Part II., and need not now detain us.\u003c/p\u003e\n\n\u003cp\u003eThe apparatus so far assumed, apart from qualities, has been:\nco-punctuality, cause-and-effect, and the quantum laws. I say\n\"cause-and-effect\" because it is necessary to be able to distinguish\nthe earlier from the later event in a transaction, and this is a\nsmaller assumption than that of a general time-order among events\nin one causal series. The above apparatus sufficed except for one\npurpose: that of defining \"repetition.\" The possibility of repetition\nis at the bottom of the common-sense distinction between space and\ntime; the substitution of space-time should, one might suppose, make\nrepetition impossible, and yet the whole of what is distinctive in\nquantum physics, and the theories of light and sound, not to mention\nother matters, depend upon periodicity, which involves repetition.\nSo long as we had billiard-balls moving in an unchanging space, we\ncould be content with repetition of configuration. But now spatial\ndistance, which is essential to configuration, has to be analyzed\ninto an elaborate indirect relation depending upon the existence of\ncommon causal ancestors or descendants. We must, therefore, be able\nto distinguish among events by means additional to their space-time\nrelations.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, a considerable difficulty in finding laws\ngoverning what we are calling \"qualities.\" In a world of continuous\nprocesses, one would say that qualities must\u003cspan class=\"pagenum\" id=\"Page_347\"\u003e[Pg 347]\u003c/span\u003e change gradually.\nBut in a quantum process they apparently change suddenly. Perhaps,\nhowever, this suddenness does not exist in a steady rhythmic process;\nor perhaps, even if it does, it may involve small changes producing\na serial character in the successive qualities. Take, for example,\nthe revolution of an electron about a nucleus. In the newer quantum\ntheory this does not really occur, but we may consider how it could be\ninterpreted if it were necessary to assume it. Let us make a fantastic\nhypothesis, purely for illustrative purposes: let us suppose that the\nelectron and the nucleus can see each other, and that neither rotates\non its own axis. Then they will get pictures of each other which\nchange during each revolution, and repeat the cycle of changes each\ntime. Now let us turn this hypothesis round, and begin by assuming the\nrecurrent series of pictures. From this we can infer the revolution\nof the electron, provided we are free to construct space as we like,\nsubject to certain formal laws. Now in fact we have this freedom:\nthe \"space\" in which the electron revolves need only have certain\nabstract mathematical properties, and, so long as it has them, it may\nbe constructed out of any material available. So long as the electron\ncontinues in one orbit, we may conceive, at any rate as a schematic\nsimplification, that there is a persistent event \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e which may be\ntaken as representative of it, and in like manner that there is a\npersistent event P representative of the proton. Now let us suppose\nthat, compresent with E but not with each other, there are successive\nevents \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-130.png\" alt=\"\" data-tex=\"\\(p_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-131.png\" alt=\"\" data-tex=\"\\(p_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-132.png\" alt=\"\" data-tex=\"\\(p_3\\)\"\u003e, … which may be regarded as\n\"aspects\" of the proton, and are related to each other more or less in\nthe way in which the appearances of the proton from different places\nwould be related if the electron could see. Similarly let us assume a\nseries of events \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.042ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-446.png\" alt=\"\" data-tex=\"\\(e_3\\)\"\u003e, … compresent with \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e\nbut not with each other, analogous to what would be appearances of the\nelectron to the proton if the proton could see. And let us further\nsuppose that, after a certain set\u003cspan class=\"pagenum\" id=\"Page_348\"\u003e[Pg 348]\u003c/span\u003e of such events, an exactly similar\nset recurs, or a very approximately similar set. This supposition\nprovides us with the material required for a periodic relative motion.\nWe shall say, therefore, not that perspectives differ because spatial\nrelations change, but that change in spatial relations consists of\nsystematic alteration in perspectives. Such a view is feasible, but it\nmakes similarity and difference of quality essential. It ceases to be\nfantastic if we drop the analogy with vision except as regards purely\nformal characteristics.\u003c/p\u003e\n\n\u003cp\u003eLet us now set forth the analysis of a periodic process suggested by\nthe above, bringing it into relation with the construction of points\nin Chapter XXVIII. Let us assume, to begin with, that the process is\ndiscrete; this hypothesis can be dropped later, but simplifies the\ninitial statement. Suppose, for the sake of illustration, that there\nare ten qualities, \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-447.png\" alt=\"\" data-tex=\"\\(q_0\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-448.png\" alt=\"\" data-tex=\"\\(q_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-449.png\" alt=\"\" data-tex=\"\\(q_2\\)\"\u003e … \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-450.png\" alt=\"\" data-tex=\"\\(q_9\\)\"\u003e, and that\nthere exist events\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 39.374ex; height: 1.437ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-451.png\" alt=\"\" data-tex=\"\\[\na_{10}, a_{11}, a_{12} \\ldots a_{19}, a_{20}, a_{21} \\ldots a_{29}, a_{30}, \\ldots\n\\]\"\u003e\u003c/span\u003e\nwhich are subject to the following conditions:\u003c/p\u003e\n\n\u003cp\u003e(1) \u003cimg style=\"vertical-align: -0.375ex; width: 2.984ex; height: 1.372ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-452.png\" alt=\"\" data-tex=\"\\(a_{10}\\)\"\u003e, a\u003cimg style=\"vertical-align: -0.375ex; width: 2.984ex; height: 1.372ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-453.png\" alt=\"\" data-tex=\"\\(a_{20}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.984ex; height: 1.372ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-454.png\" alt=\"\" data-tex=\"\\(a_{30}\\)\"\u003e, … have the quality \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-447.png\" alt=\"\" data-tex=\"\\(q_0\\)\"\u003e:\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.984ex; height: 1.337ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-455.png\" alt=\"\" data-tex=\"\\(a_{11}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.984ex; height: 1.337ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-456.png\" alt=\"\" data-tex=\"\\(a_{21}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.984ex; height: 1.372ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-457.png\" alt=\"\" data-tex=\"\\(a_{31}\\)\"\u003e … have the quality \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-448.png\" alt=\"\" data-tex=\"\\(q_1\\)\"\u003e etc.\u003c/p\u003e\n\n\u003cp\u003e(2) Each of the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s is compresent with its immediate neighbour to\nleft and right, but with none of the other \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s;\u003c/p\u003e\n\n\u003cp\u003e(3) If \u003cimg style=\"vertical-align: -0.09ex; width: 6.361ex; height: 1.312ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-458.png\" alt=\"\" data-tex=\"\\(m \u003c n\\)\"\u003e, any point of space-time of which \u003cimg style=\"vertical-align: -0.357ex; width: 2.789ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-459.png\" alt=\"\" data-tex=\"\\(a_m\\)\"\u003e but not\n\u003cimg style=\"vertical-align: -0.357ex; width: 2.344ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-460.png\" alt=\"\" data-tex=\"\\(a_n\\)\"\u003e is a member has a time-like interval from any point of which\n\u003cimg style=\"vertical-align: -0.357ex; width: 2.344ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-460.png\" alt=\"\" data-tex=\"\\(a_n\\)\"\u003e but not \u003cimg style=\"vertical-align: -0.357ex; width: 2.789ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-459.png\" alt=\"\" data-tex=\"\\(a_m\\)\"\u003e is a member.\u003c/p\u003e\n\n\u003cp\u003eIn that case, the series of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s constitutes a periodic process,\nhaving ten \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s in each period. The last digit in the suffix of an\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e indicates the quality of the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e—\u003ci\u003ei.e.\u003c/i\u003e if the last digit\nis \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e, the quality is \u003cimg style=\"vertical-align: -0.439ex; width: 1.918ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-461.png\" alt=\"\" data-tex=\"\\(q_r\\)\"\u003e—while the remaining digits indicate\nthe number of the period.\u003c/p\u003e\n\n\u003cp\u003eIf all the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s are events in the history of one piece of matter,\nthat piece of matter is undergoing the periodic process. If there is\na correlative series of \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e\u0027s in another piece of matter,\u003cspan class=\"pagenum\" id=\"Page_349\"\u003e[Pg 349]\u003c/span\u003e the two\nperiodic processes together make up one relative motion of a periodic\ncharacter, such as the revolution of an electron about a proton.\u003c/p\u003e\n\n\u003cp\u003eGeneralizing the above, while still assuming that the process is\ndiscrete, suppose we have \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-73.png\" alt=\"\" data-tex=\"\\(r\\)\"\u003e qualities \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-448.png\" alt=\"\" data-tex=\"\\(q_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-449.png\" alt=\"\" data-tex=\"\\(q_2\\)\"\u003e … \u003cimg style=\"vertical-align: -0.439ex; width: 1.918ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-461.png\" alt=\"\" data-tex=\"\\(q_r\\)\"\u003e,\nand a set of events\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 35.856ex; height: 1.437ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-462.png\" alt=\"\" data-tex=\"\\[\na_{11}, a_{12} \\ldots a_{1r}, a_{21}, a_{22}, \\ldots a_{2r}, a_{31}, \\ldots\n\\]\"\u003e\u003c/span\u003e\nwhere, as before, the last suffix indicates the quality, \u003ci\u003ei.e.\u003c/i\u003e\n\u003cimg style=\"vertical-align: -0.357ex; width: 3.066ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-463.png\" alt=\"\" data-tex=\"\\(a_{nr}\\)\"\u003e has the quality \u003cimg style=\"vertical-align: -0.439ex; width: 2.157ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-464.png\" alt=\"\" data-tex=\"\\(q_n\\)\"\u003e (\u003cimg style=\"vertical-align: -0.312ex; width: 5.395ex; height: 1.751ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-465.png\" alt=\"\" data-tex=\"\\(n \\leq r\\)\"\u003e). Suppose, also, that\neach \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is compresent with \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e of its predecessors and \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-66.png\" alt=\"\" data-tex=\"\\(p\\)\"\u003e of\nits successors, where \u003cimg style=\"vertical-align: -0.439ex; width: 10.204ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-466.png\" alt=\"\" data-tex=\"\\(2p +1 \u003c r\\)\"\u003e; but that no \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is compresent\nwith any \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e except these. The remaining assumptions are to be as\nbefore. Then again we obtain a rhythm which may be regarded as an\nanalysis of periodic processes in physics.\u003c/p\u003e\n\n\u003cp\u003eIf we suppose that the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s are not compresent with any events\nexcept the other specified \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s, then the group of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s with\nwhich a given \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is compresent constitutes a point, which may be\ntaken as the middle point in the duration of the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e in question. We\ncan take this point as representative of the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e in question, since\ntheir relation is one-one. Thus the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e in question is associated\nwith a point, in spite of the fact that it lasts for a finite time,\n\u003ci\u003ei.e.\u003c/i\u003e is compresent with events not compresent with each other.\u003c/p\u003e\n\n\u003cp\u003eIt is to be observed that, according to the theory of space-time in\nChapters XXVIII. and XXIX., it is quite possible for some parts of\nspace-time to be continuous and others discrete. I am supposing, at the\nmoment, that we are considering a periodic process in a discrete part\nof space-time; this does not involve the hypothesis that \u003ci\u003eall\u003c/i\u003e\nspace-time is discrete.\u003c/p\u003e\n\n\u003cp\u003eIf the \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s in one periodic process, as we supposed a moment ago,\nare not compresent with any events except certain neighbouring \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s\n(which must be fewer than the whole of one period), then the number of\npoints in a period is the same as\u003cspan class=\"pagenum\" id=\"Page_350\"\u003e[Pg 350]\u003c/span\u003e the number of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s, and either\naffords a measure of the duration of the period, measured by its proper\ntime. It is obvious that, in a discrete part of space-time, the natural\nmeasure of distance will be number of intermediate points. We see also\nhow the proper time of one process can differ from that of another. Let\nus suppose that our \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s form an \"isolated\" process (\u003ci\u003ei.e.\u003c/i\u003e\nare not compresent with anything except other \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s), except at the\nbeginning and end; the first and last \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s are to be compresent\nwith the first and last terms of another periodic process composed of\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e\u0027s, which also is to be isolated except at its ends. Then the\nproper time of the \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e-process is measured by the number of \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e\u0027s\nbetween the two ends, which need not have any relation to the number\nof \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s. This illustrates, what of course follows from relativity,\nthat periodicity must be measured by standards intrinsic to the process\nconcerned, not by standards appropriate to other periodic processes.\nSuch remarks would hardly be necessary but for the fact that relativity\nand quantum theory at present stand apart from each other, and have not\nyet been brought into one whole by the physicists.\u003c/p\u003e\n\n\u003cp\u003eThe above can be stated in the language of mathematical logic, thereby\nmaking the character of the assumptions clearer and the generalization\nto continuous processes easier. Let \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e be the series of qualities,\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e the series of events in the rhythmic process. Let us imagine\nthe events arranged in rows and columns, so that each row consists of\none period and each column consists of all the events having a given\nquality. We assume a one-many relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e, whose domain is the field\nof \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e and whose converse domain is the field of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. When \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e\nhas the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, we say \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e has the quality \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e.\"\nIf \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is any term in the field of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, let \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e be the term which\nhas the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e; then the next term below a in the same\ncolumn (\u003ci\u003ei.e.\u003c/i\u003e the corresponding \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e in the next period) is\nthe first term \u003cimg style=\"vertical-align: -0.023ex; width: 1.825ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-361.png\" alt=\"\" data-tex=\"\\(a\u0027\\)\"\u003e in the \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e series which is after \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and to\nwhich \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e.\u003cspan class=\"pagenum\" id=\"Page_351\"\u003e[Pg 351]\u003c/span\u003e The \"row of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\" consists of all\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s earlier than \u003cimg style=\"vertical-align: -0.023ex; width: 1.825ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-361.png\" alt=\"\" data-tex=\"\\(a\u0027\\)\"\u003e and not earlier than \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e. The \"column of\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\" consists of all \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e\u0027s to which \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-89.png\" alt=\"\" data-tex=\"\\(q\\)\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e.\nWe assume that \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e with its converse domain limited to one row is\none-one, so that each row (\u003ci\u003ei.e.\u003c/i\u003e each period) is a series which\nis similar (in the technical sense) to the series \u003cimg style=\"vertical-align: -0.439ex; width: 1.79ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-306.png\" alt=\"\" data-tex=\"\\(Q\\)\"\u003e.\u003c/p\u003e\n\n\u003cfigure class=\"figcenter width500\" id=\"i_351\" style=\"width: 300px;\"\u003e\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-i-351.jpg\" width=\"300\" height=\"158\" alt=\"A\nmathematical diagram showing two horizontal sequences with arrows\npointing right to Q and A. Vertical arrows labeled S connect points\nq₁, q₂, and qᵣ above to points a and a\u0027 below, illustrating a mapping\nrelationship.\"\u003e\n\u003c/figure\u003e\n\n\u003cp\u003eThere is no difficulty in adapting the above analysis of periodicity\nto continuous processes. Instead of an enumerated set of qualities\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-448.png\" alt=\"\" data-tex=\"\\(q_1\\)\"\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.997ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-449.png\" alt=\"\" data-tex=\"\\(q_2\\)\"\u003e, …, we shall have to take some continuous series of\nqualities, such as the colours of the rainbow, or the notes produced\non a violin by running one\u0027s finger up and down the string. The number\nof events compresent with a given event must now be infinite, but must\nstill be less than the whole of one period (ignoring events outside\nthe process concerned). The number of points in one period, or in any\nfinite portion of it, is now infinite, and cannot therefore be used\nas a measure of distance. Thus in regard to metrical properties there\nare important differences between continuous and discrete processes.\nHowever, I shall not enlarge upon these, as I propose to consider the\nanalysis of \"interval\" in a later chapter.\u003c/p\u003e\n\n\u003cp\u003eHitherto I have been considering processes which may be regarded as\ntaking place in matter, or which, at any rate, do not move with the\nvelocity of light. But light, also, is commonly regarded as consisting\nof a periodic process. Accepting the wave-theory of light, let us\nproceed to analyze its periodic character. We shall find that it\ndiffers in important respects from that of periodic processes in\nmatter.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_352\"\u003e[Pg 352]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe periodic character of a light-wave cannot exist from its own\npoint of view, but only from that of the matter which it encounters\nor from which it radiates. We may suppose that when light radiates\nfrom an atom at the time of a quantum change, there is, from the point\nof view of the atom, a temporal series of what we may call \"luminous\nevents,\" and that this series is periodic in the sense which we have\nbeen considering. One period of such luminous events constitutes the\nemission of one light-wave. If we suppose that the light is absorbed\nby another atom, we may suppose that each of the luminous events is\ncompresent with certain events in the absorbing atom, as well as with\ncertain events in the emitting atom. As measured by the proper times\nof the atoms, the time-order of the luminous events is the same for\nthe two atoms. But from the point of view of the luminous events\nthemselves, there is no periodicity. So long as the light does not\nencounter matter, it consists of separated events which at most \"touch\"\none other event at each boundary; the traveller who accompanies one of\nthe events can have no cognizance of any of the other events, since\nthey cannot catch each other up. If we could imagine a homunculus\nfloating on the crest of a light-wave, he would have no means of\ndiscovering that anything periodic was occurring, since he could not\n\"see\" the other parts of the wave. The different parts of a light-wave\ncannot, in a word, interact causally in any way, because no causal\naction can travel faster than light.\u003c/p\u003e\n\n\u003cp\u003eWe cannot even properly speak of a periodicity in the light-wave for\nan observer who watches it pass. We can only see light by stopping it.\nThis applies to such phenomena as interference, which is only made\nvisible by allowing light to meet matter. It is true that interference\ngives us a ground for inferring structure: two processes can neutralize\neach other, but two \"things\" cannot. If \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e owes \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e a pound,\nand \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e owes \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e a like sum, the result is zero; but if \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nhas a pound in\u003cspan class=\"pagenum\" id=\"Page_353\"\u003e[Pg 353]\u003c/span\u003e his hand to give to \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e has a pound in\nhis hand to give to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, there are two pounds. Wherever the sum of\ntwo occurrences can be null, both occurrences must have a relational\ncharacter. Thus we are justified, by such facts as interference\npatterns, in supposing that, when light falls on a body, the body\nexperiences a series of events whose effects upon it are of opposite\nkinds, as if some pushed it one way and some another. But all this\nis from the point of view of the body, not of the light. Thus the\nfrequency of light is a characteristic which exists for a body which\nemits light, and for a body which absorbs it (\u003ci\u003ee.g.\u003c/i\u003e the body of\na scientific observer), but not for the light itself while it is \u003ci\u003ein\nvacuo\u003c/i\u003e.\u003c/p\u003e\n\n\u003cp\u003eWhen light is emitted and absorbed, we may therefore suppose that what\nhappens is according to the following scheme. We have a temporal series\nof events in the emitting body, and, compresent with each of these, a\nluminous event. These luminous events, arranged in the time-order of\nthe compresent events in the emitting body, form a periodic process\nin the previous sense. Each of the luminous events is also compresent\nwith some event in the absorbing body. The time-order of the events in\nthe absorbing body is the same as that of the events in the emitting\nbody; \u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.682ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-467.png\" alt=\"\" data-tex=\"\\(e\u0027\\)\"\u003e are events in the emitting body,\ncompresent respectively with \u003cimg style=\"vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-358.png\" alt=\"\" data-tex=\"\\(l\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.302ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-468.png\" alt=\"\" data-tex=\"\\(l\u0027\\)\"\u003e, two luminous events; and\nif \u003cimg style=\"vertical-align: -0.025ex; width: 0.674ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-358.png\" alt=\"\" data-tex=\"\\(l\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.302ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-468.png\" alt=\"\" data-tex=\"\\(l\u0027\\)\"\u003e, are respectively compresent with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.825ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-361.png\" alt=\"\" data-tex=\"\\(a\u0027\\)\"\u003e, two\nevents in the absorbing body, then if \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e is earlier than \u003cimg style=\"vertical-align: -0.025ex; width: 1.682ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-467.png\" alt=\"\" data-tex=\"\\(e\u0027\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e is earlier than \u003cimg style=\"vertical-align: -0.023ex; width: 1.825ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-361.png\" alt=\"\" data-tex=\"\\(a\u0027\\)\"\u003e. What happens to light-waves which are\nemitted but not re-absorbed we cannot tell, since, by the nature of the\ncase, there can never conceivably be any evidence on the point.\u003c/p\u003e\n\n\u003cp\u003eAccording to the above, the frequency of a light-wave is a\ncharacteristic which it has in relation to matter, not in relation\nto itself. In this it differs from, \u003ci\u003ee.g.\u003c/i\u003e, the periodicity in\nthe revolution of an electron, which may be supposed to exist for the\nelectron itself.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_354\"\u003e[Pg 354]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThe chief point of the above hypothesis is the suggestion that single\n\"luminous events\" extend from the emitting to the absorbing body. I do\nnot advance it as anything more than a possible hypothesis. One of its\nmain purposes is to account for the fact that the interval between two\nparts of a light-ray is zero; but this part of the argument belongs to\na later chapter.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_70\" href=\"#FNanchor_70\" class=\"label\"\u003e[70]\u003c/a\u003e\nM. Born and P. Jordan, \u003ci\u003eZur Quantenmechanik,\nZeitschrift für Physik\u003c/i\u003e, 34, p. 871. Also M. Born, W. Heisenberg,\nand P. Jordan, 35. p. 562.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_355\"\u003e[Pg 355]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXIV\"\u003eCHAPTER XXXIV\u003cbr\u003e\nTYPES OF PHYSICAL OCCURRENCES\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN this chapter, I propose to advocate a division of physical\noccurrences into three types, which I shall call respectively steady\nevents, rhythms, and transactions. The phrase \"steady events\" is formed\non the analogy of \"steady motions,\" though the events concerned are\nnot supposed to be motions. Rhythms are periodic processes, such as we\nconsidered in the preceding chapter. Transactions are quantum changes,\nin which energy passes from one system to another. The laws governing\ndifferent types of occurrences are different, and it is necessary to\nseparate them before embarking upon a general discussion of physical\ncausality.\u003c/p\u003e\n\n\u003cp\u003eThe traditional view, that physics is concerned exclusively with\nmatter in motion, cannot be maintained, for a number of reasons. In\nthe first place, the æther, even if it can be said to exist, can\nhardly be regarded as having a granular structure, and events in it,\nsuch as the passage of light, cannot be explained as movements of\nparticles of æther. In the second place, quantum changes, if they\nreally are sudden, violate the continuity of motion, and thus destroy\nits advantages as an imaginative picture. In the third place—and this\nis philosophically the most important point—the conception of motion\ndepends upon that of persistent material substances, which we have seen\nreason to regard as merely an approximate empirical generalization.\nBefore we can say that one piece of matter has moved, we must decide\nthat two events at different times belong to one \"biography,\" and a\n\"biography\" is defined by certain causal laws, not by persistence of\nsubstance. Consequently motion is something constructed in accordance\u003cspan class=\"pagenum\" id=\"Page_356\"\u003e[Pg 356]\u003c/span\u003e\nwith the laws of physics, or—we might say—as a convenience in stating\nthem; it cannot be one of the fundamental concepts of physics. Lastly,\nthere is an argument which is difficult to state precisely, but which\nnevertheless has some weight. For Newton, motion was absolute, and a\nbody in motion might be regarded as in a different state from a body\nat rest. But when motion was recognized as merely relative, laws of\nmotion became laws as to relations to more or less distant bodies.\nThey thus came to involve something like action at a distance—though\nthis was disguised by the use of differential equations not always\ninterpreted according to rigid Weierstrassian methods. If we are to\navoid action at a distance, our fundamental laws must be concerned with\nterms having finite spatio-temporal extension, and thus capable of\ncontact and overlapping—in a word, with events rather than particles\nor impenetrable material units. This involves a re-interpretation of\nmotion as it occurs in physics, which will be considered in a later\nchapter. For the present, I am concerned with the materials which will\nbe required for this purpose as well as for the interpretation of other\nphysical phenomena.\u003c/p\u003e\n\n\u003cp\u003eA \"steady event,\" as I use the term, is anything which is devoid\nof physical structure and is compresent with events which are not\ncompresent with each other, but are one earlier and the other later;\nin other words, the steady event is a member of at least two points\nwhich have a time-like interval. When a steady event is contrasted\nwith a rhythm, it is assumed that the steady event is not part of a\nperiodic process; but it cannot be taken as certain that there are any\nelementary events which are not parts of such processes. It may be that\nall non-periodic changes occur by way of transactions; but this must be\nan open question in the present state of knowledge.\u003c/p\u003e\n\n\u003cp\u003eA \"rhythm,\" as already explained, is a recurring cycle of events,\nin which there is a qualitative similarity between\u003cspan class=\"pagenum\" id=\"Page_357\"\u003e[Pg 357]\u003c/span\u003e corresponding\nmembers of different periods. A rhythm may have a period consisting of\na finite number of events, or one consisting of an infinite number;\nit may be discrete or continuous. If it is discrete, the proper time\nof one period is measured by the number of events in the period, and\nthe \"frequency\" of the process is the reciprocal of this number. But\nhere we are speaking of the frequency as measured by the time proper\nto the period; by an extraneous time the frequency may be quite\ndifferent. What is commonly called the frequency of a light-wave is\nits frequency with respect to axes fixed relatively to the emitting\nbody. Its frequency relative to axes which travel with it is zero; this\nis only the extreme of the Doppler effect. There is perhaps a certain\ninconsistency in the practice of studying bodies by means of axes\nwhich move with them, while light is always treated with reference to\nmaterial axes. If we want to understand light in itself, not in its\nrelation to matter, we ought to let our axes travel with it. In that\ncase, its periodicity is spatial, not temporal; it is like that of\ncorrugated iron. From the standpoint of the light itself, each part of\na light-wave is a steady event in the sense defined above.\u003c/p\u003e\n\n\u003cp\u003eOne of the most fundamental of rhythmic processes will be the\nrevolution of an electron about a nucleus, unless we accept the view\nof the new quantum mechanics, according to which there is no reason to\nsuppose that this really occurs. In the Bohr-Sommerfeld theory, this\nrevolution goes on by itself until it is altered either by a quantum\nchange or by some more conventional chemical or electrical action.\nThe question arises: why should we suppose that there is a process\nat all? Why not suppose that there is a steady event, possessed of\na certain amount of energy, which is replaced, in a quantum change,\nby another steady event, possessed of a different amount of energy,\nthe balance being radiated or absorbed? There is a certain attraction\nabout this hypothesis, since the\u003cspan class=\"pagenum\" id=\"Page_358\"\u003e[Pg 358]\u003c/span\u003e atom gives no external indication\nof its presence while the supposed process continues, and therefore\nthere can be no direct evidence that changes are occurring, such as\na steady motion supposes. In any case, if an electron is revolving\nround a proton in a circle, and both are spherically symmetrical, it\nis not easy to see, from a relativist point of view, exactly what is\nmeant by saying that the electron is revolving. This difficulty is\nnot diminished by the hypothesis of spinning electrons. We have the\nsame difficulties as in the case of absolute rotation and Foucault\u0027s\npendulum—the difficulties, namely, which Newton advanced to prove\nthe necessity of absolute motion. Within the system consisting of the\nelectron and proton alone, nothing is changing while the electron\nrevolves in its circular orbit; the change is only with reference\nto other bodies. Why not regard the state of affairs as static, but\npossessed of a certain amount of energy? Energy may be altered in\namount by a change of axes, and is not an invariant property of the\nsystem; but reference to the outside world here is less serious, since\nthe only purpose served by the energy of the atom is to provide physics\nwith something which can be radiated into the outer world or absorbed\nfrom it. That is to say, energy is required only as something whose\nchanges govern the causal relations of the atom with the outer world.\nThis point of view is essentially that of the Heisenberg theory.\u003c/p\u003e\n\n\u003cp\u003eThere are several apparent difficulties in such a view. In the first\nplace, the formula for energy obtained on the assumption that the\nelectron revolves gives exactly the changes of energy required to\naccount for spectroscopic phenomena; the Bohr-Sommerfeld theory\nagrees with observation so minutely that its formula for energy must\nbe accepted. Of course we could say that the energy just happens to\nbe what it would be if the electron revolved in one of the quantum\norbits; but this would seem an almost miraculous coincidence. This,\nhowever, is not the strongest argument, which is that\u003cspan class=\"pagenum\" id=\"Page_359\"\u003e[Pg 359]\u003c/span\u003e derived from the\nquantum principle. The quantum principle in its older form can only be\napplied to periodic processes; if it is to apply, as we find that it\ndoes, to the interchange of energy between light and the atom, we must\nassume, if we adhere to the older theory, that within the atom there is\nsomething that can be called a \"frequency,\" \u003ci\u003ei.e.\u003c/i\u003e something which\nis periodic, which compels us to admit that, within an atom in a steady\nstate, there is a recurring process whose formal properties are those\nwhich would be exhibited by a revolution of the electron, and perhaps\nalso by a rotation.\u003c/p\u003e\n\n\u003cp\u003eIf we adhere to the Bohr theory, what can be supposed to be really\noccurring? If relative motion were all that was taking place, we should\nhave either to find an interpretation for the spinning electron, or\nelse to say that, taking axes fixed relatively to any large body, the\nline joining the electron to the proton rotates rapidly; any large body\nwill do, since none rotates with an angular velocity comparable to that\nof the electron. But why should the electron be interested in this\nfact? Why should its capacity for emitting light be connected with it?\nThere must be something happening where the electron is, if the process\nis to be intelligible. This brings us back to Maxwell\u0027s equations, as\ngoverning what is occurring in the medium. And there must be a rhythmic\ncharacter in the events occurring where the electron is, if we are to\navoid all the troubles of action at a distance.\u003c/p\u003e\n\n\u003cp\u003eWe suppose, therefore, that throughout an electromagnetic field there\nare events whose formal properties we know more or less, and that they,\nnot the change of spatial configuration, are the \u003ci\u003eimmediate\u003c/i\u003e\ncauses of what takes place. This brings us back to the cycle of events\nwhich we used in the preceding chapter to define a rhythm. The point is\nthat a rhythm can never consist merely in periodic changes of spatial\nrelation between two or more bodies, but must consist of qualitative\ncycles of events. We have experience of such cycles when\u003cspan class=\"pagenum\" id=\"Page_360\"\u003e[Pg 360]\u003c/span\u003e we watch a\nlarge-scale periodic event, such as the swing of a pendulum. All that\nhappens \u003ci\u003eto\u003c/i\u003e us during the cycle happens in us, not in a number\nof different places; and any effect upon us depends upon what happens\nto us. I am suggesting that this is a proper analogy when we wish to\nunderstand how a periodic motion affects an electron.\u003c/p\u003e\n\n\u003cp\u003eI come now to what I shall call \"transactions,\" by which I mean quantum\nchanges. I call them \"transactions\" because energy is exchanged between\ndifferent processes. The processes concerned must be periodic, since\notherwise the quantum principle is unnecessary. In the simplest case,\nthat of emission of light by a hydrogen atom, we have as antecedent,\nspeaking the language of the older quantum theory, one periodic process\n(the revolution of the electron in an orbit other than the minimum\norbit) and as consequent two processes, namely: (1) The revolution\nof the electron in a smaller orbit, (2) a light-wave. The latter, as\nalready explained, is only periodic in a certain sense. The energy\nof the antecedent is the sum of the energies of the consequents. The\namount of action during one period of the antecedent is a multiple of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e, and so are the amounts of action of the consequents during one\nperiod of each. Exactly the converse occurs when light is absorbed\nby a hydrogen atom. In other cases, both the antecedent and the\nconsequent may consist of two or more rhythms; but always there will\nbe conservation of energy, and each rhythm will contain an amount of\naction which is a multiple of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eAs yet, everything concerned with quanta is more or less mysterious,\nalthough Heisenberg\u0027s theory has somewhat diminished the mystery. We\ndo not know whether quantum changes are really sudden or not; we do\nnot know whether the space concerned in atomic structure is continuous\nor discrete. If electrons always moved in circles, as in the first\nform of Bohr\u0027s theory, we could be content with a granular\u003cspan class=\"pagenum\" id=\"Page_361\"\u003e[Pg 361]\u003c/span\u003e discrete\nspace, and suppose that the intermediate orbits are geometrically\nnon-existent. But the existence of elliptic orbits in Sommerfeld\u0027s\ndevelopment of the theory makes this difficult. And in atoms with many\nplanetary electrons, the paths of some are supposed to cross those of\nothers. In spite of these difficulties, however, I do not despair of\nthe hypothesis that space-time is discrete. The older quantum theory\nuses the traditional conceptions of physics, and thinks of geometrical\norbits in a constant space. The Heisenberg theory, on the contrary,\nhas a completely new kinematics, according to which unquantized orbits\n(if we may still speak of orbits) are geometrically impossible. It is\ndifficult, as yet, to translate this theory out of its technical form.\nBut even according to the older theory, one can see that a discrete\nspace-time is possible. For when we think of the matter in terms of\nspace-time, we realize that the geometry of the neighbourhood of the\natom may be different at different times. If an electron moves in\none sort of orbit at one time and in another at another, it does not\nfollow that each sort of orbit was geometrically possible at the time\nwhen the other was being described. Perhaps it is not superfluous\nto explain what is meant by saying that an orbit is \"geometrically\npossible\" though not physically actual. What is meant is this: there is\na series of groups of events, each group being a point, and the series\nbeing one in which all the intervals of points are time-like, and in\nwhich, if a constant value is assigned to one of the co-ordinates, the\nremaining three give a curve in a three-dimensional space having the\ngeometrical properties of the orbit in question. Whenever we speak of\nan orbit geometrically, we are assuming that we can distinguish one\nof the co-ordinates as \"time,\" give it a constant value, and consider\nthe relations of the remaining three co-ordinates. Now it is always\npossible that there may be a fallacy in this procedure, since it may be\nthat such geometrical relations as we are considering are\u003cspan class=\"pagenum\" id=\"Page_362\"\u003e[Pg 362]\u003c/span\u003e impossible\namong \"simultaneous\" points. Moreover, in the general theory of\nrelativity, it may be impossible to distinguish one co-ordinate as more\nrepresentative of time than the others.\u003c/p\u003e\n\n\u003cp\u003eWhen, from a traditional point of view, two orbits cross each other,\nthis no longer happens from a relativity standpoint. We cannot assume,\nthat is to say, that there is a point from which two journeys are\npossible. Two electrons never actually collide. When their orbits are\nsaid to cross, all that is meant is this: In the system of co-ordinates\nwe have adopted, there is a point (\u003cimg style=\"vertical-align: -0.464ex; width: 7.29ex; height: 1.88ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-169.png\" alt=\"\" data-tex=\"\\(x, y, z, t\\)\"\u003e) which is part of\nthe history of one electron, and a point (\u003cimg style=\"vertical-align: -0.464ex; width: 7.917ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-469.png\" alt=\"\" data-tex=\"\\(x, y, z, t\u0027\\)\"\u003e) which is\npart of the history of the other. In another equally legitimate system\nof co-ordinates, these two points would not have three co-ordinates\nidentical. And the fact that a certain orbit passes from (\u003cimg style=\"vertical-align: -0.464ex; width: 7.29ex; height: 1.88ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-470.png\" alt=\"\" data-tex=\"\\(x, y, z,t\\)\"\u003e)\nin a certain direction does not imply that there is an orbit\npassing from (\u003cimg style=\"vertical-align: -0.464ex; width: 7.917ex; height: 2.181ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-469.png\" alt=\"\" data-tex=\"\\(x, y, z, t\u0027\\)\"\u003e) in a direction which is the same so\nfar as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-98.png\" alt=\"\" data-tex=\"\\(y\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-99.png\" alt=\"\" data-tex=\"\\(z\\)\"\u003e are concerned. Therefore the apparent\ndifficulties in the way of a discrete space are not necessarily\ninsuperable.\u003c/p\u003e\n\n\u003cp\u003eFrom our point of view, it is a difficulty in the quantum principle\nthat it is stated in a form involving energy, which, from a relativity\nstandpoint, requires re-interpretation. It is also a difficulty that\nwe do not know any laws determining \u003ci\u003ewhen\u003c/i\u003e a transaction will\ntake place, and that we do not know whether it is really sudden or\nnot. For all these reasons, we are compelled to be very tentative in\nphilosophizing. I will, however, repeat the outcome of this chapter,\nsuch as it is.\u003c/p\u003e\n\n\u003cp\u003eIn one sense, the theory of space-time points as groups of events\nrequires that all change should be discontinuous. An event e is a\nmember of a certain set of space-time points, and of no others: the\nboundaries of the region constituted by this set are the boundaries of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e, so that it comes into existence suddenly and ceases to exist\nsuddenly. Nevertheless, we can, if necessary, provide for continuity\nwithin this scheme.\u003cspan class=\"pagenum\" id=\"Page_363\"\u003e[Pg 363]\u003c/span\u003e Suppose a continuous series of qualities, like\nthe colours of the rainbow; suppose that, in some process, each of\nthese is compresent with its neighbours up to a certain distance in\neither direction, but not with more distant members of the series. Then\nthe group of qualities existing at a point will change continuously,\nalthough each separate quality changes discontinuously. We may suppose\nthis to be the nature of change between transactions, and in particular\nduring a rhythm. There is no proof that change is ever continuous, but\nthere is also no proof that it is not. We will assume, for the moment,\nthat change between transactions is continuous in the above sense, but\nthat transactions are discontinuous. This assumption is made only for\nthe sake of brevity of statement; it is not asserted to be true, or\neven more probable than the opposite assumption.\u003c/p\u003e\n\n\u003cp\u003eIf we take the above view, there will be three kinds of things to\nconsider in physics: transactions, steady events, and rhythms.\nTransactions are dominated by quantum laws. Steady events continue,\nwithout internal change, from one transaction to the next, or\nthroughout a certain portion of a continuous change; percepts are\nsteady events, or rather systems of steady events. The relation of a\nsteady event to a rhythm I conceive according to a musical analogy:\nthat of a long note on the violin while a series of chords occurs\nrepeatedly on the piano. All our life is lived to the accompaniment\nof a rhythm of breathing and heart-beating, which provides us with a\nphysiological clock by which we can roughly estimate times. I imagine,\nperhaps fancifully, something faintly analogous as an accompaniment to\nevery steady event. There are laws connecting the steady event with\nthe rhythm; these are the laws of harmony. There are laws regulating\ntransactions; these are the laws of counterpoint.\u003c/p\u003e\n\n\u003cp\u003eWe must assume periodicity as a feature of the state of affairs where\nthere are steady events, since we cannot state\u003cspan class=\"pagenum\" id=\"Page_364\"\u003e[Pg 364]\u003c/span\u003e the quantum principle\nwithout it. We have to find a meaning for \"frequency\" in order to\nconnect energy with \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. It is not altogether easy to see how one\nfrequency is to be compared with another. In the case of light, we\ncan estimate the distance between the crest of one wave and the crest\nof the next. Knowing the velocity of light, this tells us how many\nwaves pass a given place in a second. But here the periodicity exists\nfor the outside observer; for an observer travelling on the crest of\na given wave, there is no process and no periodicity. For an outside\nobserver, there is a process in the motion of the light-wave; but our\nobserver on the wave considers himself to be at rest, and presumably\ndoes not see objects flying past him. Thus for him the periodicity\nof a light-wave is spatial rather than temporal. One light-wave will\nconsist of a series \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-471.png\" alt=\"\" data-tex=\"\\(e_{11}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-472.png\" alt=\"\" data-tex=\"\\(e_{12}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.375ex; width: 2.842ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-473.png\" alt=\"\" data-tex=\"\\(e_{13}\\)\"\u003e … \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-474.png\" alt=\"\" data-tex=\"\\(e_{1n}\\)\"\u003e,\n… of steady events, the intervals between which are space-like; the\nnext will consist of a series \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-475.png\" alt=\"\" data-tex=\"\\(e_{21}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-476.png\" alt=\"\" data-tex=\"\\(e_{22}\\)\"\u003e, … \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-477.png\" alt=\"\" data-tex=\"\\(e_{2n}\\)\"\u003e,\n…, again having space-like intervals from each other and from the\nprevious series; \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-474.png\" alt=\"\" data-tex=\"\\(e_{1n}\\)\"\u003e and \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-477.png\" alt=\"\" data-tex=\"\\(e_{2n}\\)\"\u003e will have a similarity of\nquality which neither has to \u003cimg style=\"vertical-align: -0.357ex; width: 3.447ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-478.png\" alt=\"\" data-tex=\"\\(e_{1m}\\)\"\u003e or \u003cimg style=\"vertical-align: -0.357ex; width: 3.447ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-479.png\" alt=\"\" data-tex=\"\\(e_{2m}\\)\"\u003e (where \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e is\ndifferent from \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e). Each of these events is supposed to continue\nas long as the light-wave continues, \u003ci\u003ei.e.\u003c/i\u003e until there is a\ntransaction. Given any event \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e which is connected with matter,\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e may be compresent with \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-471.png\" alt=\"\" data-tex=\"\\(e_{11}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-472.png\" alt=\"\" data-tex=\"\\(e_{12}\\)\"\u003e … \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-474.png\" alt=\"\" data-tex=\"\\(e_{1n}\\)\"\u003e, …\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-475.png\" alt=\"\" data-tex=\"\\(e_{21}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-476.png\" alt=\"\" data-tex=\"\\(e_{22}\\)\"\u003e … \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-477.png\" alt=\"\" data-tex=\"\\(e_{2n}\\)\"\u003e, … successively, but not with\nall at once. This is what happens when a light-wave passes an observer\nor any other piece of matter. A series of events forming one light-wave\nare inseparably associated, in the sense that when there is one of\nthem there will be others throughout the space covered by the wave.\nSimilarly the series of events (if any) involved in the revolution of\nan electron are inseparably associated; but there is this difference,\nthat these events form a temporal series from the standpoint of the\nelectron, whereas the events constituting a light-wave form a spatial\nseries from the point of view of the light-wave.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_365\"\u003e[Pg 365]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThere are difficulties in the above which might be resolved in various\nways, but we do not know which to choose. What, for example, shall we\nsay about the transaction which consists in the absorption of energy by\nan atom from a light-wave? The correct view is supposed to be that, in\nsuch a case, a planetary electron passes suddenly from a smaller to a\nlarger orbit. But if we imagine a light-wave to consist of a number of\nevents \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-471.png\" alt=\"\" data-tex=\"\\(e_{11}\\)\"\u003e, \u003cimg style=\"vertical-align: -0.339ex; width: 2.842ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-472.png\" alt=\"\" data-tex=\"\\(e_{12}\\)\"\u003e, … \u003cimg style=\"vertical-align: -0.357ex; width: 3.002ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-474.png\" alt=\"\" data-tex=\"\\(e_{1n}\\)\"\u003e, …, one might expect\nthat at least one whole wave would be required to produce one definite\neffect, and that a part of the wave would produce only part of the\neffect, if any. But a whole wave takes a finite time to reach the atom.\nThis difficulty exists for any view which regards light as consisting\nof waves and quantum transitions as sudden, but would be obviated if\neither of these suppositions were dropped. We may therefore take it as\npart of the general unsolved problem of the relation between radiant\nenergy and energy associated with matter. This problem, though it\ninterests the philosopher, belongs to the domain of physics, and can\nonly be profitably considered by a physicist. I am therefore content to\nawait the discoveries of others.\u003c/p\u003e\n\n\u003cp\u003eAs regards quanta, let us examine once more what is implied by the fact\nthat there is an important constant \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. In the first place, \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e\nonly exists, or at any rate is only important, in the case of periodic\nprocesses, and it is a characteristic of one complete period. In the\nsecond place, only integral multiples of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e occur. In the third\nplace, when a transaction involves the loss by one system of a certain\nmultiple of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e, another system may acquire another multiple of\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e: what is transferred always unaltered in amount is energy. These\nseem to be the most significant facts about \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIt seems impossible to resist the view that \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e represents something\nof fundamental importance in the physical world, which, in turn,\ninvolves the conclusion that periodicity is an element in physical\nlaws, and that one period of a periodic\u003cspan class=\"pagenum\" id=\"Page_366\"\u003e[Pg 366]\u003c/span\u003e process must be treated as, in\nsome sense, a unit. This follows from the fact that processes arrange\nthemselves so as to secure that a period shall have an important\nproperty. This property is simplest in the case of a light-wave: the\nenergy of one light-wave multiplied by the time it takes to pass a\ngiven material point is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. If we take the velocity of light as\nunity, the time a light-wave takes to pass a given point is equal\nto the spatial distance between the beginning and end of the wave;\ntherefore this distance multiplied by the energy is \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e. This form\nmight seem preferable for our purposes, since it does not involve\nreference to an extraneous material point. At least, it does not\n\u003ci\u003eobviously\u003c/i\u003e involve such reference; but perhaps the reference is\nconcealed in the process of estimating spatial distance. We have seen\nthat this process must be indirect; one part of a light-wave cannot\ncatch up another, so that the space-like interval between them can only\nbe estimated by means of some process taking place in matter.\u003c/p\u003e\n\n\u003cp\u003eIf it should be found that quantum phenomena are not physically\nfundamental, much of what has been said in this chapter will become\nunnecessary. It should be said, however, that relativity should\nprepare our minds for the oddest feature of the quantum theory, namely\nthe existence of causal laws involving whole periods. The causal\nunit, on relativity principles, should be expected to occupy a small\nregion of space-time, not only of space; it should not therefore be\ninstantaneous, as in pre-relativity dynamics. If we combine this with\nthe hypothesis of a discrete space-time, we can imagine a theoretical\nphysics which would make the existence of the quantum no longer seem\nsurprising.\u003c/p\u003e\n\n\u003cp\u003eI have to confess, reluctantly, that the theory developed in the\npresent chapter, inadequate as it is, is the best that I know how to\nsuggest on the topic of quanta. Perhaps the progress of physics will\nmake a better philosophy of the subject possible before long. Meanwhile\nI commend the matter to the consideration of the reader.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_367\"\u003e[Pg 367]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXV\"\u003eCHAPTER XXXV\u003cbr\u003e\nCAUSALITY AND INTERVAL\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nTHE conception of \"interval,\" upon which the mathematical theory of\nrelativity depends, is very hard to translate, even approximately, into\nnon-technical terms. Yet it is difficult to resist the conviction that\nit has some connection with causality. Perhaps a discontinuous theory\nof interval might diminish the obstacles to such an interpretation. Let\nus try to discover whether this is the case.\u003c/p\u003e\n\n\u003cp\u003eThe view which naturally suggests itself as a point of departure is\nsomething like this: Given two groups of co-punctual events, it may\nhappen that at least one member of one group has a causal relation to\nat least one member of the other group; in that case, the interval\nbetween the two groups is time-like. If causality is a matter of\ndiscontinuous transitions, one might expect that the magnitude of the\ninterval would be measured by the number of intermediate transitions.\nAgain, it may happen that no member of one group has a causal relation\nto any member of the other, but that both contain members having causal\nrelations to a member of a third group. In that case, the interval\nwill be space-like, and again one might suppose that the number of\nintermediate links would determine the magnitude of the interval.\u003c/p\u003e\n\n\u003cp\u003eThis represents what might be hoped, but as it stands it is unduly\nsimple, and open to obvious objections. Let us see, therefore,\nwhether it is possible to answer the objections, or to introduce such\nmodifications as will obviate them.\u003c/p\u003e\n\n\u003cp\u003eFirst, let us be clear as to what we mean by a causal relation.\nThere is a causal relation whenever two events, or two groups of\nevents of which one at least is co-punctual, are related\u003cspan class=\"pagenum\" id=\"Page_368\"\u003e[Pg 368]\u003c/span\u003e by a law\nwhich allows something to be inferred about the one from the other.\nFormerly, one would have supposed that everything about the later\nevent could be inferred from a sufficient number of antecedents;\nbut in view of the explosive and apparently spontaneous character\nof radio-activity and quantum changes, we must be content with a\nmore modest definition so far as this point is concerned. In another\nrespect, however, our definition is less modest than it would formerly\nhave been. In classical dynamics, causal laws connect accelerations\nwith configurations, so that from the present state of a small region\nwe cannot accurately infer anything as to what will be happening there\nafter a finite time. Quanta have altered this: we can associate the\nlight radiated from an atom with its causal origin, until it hits other\nmatter; we can associate the state of the atom after the emission of\nthe light with its state before, until it undergoes another quantum\nchange. In fact, as we saw in the preceding chapter, we can analyze\nthe course of nature into a set of steady events and rhythms with\ncausal relations governing the \"transactions\" in which rhythms undergo\nchanges. The above definition was framed with these considerations in\nmind.\u003c/p\u003e\n\n\u003cp\u003eWe shall say, then, that all causal relations consist of a series of\nrhythms or steady events separated by \"transactions.\" If such a series\nconnects a rhythm or steady event \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e with a rhythm or steady event\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, we shall say that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is a \"causal ancestor\" of \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is a \"causal descendant\" of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. We may assume that, in such\na case, the number of transactions between \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is always\nfinite, since one supposes that the time between two transactions\ncannot fall below a certain minimum, or at any rate that the number of\ncausally connected transactions in a finite time is never infinite.\nPerhaps we may assume that a rhythm must last long enough to achieve an\namount of action \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e; perhaps, even, we could construct a discrete\ntheory of time from which\u003cspan class=\"pagenum\" id=\"Page_369\"\u003e[Pg 369]\u003c/span\u003e this result would follow. All this, however,\nis very speculative.\u003c/p\u003e\n\n\u003cp\u003eNow let us consider the stock case of a light-signal sent from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e to\n\u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, and reflected back from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. Only two transactions\nare involved, namely the emission and reflection of the light; perhaps\nwe ought to add the final transaction, namely the re-absorption of the\nlight by \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e. In any case, there need be only two steady events, one\nin the outward beam and one in the returning beam. But the interval\nbetween the departure and return of the light may have any magnitude.\nThis is all the more curious, as the interval between the departure\nof the light from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and its arrival at \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is zero, and so is\nthe interval between its departure from \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e and its return to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e.\nThis suggests that too much effort has been made to regard interval as\nanalogous to distance in conventional geometry and time in conventional\nkinematics. Suppose we say that, if an event \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e is a causal\nancestor of an event \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e, we take all the possible causal routes\nfrom \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e to \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e, and choose that which contains the greatest\nnumber of events: then the \"interval\" from \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e to \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e is\ndefined as the number of events in this longest route. It is obvious\nthat, if a measurable time elapses between the departure of the light\nfrom \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and its return to \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, there must have been a variety of\nevents at \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e meanwhile. When I say \"at\" \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, I have a meaning to\nbe considered shortly; but for the moment it is enough to say that\nthis meaning includes causal inheritance. Thus we have a meaning for\nthe view that the interval at \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e is quite long, and also for the\nview that the interval between the departure of the light from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e\nand its arrival at \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e is zero. This latter statement means that it\nis the very same event that starts from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and arrives at \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e,\nand moreover that there is no longer causal route connecting the two\ntransactions of starting from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and arriving at \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e. This event\nwhich starts from \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and arrives at \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e I call a \"luminous event.\"\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_370\"\u003e[Pg 370]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eBut we must deal with space-like intervals before we can decide whether\nthe above theory of time-like intervals will do. It is to be observed\nthat space-like intervals are obtained by calculation from time-like\nintervals. Let us imagine the following ideal experiment: An astronomer\non the sun sends a message to an earthly mirror, and an astronomer\non the earth sends one to a solar mirror. Each observes the time of\ndeparture and return of his own message, and the time of arrival of\nthe other\u0027s message. Each finds that the other\u0027s message is received\nat a time half-way between the arrival and departure of his own\nmessage. They compare notes, and discover this fact about each other\u0027s\nobservations. They will conclude that, according to the reckonings of\nboth, the two messages were despatched simultaneously, and that the\nmeasure of the space-like interval between the despatch of the two\nmessages is half the time between the despatch and return of either,\n\u003ci\u003ei.e.\u003c/i\u003e about eight minutes. We may re-state the general method\ninvolved as follows: Let us have two transactions \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e\nconnected by a number of causal routes, all going straight from \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e\nto \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e; and let the longest of these consist of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e events. Suppose\nthat there is another transaction \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e such that its later event\nextends to \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e, and that there is no longer causal route from \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e\nto \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e, nor any causal route at all from \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e to \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e. Here \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e\ncorresponds to the sending of the signal from the earth, \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e to the\nsending of the signal from the sun, and \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e to the arrival of the\nsolar signal at the terrestrial observatory. The question is: What is\nto be the interval between \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e? There cannot be a causal\nroute from \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e to \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e, because if there were it could be prolonged\nto \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e, and would be longer than the single event which extends from\n\u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e, \u003ci\u003econtra hyp\u003c/i\u003e. Thus no causal series connects\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e; there is a causal series connecting \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e;\nand \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e is a transaction that begins an event which ends in the\ntransaction \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e. In these circumstances, we say that the interval\nbetween \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e\u003cspan class=\"pagenum\" id=\"Page_371\"\u003e[Pg 371]\u003c/span\u003e is of a different kind from that between\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e, but has the same numerical measure. The fact that this\ndefinition works is what appears as the constant velocity of light.\u003c/p\u003e\n\n\u003cp\u003eDifficulties, however, still suggest themselves. What are we to do with\nthe bending of light in a gravitational field? And what are we to say\nabout the connected theory, according to which the velocity of light\n\u003ci\u003ein vacuo\u003c/i\u003e is not strictly constant? We have been attempting to\nregard the passage of light from one body to another as a single static\noccurrence, involving no change within itself, and therefore having\nzero for its proper time, since time must be measured by changes. If\nwe have to suppose that the light from a star alters its direction as\nit passes near the sun, we shall have to think of the journey of the\nlight as a process, not as a mere continuing event. I do not believe,\nhowever, that this would be regarded as the correct account of the\ninfluence of gravitation on light. Gravitation consists in the fact\nthat a geodesic is geometrically different from what it would be in\nthe absence of a gravitational field; the course of the light is not\n\"really\" bent, but is \"really\" the straightest course geometrically\npossible. In any case, this point arises at an advanced stage in the\ntheory of relativity, and the considerations involved are so numerous\nthat it would almost certainly be possible to find an interpretation\nconsistent with our suggestion if no other obstacle existed.\u003c/p\u003e\n\n\u003cp\u003eWhen an interval is space-like, it is always theoretically possible\nto send a light-signal from one of the events concerned to a causal\ndescendant of the other; consequently our definition of the measure of\na space-like interval is always possible.\u003c/p\u003e\n\n\u003cp\u003eTo say that the greatest velocity in nature is that of light is to say\nthat, when two transitions are the beginning and end, respectively, of\none luminous event, there is no transition which is a causal descendant\nof the one and a causal ancestor of the other. To say that a causal\nchain of transitions belongs\u003cspan class=\"pagenum\" id=\"Page_372\"\u003e[Pg 372]\u003c/span\u003e to the history of one piece of matter is\nto say that no two members of the chain can be connected by a chain\nlonger than the portion of the given chain which lies between the two\ntransitions. This is our translation of the law that the history of a\npiece of matter is a geodesic.\u003c/p\u003e\n\n\u003cp\u003eThe fact that the interval between two points of one light-ray is zero\nappears, on the above theory, to be just what might be expected. For\nwhen an event has temporal extension, that means that two events which\nare compresent with it have a causal relation to each other; while when\nan event has spatial extension, that means that two events compresent\nwith it have a common causal ancestry or posterity. Neither happens in\nthe case of a luminous event, which therefore has neither temporal nor\nspatial extension, in spite of the fact that it covers a whole region\nof space-time points.\u003c/p\u003e\n\n\u003cp\u003eIt will be seen that, according to the above, intervals are discrete,\nand are always measured by integers. There is, so far as I know, no\nempirical evidence for or against this view. If the integers concerned\nwere very large, the phenomena would be sensibly the same as if\nintervals could vary continuously. I do not put forward the theory\nwith any confidence in it as it stands, but rather to suggest to men\nwith more physical competence the possibility of great changes in our\npicture of the world without rejecting anything probably true. In\norder to bring out this point, I shall now re-state the theory without\ninterposing argumentative justifications.\u003c/p\u003e\n\n\u003cp\u003eThe world, it is suggested, consists of a number of events, each\ninvolving no change within itself, but each connected with earlier\nand later events by quantum or other laws which enable us to regard\nthe earlier as the cause and the later as the effect. The quantum\ntransition I call a \"transaction.\" A transaction is subject to laws\nas to the conservation of energy and as to action. Events may be\ncompresent, and one event may be compresent with a number of others\nwhich are\u003cspan class=\"pagenum\" id=\"Page_373\"\u003e[Pg 373]\u003c/span\u003e separated by transitions; in that case, the one event is\nsaid to last for a long time. We can even obtain a continuous time in\nour theory, if the number of events compresent with a given event is\ninfinite, and their beginnings and ends do not synchronize, \u003ci\u003ei.e.\u003c/i\u003e\none of them may be compresent with two others which are not compresent\nwith each other. But I see no reason to suppose that the number of\nevents compresent with a given event is infinite, or to desire a theory\nwhich makes time continuous; I therefore lay no stress upon this\npossibility.\u003c/p\u003e\n\n\u003cp\u003eIn a transaction, or during a rhythm, the causal antecedent may consist\nof more than one event, and so may the causal consequent; but the\nevents which constitute the causal antecedent must all be co-punctual,\nand so must those which constitute the causal consequent. Any event\nof the antecedent group will be called a \"parent\" of any event of the\nconsequent group. When two events are connected by a chain of events,\neach of which is a parent of the next, the one is said to be an\n\"ancestor\" of the other, and the other a \"descendant\" of the one. Two\nevents may be connected by many causal chains, but all will consist of\na finite number of events, and we assume that, in the case of any two\ngiven events, there is a maximum to the number of generations in the\nvarious lines of descent connecting them. This maximum number is the\nmeasure of \"interval\" when the interval is time-like. When the interval\nis space-like, the definition of interval is slightly more complicated.\u003c/p\u003e\n\n\u003cp\u003eTo define space-like intervals, we must first say a few words about\nlight. When a luminous event travels from one body to another, I\nregard the whole as one static event, involving no internal change or\nprocess. Consequently, from the standpoint of the event itself, if\none could imagine a being of whose biography it formed a part, there\nis no time between the beginning and the end. Since nothing travels\nfaster than light, it is impossible that two parts of one luminous\nevent should\u003cspan class=\"pagenum\" id=\"Page_374\"\u003e[Pg 374]\u003c/span\u003e be compresent with two events of which one is a causal\ndescendant of another; therefore there is no extraneous source from\nwhich the luminous event can discover that it is lasting a long time,\nand there is, in fact, no meaning in saying that it is lasting a long\ntime. But when we say that it is reflected back to its starting-point,\nwe mean that it has undergone a transaction which has turned it into\na new luminous event, and that this new event is compresent with\ncausal descendants of events compresent with the earlier one, these\ncompresent events being not luminous, but of the kinds associated with\nmatter. Now, given any two events \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e, neither of which\nis an ancestor of the other, it is possible to find a luminous event\ncompresent with \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e and with a descendant \u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e of \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e. We then\nsay that the events \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 2.204ex; height: 1.767ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-96.png\" alt=\"\" data-tex=\"\\(S\u0027\\)\"\u003e have a space-like separation,\nwhose measure is that of the time-like separation between \u003cimg style=\"vertical-align: -0.05ex; width: 1.459ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-97.png\" alt=\"\" data-tex=\"\\(S\\)\"\u003e and\n\u003cimg style=\"vertical-align: 0; width: 1.593ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-480.png\" alt=\"\" data-tex=\"\\(T\\)\"\u003e.\u003c/p\u003e\n\n\u003cp\u003eIn the above theory, it is assumed that, in all cases where one process\nor piece of matter has an effect upon another, there is at least one\nevent which is compresent with both. This is the form taken by the\ndenial of action at a distance.\u003c/p\u003e\n\n\u003cp\u003eIf we assume, as we have been doing, that change is discontinuous, a\nsingle period of a rhythm will contain some finite number of points.\nSuppose, now, that there are two rhythms such that the initial event\nof a period in the one is always identical with the initial event of\na period in the other, but the other events are diverse; and suppose\nthat the first rhythm contains \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e event in a period while the\nsecond contains \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. Then a period of the first rhythm will contain\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e points, and one of the second will contain \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. We said that\nthe \"interval\" between two events was to be the number of points in\nthe longest causal route from one to the other; hence the interval\nbetween the beginning and end of a period in \u003ci\u003eeither\u003c/i\u003e rhythm is\nmeasured by the greater of the two numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. Suppose\nthis is \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e. Then we may regard the \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-34.png\" alt=\"\" data-tex=\"\\(m\\)\"\u003e-rhythm as having a smaller\n\"velocity\" than the \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-5.png\" alt=\"\" data-tex=\"\\(n\\)\"\u003e-rhythm, while the frequencies\u003cspan class=\"pagenum\" id=\"Page_375\"\u003e[Pg 375]\u003c/span\u003e of the two\nrhythms would be the same. This suggests, in a certain class of cases,\na possibility of defining \"velocity\" otherwise than by relative motion.\nHow far the resulting properties of \"velocity\" would resemble those\nresulting from the usual definition, I do not know.\u003c/p\u003e\n\n\u003cp\u003eThere is no difficulty in defining what is to be meant by saying that\na steady event \"moves.\" An event \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e occupies a number of points of\nspace-time, which can be regarded as a four-dimensional tube divisible\ninto sections such that all the points in one section are simultaneous,\nand are all later or all earlier than all the points in another\nsection. We shall then regard our event \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e as moving along the tube,\nand occupying the various instantaneous sections successively. But this\ndoes not imply any process or change within \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e; it merely implies\ntransitions among events compresent with \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e but not all compresent\nwith each other. It seems, therefore, that everything essential to\ntheoretical physics can be stated in terms of our theory.\u003c/p\u003e\n\n\u003cp\u003eAccording to the above theory, motion is discontinuous. But this\nhypothesis is required for one purpose only, namely for the definition\nof interval. It is easy to introduce such axioms as shall make our\nspace-time continuous, and secure, as in current physics, that\ndiscontinuity shall be confined to quantum phenomena, \u003ci\u003ei.e.\u003c/i\u003e\nto what we have called \"transactions.\" But if this is done, our\ndefinition of interval must be abandoned, and interval resumes its\nplace as something mysterious and unaccountable. There is no logical\nreason why it should not have such a place; the laws of transactions\nhave such a place in our account. But it is always intellectually\nsatisfying when we can reduce the number of inexplicabilities. So far\nas I can discover, there is no good ground for supposing that motion\nis continuous; it is therefore worth while to develop a discontinuous\nhypothesis if we can thereby increase the unity and diminish the\narbitrariness in our account of the physical world.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_376\"\u003e[Pg 376]\u003c/span\u003e\u003c/p\u003e\n\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXVI\"\u003eCHAPTER XXXVI\u003cbr\u003e\nTHE GENESIS OF SPACE-TIME\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nSPACE-TIME, as it appears in mathematical physics, is obviously an\nartefact, \u003ci\u003ei.e.\u003c/i\u003e a structure in which materials found in the\nworld are compounded in such a manner as to be convenient for the\nmathematician. In the present chapter, I wish to collect what has\nalready been said on this subject in various parts of the book, and to\nconsider the resulting metaphysical status of space-time.\u003c/p\u003e\n\n\u003cp\u003eIn the general theory of relativity, space-time appears in two ways:\nfirst, as providing a four-dimensional order; secondly, as giving\nrise to the metrical concept of \"interval.\" Both are relations\nbetween \"points,\" but both are treated mathematically as differential\nrelations. This requires us to solve a purely mathematical problem:\nwhat is the function or process which tends towards these relations\nas a limit? This is on the assumption that space-time is continuous,\nwhich we do not know it to be. Let us begin with this hypothesis, and\nproceed afterwards to the hypothesis of discreteness. In the absence of\nevidence, it is necessary to develop both. For the present, therefore,\nI assume space-time to be continuous. This involves, or at least\nrenders natural, the assumption that there is an infinite number of\nevents compresent with any given event; I shall make this assumption\nalso so long as I assume continuity.\u003c/p\u003e\n\n\u003cp\u003e\"Compresence\" is assumed to be a symmetrical relation, which every\nterm in its field has to itself, and whose field is capable of being\nwell ordered. A group of five events is capable of a relation called\n\"co-punctuality,\" which means, in effect, that there is a region common\nto all five. A group\u003cspan class=\"pagenum\" id=\"Page_377\"\u003e[Pg 377]\u003c/span\u003e of more than five events is called \"co-punctual\"\nwhen every quintet chosen out of it is co-punctual. A \"point\" is\ndefined as a co-punctual group of events which cannot be added to\nwithout ceasing to be co-punctual. \"Events\" are defined as the field of\nthe relation of compresence. Hence, by means of not implausible axioms,\nwe arrive at the space-time order presupposed in the assignment of\nco-ordinates. This part of the theory is straightforward.\u003c/p\u003e\n\n\u003cp\u003eWhen we come to \"interval\" there is more difficulty. In the discussion\nof measurement we decided, following Eddington, that equality of two\nintervals is what has to be defined, and that this has to be defined\nas a limit when both intervals tend towards zero. For this purpose,\nwe supposed a relation of five points \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e, \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e,\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.804ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-320.png\" alt=\"\" data-tex=\"\\(d\u0027\\)\"\u003e\u0027 which we may express in the words: \"\u003cimg style=\"vertical-align: -0.025ex; width: 4.951ex; height: 1.742ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-317.png\" alt=\"\" data-tex=\"\\(abcd\u0027\\)\"\u003e is more nearly\na parallelogram than \u003cimg style=\"vertical-align: -0.025ex; width: 4.324ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-281.png\" alt=\"\" data-tex=\"\\(abcd\\)\"\u003e.\" From this, by means of a certain\napparatus of axioms, we can arrive at what seems to be metrically\nnecessary for mathematical physics. But this procedure is somewhat\nartificial. It seems natural to suppose that our relation of five\npoints arises as follows: between any two points there is a relation,\nwhich for the moment we will call \"separation,\" and the separation of\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-35.png\" alt=\"\" data-tex=\"\\(a\\)\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-68.png\" alt=\"\" data-tex=\"\\(b\\)\"\u003e is more like that of \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.804ex; height: 1.74ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-320.png\" alt=\"\" data-tex=\"\\(d\u0027\\)\"\u003e than like that of\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-78.png\" alt=\"\" data-tex=\"\\(c\\)\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-276.png\" alt=\"\" data-tex=\"\\(d\\)\"\u003e. Thus we shall have to do with degrees of resemblance\nbetween separations of point-pairs; these separations, however, cannot\nexist only for infinitesimal distances, but must exist for finite\ndistances, at any rate if they are sufficiently small.\u003c/p\u003e\n\n\u003cp\u003eWe have therefore to ask ourselves whether any physical meaning can be\nfound for \"separation,\" remembering that in the limit it is to have\nthe properties of a small interval \u003cimg style=\"vertical-align: -0.023ex; width: 2.238ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-149.png\" alt=\"\" data-tex=\"\\(ds\\)\"\u003e. This means to say that a\nseparation may be of two sorts, space-like and time-like; also that the\nseparation between two parts of a light-ray is zero. Now the separation\nwill be time-like if there is any event at the one point which is a\ncausal ancestor of an event at the other point; and the separation\nwill\u003cspan class=\"pagenum\" id=\"Page_378\"\u003e[Pg 378]\u003c/span\u003e be space-like if some event at the one point but not at the other\nand some event at the other but not at the one have a common ancestor\nor descendant, but no event at either is an ancestor or descendant of,\nor identical with, any event at the other. We shall assume that every\npair of points has some causal relation, direct or indirect; that is to\nsay, given any two events, \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e and \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e there will be somewhere\nin space-time two compresent events of which one is an ancestor or\ndescendant of \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-444.png\" alt=\"\" data-tex=\"\\(e_1\\)\"\u003e and the other of \u003cimg style=\"vertical-align: -0.339ex; width: 2.042ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-445.png\" alt=\"\" data-tex=\"\\(e_2\\)\"\u003e. This is hardly more\nthan a definition of the \"world of physics\"; for if an event had no\ncausal relation, however indirect, to the part of the world which we\nknow, it could never be inferred by us, and would in effect belong to a\ndifferent universe. It follows that if two diverse points have neither\na time-like nor a space-like separation, there is an event which is a\nmember of both, but nothing at either is an effect of anything at the\nother. This happens with parts of a light-ray, if we suppose, as we\nhave done, that it consists of steady events which persist until the\nlight-ray is transformed into some other form of energy.\u003c/p\u003e\n\n\u003cp\u003eThus we are led to the view that the relation of separation is somehow\nconnected with the amount of causal action intervening between the two\npoints concerned. It is easy to give a precise meaning to this idea\nwhen we assume a discrete space-time, but it is much more difficult in\na continuous space-time. Nevertheless, it is perhaps not impossible.\u003c/p\u003e\n\n\u003cp\u003eCausality, for these purposes, may be confined to rhythms and\ntransactions; mere relative motion, whether accelerated or uniform,\nwill be regarded as not involving causality in the sense in which\nwe mean it. Indirectly causality will be involved, since there will\nbe a change of space-like separation; but the causality will be\nprimarily concerned with other events, not with those constituting the\nbiographies of the bodies in relative motion. In saying this, we are, I\nthink, only interpreting the Einsteinian theory of gravitation.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_379\"\u003e[Pg 379]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eIn the preceding chapter, when we were considering a discrete\nspace-time, we defined a time-like interval as the number of\nintervening points on the longest causal route connecting the two given\npoints. The natural way to generalize this so as to become applicable\nto a continuous space-time would be to regard the number of points as\nthe measure of geodesic distance; this would enable us to say that the\ngeodesic distance traversed by a unit of matter measures the amount of\ncausal action which it has undergone. If we further assume that, in\ncomparing different units of matter, we must multiply by the mass to\nobtain a measure of the amount of causal action, then the amount in a\nfinite motion is the integral of \u003cimg style=\"vertical-align: -0.025ex; width: 4.224ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-481.png\" alt=\"\" data-tex=\"\\(mds\\)\"\u003e. But this is the amount of\n\"action\" in the technical sense.\u003ca id=\"FNanchor_71\" href=\"#Footnote_71\" class=\"fnanchor\"\u003e[71]\u003c/a\u003e\u003c/p\u003e\n\n\u003cp\u003eIt seems therefore—though this is only a tentative suggestion—that we\ncan regard a time-like separation as the measure of the maximum amount\nof causal action on the various causal routes which lead from one\npoint to another. It is to be observed that, since points are classes\nof events, motion from one point to another consists in the cessation\nof certain events and the coming into existence of others; every such\nchange is causal when it happens along the route of a piece of matter,\nsince the unity of a piece of matter at different times is defined by\nmeans of the concept of a causal route. There is, therefore, so far as\nI can see, no fundamental objection to regarding time-like separations\nas measuring amounts of intervening causal action, and small time-like\nintervals as limits of separations. Space-like intervals, as we have\nseen, are derivative from time-like intervals; hence they, also, depend\nupon amount of causal action.\u003c/p\u003e\n\n\u003cp\u003ePassing now to the hypothesis of a discrete space-time, in which each\npoint consists of a finite number of events, we find that a similar\nanalysis to the above is still possible, and is in fact considerably\neasier than when we assume continuity.\u003cspan class=\"pagenum\" id=\"Page_380\"\u003e[Pg 380]\u003c/span\u003e In a discrete space-time, if\n\u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e are two points containing events which belong to the\nbiography of one material unit, the number of points on the route\nof this unit between \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e is always finite. If several\ngeodesic routes lead from \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e, there will be a maximum to\nthe number of points on such routes; this maximum will be the measure\nof the interval between \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e, which will therefore always\nbe an integer. A longer route means a greater number of intermediate\nevents, and therefore a greater amount of causal action. Thus again the\ninterval measures the greatest amount of causal action on any causal\nroute from \u003cimg style=\"vertical-align: 0; width: 1.699ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-258.png\" alt=\"\" data-tex=\"\\(P\\)\"\u003e to \u003cimg style=\"vertical-align: 0; width: 2.452ex; height: 1.717ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-259.png\" alt=\"\" data-tex=\"\\(P\u0027\\)\"\u003e. And causal routes consist of a succession\nof rhythms or steady events separated by transactions.\u003c/p\u003e\n\n\u003cp\u003eIt will be observed that, in our theory, spatial distance does not\ndirectly represent any physical fact, but is a rather complicated\nway of speaking about the possibility of a common causal ancestry or\nposterity. For example, while a light-wave is supposed to be travelling\naway from an atom, it has no physical relation to anything in the atom\nsubsequent to its emission. It may be reflected back to the atom after\nreaching some other atom, and then half the time of the double journey\n(as measured at the first atom) is called the spatial distance between\nthe two atoms (taking the velocity of light as unity). But there is no\nadequate ground for asserting that at every moment of the intervening\ntime the light-ray is at a certain spatial distance from the atom;\nindeed, the theory of relativity vetoes such a suggestion. There is\ntherefore, so far as I can see, no reason in physics for believing in\ncontinuous motion, except as a convenient symbolic device for dealing\nwith the time-relations of various discontinuous changes. And whether\nwe regard space-time as continuous or discontinuous, motion loses\nits fundamental character, being replaced by successions of events\nbelonging to the biographies of bits of matter. This is inevitable if\nwe are to hold that motion is relative and action at a distance is a\nfiction.\u003c/p\u003e\n\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_381\"\u003e[Pg 381]\u003c/span\u003e\u003c/p\u003e\n\n\u003cp\u003eThere remains a question which is of some interest. Can time be derived\nfrom causality, or must we retain temporal order as fundamental,\nand distinguish cause and effect as the earlier and later terms in\na causal relation?\u003ca id=\"FNanchor_72\" href=\"#Footnote_72\" class=\"fnanchor\"\u003e[72]\u003c/a\u003e This question is bound up with that as to\nthe reversibility of physical processes. If causal relations are\nsymmetrical, so that whenever \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e are related as cause\nand effect it is physically possible that, on another occasion, \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e\nand \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e may be so related, then we must regard the time-order as\nsomething additional to the causal relation, not derivative from it.\nIf, on the other hand, causal laws are irreversible, then we can\ndefine the time-order in terms of them, and need not introduce it as\na logically separate factor. The question of reversibility is still\n\u003ci\u003esub judice\u003c/i\u003e, and I will not venture an opinion. The second law\nof thermodynamics asserts an irreversible process, but is purely\nstatistical. All radiation of energy in spherical waves is prima\nfacie irreversible, but we do not know that it really takes place.\nDr Jeans suggests that there may also be converging spherical waves,\nand that these can be used to explain quantum phenomena.\u003ca id=\"FNanchor_73\" href=\"#Footnote_73\" class=\"fnanchor\"\u003e[73]\u003c/a\u003e For him,\nreversibility is a fundamental postulate.\u003ca id=\"FNanchor_74\" href=\"#Footnote_74\" class=\"fnanchor\"\u003e[74]\u003c/a\u003e I do not know whether he\nwould maintain that the ejection of an electron or helium nucleus from\na radio-active atom is a reversible process; but it must be confessed\nthat, if it is not, the existence of radio-active elements becomes a\nmystery. Quantum theory has, on the whole, increased the arguments in\nfavour of reversibility; but it cannot be said that there is as yet\nconclusive evidence on either side. We must, therefore, leave open the\nquestion whether the time-order of events in one causal route can be\ndefined in terms of causal laws.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_71\" href=\"#FNanchor_71\" class=\"label\"\u003e[71]\u003c/a\u003e\nEddington, \u003ci\u003eop. cit.\u003c/i\u003e, p. 137.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_72\" href=\"#FNanchor_72\" class=\"label\"\u003e[72]\u003c/a\u003e\nThis question (as well as various others) is ably\ndiscussed in a valuable article by Hans Reichenbach, \u003ci\u003eKausalstruktur\nder Welt und der Unterschied von Vergangenheit und Zukunft\u003c/i\u003e,\nSitzungsberichte der Bayerischen Akademie der Wissenschaften,\nmathematisch-natur-wissenschaftliche Abteilung, 1925, pp. 133-175.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_73\" href=\"#FNanchor_73\" class=\"label\"\u003e[73]\u003c/a\u003e\n\u003ci\u003eOp. cit.\u003c/i\u003e, pp. 52-3.\u003c/p\u003e\n\n\u003c/div\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_74\" href=\"#FNanchor_74\" class=\"label\"\u003e[74]\u003c/a\u003e\n\u003ci\u003elb.\u003c/i\u003e, p. 33.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_382\"\u003e[Pg 382]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXVII\"\u003eCHAPTER XXXVII\u003cbr\u003e\nPHYSICS AND NEUTRAL MONISM\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN this chapter, I wish to define the outcome of our analysis in regard\nto the old controversy between materialism and idealism, and to make it\nclear wherein our theory differs from both. So long as the views set\nforth in previous chapters are supposed to be either materialistic or\nidealistic, they will seem to involve inconsistencies, since some seem\nto tend in the one direction, some in the other. For example, when I\nsay that my percepts are in my head, I shall be thought materialistic;\nwhen I say that my head consists of my percepts and other similar\nevents, I shall be thought idealistic. Yet the former statement is a\nlogical consequence of the latter.\u003c/p\u003e\n\n\u003cp\u003eBoth materialism and idealism have been guilty, unconsciously and in\nspite of explicit disavowals, of a confusion in their imaginative\npicture of matter. They have thought of the matter in the external\nworld as being represented by their percepts when they see and\ntouch, whereas these percepts are really part of the matter of the\npercipient\u0027s brain. By examining our percepts it is possible—so I\nhave contended—to infer certain formal mathematical properties of\nexternal matter, though the inference is not demonstrative or certain.\nBut by examining our percepts we obtain knowledge which is not purely\nformal as to the matter of our brains. This knowledge, it is true, is\nfragmentary, but so far as it goes it has merits surpassing those of\nthe knowledge given by physics.\u003c/p\u003e\n\n\u003cp\u003eThe usual view would be that by psychology we acquire knowledge of our\n\"minds,\" but that the only way to acquire knowledge of our brains is to\nhave them examined by a physiologist, usually after we are dead, which\nseems somewhat\u003cspan class=\"pagenum\" id=\"Page_383\"\u003e[Pg 383]\u003c/span\u003e unsatisfactory. I should say that what the physiologist\nsees when he looks at a brain is part of his own brain, not part of the\nbrain he is examining. The feeling of paradox about this view comes, I\nshould say, from wrong views of space. It is true that what we see is\nnot located where our percept of our own brain would be located if we\ncould see our own brain; but this is a question of perceptual space,\nnot of the space of physics. The space of physics is connected with\ncausation in a manner which compels us to hold that our percepts are in\nour brains, if we accept the causal theory of perception, as I think\nwe are bound to do. To say that two events have no spatio-temporal\nseparation is to say that they are compresent; to say that they have\na small separation is to say that they are connected by causal chains\nall of which are short. The percept must therefore be nearer to the\nsense-organ than to the physical object, nearer to the nerve than to\nthe sense-organ, and nearer to the cerebral end of the nerve than to\nthe other end. This is inevitable, unless we are going to say that the\npercept is not in space-time at all. It is usual to hold that \"mental\"\nevents are in time but not in space; let us ask ourselves whether there\nis any ground for this view as regards percepts.\u003c/p\u003e\n\n\u003cp\u003eThe question whether percepts are located in physical space is the\nsame as the question of their causal connection with physical events.\nIf they can be effects and causes of physical events, we are bound to\ngive them a position in physical space-time in so far as interval is\nconcerned, since interval was defined in causal terms. But the real\nquestion is as to \"compresence\" in the sense of Chapter XXVIII. Can a\nmental event be compresent with a physical event? If yes, then a mental\nevent has a position in the space-time order; if no, then it has no\nsuch position. This, therefore, is the crucial question.\u003c/p\u003e\n\n\u003cp\u003eWhen I maintain that a percept and a physical event can\u003cspan class=\"pagenum\" id=\"Page_384\"\u003e[Pg 384]\u003c/span\u003e be compresent,\nI am not maintaining that a percept can have to a piece of matter\nthe sort of relation which another piece of matter would have. The\nrelation of compresence is between a percept and a physical event,\nand physical events are not to be confounded with pieces of matter. A\npiece of matter is a logical structure composed of events; the causal\nlaws of the events concerned, and the abstract logical properties of\ntheir spatio-temporal relations, are more or less known, but their\nintrinsic character is not known. Percepts fit into the same causal\nscheme as physical events, and are not known to have any intrinsic\ncharacter which physical events cannot have, since we do not know of\nany intrinsic character which could be incompatible with the logical\nproperties that physics assigns to physical events. There is therefore\nno ground for the view that percepts cannot be physical events, or for\nsupposing that they are never compresent with other physical events.\u003c/p\u003e\n\n\u003cp\u003eThe fact that mental events admittedly have temporal relations has\nmuch force, now that time and space are so much less distinct than\nthey were. It has become difficult to hold that mental events,\nthough in time, are not in space. The fact that their relations to\neach other can be viewed as only temporal is a fact which they share\nwith any set of events forming the biography of one piece of matter.\nRelatively to axes moving with the percipient\u0027s brain, the interval\nbetween two percepts of his which are not compresent should always be\ntemporal, if his percepts are in his head. But the interval between\nsimultaneous percepts of different percipients is of a different kind;\nand their whole causal environment is such as to make us call this\ninterval space-like. I conclude, then, that there is no good ground for\nexcluding percepts from the physical world, but several strong reasons\nfor including them. The difficulties that have been supposed to stand\nin the way seem to me to be entirely due to wrong\u003cspan class=\"pagenum\" id=\"Page_385\"\u003e[Pg 385]\u003c/span\u003e views as to the\nphysical world, and more particularly as to physical space. The wrong\nviews as to physical space have been encouraged by the notion that the\nprimary qualities are objective, which has been held imaginatively by\nmany men who would have emphatically repudiated it so far as their\nexplicit thought was concerned.\u003c/p\u003e\n\n\u003cp\u003eI hold, therefore, that two simultaneous percepts of one percipient\nhave the relation of compresence out of which spatio-temporal order\narises. It is almost irresistible to go a step further, and say that\nany two simultaneous perceived contents of a mind are compresent, so\nthat all our conscious mental states are in our heads. I see as little\nreason against this extension as against the view that percepts can be\ncompresent. A percept differs from another mental state, I should say,\nonly in the nature of its causal relation to an external stimulus. Some\nrelation of this kind no doubt always exists, but with other mental\nstates the relation may be more indirect, or may be only to some state\nof the body, more particularly the brain. \"Unconscious\" mental states\nwill be events compresent with certain other mental states, but not\nhaving those effects which constitute what is called awareness of a\nmental state. However, I have no wish to go further into psychology\nthan is necessary, and I will pursue this topic no longer, but return\nto matters of more concern to physics.\u003c/p\u003e\n\n\u003cp\u003eThe point which concerns the philosophy of matter is that the events\nout of which we have been constructing the physical world are very\ndifferent from matter as traditionally conceived. Matter was expected\nto be impenetrable and indestructible. The matter that we construct is\nimpenetrable as a result of definition: the matter in a place is all\nthe events that are there, and consequently no other event or piece\nof matter can be there. This is a tautology, not a physical fact; one\nmight as well argue that London is impenetrable because nobody can\nlive in it except one of its inhabitants. Indestructibility,\u003cspan class=\"pagenum\" id=\"Page_386\"\u003e[Pg 386]\u003c/span\u003e on the\nother hand, is an empirical property, believed to be approximately\nbut not exactly possessed by matter. I mean by indestructibility,\nnot conservation of mass, which is known to be only approximate, but\nconservation of electrons and protons. At present it is not known\nwhether an electron and a proton sometimes enter into a suicide pact\nor not,\u003ca id=\"FNanchor_75\" href=\"#Footnote_75\" class=\"fnanchor\"\u003e[75]\u003c/a\u003e but there is certainly no known reason why electrons and\nprotons should be indestructible.\u003c/p\u003e\n\n\u003cp\u003eElectrons and protons, however, are not the stuff of the physical\nworld: they are elaborate logical structures composed of events, and\nultimately of particulars, in the sense of Chapter XXVII. As to what\nthe events are that compose the physical world, they are, in the first\nplace, percepts, and then whatever can be inferred from percepts by\nthe methods considered in Part II. But on various inferential grounds\nwe are led to the view that a percept in which we cannot perceive\na structure nevertheless often has a structure, \u003ci\u003ei.e.\u003c/i\u003e that\nthe apparently simple is often complex. We cannot therefore treat\nthe \u003ci\u003eminimum visible\u003c/i\u003e as a particular, for both physical and\npsychological facts may lead us to attribute a structure to it—not\nmerely a structure in general, but such and such a structure.\u003c/p\u003e\n\n\u003cp\u003eEvents are neither impenetrable nor indestructible. Space-time is\nconstructed by means of co-punctuality, which is the same thing as\nspatio-temporal interpenetration. Perhaps it is not unnecessary to\nexplain that spatio-temporal interpenetration is quite a different\nthing from logical interpenetration, though it may be suspected that\nsome philosophers have been led to favour the latter as a result of the\narguments for the former. We are accustomed to imagining that numerical\ndiversity involves spatio-temporal separation; hence we tend to think\nthat, if two diverse entities are in one place, they\u003cspan class=\"pagenum\" id=\"Page_387\"\u003e[Pg 387]\u003c/span\u003e cannot be wholly\ndiverse, but must be also in some sense one. It is this combination\nthat is supposed to constitute logical interpenetration. For my part,\nI do not think that logical interpenetration can be defined without\nobvious self-contradiction; Bergson, who advocates it, does not\ndefine it. The only author I know of who has dealt seriously with its\ndifficulties is Bradley, in whom, quite consistently, it led to a\nthorough-going monism, combined with the avowal that, in the end, all\ntruth is self-contradictory. I should myself regard this latter result\nas a refutation of the logic from which it follows. Therefore, while\nI respect Bradley more than any other advocate of interpenetration,\nhe seems to me, in virtue of his ability, to have done more than any\nother philosopher to disprove the kind of system which he advocated.\nHowever that may be, the spatio-temporal interpenetration which is\nused in constructing space-time order is quite different from logical\ninterpenetration. Philosophers have been slaves of space and time in\nthe imaginative application of their logic. This is partly due to\nEuler\u0027s diagrams and the notion that the traditional \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e, \u003cimg style=\"vertical-align: 0; width: 1.729ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-154.png\" alt=\"\" data-tex=\"\\(E\\)\"\u003e,\n\u003cimg style=\"vertical-align: 0; width: 1.14ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-482.png\" alt=\"\" data-tex=\"\\(I\\)\"\u003e, \u003cimg style=\"vertical-align: -0.05ex; width: 1.726ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-315.png\" alt=\"\" data-tex=\"\\(O\\)\"\u003e were elementary forms of propositions and the confounding\nof \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-10.png\" alt=\"\" data-tex=\"\\(x\\)\"\u003e is a \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e\" with \"all \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-201.png\" alt=\"\" data-tex=\"\\(\\alpha\\)\"\u003e\u0027s are \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-111.png\" alt=\"\" data-tex=\"\\(\\beta\\)\"\u003e\u0027s.\"\nAll this led to a confusion between classes and individuals, and to\nthe inference that individuals can interpenetrate because classes can\noverlap. I do not suggest explicit confusions of this sort, but only\nthat traditional elementary logic, taught in youth, is an almost fatal\nbarrier to clear thinking in later years, unless much time is spent in\nacquiring a new technique.\u003c/p\u003e\n\n\u003cp\u003eOn the question of the material out of which the physical world is\nconstructed, the views advocated in this volume have, perhaps, more\naffinity with idealism than with materialism. What are called \"mental\"\nevents, if we have been right, are part of the material of the physical\nworld, and what is in our heads is the mind (with additions) rather\nthan what the physiologist sees through his microscope. It is true\nthat we\u003cspan class=\"pagenum\" id=\"Page_388\"\u003e[Pg 388]\u003c/span\u003e have not suggested that all reality is mental. The positive\narguments in favour of such a view, whether Berkeleyan or German,\nappear to me fallacious. The sceptical argument of the phenomenalists,\nthat, whatever else there may be, we cannot know it, is much more\nworthy of respect. There are, in fact, if we have been right, three\ngrades of certainty. The highest grade belongs to my own percepts;\nthe second grade to the percepts of other people; the third to events\nwhich are not percepts of anybody. It is to be observed, however,\nthat the second grade belongs only to the percepts of those who can\ncommunicate with me, directly or indirectly, and of those who are known\nto be closely analogous to people who can communicate with me. The\npercepts of minds, if such there be, which are not related to mine by\ncommunication—\u003ci\u003ee.g.\u003c/i\u003e minds in other planets—can have, at best,\nonly the third grade of certainty, that, namely, which belongs to the\napparently lifeless physical world.\u003c/p\u003e\n\n\u003cp\u003eThe events which are not perceived by any person who can communicate\nwith me, supposing they have been rightly inferred, have a causal\nconnection with percepts, and are inferred by means of this connection.\nMuch is known about their structure, but nothing about their quality.\u003c/p\u003e\n\n\u003cp\u003eWhile, on the question of the stuff of the world, the theory of the\nforegoing pages has certain affinities with idealism—namely, that\nmental events are part of that stuff, and that the rest of the stuff\nresembles them more than it resembles traditional billiard-balls—the\nposition advocated as regards scientific laws has more affinity\nwith materialism than with idealism. Inference from one event to\nanother, where possible, seems only to acquire exactness when it can\nbe stated in terms of the laws of physics. There are psychological\nlaws, physiological laws, and chemical laws, which cannot at\npresent be reduced to physical laws. But none of them is exact and\nwithout exceptions; they state tendencies and averages rather\u003cspan class=\"pagenum\" id=\"Page_389\"\u003e[Pg 389]\u003c/span\u003e than\nmathematical laws governing minimum events. Take, for example, the\npsychological laws of memory. We cannot say: At 12.55 G.M.T. on such\nand such a day, \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e will remember the event \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-283.png\" alt=\"\" data-tex=\"\\(e\\)\"\u003e—unless, indeed, we\nare in a position to remind him of it at that moment. The known laws\nof memory belong to an early stage of science—earlier than Kepler\u0027s\nlaws or Boyle\u0027s law. We can say that, if \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e have been\nexperienced together, the recurrence of \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-188.png\" alt=\"\" data-tex=\"\\(A\\)\"\u003e \u003ci\u003etends\u003c/i\u003e to cause a\nrecollection of \u003cimg style=\"vertical-align: 0; width: 1.717ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-189.png\" alt=\"\" data-tex=\"\\(B\\)\"\u003e, but we cannot say that it is sure to do so, or\nthat it will do so in one assignable class of cases and not in another.\nOne supposes that, to obtain an exact causal theory of memory, it would\nbe necessary to know more about the structure of the brain. The ideal\nto be aimed at would be something like the physical explanation of\nfluorescence, which is a phenomenon in many ways analogous to memory.\nSo far as causal laws go, therefore, physics seems to be supreme among\nthe sciences, not only as against other sciences of matter, but also as\nagainst the sciences that deal with life and mind.\u003c/p\u003e\n\n\u003cp\u003eThere is, however, one important limitation to this. We need to know\nin what physical circumstances such-and-such a percept will arise,\nand we must not neglect the more intimate qualitative knowledge which\nwe possess concerning mental events. There will thus remain a certain\nsphere which will be outside physics. To take a simple instance:\nphysics might, ideally, be able to predict that at such a time my eye\nwould receive a stimulus of a certain sort; it might be able to trace\nthe physical properties of the resulting events in the eye and the\nbrain, one of which is, in fact, a visual percept; but it could not\nitself give us the knowledge that one of them is a visual percept. It\nis obvious that a man who can see knows things which a blind man cannot\nknow; but a blind man can know the whole of physics. Thus the knowledge\nwhich other men have and he has not is not part of physics.\u003c/p\u003e\n\n\u003cp\u003eAlthough there is thus a sphere excluded from physics,\u003cspan class=\"pagenum\" id=\"Page_390\"\u003e[Pg 390]\u003c/span\u003e yet physics,\ntogether with a \"dictionary,\" gives, apparently, all causal knowledge.\nOne supposes that, given the physical characteristics of the events\nin my head, the \"dictionary\" gives the \"mental\" events in my head.\nThis is by no means a matter of course. The whole of the foregoing\ntheory of physics might be true without entailing this consequence.\nSo far as physics can show, it might be possible for different groups\nof events having the same structure to have the same part in causal\nseries. That is to say, given the physical causal laws, and given\nenough knowledge of an initial group of events to determine the purely\nphysical properties of their effects, it might nevertheless be the\ncase that these effects could be qualitatively of different sorts.\nIf that were so, physical determinism would not entail psychological\ndeterminism, since, given two percepts of identical structure but\ndiverse quality, we could not tell which would result from a stimulus\nknown only as to its physical, \u003ci\u003ei.e.\u003c/i\u003e structural, properties.\nThis is an unavoidable consequence of the abstractness of physics. If\nphysics is concerned only with structure, it cannot, \u003ci\u003eper se\u003c/i\u003e,\nwarrant inferences to any but the structural properties of events. Now\nit may be a fact that (\u003ci\u003ee.g.\u003c/i\u003e) the structure of visual percepts\nis very different from that of tactual percepts; but I do not think\nsuch differences could be established with sufficient strictness and\ngenerality to enable us to say that such-and-such a stimulus must\nproduce a visual percept, while such another must produce a tactual\npercept.\u003c/p\u003e\n\n\u003cp\u003eOn this matter, we must, I think, appeal to evidence which is partly\npsychological. We do know, as a matter of fact, that we can, in normal\ncircumstances, more or less infer the percept from the stimulus. If\nthis were not the case, speaking and writing would be useless. When the\nlessons are read, the congregation can follow the words in their own\nBibles. The differences in their \"thoughts\" meanwhile can be connected\ncausally, at least in part, with differences\u003cspan class=\"pagenum\" id=\"Page_391\"\u003e[Pg 391]\u003c/span\u003e in their past experience,\nand these are supposed to make themselves effective by causing\ndifferences in the structure of brains. All this seems sufficiently\nprobable to be worth taking seriously; but it lies outside physics, and\ndoes not follow from the causal autonomy of physics, supposing this to\nbe established even for human bodies. It will be observed that what\nwe are now considering is the converse of what is required for the\ninference from perception to physics. What is wanted there is that,\ngiven the percept, we should be able to infer, at least partially, the\n\u003ci\u003estructure\u003c/i\u003e of the stimulus—or at any rate that this should be\npossible when a sufficient number of percepts are given. What we want\nnow is that, given the structure of the stimulus (which is all that\nphysics can give), we should be able to infer the \u003ci\u003equality\u003c/i\u003e of the\npercept—with the same limitations as before. Whether this is the case\nor not, is a question lying outside physics; but there is reason to\nthink that it is the case.\u003c/p\u003e\n\n\u003cp\u003eThe aim of physics, consciously or unconsciously, has always been\nto discover what we may call the causal skeleton of the world. It\nis perhaps surprising that there should be such a skeleton, but\nphysics seems to prove that there is, particularly when taken in\nconjunction with the evidence that percepts are determined by the\nphysical character of their stimuli. There is reason—though not quite\nconclusive reason—for regarding physics as causally dominant, in the\nsense that, given the physical structure of the world, the qualities of\nits events, in so far as we are acquainted with them, can be inferred\nby means of correlations. We have thus in effect a psycho-cerebral\nparallelism, although the interpretation to be put upon it is not\nthe usual one. We suppose that, given sufficient knowledge, we could\ninfer the qualities of the events in our heads from their physical\nproperties. This is what is really meant when it is said, loosely, that\nthe state of the mind can be inferred from the state of the brain.\nAlthough\u003cspan class=\"pagenum\" id=\"Page_392\"\u003e[Pg 392]\u003c/span\u003e I think that this is probably true, I am less anxious to\nassert it than to assert, what seems to me much more certain, that\nits truth does not follow from the causal autonomy of physics or from\nphysical determinism as applied to all matter, including that of living\nbodies. This latter result flows from the abstractness of physics, and\nbelongs to the philosophy of physics. The other proposition, if true,\ncannot be established by considering physics alone, but only by a study\nof percepts for their own sakes, which belongs to psychology. Physics\nstudies percepts only in their cognitive aspect; their other aspects\nlie outside its purview.\u003c/p\u003e\n\n\u003cp\u003eEven if we reject the view that the quality of events in our heads can\nbe inferred from their structure, the view that physical determinism\napplies to human bodies brings us very near to what is most disliked\nin materialism. Physics may be unable to tell us what we shall hear\nor see or \"think,\" but it can, on the view advocated in these pages,\ntell us what we shall say or write, where we shall go, whether we shall\ncommit murder or theft, and so on. For all these are bodily movements,\nand thus come within the scope of physical laws. We are often asked\nto concede that the beauties of poetry or music cannot result from\nphysical laws. I should concede that the beauty does not result from\nphysics, since beauty depends in part upon intrinsic quality; if it\nwere, as some writers on æsthetics contend, solely a matter of form, it\nwould come within the scope of physics, but I think these writers do\nnot realize what an abstract affair form really is. I should concede\nalso that the \u003ci\u003ethoughts\u003c/i\u003e of Shakespeare or Bach do not come\nwithin the scope of physics. But their thoughts are of no importance\nto us: their whole social efficacy depended upon certain black marks\nwhich they made on white paper. Now there seems no reason to suppose\nthat physics does not apply to the making of these marks, which was\na movement of matter just as truly as the revolution of the earth in\nits orbit. In\u003cspan class=\"pagenum\" id=\"Page_393\"\u003e[Pg 393]\u003c/span\u003e any case, it is undeniable that the socially important\npart of their thought had a one-one relation to certain purely physical\nevents, namely the occurrence of the black marks on the white paper.\nAnd no one can doubt that the causes of our emotions when we read\nShakespeare or hear Bach are purely physical. Thus we cannot escape\nfrom the universality of physical causation.\u003c/p\u003e\n\n\u003cp\u003eThis, however, is perhaps not quite the last word on the subject. We\nhave seen that, on the basis of physics itself, there may be limits\nto physical determinism. We know of no laws as to when a quantum\ntransaction will take place or a radio-active atom will break down.\nWe know fairly well what will happen \u003ci\u003eif\u003c/i\u003e anything happens, and\nwe know statistical averages, which suffice to determine macroscopic\nphenomena. But if mind and brain are causally interconnected, very\nsmall cerebral differences must be correlated with noticeable mental\ndifferences. Thus we are perhaps forced to descend into the region\nof quantum transactions, and to desert the macroscopic level where\nstatistical averages obtain. Perhaps the electron jumps when it likes;\nperhaps the minute phenomena in the brain which make all the difference\nto mental phenomena belong to the region where physical laws no longer\ndetermine definitely what must happen. This, of course, is merely a\nspeculative possibility; but it interposes a veto upon materialistic\ndogmatism. It may be that the progress of physics will decide the\nmatter one way or other; for the present, as in so many other matters,\nthe philosopher must be content to await the progress of science.\u003c/p\u003e\n\n\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\n\n\u003cdiv class=\"footnote\"\u003e\n\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_75\" href=\"#FNanchor_75\" class=\"label\"\u003e[75]\u003c/a\u003e\nIt is thought highly probable that they do. See Dr Jeans,\n\"Recent Developments of Cosmical Physics,\" \u003ci\u003eNature\u003c/i\u003e, December 4,\n1926.\u003c/p\u003e\n\n\u003c/div\u003e\n\u003c/div\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_394\"\u003e[Pg 394]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"CHAPTER_XXXVIII\"\u003eCHAPTER XXXVIII\u003cbr\u003e\nSUMMARY AND CONCLUSION\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cp class=\"nind\"\u003e\nIN the present state of physics, many questions of considerable\nphilosophical importance cannot be answered, although they are such as\nscience may hope to answer, and largely such as were formerly supposed\nto have been already answered. This makes the task of the philosopher\nmore difficult; it is necessary to develop various hypotheses, so as\nto be prepared for whatever decision science may arrive at. Certain\nthings, it is true, may be taken as definitely ascertained; these\nthings, so far as they are relevant to philosophy, were considered\nin Part I. It is clear that, in some sense, there are electrons and\nprotons, and we cannot well doubt the substantial accuracy of their\nestimated masses and electric charge. That is to say, these constants\nevidently represent something of importance in the physical world,\nthough it would be rash to say that they represent exactly what is\nat present supposed. In like manner there seems to be no reasonable\ndoubt that there is a constant \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e, whose dimensions are those of\naction or angular momentum, and whose magnitude is substantially what\nit has been estimated to be. It would seem clear also that \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e is a\nconstant which is characteristic of periodic processes. Moreover, the\nchange from one such process to another, which is what we have called a\ntransaction, is governed by principles connected with h in addition to\nthe conservation of energy.\u003c/p\u003e\n\n\u003cp\u003eBut it would be very rash to maintain that the current mathematical\nformulation of the quantum principle is the best possible; indeed,\nthere are reasons for dissatisfaction with it. Perhaps the most\nimportant of these is that in expressing the kinetic energy we have\nto employ the method of separation of\u003cspan class=\"pagenum\" id=\"Page_395\"\u003e[Pg 395]\u003c/span\u003e variables, and that we do not\nknow whether separation of variables is always possible, or whether all\nways of separating the variables give equivalent results. Apart from\nthese rather technical difficulties, there are others that are less\ndefinite but perhaps not less important. No one has succeeded in making\nthe existence of quanta seem at all \"reasonable\"; that is to say, it\nremains isolated and separate from other physical ideas. And whereas\nit involves discontinuity, the whole effect of relativity has been to\nemphasize continuity. Moreover, no one has yet succeeded in explaining\ninterference and diffraction by means of light-quanta, or in explaining\nthe photo-electric effect without them. For these reasons, the time has\nnot yet come when the philosopher can deal confidently with quantum\ntheory; he can only suggest what would be his philosophy if this or\nthat view had prevailed in physics.\u003c/p\u003e\n\n\u003cp\u003eIn relativity, we are on surer ground. The advance on the physics\nof the past, where relativity is concerned, is mainly logical and\nphilosophical. It is true that facts led to the theory, and that\nthe theory in turn led to the discovery of new facts. But the facts\nwere small and only just within the limits of observation; and they\nhad not, as facts, the revolutionary importance of the facts about\nquanta. And now that the theory is fairly complete, one can see that,\ntheoretically, it ought to have been discovered by Galileo, or at any\nrate as soon as the velocity of light became known. It represents in\nits technique a better philosophy than that of Newton; indeed, one of\nits most remarkable features is the adaptation of the technique to the\nphilosophy.\u003c/p\u003e\n\n\u003cp\u003eThe theory of relativity, to my mind, is most remarkable when\nconsidered as a logical deductive system. That is the reason, or one\nof the reasons, why I have found occasion to allude so constantly to\nEddington. He, more than Einstein or Weyl, has expounded the theory in\nthe form most apt for the purposes of the philosopher. Minkowski had\nthe same\u003cspan class=\"pagenum\" id=\"Page_396\"\u003e[Pg 396]\u003c/span\u003e quality, but he did not live to see the general theory. For\nphilosophical purposes, therefore, I have allowed myself to be guided\nalmost entirely by Eddington.\u003c/p\u003e\n\n\u003cp\u003eIn the general theory of relativity, we start with a four-dimensional\ncontinuum of points, whose properties, to begin with, are purely\nordinal. We then assign four co-ordinates to each point on any\nprinciple such that the ordinal properties of the co-ordinates are the\nsame as those of the points. We then assume that, if two points are\nvery close together, there is a quadratic function of the co-ordinates\nwhich has the same value however the co-ordinates may be assigned,\nsubject to the above ordinal condition. If this function is positive,\nits square root is called the (time-like) interval; if negative,\nthe square root of the function with its sign changed is called the\n(space-like) interval. Omitting niceties, we may say that the remainder\nof the theory turns mainly on geodesics. A geodesic is a route between\ntwo space-time points such that the integral of the interval along\nthis route is stationary. In the important routes, it is a maximum.\nIt appears that energy can be divided into parcels which move in\ngeodesics; when these parcels move with a velocity less than that of\nlight, they are regarded as pieces of matter. Weyl, by imposing certain\nlimitations on measurement, succeeds in including electromagnetic\nphenomena in this scheme. Thus we have a comprehensive theory which may\nbe taken to embrace everything except quantum phenomena.\u003c/p\u003e\n\n\u003cp\u003eBut although there is so much to give pleasure to the logician in\nthis scheme—more especially the method of tensors and Hamiltonian\nderivatives—yet the philosopher cannot but feel dissatisfaction\nwith the apparently arbitrary assumption about intervals. This\nassumption seemed less arbitrary than it is, because of its connection,\nhistorically, with the theorem of Pythagoras and its modifications in\nnon-Euclidean geometry. But the theorem was believed formerly because\u003cspan class=\"pagenum\" id=\"Page_397\"\u003e[Pg 397]\u003c/span\u003e\nit had been proved; when the proof was found to have no value, it was\nbelieved because empirical evidence was thought to show its approximate\ntruth. This empirical evidence, of course, remains, but the theory of\nrelativity has made its value much more problematical than it formerly\nseemed. And it is customary to carry out measurements carefully,\ntaking trouble to secure bodies that are as nearly rigid as possible,\nand optical instruments that are accurate. If our co-ordinates are to\nbe arbitrary, as they are in the general theory of relativity, it is\ndoubtful whether we still have a right to expect that they will verify\nanything analogous to the theorem of Pythagoras.\u003c/p\u003e\n\n\u003cp\u003eAs against these doubts, it may be said that the general theory has\njustified itself by the correctness of all its verifiable consequences.\nThis is true, and I do not wish to minimize the force of the argument.\nBut I seem to observe that, in obtaining these results, the theory\ndoes not make use of the full liberty in assignment of co-ordinates\nwhich it claims at the start. In astronomy, its co-ordinates are still\nassigned by the usual careful methods, and it is not clear that this\ncare is useless. From the method of tensors, it \u003ci\u003eseems\u003c/i\u003e to follow\nthat we can employ any co-ordinates subject to the ordinal condition.\nBut the method of tensors, as used, assumes the formula for interval;\nfor this reason, Dr Whitehead found it necessary, in his \u003ci\u003ePrinciple\nof Relativity\u003c/i\u003e, to give a theory of tensors independent of the\nformula for interval. There is thus still legitimate room for doubt as\nto whether the formula for interval is really quite independent of the\nchoice of co-ordinates.\u003c/p\u003e\n\n\u003cp\u003eAnd, apart from this question, there is great difficulty in suggesting\nany non-technical meaning for interval; yet such a meaning ought\nto exist, if interval is as fundamental as it appears to be in the\ntheory of relativity. There is difficulty also as to what is meant by\nmeasurement. And there is the feeling that, perhaps, tensor equations\nrepresent purely ordinal properties of the space-time continuum, and\ncould, by a better\u003cspan class=\"pagenum\" id=\"Page_398\"\u003e[Pg 398]\u003c/span\u003e technique, be set forth without the use of any\nco-ordinates at all. The necessary technique does not exist at present,\nbut it is not impossible that it may be created before long.\u003c/p\u003e\n\n\u003cp\u003eIn Part II., we approached a different type of question: the question\nof the evidence for the truth of physics, \u003ci\u003ei.e.\u003c/i\u003e of the relation\nof physics to perception. For the purposes of this inquiry, it is\nconvenient to use \"perception\" somewhat more narrowly than it would\nbe used in psychology. Our purpose is epistemological, and therefore\nperception is only relevant in so far as it is explicit and the\npercept is observed: percepts which pass unnoticed cannot be made\ninto premisses for physics. The use of percepts for inference as to\nthe physical world rests upon the causal theory of perception, since\nthe naive realism of common sense turns out to be self-contradictory.\nThe serious alternatives to the causal theory of perception are not\ncommon sense, but solipsism and phenomenalism. Solipsism, as an\nepistemologically serious theory, must mean the view that from the\nevents which I experience there is no valid method of inferring the\ncharacter, or even the existence, of events which I do not experience.\nIf inference is taken in the sense of strict deductive logic, there\nis, so far as I can see, no escape from the solipsist position. And it\nshould be observed that this position cannot admit unconscious events\nin me, any more than events outside me: its basis is epistemological,\nand therefore, for it, the important distinction is between what I\nexperience and what I do not experience, not between what is mine and\nwhat is not mine in some metaphysical or physical sense. We cannot\nescape from the solipsist position without bringing in induction and\ncausality, which are still subject to the doubts resulting from Hume\u0027s\nsceptical criticism.\u003c/p\u003e\n\n\u003cp\u003eSince, however, all science rests upon induction and causality, it\nseems justifiable, at least pragmatically, to assume that, when\nproperly employed, they can give at least a probability. In the\npresent work, I have made this assumption baldly,\u003cspan class=\"pagenum\" id=\"Page_399\"\u003e[Pg 399]\u003c/span\u003e without attempting\nto justify it; I have done this because I do not believe that a\njustification could be much briefer than Mr Keynes\u0027s \u003ci\u003eTreatise\non Probability\u003c/i\u003e, and also because, while I am convinced that a\njustification is possible, I am not satisfied with those put forward by\nothers or with any that I have been able to invent myself. It seemed\nbest, therefore, to make the assumption as stark as possible, without\nany attempt at artificial plausibility.\u003c/p\u003e\n\n\u003cp\u003eIntermediate between solipsism and the ordinary scientific view, there\nis a half-way house called \"phenomenalism.\" This admits events other\nthan those which I experience, but holds that all of them are percepts\nor other mental events. Practically, it means, when advocated by\nscientific men, that they will accept the testimony of other observers\nas to what they have actually experienced, but that they will not infer\nthence anything which no observer has experienced. It may be said, in\njustification of this position, that, while it employs analogy and\ninduction, it refrains from assuming causality. But it may be doubted\nwhether it can really abstain from causality. Phenomenalists appear to\ntake testimony for granted, \u003ci\u003ei.e.\u003c/i\u003e to assume that the words which\nthey see and hear express what they themselves would express if they\nused them. But this involves causality, and involves it in the form\nin which the cause is in one person and the effect in another. There\ndoes not seem, therefore, to be any substantial justification for this\nhalf-way house.\u003c/p\u003e\n\n\u003cp\u003eWe therefore assume, though with less than demonstrative certainty,\nthat percepts have causes which may be not percepts, and, in\nparticular, that when a number of people have similar percepts\nsimultaneously, there is what may be called a \"field\" of causally\nconnected events, which, it is found, have relations that often enable\nus to arrange them in a spherical order about a centre. We thus\narrive at a space-time order of events, which is found to be the same\nwhichever of many\u003cspan class=\"pagenum\" id=\"Page_400\"\u003e[Pg 400]\u003c/span\u003e possible methods of arriving at it we adopt; in this\norder, a percept is located in the head of the percipient. In drawing\ninferences from percepts to their causes, we assume that the stimulus\nmust possess whatever structure is possessed by the percept, though it\nmay also have structural properties not possessed by the percept. The\nassumption that the structural properties of the percept must exist\nin the stimulus follows from the maxim \"same cause, same effect\" in\nthe inverted form \"different effects, different causes,\" from which\nit follows that if, \u003ci\u003ee.g.\u003c/i\u003e, we see red and green side by side,\nthere is some difference between the stimulus to the red percept and\nthe stimulus to the green percept. The structural features possessed\nby the stimulus but not by the percept, when they can be inferred, are\ninferred by means of general laws—\u003ci\u003ee.g.\u003c/i\u003e when two objects look\nsimilar to the naked eye but dissimilar under the microscope, we assume\nthat there are differences in the stimuli to the naked-eye percepts\nwhich produce either no differences, or no perceptible differences, in\nthe corresponding percepts.\u003c/p\u003e\n\n\u003cp\u003eThese principles enable us to infer a great deal as to the structure\nof the physical world, but not as to its intrinsic character. They put\npercepts in their place as occurrences analogous to and connected with\nother events in the physical world, and they enable us to regard a\ndictaphone or a photographic plate as having something which, from the\nstandpoint of physics, is not very dissimilar from perception. We no\nlonger have to contend with what used to seem mysterious in the causal\ntheory of perception: a series of light-waves or sound-waves or what\nnot suddenly producing a mental event apparently totally different from\nthemselves in character. As to intrinsic character, we do not know\nenough about it in the physical world to have a right to say that it\nis very different from that of percepts; while as to structure we have\nreason to hold that it is similar in the stimulus and the percept. This\nhas become possible owing to the facts that \"matter\" can be\u003cspan class=\"pagenum\" id=\"Page_401\"\u003e[Pg 401]\u003c/span\u003e regarded\nas a system of events, not as part of the stuff of the world, and that\nspace-time, as it occurs in physics, has been found to be much more\ndifferent from perceptual space than was formerly imagined.\u003c/p\u003e\n\n\u003cp\u003eThis brings us to Part III., in which we endeavour to discover a\npossible structure of the physical world which shall at once justify\nphysics and take account of the connection with perception demanded by\nthe necessity for an empirical basis for physics. Here we are concerned\nfirst with the construction of points as systems of events which\noverlap, or are \"co-punctual,\" in space-time, and then with the purely\nordinal properties of space-time. The method employed is very general,\nand can be adapted to a discrete or to a continuous order; it is proved\nthat \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-313.png\" alt=\"\" data-tex=\"\\(\\aleph_0\\)\"\u003e, events are sufficient to generate a continuum of\npoints, given certain laws as to the manner of their overlapping.\nThe whole of this theory, however, aims only at constructing such\nproperties of space-time as belong to \u003ci\u003eanalysis situs\u003c/i\u003e; everything\nappertaining to intervals and metrics is omitted at this stage, since\ncausal considerations are required for the theory of intervals.\u003c/p\u003e\n\n\u003cp\u003eThe conception of one unit of matter—say one electron—as a\n\"substance,\" \u003ci\u003ei.e.\u003c/i\u003e a single simple entity persisting through\ntime, is not one which we are justified in adopting, since we have no\nevidence whatever as to whether it is false or true. We define a single\nmaterial unit as a \"causal line,\" \u003ci\u003ei.e.\u003c/i\u003e as a series of events\nconnected with each other by an intrinsic differential causal law which\ndetermines first-order changes, leaving second-order changes to be\ndetermined by extrinsic causal laws. (In this we are for the moment\nignoring quantum phenomena.) If there are light-quanta, these will more\nor less fulfil this definition of matter, and we shall have returned to\na corpuscular theory of light; but this is at present an open question.\nThe whole conception of matter is less fundamental to physics than it\nused to be, since energy has more and more\u003cspan class=\"pagenum\" id=\"Page_402\"\u003e[Pg 402]\u003c/span\u003e taken its place. We find\nthat under terrestrial conditions electrons and protons persist, but\nthere is nothing in theoretical physics to lead us to expect this, and\nphysicists are quite prepared to find that matter can be annihilated.\nThis view is even, put forward to account for the energy of the stars.\u003c/p\u003e\n\n\u003cp\u003eThe question of interval presents great difficulties, when we attempt\nto construct a picture of the world which shall make its importance\nseem not surprising. The same may be said of the quantum. I have\nendeavoured, not, I fear, with much success, to suggest hypotheses\nwhich would link these two curious facts into one whole. I suggest\nthat the world consists of steady events accompanied by rhythms, like\na long note on the violin while arpeggios are played on the piano,\nor of rhythms alone. Steady events are of various sorts, and many\nsorts have their appropriate rhythmic accompaniments. Quantum changes\nconsist of \"transactions,\" \u003ci\u003ei.e.\u003c/i\u003e of the substitution, suddenly,\nof one rhythm for another. When two events have a time-like interval,\nif space-time is discrete, this interval is the greatest number of\ntransitions on any causal route leading from the one event to the\nother. The definition of space-like intervals is derived from that of\ntime-like intervals. The whole process of nature may, so far as present\nevidence goes, be conceived as discontinuous; even the periodic rhythms\nmay consist of a finite number of events per period. The periodic\nrhythms are required in order to give an account of the uses of the\nquantum principle. A percept, at any rate when it is visual, will be a\nsteady event, or system of steady events, following upon a transaction.\nPercepts are the only part of the physical world that we know otherwise\nthan abstractly. As regards the world in general, both physical and\nmental, everything that we know of its intrinsic character is derived\nfrom the mental side, and almost everything that we know of its causal\nlaws is derived from the physical side. But from the standpoint of\nphilosophy the distinction between physical and mental is superficial\nand unreal.\u003c/p\u003e\n\n\n\u003chr class=\"chap x-ebookmaker-drop\"\u003e\n\n\u003cdiv class=\"chapter\"\u003e\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_403\"\u003e[Pg 403]\u003c/span\u003e\u003c/p\u003e\n\u003ch2 class=\"nobreak\" id=\"INDEX\"\u003eINDEX\u003c/h2\u003e\n\u003c/div\u003e\n\n\u003cul class=\"index\"\u003e\n\u003cli class=\"ifrst\"\u003e\u003ci\u003eA priori\u003c/i\u003e and empirical knowledge:\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAbstract geometry: \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAcceleration: \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ absolute: \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_67\"\u003e67\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAccuracy: \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAction: \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e, \u003ca href=\"#Page_379\"\u003e379\u003c/a\u003e, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ at a distance: \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_108\"\u003e108\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_162\"\u003e162\u003c/a\u003e, \u003ca href=\"#Page_327\"\u003e327\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e, \u003ca href=\"#Page_374\"\u003e374\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_380\"\u003e380\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAction, principle of least: \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAdjectives: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eÆther: \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_121\"\u003e121\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e, \u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAnalogy: \u003ca href=\"#Page_399\"\u003e399\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003eAnalysis: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ logical: \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003e\u003ci\u003eAnalysis situs\u003c/i\u003e: \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e, \u003ca href=\"#Page_113\"\u003e113\u003c/a\u003e, \u003ca href=\"#Page_114\"\u003e114\u003c/a\u003e, \u003ca href=\"#Page_295\"\u003e295\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_298\"\u003e298\u003c/a\u003e, \u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e, \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAnalytic and synthetic propositions: \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAncestor: \u003ca href=\"#Page_373\"\u003e373\u003c/a\u003e, \u003ca href=\"#Page_378\"\u003e378\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eApes: \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAristotle: \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eArithmetic: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAspects: \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAstronomy: \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_317\"\u003e317\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAtom: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ Heisenberg\u0027s: \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ nucleus of: \u003ca href=\"#Page_25\"\u003e25\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ Rutherford-Bohr: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_27\"\u003e27\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAtomic number: \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ weight: \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAtomism, spatial: \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eAxioms: \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eBach: \u003ca href=\"#Page_392\"\u003e392\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBeauty: \u003ca href=\"#Page_392\"\u003e392\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBehaviourism: \u003ca href=\"#Page_142\"\u003e142\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBeliefs: \u003ca href=\"#Page_142\"\u003e142\u003c/a\u003e, \u003ca href=\"#Page_174\"\u003e174\u003c/a\u003e, \u003ca href=\"#Page_178\"\u003e178\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBergson: \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e, \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e, \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBerkeley: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_7\"\u003e7\u003c/a\u003e, \u003ca href=\"#Page_136\"\u003e136\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_213\"\u003e213\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBetween: \u003ca href=\"#Page_305\"\u003e305\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBifurcation of nature: \u003ca href=\"#Page_257\"\u003e257\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBiography: \u003ca href=\"#Page_212\"\u003e212\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBlack-body radiation: \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBohr: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_33\"\u003e33\u003c/a\u003e, \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e, \u003ca href=\"#Page_133\"\u003e133\u003c/a\u003e, \u003ca href=\"#Page_134\"\u003e134\u003c/a\u003e, \u003ca href=\"#Page_195\"\u003e195\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBolyai: \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBorn: \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBoscovitch: \u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBradley: \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBrain: \u003ca href=\"#Page_382\"\u003e382\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBraithwaite: \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eBroad: \u003ca href=\"#Page_197\"\u003e197\u003c/a\u003e, \u003ca href=\"#Page_259\"\u003e259\u003c/a\u003e, \u003ca href=\"#Page_260\"\u003e260\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eCamera: \u003ca href=\"#Page_209\"\u003e209\u003c/a\u003e, \u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCantor, Georg: \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCantorian continuum: \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCartesianism: \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCategory: \u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e, \u003ca href=\"#Page_243\"\u003e243\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCausal ancestor: \u003ca href=\"#Page_314\"\u003e314\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ chain: \u003ca href=\"#Page_285\"\u003e285\u003c/a\u003e, \u003ca href=\"#Page_314\"\u003e314\u003c/a\u003e, \u003ca href=\"#Page_320\"\u003e320\u003c/a\u003e, \u003ca href=\"#Page_371\"\u003e371\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ laws: \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e, \u003ca href=\"#Page_121\"\u003e121\u003c/a\u003e, \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e, \u003ca href=\"#Page_191\"\u003e191\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_200\"\u003e200\u003c/a\u003e, \u003ca href=\"#Page_214\"\u003e214\u003c/a\u003e, \u003ca href=\"#Page_245\"\u003e245\u003c/a\u003e, \u003ca href=\"#Page_285\"\u003e285\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_340\"\u003e340\u003c/a\u003e, \u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_366\"\u003e366\u003c/a\u003e, \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ „ extrinsic: \u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ lines: \u003ca href=\"#Page_313\"\u003e313\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ relation: \u003ca href=\"#Page_367\"\u003e367\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ route: \u003ca href=\"#Page_379\"\u003e379\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ unit: \u003ca href=\"#Page_315\"\u003e315\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCausality: \u003ca href=\"#Page_313\"\u003e313\u003c/a\u003e, \u003ca href=\"#Page_367\"\u003e367\u003c/a\u003e, \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e, \u003ca href=\"#Page_399\"\u003e399\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCausation: \u003ca href=\"#Page_108\"\u003e108\u003c/a\u003e, \u003ca href=\"#Page_383\"\u003e383\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ at a distance: \u003ca href=\"#Page_214\"\u003e214\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCause: \u003ca href=\"#Page_150\"\u003e150\u003c/a\u003e, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCause-and-effect: \u003ca href=\"#Page_346\"\u003e346\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eChange: \u003ca href=\"#Page_281\"\u003e281\u003c/a\u003e, \u003ca href=\"#Page_362\"\u003e362\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eChuang-Tze: \u003ca href=\"#Page_180\"\u003e180\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eClarke: \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eClasses: \u003ca href=\"#Page_116\"\u003e116\u003c/a\u003e, \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCollinear: \u003ca href=\"#Page_306\"\u003e306\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCollinearity: \u003ca href=\"#Page_308\"\u003e308\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eColour: \u003ca href=\"#Page_288\"\u003e288\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eColour-similarity: \u003ca href=\"#Page_288\"\u003e288\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eColours: \u003ca href=\"#Page_253\"\u003e253\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCommon sense: \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eComparison: \u003ca href=\"#Page_278\"\u003e278\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCompresent: \u003ca href=\"#Page_294\"\u003e294\u003c/a\u003e, \u003ca href=\"#Page_336\"\u003e336\u003c/a\u003e, \u003ca href=\"#Page_349\"\u003e349\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_372\"\u003e372\u003c/a\u003e, \u003ca href=\"#Page_374\"\u003e374\u003c/a\u003e, \u003ca href=\"#Page_376\"\u003e376\u003c/a\u003e, \u003ca href=\"#Page_378\"\u003e378\u003c/a\u003e, \u003ca href=\"#Page_383\"\u003e383\u003c/a\u003e, \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eConjunctions: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eConstants: \u003ca href=\"#Page_172\"\u003e172\u003c/a\u003e, \u003ca href=\"#Page_251\"\u003e251\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eContinuity: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e, \u003ca href=\"#Page_214\"\u003e214\u003c/a\u003e, \u003ca href=\"#Page_217\"\u003e217\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e, \u003ca href=\"#Page_322\"\u003e322\u003c/a\u003e, \u003ca href=\"#Page_362\"\u003e362\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ perceptual: \u003ca href=\"#Page_280\"\u003e280\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ qualitative: \u003ca href=\"#Page_81\"\u003e81\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eContinuous, space-time is: \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eContinuum: \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ mathematical: \u003ca href=\"#Page_280\"\u003e280\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eContravariant: \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-ordinates: \u003ca href=\"#Page_56\"\u003e56\u003c/a\u003e, \u003ca href=\"#Page_57\"\u003e57\u003c/a\u003e, \u003ca href=\"#Page_60\"\u003e60\u003c/a\u003e, \u003ca href=\"#Page_63\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003cspan class=\"pagenum\" id=\"Page_404\"\u003e[Pg 404]\u003c/span\u003e110, \u003ca href=\"#Page_334\"\u003e334\u003c/a\u003e, \u003ca href=\"#Page_337\"\u003e337\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-punctual: \u003ca href=\"#Page_307\"\u003e307\u003c/a\u003e, \u003ca href=\"#Page_314\"\u003e314\u003c/a\u003e, \u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e, \u003ca href=\"#Page_373\"\u003e373\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ events: \u003ca href=\"#Page_315\"\u003e315\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-punctuality: \u003ca href=\"#Page_299\"\u003e299\u003c/a\u003e, \u003ca href=\"#Page_376\"\u003e376\u003c/a\u003e, \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-regional: \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCorrelation: \u003ca href=\"#Page_188\"\u003e188\u003c/a\u003e, \u003ca href=\"#Page_204\"\u003e204\u003c/a\u003e, \u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCorrespondence: \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-superficial: \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCo-variant: \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ derivative: \u003ca href=\"#Page_96\"\u003e96\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eCurve of pursuit: \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eData: \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e, \u003ca href=\"#Page_189\"\u003e189\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDe Broglie: \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDebye: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDescartes: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDeterminism: \u003ca href=\"#Page_390\"\u003e390\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physical: \u003ca href=\"#Page_214\"\u003e214\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDictaphone: \u003ca href=\"#Page_209\"\u003e209\u003c/a\u003e, \u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e, \u003ca href=\"#Page_400\"\u003e400\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDifferential equations: \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ law: \u003ca href=\"#Page_245\"\u003e245\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ „ intrinsic: \u003ca href=\"#Page_323\"\u003e323\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDiffraction: \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDimensions: \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDirac: \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_43\"\u003e43\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDiscontinuity: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDistance: \u003ca href=\"#Page_166\"\u003e166\u003c/a\u003e, \u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e, \u003ca href=\"#Page_350\"\u003e350\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDiversity: \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDoppler effect: \u003ca href=\"#Page_165\"\u003e165\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eDreams: \u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e, \u003ca href=\"#Page_222\"\u003e222\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eEddington: \u003ca href=\"#Page_57\"\u003e57\u003c/a\u003e, \u003ca href=\"#Page_64\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_75\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_78\"\u003e78\u003c/a\u003e, \u003ca href=\"#Page_82\"\u003e82\u003c/a\u003e, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_91\"\u003e91\u003c/a\u003e, \u003ca href=\"#Page_96\"\u003e96\u003c/a\u003e, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_106\"\u003e106\u003c/a\u003e, \u003ca href=\"#Page_110\"\u003e110\u003c/a\u003e, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_122\"\u003e122\u003c/a\u003e, \u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_136\"\u003e136\u003c/a\u003e, \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEinstein: \u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e, \u003ca href=\"#Page_75\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e, \u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e, \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e, \u003ca href=\"#Page_323\"\u003e323\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eElectromagnetism: \u003ca href=\"#Page_328\"\u003e328\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eElectrons: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e, \u003ca href=\"#Page_90\"\u003e90\u003c/a\u003e, \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e, \u003ca href=\"#Page_153\"\u003e153\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e, \u003ca href=\"#Page_287\"\u003e287\u003c/a\u003e, \u003ca href=\"#Page_319\"\u003e319\u003c/a\u003e, \u003ca href=\"#Page_320\"\u003e320\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_325\"\u003e325\u003c/a\u003e, \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e, \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEllis: \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_125\"\u003e125\u003c/a\u003e, \u003ca href=\"#Page_126\"\u003e126\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEmergent: \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEmpirical: \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_176\"\u003e176\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEmpiricists: \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEnclosure-series: \u003ca href=\"#Page_291\"\u003e291\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEnergy: \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e, \u003ca href=\"#Page_122\"\u003e122\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_318\"\u003e318\u003c/a\u003e, \u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e, \u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e362, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ kinetic: \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ radiant: \u003ca href=\"#Page_365\"\u003e365\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEntities, unperceived: \u003ca href=\"#Page_191\"\u003e191\u003c/a\u003e, \u003ca href=\"#Page_206\"\u003e206\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ unperceivable: \u003ca href=\"#Page_216\"\u003e216\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEpistemology: \u003ca href=\"#Page_179\"\u003e179\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eError: \u003ca href=\"#Page_183\"\u003e183\u003c/a\u003e, \u003ca href=\"#Page_184\"\u003e184\u003c/a\u003e, \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEuclid: \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEuclid\u0027s first axiom: \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e, \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEuler\u0027s diagrams: \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEvents: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_243\"\u003e243\u003c/a\u003e, \u003ca href=\"#Page_247\"\u003e247\u003c/a\u003e, \u003ca href=\"#Page_275\"\u003e275\u003c/a\u003e, \u003ca href=\"#Page_284\"\u003e284\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e, \u003ca href=\"#Page_384\"\u003e384\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEvents: group of: \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ luminous: \u003ca href=\"#Page_352\"\u003e352\u003c/a\u003e, \u003ca href=\"#Page_333\"\u003e333\u003c/a\u003e, \u003ca href=\"#Page_354\"\u003e354\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_369\"\u003e369\u003c/a\u003e, \u003ca href=\"#Page_371\"\u003e371\u003c/a\u003e, \u003ca href=\"#Page_373\"\u003e373\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ maximum: \u003ca href=\"#Page_293\"\u003e293\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ minimum: \u003ca href=\"#Page_292\"\u003e292\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ persistent: \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ steady: \u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_363\"\u003e363\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e, \u003ca href=\"#Page_402\"\u003e402\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eEvidence, empirical: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eExistents, unperceived: \u003ca href=\"#Page_213\"\u003e213\u003c/a\u003e, \u003ca href=\"#Page_226\"\u003e226\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eExpectations: \u003ca href=\"#Page_181\"\u003e181\u003c/a\u003e, \u003ca href=\"#Page_182\"\u003e182\u003c/a\u003e, \u003ca href=\"#Page_183\"\u003e183\u003c/a\u003e, \u003ca href=\"#Page_189\"\u003e189\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eExperience: \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eFacts: \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFallacy of simple location: \u003ca href=\"#Page_340\"\u003e340\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFaraday: \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFijians: \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFitzGerald contraction: \u003ca href=\"#Page_50\"\u003e50\u003c/a\u003e, \u003ca href=\"#Page_100\"\u003e100\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eForce: \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e, \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eForm: \u003ca href=\"#Page_172\"\u003e172\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFoucault\u0027s pendulum: \u003ca href=\"#Page_358\"\u003e358\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFowler: \u003ca href=\"#Page_33\"\u003e33\u003c/a\u003e, \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFranck: \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFreewill: \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFrege: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eFrequency: \u003ca href=\"#Page_357\"\u003e357\u003c/a\u003e, \u003ca href=\"#Page_364\"\u003e364\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eGalileo: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e, \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e, \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGauge-system: \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGauss: \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e, \u003ca href=\"#Page_113\"\u003e113\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGeodesics: \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e, \u003ca href=\"#Page_62\"\u003e62\u003c/a\u003e, \u003ca href=\"#Page_69\"\u003e69\u003c/a\u003e, \u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e, \u003ca href=\"#Page_313\"\u003e313\u003c/a\u003e, \u003ca href=\"#Page_327\"\u003e327\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_371\"\u003e371\u003c/a\u003e, \u003ca href=\"#Page_372\"\u003e372\u003c/a\u003e, \u003ca href=\"#Page_379\"\u003e379\u003c/a\u003e, \u003ca href=\"#Page_396\"\u003e396\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGeometry: \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e, \u003ca href=\"#Page_3\"\u003e3\u003c/a\u003e, \u003ca href=\"#Page_21\"\u003e21\u003c/a\u003e, \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e, \u003ca href=\"#Page_174\"\u003e174\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_227\"\u003e227\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ abstract: \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ affine: \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ descriptive: \u003ca href=\"#Page_210\"\u003e210\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ Euclidean, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ natural: \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ non-Euclidean: \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_396\"\u003e396\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ projective: \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_109\"\u003e109\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003e\u003ci\u003eGestalt-psychologie\u003c/i\u003e: \u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGramophone: \u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e, \u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGravitation: \u003ca href=\"#Page_17\"\u003e17\u003c/a\u003e, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e, \u003ca href=\"#Page_60\"\u003e60\u003c/a\u003e, \u003ca href=\"#Page_74\"\u003e74\u003c/a\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e, \u003ca href=\"#Page_85\"\u003e85\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_89\"\u003e89\u003c/a\u003e, \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e, \u003ca href=\"#Page_176\"\u003e176\u003c/a\u003e, \u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e, \u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e, \u003ca href=\"#Page_326\"\u003e326\u003c/a\u003e, \u003ca href=\"#Page_371\"\u003e371\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eGroup, ideal: \u003ca href=\"#Page_212\"\u003e212\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e: \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e, \u003ca href=\"#Page_33\"\u003e33\u003c/a\u003e, \u003ca href=\"#Page_45\"\u003e45\u003c/a\u003e, \u003ca href=\"#Page_317\"\u003e317\u003c/a\u003e, \u003ca href=\"#Page_330\"\u003e330\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e, \u003ca href=\"#Page_364\"\u003e364\u003c/a\u003e, \u003ca href=\"#Page_365\"\u003e365\u003c/a\u003e, \u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHabit: \u003ca href=\"#Page_184\"\u003e184\u003c/a\u003e, \u003ca href=\"#Page_191\"\u003e191\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHaldane, J. B.: \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHallucinations: \u003ca href=\"#Page_222\"\u003e222\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHand-eye co-ordination: \u003ca href=\"#Page_143\"\u003e143\u003c/a\u003e, \u003ca href=\"#Page_147\"\u003e147\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_185\"\u003e185\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHausdorff: \u003ca href=\"#Page_296\"\u003e296\u003c/a\u003e, \u003ca href=\"#Page_297\"\u003e297\u003c/a\u003e, \u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e, \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHearing: \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHegel: \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e, \u003ca href=\"#Page_199\"\u003e199\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHeisenberg: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_27\"\u003e27\u003c/a\u003e, \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_43\"\u003e43\u003c/a\u003e, \u003ca href=\"#Page_45\"\u003e45\u003c/a\u003e, \u003ca href=\"#Page_246\"\u003e246\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_283\"\u003e283\u003c/a\u003e, \u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e, \u003ca href=\"#Page_322\"\u003e322\u003c/a\u003e, \u003ca href=\"#Page_326\"\u003e326\u003c/a\u003e, \u003ca href=\"#Page_332\"\u003e332\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e, \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003cspan class=\"pagenum\" id=\"Page_405\"\u003e[Pg 405]\u003c/span\u003e\u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e, \u003ca href=\"#Page_361\"\u003e361\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHelium: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_26\"\u003e26\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHelmholtz: \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHeraclitus: \u003ca href=\"#Page_62\"\u003e62\u003c/a\u003e, \u003ca href=\"#Page_176\"\u003e176\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHertz: \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003eHolt: \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHume: \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e, \u003ca href=\"#Page_190\"\u003e190\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHydrogen: \u003ca href=\"#Page_25\"\u003e25\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ spectrum: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eHypothesis: \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eIdealism: \u003ca href=\"#Page_7\"\u003e7\u003c/a\u003e, \u003ca href=\"#Page_28\"\u003e28\u003c/a\u003e, \u003ca href=\"#Page_215\"\u003e215\u003c/a\u003e, \u003ca href=\"#Page_382\"\u003e382\u003c/a\u003e, \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e, \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIdealists: \u003ca href=\"#Page_215\"\u003e215\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eImpenetrable: \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eImplication, strict: \u003ca href=\"#Page_199\"\u003e199\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIndestructible: \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIndistinguishability: \u003ca href=\"#Page_282\"\u003e282\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIndividuals: \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInduction: \u003ca href=\"#Page_153\"\u003e153\u003c/a\u003e, \u003ca href=\"#Page_154\"\u003e154\u003c/a\u003e, \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_175\"\u003e175\u003c/a\u003e, \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_230\"\u003e230\u003c/a\u003e, \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ mathematical: \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInference: \u003ca href=\"#Page_150\"\u003e150\u003c/a\u003e, \u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e, \u003ca href=\"#Page_190\"\u003e190\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physiological: \u003ca href=\"#Page_150\"\u003e150\u003c/a\u003e, \u003ca href=\"#Page_190\"\u003e190\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInfinite divisibility: \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInfinitesimals: \u003ca href=\"#Page_104\"\u003e104\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInstants: \u003ca href=\"#Page_294\"\u003e294\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIntegers, finite: \u003ca href=\"#Page_3\"\u003e3\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInterference: \u003ca href=\"#Page_352\"\u003e352\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003eInterpenetration, logical: \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ spatio-temporal: \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInterpretation: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_88\"\u003e88\u003c/a\u003e, \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e, \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_188\"\u003e188\u003c/a\u003e, \u003ca href=\"#Page_225\"\u003e225\u003c/a\u003e, \u003ca href=\"#Page_288\"\u003e288\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInterval: \u003ca href=\"#Page_51\"\u003e51\u003c/a\u003e, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_69\"\u003e69\u003c/a\u003e, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e, \u003ca href=\"#Page_88\"\u003e88\u003c/a\u003e, \u003ca href=\"#Page_110\"\u003e110\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e, \u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_314\"\u003e314\u003c/a\u003e, \u003ca href=\"#Page_327\"\u003e327\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_330\"\u003e330\u003c/a\u003e, \u003ca href=\"#Page_354\"\u003e354\u003c/a\u003e, \u003ca href=\"#Page_367\"\u003e367\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_396\"\u003e396\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e, \u003ca href=\"#Page_402\"\u003e402\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ discontinuous: \u003ca href=\"#Page_367\"\u003e367\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ time-like; space-like: \u003ca href=\"#Page_51\"\u003e51\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eIntrospection: \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eInvariants: \u003ca href=\"#Page_82\"\u003e82\u003c/a\u003e, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eJeans: \u003ca href=\"#Page_31\"\u003e31\u003c/a\u003e, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e, \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eJeremiah: \u003ca href=\"#Page_240\"\u003e240\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eJohnson, Dr.: \u003ca href=\"#Page_136\"\u003e136\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eJordan, P.: \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e, \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eKant: \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e, \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e, \u003ca href=\"#Page_78\"\u003e78\u003c/a\u003e, \u003ca href=\"#Page_157\"\u003e157\u003c/a\u003e, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e, \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e, \u003ca href=\"#Page_174\"\u003e174\u003c/a\u003e, \u003ca href=\"#Page_175\"\u003e175\u003c/a\u003e, \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKeynes: \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e, \u003ca href=\"#Page_399\"\u003e399\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKinematics: \u003ca href=\"#Page_361\"\u003e361\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKing, L. V.: \u003ca href=\"#Page_47\"\u003e47\u003c/a\u003e, \u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKirchoff: \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKnowledge: \u003ca href=\"#Page_174\"\u003e174\u003c/a\u003e, \u003ca href=\"#Page_178\"\u003e178\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ subjective factor in:\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_334\"\u003e334\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKöhler: \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eKropotkin: \u003ca href=\"#Page_74\"\u003e74\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eLanguage: \u003ca href=\"#Page_152\"\u003e152\u003c/a\u003e, \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLaws, causal: \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLaws: differential: \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ general: \u003ca href=\"#Page_229\"\u003e229\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ of contradiction: \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ of motion, first: \u003ca href=\"#Page_323\"\u003e323\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ simple: \u003ca href=\"#Page_232\"\u003e232\u003c/a\u003e, \u003ca href=\"#Page_236\"\u003e236\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ scientific: \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ statistical: \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_235\"\u003e235\u003c/a\u003e, \u003ca href=\"#Page_236\"\u003e236\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLearned reactions: \u003ca href=\"#Page_154\"\u003e154\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLeibniz: \u003ca href=\"#Page_8\"\u003e8\u003c/a\u003e, \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e, \u003ca href=\"#Page_17\"\u003e17\u003c/a\u003e, \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_157\"\u003e157\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_200\"\u003e200\u003c/a\u003e, \u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e, \u003ca href=\"#Page_328\"\u003e328\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLeonardo: \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLewis, G. I.: \u003ca href=\"#Page_199\"\u003e199\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003e„ G. N.: \u003ca href=\"#Page_125\"\u003e125\u003c/a\u003e, \u003ca href=\"#Page_126\"\u003e126\u003c/a\u003e, \u003ca href=\"#Page_134\"\u003e134\u003c/a\u003e, \u003ca href=\"#Page_164\"\u003e164\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLight: \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_155\"\u003e155\u003c/a\u003e, \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e, \u003ca href=\"#Page_216\"\u003e216\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_235\"\u003e235\u003c/a\u003e, \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_267\"\u003e267\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e, \u003ca href=\"#Page_314\"\u003e314\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_317\"\u003e317\u003c/a\u003e, \u003ca href=\"#Page_328\"\u003e328\u003c/a\u003e, \u003ca href=\"#Page_339\"\u003e339\u003c/a\u003e, \u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e, \u003ca href=\"#Page_351\"\u003e351\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_364\"\u003e364\u003c/a\u003e, \u003ca href=\"#Page_369\"\u003e369\u003c/a\u003e, \u003ca href=\"#Page_373\"\u003e373\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ frequency of: \u003ca href=\"#Page_353\"\u003e353\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLight-quantum: \u003ca href=\"#Page_124\"\u003e124\u003c/a\u003e, \u003ca href=\"#Page_268\"\u003e268\u003c/a\u003e, \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_318\"\u003e318\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLight-ray: \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLimit: \u003ca href=\"#Page_309\"\u003e309\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLimitation of variety: \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLimits: \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLine: \u003ca href=\"#Page_306\"\u003e306\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ spectra: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ \"straight\": \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLobatchevsky: \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLocation: \u003ca href=\"#Page_258\"\u003e258\u003c/a\u003e, \u003ca href=\"#Page_320\"\u003e320\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLocke: \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e, \u003ca href=\"#Page_257\"\u003e257\u003c/a\u003e, \u003ca href=\"#Page_339\"\u003e339\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLogic: \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e, \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e, \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e, \u003ca href=\"#Page_247\"\u003e247\u003c/a\u003e, \u003ca href=\"#Page_250\"\u003e250\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ mathematical: \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e, \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e, \u003ca href=\"#Page_350\"\u003e350\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ traditional: \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLogical property: \u003ca href=\"#Page_251\"\u003e251\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ symbols: \u003ca href=\"#Page_289\"\u003e289\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eLorentz: \u003ca href=\"#Page_124\"\u003e124\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ transformation: \u003ca href=\"#Page_49\"\u003e49\u003c/a\u003e, \u003ca href=\"#Page_54\"\u003e54\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eMach: \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_19\"\u003e19\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMass: \u003ca href=\"#Page_53\"\u003e53\u003c/a\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_122\"\u003e122\u003c/a\u003e, \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e, \u003ca href=\"#Page_284\"\u003e284\u003c/a\u003e, \u003ca href=\"#Page_318\"\u003e318\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ gravitational: \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ invariant: \u003ca href=\"#Page_122\"\u003e122\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ relative: \u003ca href=\"#Page_122\"\u003e122\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMaterial energy-tensor: \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMaterialism: \u003ca href=\"#Page_7\"\u003e7\u003c/a\u003e, \u003ca href=\"#Page_162\"\u003e162\u003c/a\u003e, \u003ca href=\"#Page_383\"\u003e383\u003c/a\u003e, \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e, \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMaterialists: \u003ca href=\"#Page_215\"\u003e215\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMathematics: \u003ca href=\"#Page_174\"\u003e174\u003c/a\u003e, \u003ca href=\"#Page_176\"\u003e176\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ applied: \u003ca href=\"#Page_5\"\u003e5\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMatrices: \u003ca href=\"#Page_44\"\u003e44\u003c/a\u003e, \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMatter: \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_134\"\u003e134\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_182\"\u003e182\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e, \u003ca href=\"#Page_215\"\u003e215\u003c/a\u003e, \u003ca href=\"#Page_243\"\u003e243\u003c/a\u003e, \u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e, \u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_372\"\u003e372\u003c/a\u003e, \u003ca href=\"#Page_379\"\u003e379\u003c/a\u003e, \u003ca href=\"#Page_384\"\u003e384\u003c/a\u003e, \u003ca href=\"#Page_392\"\u003e392\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ atomicity of: \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ indestructibility of: \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ unperceived: \u003ca href=\"#Page_232\"\u003e232\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eMatters of fact: \u003ca href=\"#Page_176\"\u003e176\u003c/a\u003e, \u003ca href=\"#Page_178\"\u003e178\u003c/a\u003e\u003cspan class=\"pagenum\" id=\"Page_406\"\u003e[Pg 406]\u003c/span\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMaxwell, Clerk: \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMaxwell\u0027s equations: \u003ca href=\"#Page_49\"\u003e49\u003c/a\u003e, \u003ca href=\"#Page_54\"\u003e54\u003c/a\u003e, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_121\"\u003e121\u003c/a\u003e, \u003ca href=\"#Page_258\"\u003e258\u003c/a\u003e, \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMeaning: \u003ca href=\"#Page_240\"\u003e240\u003c/a\u003e, \u003ca href=\"#Page_243\"\u003e243\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMeasurement: \u003ca href=\"#Page_90\"\u003e90\u003c/a\u003e, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e, \u003ca href=\"#Page_109\"\u003e109\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_337\"\u003e337\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMemory: \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e, \u003ca href=\"#Page_266\"\u003e266\u003c/a\u003e, \u003ca href=\"#Page_267\"\u003e267\u003c/a\u003e, \u003ca href=\"#Page_389\"\u003e389\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMendelians: \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMenger, Karl: \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMental states: \u003ca href=\"#Page_320\"\u003e320\u003c/a\u003e, \u003ca href=\"#Page_322\"\u003e322\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMercury: \u003ca href=\"#Page_40\"\u003e40\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMetre: \u003ca href=\"#Page_91\"\u003e91\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMill, J. S.: \u003ca href=\"#Page_21\"\u003e21\u003c/a\u003e, \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e, \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMind: \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e, \u003ca href=\"#Page_336\"\u003e336\u003c/a\u003e, \u003ca href=\"#Page_382\"\u003e382\u003c/a\u003e, \u003ca href=\"#Page_387\"\u003e387\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMinkowski: \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMnemic phenomena: \u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e, \u003ca href=\"#Page_315\"\u003e315\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eModality: \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMomentum: \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e, \u003ca href=\"#Page_162\"\u003e162\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMonad: \u003ca href=\"#Page_157\"\u003e157\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMoore, G. E.: \u003ca href=\"#Page_210\"\u003e210\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMotion: \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_246\"\u003e246\u003c/a\u003e, \u003ca href=\"#Page_278\"\u003e278\u003c/a\u003e, \u003ca href=\"#Page_317\"\u003e317\u003c/a\u003e, \u003ca href=\"#Page_326\"\u003e326\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ absolute: \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_358\"\u003e358\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ continuous: \u003ca href=\"#Page_380\"\u003e380\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ percept of: \u003ca href=\"#Page_279\"\u003e279\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ periodic: \u003ca href=\"#Page_344\"\u003e344\u003c/a\u003e, \u003ca href=\"#Page_348\"\u003e348\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ relative: \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e, \u003ca href=\"#Page_378\"\u003e378\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMotor habits: \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eMuscular physics: \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eNaive realism: \u003ca href=\"#Page_155\"\u003e155\u003c/a\u003e, \u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e, \u003ca href=\"#Page_262\"\u003e262\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNames: \u003ca href=\"#Page_152\"\u003e152\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNapoleon: \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNecessary and contingent propositions: \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNeighbourhoods: \u003ca href=\"#Page_295\"\u003e295\u003c/a\u003e ff., \u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e, \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNeutral monism: \u003ca href=\"#Page_382\"\u003e382\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ stuff: \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNewman, M. H. A.: \u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNewton: \u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e, \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_358\"\u003e358\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNicod: \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNisbet: \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNucleus: \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eNumbers: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ ordinal: \u003ca href=\"#Page_250\"\u003e250\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ real: \u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eObject: \u003ca href=\"#Page_142\"\u003e142\u003c/a\u003e, \u003ca href=\"#Page_181\"\u003e181\u003c/a\u003e, \u003ca href=\"#Page_182\"\u003e182\u003c/a\u003e, \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e, \u003ca href=\"#Page_204\"\u003e204\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_207\"\u003e207\u003c/a\u003e, \u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eObjectivity: \u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e, \u003ca href=\"#Page_222\"\u003e222\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eObjects, external: \u003ca href=\"#Page_198\"\u003e198\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOccam: \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOdour: \u003ca href=\"#Page_133\"\u003e133\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOntology: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOrbits, elliptical: \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOrder: \u003ca href=\"#Page_109\"\u003e109\u003c/a\u003e, \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ macroscopic: \u003ca href=\"#Page_304\"\u003e304\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ microscopic: \u003ca href=\"#Page_305\"\u003e305\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eOverlapping: \u003ca href=\"#Page_294\"\u003e294\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003ePaneth: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eParallel displacement: \u003ca href=\"#Page_96\"\u003e96\u003c/a\u003e, \u003ca href=\"#Page_106\"\u003e106\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eParallelism, psycho-cerebral: \u003ca href=\"#Page_391\"\u003e391\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eParallelogram: \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e, \u003ca href=\"#Page_104\"\u003e104\u003c/a\u003e, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eParticulars: \u003ca href=\"#Page_275\"\u003e275\u003c/a\u003e, \u003ca href=\"#Page_319\"\u003e319\u003c/a\u003e, \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePasch: \u003ca href=\"#Page_210\"\u003e210\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePeano: \u003ca href=\"#Page_3\"\u003e3\u003c/a\u003e, \u003ca href=\"#Page_240\"\u003e240\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePerception: \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e, \u003ca href=\"#Page_164\"\u003e164\u003c/a\u003e, \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e, \u003ca href=\"#Page_175\"\u003e175\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_177\"\u003e177\u003c/a\u003e, \u003ca href=\"#Page_178\"\u003e178\u003c/a\u003e, \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e, \u003ca href=\"#Page_187\"\u003e187\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_188\"\u003e188\u003c/a\u003e, \u003ca href=\"#Page_189\"\u003e189\u003c/a\u003e, \u003ca href=\"#Page_215\"\u003e215\u003c/a\u003e, \u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_219\"\u003e219\u003c/a\u003e, \u003ca href=\"#Page_247\"\u003e247\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ causal theory of: \u003ca href=\"#Page_181\"\u003e181\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_193\"\u003e193\u003c/a\u003e, \u003ca href=\"#Page_197\"\u003e197\u003c/a\u003e, \u003ca href=\"#Page_320\"\u003e320\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ causes of: \u003ca href=\"#Page_8\"\u003e8\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ cognitive efficacy of:\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_262\"\u003e262\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePercepts: \u003ca href=\"#Page_218\"\u003e218\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ \"ideal\": \u003ca href=\"#Page_211\"\u003e211\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physical status of: \u003ca href=\"#Page_257\"\u003e257\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ structure of: \u003ca href=\"#Page_281\"\u003e281\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePercipients, \"ideal\": \u003ca href=\"#Page_210\"\u003e210\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePeriodicity: \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e, \u003ca href=\"#Page_363\"\u003e363\u003c/a\u003e, \u003ca href=\"#Page_365\"\u003e365\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePeriodic process: \u003ca href=\"#Page_267\"\u003e267\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e, \u003ca href=\"#Page_348\"\u003e348\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_208\"\u003e208\u003c/a\u003e, \u003ca href=\"#Page_216\"\u003e216\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePerspective: \u003ca href=\"#Page_198\"\u003e198\u003c/a\u003e, \u003ca href=\"#Page_208\"\u003e208\u003c/a\u003e, \u003ca href=\"#Page_216\"\u003e216\u003c/a\u003e, \u003ca href=\"#Page_221\"\u003e221\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_323\"\u003e323\u003c/a\u003e, \u003ca href=\"#Page_333\"\u003e333\u003c/a\u003e, \u003ca href=\"#Page_334\"\u003e334\u003c/a\u003e, \u003ca href=\"#Page_348\"\u003e348\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePeters: \u003ca href=\"#Page_24\"\u003e324\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhenomenalism: \u003ca href=\"#Page_209\"\u003e209\u003c/a\u003e, \u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e, \u003ca href=\"#Page_333\"\u003e333\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e, \u003ca href=\"#Page_399\"\u003e399\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhilosophy: \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhoto-electric effect: \u003ca href=\"#Page_31\"\u003e31\u003c/a\u003e, \u003ca href=\"#Page_329\"\u003e329\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhotographs: \u003ca href=\"#Page_334\"\u003e334\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhoton: \u003ca href=\"#Page_126\"\u003e126\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhysical time: \u003ca href=\"#Page_254\"\u003e254\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhysics, abstractness of: \u003ca href=\"#Page_130\"\u003e130\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ and perception: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ and pure mathematics: \u003ca href=\"#Page_1\"\u003e1\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePhysiology: \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePlace: \u003ca href=\"#Page_217\"\u003e217\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePlanck: \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePlanck\u0027s constant. See \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-the-analysis-of-matter-21.png\" alt=\"\" data-tex=\"\\(h\\)\"\u003e\"\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePluralism: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePoincaré: \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e, \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePoints: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e, \u003ca href=\"#Page_299\"\u003e299\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e, \u003ca href=\"#Page_348\"\u003e348\u003c/a\u003e, \u003ca href=\"#Page_376\"\u003e376\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ \"connected\": \u003ca href=\"#Page_304\"\u003e304\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ material: \u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePossibility: \u003ca href=\"#Page_170\"\u003e170\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePostulates: \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePredicate: \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e, \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePrepositions: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePrimitive religion: \u003ca href=\"#Page_149\"\u003e149\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePrinciple of identification: \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eProbability: \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_170\"\u003e170\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eProgression: \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eProper names: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePropositional functions: \u003ca href=\"#Page_170\"\u003e170\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePropositions, general: \u003ca href=\"#Page_185\"\u003e185\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eProtons: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e, \u003ca href=\"#Page_153\"\u003e153\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e\u003cspan class=\"pagenum\" id=\"Page_407\"\u003e[Pg 407]\u003c/span\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePsychologist\u0027s fallacy: \u003ca href=\"#Page_179\"\u003e179\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePsychology: \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003ePythagoras: \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e, \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e, \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_235\"\u003e235\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eQualitative cycles: \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ differences: \u003ca href=\"#Page_346\"\u003e346\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ series: \u003ca href=\"#Page_9\"\u003e9\u003c/a\u003e, \u003ca href=\"#Page_115\"\u003e115\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ similarity: \u003ca href=\"#Page_120\"\u003e120\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eQualities: \u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ primary, secondary:\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eQuality: \u003ca href=\"#Page_288\"\u003e288\u003c/a\u003e, \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e, \u003ca href=\"#Page_391\"\u003e391\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ intrinsic: \u003ca href=\"#Page_264\"\u003e264\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eQuanta: \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_26\"\u003e26\u003c/a\u003e, \u003ca href=\"#Page_30\"\u003e30\u003c/a\u003e, \u003ca href=\"#Page_329\"\u003e329\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eQuantities, extensive: \u003ca href=\"#Page_116\"\u003e116\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ intensive: \u003ca href=\"#Page_116\"\u003e116\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eQuantum: \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e, \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e, \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_234\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_268\"\u003e268\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_326\"\u003e326\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e, \u003ca href=\"#Page_345\"\u003e345\u003c/a\u003e, \u003ca href=\"#Page_352\"\u003e352\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e, \u003ca href=\"#Page_366\"\u003e366\u003c/a\u003e, \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_393\"\u003e393\u003c/a\u003e, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e, \u003ca href=\"#Page_402\"\u003e402\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ laws: \u003ca href=\"#Page_331\"\u003e331\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ number: \u003ca href=\"#Page_33\"\u003e33\u003c/a\u003e, \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ phenomena: \u003ca href=\"#Page_322\"\u003e322\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRadiations: \u003ca href=\"#Page_283\"\u003e283\u003c/a\u003e, \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e, \u003ca href=\"#Page_326\"\u003e326\u003c/a\u003e, \u003ca href=\"#Page_331\"\u003e331\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRadio-activity: \u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e, \u003ca href=\"#Page_393\"\u003e393\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRamsey, F. P.: \u003ca href=\"#Page_299\"\u003e299\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRationalists: \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRealism, naive: \u003ca href=\"#Page_149\"\u003e149\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRecognition: \u003ca href=\"#Page_151\"\u003e151\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eReflexes: \u003ca href=\"#Page_143\"\u003e143\u003c/a\u003e, \u003ca href=\"#Page_184\"\u003e184\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRegion: \u003ca href=\"#Page_302\"\u003e302\u003c/a\u003e, \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eReichenbach: \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRelation-number: \u003ca href=\"#Page_250\"\u003e250\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRelations: \u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e, \u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ atomic; molecular: \u003ca href=\"#Page_116\"\u003e116\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRelativity: \u003ca href=\"#Page_337\"\u003e337\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ special theory of: \u003ca href=\"#Page_39\"\u003e39\u003c/a\u003e, \u003ca href=\"#Page_48\"\u003e48\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ theory of: \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e, \u003ca href=\"#Page_17\"\u003e17\u003c/a\u003e, \u003ca href=\"#Page_53\"\u003e53\u003c/a\u003e, \u003ca href=\"#Page_100\"\u003e100\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRepetition: \u003ca href=\"#Page_346\"\u003e346\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eReversibility: \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRhythms: \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e, \u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e, \u003ca href=\"#Page_363\"\u003e363\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e, \u003ca href=\"#Page_374\"\u003e374\u003c/a\u003e, \u003ca href=\"#Page_378\"\u003e378\u003c/a\u003e, \u003ca href=\"#Page_402\"\u003e402\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRiemann: \u003ca href=\"#Page_21\"\u003e21\u003c/a\u003e, \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRobb, A. A.: \u003ca href=\"#Page_313\"\u003e313\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRotation, absolute: \u003ca href=\"#Page_17\"\u003e17\u003c/a\u003e, \u003ca href=\"#Page_358\"\u003e358\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRutherford: \u003ca href=\"#Page_285\"\u003e285\u003c/a\u003e, \u003ca href=\"#Page_325\"\u003e325\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eRydberg\u0027s constant: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e, \u003ca href=\"#Page_195\"\u003e195\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eSchrödinger: \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSemi-similarity: \u003ca href=\"#Page_254\"\u003e254\u003c/a\u003e, \u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e, \u003ca href=\"#Page_269\"\u003e269\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSensation: \u003ca href=\"#Page_189\"\u003e189\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSense organs: \u003ca href=\"#Page_259\"\u003e259\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSeparation: \u003ca href=\"#Page_118\"\u003e118\u003c/a\u003e, \u003ca href=\"#Page_377\"\u003e377\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eShakespeare: \u003ca href=\"#Page_392\"\u003e392\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eShapes: \u003ca href=\"#Page_202\"\u003e202\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSheffer, H. M.: \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e, \u003ca href=\"#Page_299\"\u003e299\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSight: \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physics: \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSimilar: \u003ca href=\"#Page_265\"\u003e265\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSimilarity: \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e, \u003ca href=\"#Page_120\"\u003e120\u003c/a\u003e, \u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSimultaneity: \u003ca href=\"#Page_55\"\u003e52\u003c/a\u003e, \u003ca href=\"#Page_63\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_278\"\u003e278\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSkeleton, causal: \u003ca href=\"#Page_391\"\u003e391\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSmell: \u003ca href=\"#Page_165\"\u003e165\u003c/a\u003e, \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSolipsism: \u003ca href=\"#Page_8\"\u003e8\u003c/a\u003e, \u003ca href=\"#Page_28\"\u003e28\u003c/a\u003e, \u003ca href=\"#Page_158\"\u003e158\u003c/a\u003e, \u003ca href=\"#Page_325\"\u003e325\u003c/a\u003e, \u003ca href=\"#Page_398\"\u003e398\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSommerfeld: \u003ca href=\"#Page_34\"\u003e34\u003c/a\u003e, \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e, \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e, \u003ca href=\"#Page_39\"\u003e39\u003c/a\u003e, \u003ca href=\"#Page_40\"\u003e40\u003c/a\u003e, \u003ca href=\"#Page_41\"\u003e41\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e, \u003ca href=\"#Page_361\"\u003e361\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSound: \u003ca href=\"#Page_133\"\u003e133\u003c/a\u003e, \u003ca href=\"#Page_165\"\u003e165\u003c/a\u003e, \u003ca href=\"#Page_209\"\u003e209\u003c/a\u003e, \u003ca href=\"#Page_253\"\u003e253\u003c/a\u003e, \u003ca href=\"#Page_261\"\u003e261\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_267\"\u003e267\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpace: \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_143\"\u003e143\u003c/a\u003e, \u003ca href=\"#Page_198\"\u003e198\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ absolute: \u003ca href=\"#Page_321\"\u003e321\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ empty: \u003ca href=\"#Page_324\"\u003e324\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ Euclidean: \u003ca href=\"#Page_75\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_298\"\u003e298\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ metrical: \u003ca href=\"#Page_296\"\u003e296\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ non-Euclidean: \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ perceptual: \u003ca href=\"#Page_252\"\u003e252\u003c/a\u003e, \u003ca href=\"#Page_335\"\u003e335\u003c/a\u003e, \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_383\"\u003e386\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physical: \u003ca href=\"#Page_252\"\u003e252\u003c/a\u003e, \u003ca href=\"#Page_336\"\u003e336\u003c/a\u003e, \u003ca href=\"#Page_383\"\u003e383\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ tactual; visual: \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e, \u003ca href=\"#Page_253\"\u003e253\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ topological: \u003ca href=\"#Page_296\"\u003e296\u003c/a\u003e, \u003ca href=\"#Page_312\"\u003e312\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpace-time: \u003ca href=\"#Page_49\"\u003e49\u003c/a\u003e, \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_340\"\u003e340\u003c/a\u003e, \u003ca href=\"#Page_341\"\u003e341\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ genesis of: \u003ca href=\"#Page_376\"\u003e376\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ order: \u003ca href=\"#Page_303\"\u003e303\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physical and perceptual: \u003ca href=\"#Page_333\"\u003e333\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpace and time, absolute: \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ „ common: \u003ca href=\"#Page_207\"\u003e207\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpatial relations:\u003ca href=\"#Page_348\"\u003e348\u003c/a\u003e, \u003ca href=\"#Page_359\"\u003e359\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpecific heat: \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_195\"\u003e195\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpecious present: \u003ca href=\"#Page_278\"\u003e278\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpeech, parts of: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSpinoza: \u003ca href=\"#Page_157\"\u003e157\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"ifrst\"\u003eStatistical averages: \u003ca href=\"#Page_191\"\u003e191\u003c/a\u003e, \u003ca href=\"#Page_393\"\u003e393\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ fact: \u003ca href=\"#Page_339\"\u003e339\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eStimulus: \u003ca href=\"#Page_227\"\u003e227\u003c/a\u003e, \u003ca href=\"#Page_337\"\u003e337\u003c/a\u003e, \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e, \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e, \u003ca href=\"#Page_390\"\u003e390\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_400\"\u003e400\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eStraightness: \u003ca href=\"#Page_307\"\u003e307\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eStructure: \u003ca href=\"#Page_227\"\u003e227\u003c/a\u003e, \u003ca href=\"#Page_249\"\u003e249\u003c/a\u003e, \u003ca href=\"#Page_276\"\u003e276\u003c/a\u003e, \u003ca href=\"#Page_282\"\u003e282\u003c/a\u003e, \u003ca href=\"#Page_285\"\u003e285\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_335\"\u003e335\u003c/a\u003e, \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e, \u003ca href=\"#Page_340\"\u003e340\u003c/a\u003e, \u003ca href=\"#Page_386\"\u003e386\u003c/a\u003e, \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e, \u003ca href=\"#Page_390\"\u003e390\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_400\"\u003e400\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eStuff: \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_388\"\u003e388\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSubject: \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e, \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSubject-predicate logic: \u003ca href=\"#Page_287\"\u003e287\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSubjectivity: \u003ca href=\"#Page_223\"\u003e223\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ cerebral (or psychological): \u003ca href=\"#Page_225\"\u003e225\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physical: \u003ca href=\"#Page_224\"\u003e224\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physiological (or sensory): \u003ca href=\"#Page_224\"\u003e224\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSubstance: \u003ca href=\"#Page_121\"\u003e121\u003c/a\u003e, \u003ca href=\"#Page_151\"\u003e151\u003c/a\u003e, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e, \u003ca href=\"#Page_231\"\u003e231\u003c/a\u003e, \u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_277\"\u003e277\u003c/a\u003e, \u003ca href=\"#Page_283\"\u003e283\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_318\"\u003e318\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_401\"\u003e401\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ indestructible: \u003ca href=\"#Page_238\"\u003e238\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ permanent: \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSuddenness: \u003ca href=\"#Page_347\"\u003e347\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSurface: \u003ca href=\"#Page_311\"\u003e311\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSurprise: \u003ca href=\"#Page_185\"\u003e185\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSymbols: \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e\u003cspan class=\"pagenum\" id=\"Page_408\"\u003e[Pg 408]\u003c/span\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eSyntax: \u003ca href=\"#Page_151\"\u003e151\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"indx\"\u003eTautologies: \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTensor equation: \u003ca href=\"#Page_233\"\u003e233\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTensors: \u003ca href=\"#Page_18\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_63\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_96\"\u003e96\u003c/a\u003e, \u003ca href=\"#Page_136\"\u003e136\u003c/a\u003e, \u003ca href=\"#Page_396\"\u003e396\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTestimony: \u003ca href=\"#Page_193\"\u003e193\u003c/a\u003e, \u003ca href=\"#Page_203\"\u003e203\u003c/a\u003e, \u003ca href=\"#Page_206\"\u003e206\u003c/a\u003e, \u003ca href=\"#Page_399\"\u003e399\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTheory: \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e, \u003ca href=\"#Page_195\"\u003e195\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eThermodynamics, second law of:\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eThing, ideal: \u003ca href=\"#Page_213\"\u003e213\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003eThomas, St.: \u003ca href=\"#Page_55\"\u003e55\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTime: \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_208\"\u003e208\u003c/a\u003e, \u003ca href=\"#Page_253\"\u003e253\u003c/a\u003e, \u003ca href=\"#Page_286\"\u003e286\u003c/a\u003e, \u003ca href=\"#Page_381\"\u003e381\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ perceptual: \u003ca href=\"#Page_338\"\u003e338\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ proper: \u003ca href=\"#Page_350\"\u003e350\u003c/a\u003e, \u003ca href=\"#Page_357\"\u003e357\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ psychological: \u003ca href=\"#Page_254\"\u003e254\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTopography: \u003ca href=\"#Page_177\"\u003e177\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTouch: \u003ca href=\"#Page_260\"\u003e260\u003c/a\u003e, \u003ca href=\"#Page_316\"\u003e316\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ physics: \u003ca href=\"#Page_160\"\u003e160\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTransactions: \u003ca href=\"#Page_355\"\u003e355\u003c/a\u003e, \u003ca href=\"#Page_356\"\u003e356\u003c/a\u003e, \u003ca href=\"#Page_360\"\u003e360\u003c/a\u003e, \u003ca href=\"#Page_362\"\u003e362\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_363\"\u003e363\u003c/a\u003e, \u003ca href=\"#Page_368\"\u003e368\u003c/a\u003e, \u003ca href=\"#Page_372\"\u003e372\u003c/a\u003e, \u003ca href=\"#Page_378\"\u003e378\u003c/a\u003e, \u003ca href=\"#Page_394\"\u003e394\u003c/a\u003e, \u003ca href=\"#Page_402\"\u003e402\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTransitiveness: \u003ca href=\"#Page_251\"\u003e251\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTruth: \u003ca href=\"#Page_8\"\u003e8\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eTruth-functions: \u003ca href=\"#Page_243\"\u003e243\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eUnconscious: \u003ca href=\"#Page_385\"\u003e385\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eUniversals: \u003ca href=\"#Page_152\"\u003e152\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eUrysohn: \u003ca href=\"#Page_297\"\u003e297\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eVagueness: \u003ca href=\"#Page_220\"\u003e220\u003c/a\u003e, \u003ca href=\"#Page_223\"\u003e223\u003c/a\u003e, \u003ca href=\"#Page_224\"\u003e224\u003c/a\u003e, \u003ca href=\"#Page_280\"\u003e280\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVariables: \u003ca href=\"#Page_172\"\u003e172\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVariables, separation of: \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e, \u003ca href=\"#Page_393\"\u003e393\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVelocities, composition of: \u003ca href=\"#Page_53\"\u003e53\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVelocity: \u003ca href=\"#Page_323\"\u003e323\u003c/a\u003e, \u003ca href=\"#Page_374\"\u003e374\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVerbs: \u003ca href=\"#Page_242\"\u003e242\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eVietoris, L.: \u003ca href=\"#Page_296\"\u003e296\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003e\u003ci\u003eVis viva\u003c/i\u003e: \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eWatson, T. B.: \u003ca href=\"#Page_154\"\u003e154\u003c/a\u003e, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWaves, electromagnetic: \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWeierstrass: \u003ca href=\"#Page_3\"\u003e3\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWeight: \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWells: \u003ca href=\"#Page_226\"\u003e226\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWeyl: \u003ca href=\"#Page_87\"\u003e87\u003c/a\u003e, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e, \u003ca href=\"#Page_95\"\u003e95\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_106\"\u003e106\u003c/a\u003e, \u003ca href=\"#Page_119\"\u003e119\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_328\"\u003e328\u003c/a\u003e, \u003ca href=\"#Page_395\"\u003e395\u003c/a\u003e, \u003ca href=\"#Page_396\"\u003e396\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWhitehead: \u003ca href=\"#Page_6\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e, \u003ca href=\"#Page_57\"\u003e57\u003c/a\u003e, \u003ca href=\"#Page_75\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e, \u003ca href=\"#Page_78\"\u003e78\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_90\"\u003e90\u003c/a\u003e, \u003ca href=\"#Page_130\"\u003e130\u003c/a\u003e, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e, \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e, \u003ca href=\"#Page_158\"\u003e158\u003c/a\u003e, \u003ca href=\"#Page_199\"\u003e199\u003c/a\u003e, \u003ca href=\"#Page_210\"\u003e210\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_230\"\u003e230\u003c/a\u003e, \u003ca href=\"#Page_247\"\u003e247\u003c/a\u003e, \u003ca href=\"#Page_237\"\u003e237\u003c/a\u003e, \u003ca href=\"#Page_290\"\u003e290\u003c/a\u003e, \u003ca href=\"#Page_291\"\u003e291\u003c/a\u003e, \u003ca href=\"#Page_292\"\u003e292\u003c/a\u003e, \u003ca href=\"#Page_294\"\u003e294\u003c/a\u003e,\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e\u003ca href=\"#Page_340\"\u003e340\u003c/a\u003e, \u003ca href=\"#Page_397\"\u003e397\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWilson: \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e, \u003ca href=\"#Page_41\"\u003e41\u003c/a\u003e, \u003ca href=\"#Page_343\"\u003e343\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWittgenstein: \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e, \u003ca href=\"#Page_239\"\u003e239\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWords: \u003ca href=\"#Page_151\"\u003e151\u003c/a\u003e\u003c/li\u003e\n\u003cli class=\"isub1\"\u003e„ written: \u003ca href=\"#Page_241\"\u003e241\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWorld-lines: \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e, \u003ca href=\"#Page_244\"\u003e244\u003c/a\u003e, \u003ca href=\"#Page_253\"\u003e253\u003c/a\u003e, \u003ca href=\"#Page_317\"\u003e317\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eWriting: \u003ca href=\"#Page_241\"\u003e241\u003c/a\u003e\u003c/li\u003e\n\n\n\u003cli class=\"ifrst\"\u003eZeeman effect: \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eZermelo\u0027s axiom: \u003ca href=\"#Page_299\"\u003e299\u003c/a\u003e\u003c/li\u003e\n\n\u003cli class=\"indx\"\u003eZones: \u003ca href=\"#Page_304\"\u003e304\u003c/a\u003e\u003c/li\u003e\n\u003c/ul\u003e\n\n\n\u003cp class=\"nindc space-above2 space-below2\"\u003e\nPRINTED IN GREAT BRITAIN BY\u003cbr\u003e\nBILLING AND SONS, LTD., GUILDFORD AND ESHER\n\u003c/p\u003e\r\n\u003c/article\u003e"}],"SectionSequence":["Back Link","Work Title","Deck","Author","Period","Era","Composition","Date Note","Region","Terra Avita","Terra Avita Region","Modern Country","Original Title","Language","Primary Discipline","Secondary Discipline","Tradition","Full Versions","Core Thesis","Classification","Arguments","Influence","Significance","Evidence Note","Full Text"],"Counts":{"ContextCards":3,"GeoCards":4,"DisciplineCards":2,"Links":11,"Sections":25,"Styles":3,"Scripts":1}}