Our Knowledge of the External World
{"WorkMasterId":5227,"WpPageId":252986,"ParentWpPageId":189742,"Slug":"our-knowledge-of-the-external-world","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/our-knowledge-of-the-external-world/","RelativeUrl":"theos/humanities/philosophy/philosophers/bertrand-russell/our-knowledge-of-the-external-world/","HasFullText":true,"RawHtmlLength":561844,"CleanHtmlLength":505734,"Kicker":"Philosophy Work","Title":"Our Knowledge of the External World","Deck":"Russell develops logical analysis as a method for reconstructing knowledge of the external world and scientific objects.","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":"1914 CE","Url":"","DateText":""}],"DateNote":"Displayed year is the publication year of the Lowell Lectures.","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:epistemology"},{"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 #37090 .","Url":"","Label":"","Kicker":"","Cards":[]},"CoreThesis":["Russell develops logical analysis as a method for reconstructing knowledge of the external world and scientific objects."],"Classification":{"AlternateTitles":"","KeyConcepts":"Our Knowledge of the External World; 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 logical-analytic epistemology."],"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 #37090\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/37090\"\u003eOpen full version\u003c/a\u003e\n \u003c/article\u003e\n \u003c/div\u003e"},{"Kind":"TextSection","Title":"Core Thesis","Paragraphs":["Russell develops logical analysis as a method for reconstructing knowledge of the external world and scientific objects."]},{"Kind":"FieldSection","Title":"Classification","Fields":[{"Label":"Alternate Titles","Value":""},{"Label":"Key Concepts","Value":"Our Knowledge of the External World; 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 logical-analytic epistemology."]},{"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/37090\"\u003eProject Gutenberg eBook #37090\u003c/a\u003e.\u003c/p\u003e\n \u003carticle class=\"dz-philo__full-text-body\"\u003e\r\n\u003cpre\u003e\u003c/pre\u003e\r\n\u003cdiv id=\"tnote\"\u003e\r\n\u003cp class=\"center\"\u003e\u003cb\u003eTranscriber\u0027s Notes:\u003c/b\u003e\u003c/p\u003e\r\n\u003cp class=\"no-indent\"\u003eEvery effort has been made to replicate this text as\r\nfaithfully as possible, including inconsistencies in spelling and hyphenation.\u003c/p\u003e\r\n\u003cp\u003eSome corrections of spelling and punctuation have been made.\r\n\u003cspan class=\"screen\"\u003eThey are marked\r\n\u003cins title=\"transcriber\u0027s note\"\u003elike this\u003c/ins\u003e in the text. The original\r\ntext appears when hovering the cursor over the marked text.\u003c/span\u003e\r\nA \u003ca href=\"#tn-bottom\" class=\"pginternal\"\u003elist of amendments\u003c/a\u003e is at the end of the text.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp class=\"center page-break\" style=\"font-size: x-large;\"\u003eOUR KNOWLEDGE OF THE\u003cbr\u003e\r\nEXTERNAL WORLD\u003cbr\u003e\r\n\u003csmall\u003eAS A FIELD FOR SCIENTIFIC METHOD\u003cbr\u003e\r\nIN PHILOSOPHY\u003c/small\u003e\u003c/p\u003e\r\n\u003cdiv id=\"same-author\"\u003e\r\n\u003cp class=\"center italic\" style=\"font-size: large;\"\u003e\u003ca id=\"advertisement\"\u003eBY THE SAME AUTHOR\u003c/a\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003eINTRODUCTION TO MATHEMATICAL\r\nPHILOSOPHY\u003c/p\u003e\r\n\u003cp\u003e\u003ci\u003eSecond Edition.\u003c/i\u003e \u003cspan class=\"float-right\"\u003eDemy 8vo, 12s. 6d. net.\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003eTHE ANALYSIS OF MIND\u003c/p\u003e\r\n\u003cp\u003e \u003cspan class=\"float-right\"\u003eDemy 8vo, 16s. net.\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003ePRINCIPLES OF SOCIAL\r\nRECONSTRUCTION\u003c/p\u003e\r\n\u003cp\u003e\u003ci\u003eSeventh Impression.\u003c/i\u003e \u003cspan class=\"float-right\"\u003eCr. 8vo, 5s. net; Limp, 3s. 6d. net.\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003eROADS TO FREEDOM: SOCIALISM,\r\nANARCHISM AND SYNDICALISM\u003c/p\u003e\r\n\u003cp\u003e\u003ci\u003eFourth Impression.\u003c/i\u003e \u003cspan class=\"float-right\"\u003eCr. 8vo, 5s. net; Limp, 3s. 6d. net.\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003eTHE PRACTICE AND THEORY OF\r\nBOLSHEVISM\u003c/p\u003e\r\n\u003cp\u003e\u003ci\u003eSecond \u003cins title=\"Impression\"\u003eImpression.\u003c/ins\u003e\u003c/i\u003e \u003cspan class=\"float-right\"\u003eCr. 8vo, 6s. net.\u003c/span\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003ch1\u003eOUR KNOWLEDGE OF\u003cbr\u003e\r\nTHE EXTERNAL WORLD\u003cbr\u003e\r\n\u003csmall\u003eAS A FIELD FOR SCIENTIFIC METHOD\u003cbr\u003e\r\nIN PHILOSOPHY\u003c/small\u003e\u003c/h1\u003e\r\n\u003cp class=\"center\" style=\"line-height: 1.4em;\"\u003e\u003csmall\u003eBY\u003c/small\u003e\u003cbr\u003e\r\nBERTRAND RUSSELL, F.R.S\u003c/p\u003e\r\n\u003cdiv class=\"figcenter\" style=\"width: 150px;\"\u003e\r\n\u003cimg alt=\"\" height=\"151\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-our-knowledge-of-the-external-world-bertrand-russell-logo.png\" width=\"150\" id=\"img_images_logo.png\"\u003e\r\n\u003c/div\u003e\r\n\u003cp class=\"center\" style=\"line-height: 1.4em;\"\u003eLONDON: GEORGE ALLEN \u0026amp; UNWIN LTD\u003cbr\u003e\r\nRUSKIN HOUSE, 40 MUSEUM STREET, W.C. 1\u003c/p\u003e\r\n\u003cp class=\"center page-break\"\u003eFirst published in 1914 by\u003cbr\u003e\r\n\u003cspan class=\"small-caps\"\u003eThe Open Court Publishing Company\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"center\"\u003eReissued by \u003cspan class=\"small-caps\"\u003eGeorge Allen \u0026amp; Unwin Ltd.\u003c/span\u003e\u003cbr\u003e\r\n1922\u003c/p\u003e\r\n\u003ch2\u003e\u003ca class=\"pagenum\" title=\"v\" id=\"Page_v\"\u003e \u003c/a\u003ePREFACE\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eThe\u003c/span\u003e following lectures\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_1\" id=\"FNanchor_1\"\u003e[1]\u003c/a\u003e are an attempt to show, by means\r\nof examples, the nature, capacity, and limitations of the\r\nlogical-analytic method in philosophy. This method, of\r\nwhich the first complete example is to be found in the\r\nwritings of Frege, has gradually, in the course of actual\r\nresearch, increasingly forced itself upon me as something\r\nperfectly definite, capable of embodiment in maxims, and\r\nadequate, in all branches of philosophy, to yield whatever\r\nobjective scientific knowledge it is possible to obtain.\r\nMost of the methods hitherto practised have professed\r\nto lead to more ambitious results than any that logical\r\nanalysis can claim to reach, but unfortunately these results\r\nhave always been such as many competent philosophers\r\nconsidered inadmissible. Regarded merely as hypotheses\r\nand as aids to imagination, the great systems of the past\r\nserve a very useful purpose, and are abundantly worthy\r\nof study. But something different is required if philosophy\r\nis to become a science, and to aim at results independent\r\nof the tastes and temperament of the philosopher\r\nwho advocates them. In what follows, I have endeavoured\r\nto show, however imperfectly, the way by which I believe\r\nthat this \u003ci lang=\"la\"\u003edesideratum\u003c/i\u003e is to be found.\u003c/p\u003e\r\n\u003cp\u003eThe central problem by which I have sought to illustrate\r\nmethod is the problem of the relation between the\r\ncrude data of sense and the space, time, and matter of\r\n\u003ca class=\"pagenum\" title=\"vi\" id=\"Page_vi\"\u003e \u003c/a\u003e\r\nmathematical physics. I have been made aware of the\r\nimportance of this problem by my friend and collaborator\r\nDr Whitehead, to whom are due almost all the differences\r\nbetween the views advocated here and those suggested in\r\n\u003ccite\u003eThe Problems of Philosophy\u003c/cite\u003e.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_2\" id=\"FNanchor_2\"\u003e[2]\u003c/a\u003e I owe to him the definition of\r\npoints, the suggestion for the treatment of instants and\r\n“things,” and the whole conception of the world of\r\nphysics as a \u003cem\u003econstruction\u003c/em\u003e rather than an \u003cem\u003einference\u003c/em\u003e. What is\r\nsaid on these topics here is, in fact, a rough preliminary\r\naccount of the more precise results which he is giving in\r\nthe fourth volume of our \u003ccite lang=\"la\"\u003ePrincipia Mathematica\u003c/cite\u003e.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_3\" id=\"FNanchor_3\"\u003e[3]\u003c/a\u003e It will\r\nbe seen that if his way of dealing with these topics is capable\r\nof being successfully carried through, a wholly new light\r\nis thrown on the time-honoured controversies of realists\r\nand idealists, and a method is obtained of solving all that\r\nis soluble in their problem.\u003c/p\u003e\r\n\u003cp\u003eThe speculations of the past as to the reality or unreality\r\nof the world of physics were baffled, at the outset,\r\nby the absence of any satisfactory theory of the mathematical\r\ninfinite. This difficulty has been removed by the\r\nwork of Georg Cantor. But the positive and detailed\r\nsolution of the problem by means of mathematical constructions\r\nbased upon sensible objects as data has only\r\nbeen rendered possible by the growth of mathematical\r\nlogic, without which it is practically impossible to manipulate\r\nideas of the requisite abstractness and complexity.\r\nThis aspect, which is somewhat obscured in a merely\r\npopular outline such as is contained in the following\r\nlectures, will become plain as soon as Dr Whitehead\u0027s\r\nwork is published. In pure logic, which, however, will\r\nbe very briefly discussed in these lectures, I have had\r\n\u003ca class=\"pagenum\" title=\"vii\" id=\"Page_vii\"\u003e \u003c/a\u003e\r\nthe benefit of vitally important discoveries, not yet\r\npublished, by my friend Mr Ludwig Wittgenstein.\u003c/p\u003e\r\n\u003cp\u003eSince my purpose was to illustrate method, I have\r\nincluded much that is tentative and incomplete, for it is\r\nnot by the study of finished structures alone that the\r\nmanner of construction can be learnt. Except in regard\r\nto such matters as Cantor\u0027s theory of infinity, no finality\r\nis claimed for the theories suggested; but I believe that\r\nwhere they are found to require modification, this will\r\nbe discovered by substantially the same method as that\r\nwhich at present makes them appear probable, and it is\r\non this ground that I ask the reader to be tolerant of\r\ntheir incompleteness.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"small-caps\"\u003eCambridge\u003c/span\u003e,\u003cbr\u003e\r\n\u003ci style=\"margin-left: 3em;\"\u003eJune 1914\u003c/i\u003e.\u003c/p\u003e\r\n\u003ch2\u003e\u003ca class=\"pagenum\" title=\"ix\" id=\"Page_ix\"\u003e \u003c/a\u003eCONTENTS\u003c/h2\u003e\r\n\u003ctable id=\"toc\" data-summary=\"Contents\"\u003e\r\n\u003ctbody\u003e\u003ctr\u003e\r\n\u003cth\u003eLECTURE\u003c/th\u003e\r\n\u003cth colspan=\"2\"\u003ePAGE\u003c/th\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eI.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eCurrent Tendencies\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eII.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eLogic as the Essence of Philosophy\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_33\" class=\"pginternal\"\u003e33\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eIII.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eOn our Knowledge of the External World\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eIV.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eThe World of Physics and the World of Sense\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_101\" class=\"pginternal\"\u003e101\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eV.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eThe Theory of Continuity\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_129\" class=\"pginternal\"\u003e129\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eVI.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eThe Problem of Infinity considered Historically\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eVII.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eThe Positive Theory of Infinity\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd class=\"chapnum\"\u003eVIII.\u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eOn the Notion of Cause, with Applications to the Free-will Problem\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_211\" class=\"pginternal\"\u003e211\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd class=\"small-caps\"\u003eIndex\u003c/td\u003e\r\n\u003ctd class=\"right\"\u003e\u003ca href=\"#Page_243\" class=\"pginternal\"\u003e243\u003c/a\u003e\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003c/tbody\u003e\u003c/table\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"1\" id=\"Page_1\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE I\u003c/small\u003e\u003cbr\u003e\r\nCURRENT TENDENCIES\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"3\" id=\"Page_3\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_1\"\u003eLECTURE I\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nCURRENT TENDENCIES\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003ePhilosophy\u003c/span\u003e, from the earliest times, has made greater\r\nclaims, and achieved fewer results, than any other branch\r\nof learning. Ever since Thales said that all is water,\r\nphilosophers have been ready with glib assertions about\r\nthe sum-total of things; and equally glib denials have\r\ncome from other philosophers ever since Thales was contradicted\r\nby Anaximander. I believe that the time has\r\nnow arrived when this unsatisfactory state of things can\r\nbe brought to an end. In the following course of lectures\r\nI shall try, chiefly by taking certain special problems as\r\nexamples, to indicate wherein the claims of philosophers\r\nhave been excessive, and why their achievements have not\r\nbeen greater. The problems and the method of philosophy\r\nhave, I believe, been misconceived by all schools,\r\nmany of its traditional problems being insoluble with our\r\nmeans of knowledge, while other more neglected but not\r\nless important problems can, by a more patient and more\r\nadequate method, be solved with all the precision and\r\ncertainty to which the most advanced sciences have\r\nattained.\u003c/p\u003e\r\n\u003cp\u003eAmong present-day philosophies, we may distinguish\r\nthree principal types, often combined in varying proportions\r\nby a single philosopher, but in essence and tendency\r\ndistinct. The first of these, which I shall call the classical\r\ntradition, descends in the main from Kant and Hegel;\r\n\u003ca class=\"pagenum\" title=\"4\" id=\"Page_4\"\u003e \u003c/a\u003e\r\nit represents the attempt to adapt to present needs the\r\nmethods and results of the great constructive philosophers\r\nfrom Plato downwards. The second type, which may\r\nbe called evolutionism, derived its predominance from\r\nDarwin, and must be reckoned as having had Herbert\r\nSpencer for its first philosophical representative; but in\r\nrecent times it has become, chiefly through William\r\nJames and M. Bergson, far bolder and far more searching\r\nin its innovations than it was in the hands of Herbert\r\nSpencer. The third type, which may be called “logical\r\natomism” for want of a better name, has gradually crept\r\ninto philosophy through the critical scrutiny of mathematics.\r\nThis type of philosophy, which is the one that\r\nI wish to advocate, has not as yet many whole-hearted\r\nadherents, but the “new realism” which owes its inception\r\nto Harvard is very largely impregnated with its spirit.\r\nIt represents, I believe, the same kind of advance as was\r\nintroduced into physics by Galileo: the substitution of\r\npiecemeal, detailed, and verifiable results for large untested\r\ngeneralities recommended only by a certain appeal to\r\nimagination. But before we can understand the changes\r\nadvocated by this new philosophy, we must briefly\r\nexamine and criticise the other two types with which it\r\nhas to contend.\u003c/p\u003e\r\n\u003ch3\u003eA. The Classical Tradition\u003c/h3\u003e\r\n\u003cp\u003eTwenty years ago, the classical tradition, having vanquished\r\nthe opposing tradition of the English empiricists,\r\nheld almost unquestioned sway in all Anglo-Saxon\r\nuniversities. At the present day, though it is losing\r\nground, many of the most prominent teachers still adhere\r\nto it. In academic France, in spite of M. Bergson, it is\r\nfar stronger than all its opponents combined; and in\r\nGermany it has many vigorous advocates. Nevertheless,\r\n\u003ca class=\"pagenum\" title=\"5\" id=\"Page_5\"\u003e \u003c/a\u003e\r\nit represents on the whole a decaying force, and it has\r\nfailed to adapt itself to the temper of the age. Its\r\nadvocates are, in the main, those whose extra-philosophical\r\nknowledge is literary, rather than those who have felt\r\nthe inspiration of science. There are, apart from reasoned\r\narguments, certain general intellectual forces against it—the\r\nsame general forces which are breaking down the\r\nother great syntheses of the past, and making our age\r\none of bewildered groping where our ancestors walked\r\nin the clear daylight of unquestioning certainty.\u003c/p\u003e\r\n\u003cp\u003eThe original impulse out of which the classical tradition\r\ndeveloped was the naïve faith of the Greek philosophers\r\nin the omnipotence of reasoning. The discovery of\r\ngeometry had intoxicated them, and its \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e deductive\r\nmethod appeared capable of universal application. They\r\nwould prove, for instance, that all reality is one, that\r\nthere is no such thing as change, that the world of sense\r\nis a world of mere illusion; and the strangeness of their\r\nresults gave them no qualms because they believed in\r\nthe correctness of their reasoning. Thus it came to be\r\nthought that by mere thinking the most surprising and\r\nimportant truths concerning the whole of reality could be\r\nestablished with a certainty which no contrary observations\r\ncould shake. As the vital impulse of the early philosophers\r\ndied away, its place was taken by authority and\r\ntradition, reinforced, in the Middle Ages and almost to\r\nour own day, by systematic theology. Modern philosophy,\r\nfrom Descartes onwards, though not bound by authority\r\nlike that of the Middle Ages, still accepted more or less\r\nuncritically the Aristotelian logic. Moreover, it still\r\nbelieved, except in Great Britain, that \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e reasoning\r\ncould reveal otherwise undiscoverable secrets about the\r\nuniverse, and could prove reality to be quite different\r\nfrom what, to direct observation, it appears to be. It is\r\n\u003ca class=\"pagenum\" title=\"6\" id=\"Page_6\"\u003e \u003c/a\u003e\r\nthis belief, rather than any particular tenets resulting from\r\nit, that I regard as the distinguishing characteristic of the\r\nclassical tradition, and as hitherto the main obstacle to a\r\nscientific attitude in philosophy.\u003c/p\u003e\r\n\u003cp\u003eThe nature of the philosophy embodied in the classical\r\ntradition may be made clearer by taking a particular\r\nexponent as an illustration. For this purpose, let us\r\nconsider for a moment the doctrines of Mr Bradley, who\r\nis probably the most distinguished living representative\r\nof this school. Mr Bradley\u0027s \u003ccite\u003eAppearance and Reality\u003c/cite\u003e is a\r\nbook consisting of two parts, the first called \u003ci\u003eAppearance\u003c/i\u003e,\r\nthe second \u003ci\u003eReality\u003c/i\u003e. The first part examines and condemns\r\nalmost all that makes up our everyday world: things\r\nand qualities, relations, space and time, change, causation,\r\nactivity, the self. All these, though in some sense facts\r\nwhich qualify reality, are not real as they appear. What\r\nis real is one single, indivisible, timeless whole, called the\r\nAbsolute, which is in some sense spiritual, but does not\r\nconsist of souls, or of thought and will as we know them.\r\nAnd all this is established by abstract logical reasoning\r\nprofessing to find self-contradictions in the categories\r\ncondemned as mere appearance, and to leave no tenable\r\nalternative to the kind of Absolute which is finally\r\naffirmed to be real.\u003c/p\u003e\r\n\u003cp\u003eOne brief example may suffice to illustrate Mr Bradley\u0027s\r\nmethod. The world appears to be full of many things\r\nwith various relations to each other—right and left,\r\nbefore and after, father and son, and so on. But relations,\r\naccording to Mr Bradley, are found on examination\r\nto be self-contradictory and therefore impossible. He\r\nfirst argues that, if there are relations, there must be\r\nqualities between which they hold. This part of his\r\nargument need not detain us. He then proceeds:\u003c/p\u003e\r\n\u003cp\u003e“But how the relation can stand to the qualities is,\r\n\u003ca class=\"pagenum\" title=\"7\" id=\"Page_7\"\u003e \u003c/a\u003e\r\non the other side, unintelligible. If it is nothing to the\r\nqualities, then they are not related at all; and, if so, as\r\nwe saw, they have ceased to be qualities, and their relation\r\nis a nonentity. But if it is to be something to them,\r\nthen clearly we shall require a \u003cem\u003enew\u003c/em\u003e connecting relation.\r\nFor the relation hardly can be the mere adjective of one\r\nor both of its terms; or, at least, as such it seems\r\nindefensible. And, being something itself, if it does not\r\nitself bear a relation to the terms, in what intelligible way\r\nwill it succeed in being anything to them? But here\r\nagain we are hurried off into the eddy of a hopeless\r\nprocess, since we are forced to go on finding new relations\r\nwithout end. The links are united by a link, and this\r\nbond of union is a link which also has two ends; and\r\nthese require each a fresh link to connect them with\r\nthe old. The problem is to find how the relation can\r\nstand to its qualities, and this problem is insoluble.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_4\" id=\"FNanchor_4\"\u003e[4]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eI do not propose to examine this argument in detail,\r\nor to show the exact points where, in my opinion, it\r\nis fallacious. I have quoted it only as an example of\r\nmethod. Most people will admit, I think, that it is\r\ncalculated to produce bewilderment rather than conviction,\r\nbecause there is more likelihood of error in a very\r\nsubtle, abstract, and difficult argument than in so patent\r\na fact as the interrelatedness of the things in the world.\r\nTo the early Greeks, to whom geometry was practically\r\nthe only known science, it was possible to follow reasoning\r\nwith assent even when it led to the strangest conclusions.\r\nBut to us, with our methods of experiment and observation,\r\nour knowledge of the long history of \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e errors\r\nrefuted by empirical science, it has become natural to\r\nsuspect a fallacy in any deduction of which the conclusion\r\nappears to contradict patent facts. It is easy to carry\r\n\u003ca class=\"pagenum\" title=\"8\" id=\"Page_8\"\u003e \u003c/a\u003e\r\nsuch suspicion too far, and it is very desirable, if possible,\r\nactually to discover the exact nature of the error when\r\nit exists. But there is no doubt that what we may call\r\nthe empirical outlook has become part of most educated\r\npeople\u0027s habit of mind; and it is this, rather than any\r\ndefinite argument, that has diminished the hold of the\r\nclassical tradition upon students of philosophy and the\r\ninstructed public generally.\u003c/p\u003e\r\n\u003cp\u003eThe function of logic in philosophy, as I shall try to\r\nshow at a later stage, is all-important; but I do not\r\nthink its function is that which it has in the classical\r\ntradition. In that tradition, logic becomes constructive\r\nthrough negation. Where a number of alternatives seem,\r\nat first sight, to be equally possible, logic is made to\r\ncondemn all of them except one, and that one is then\r\npronounced to be realised in the actual world. Thus\r\nthe world is constructed by means of logic, with little\r\nor no appeal to concrete experience. The true function\r\nof logic is, in my opinion, exactly the opposite of this.\r\nAs applied to matters of experience, it is analytic rather\r\nthan constructive; taken \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e, it shows the possibility\r\nof hitherto unsuspected alternatives more often than the\r\nimpossibility of alternatives which seemed \u003ci lang=\"la\"\u003e\u003cins title=\"prima\"\u003eprimâ\u003c/ins\u003e facie\u003c/i\u003e\r\npossible. Thus, while it liberates imagination as to what\r\nthe world \u003cem\u003emay\u003c/em\u003e be, it refuses to legislate as to what the\r\nworld \u003cem\u003eis\u003c/em\u003e. This change, which has been brought about\r\nby an internal revolution in logic, has swept away the\r\nambitious constructions of traditional metaphysics, even\r\nfor those whose faith in logic is greatest; while to the many\r\nwho regard logic as a chimera the paradoxical systems\r\nto which it has given rise do not seem worthy even of\r\nrefutation. Thus on all sides these systems have ceased\r\nto attract, and even the philosophical world tends more\r\nand more to pass them by.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"9\" id=\"Page_9\"\u003e \u003c/a\u003eOne or two of the favourite doctrines of the school\r\nwe are considering may be mentioned to illustrate the\r\nnature of its claims. The universe, it tells us, is an\r\n“organic unity,” like an animal or a perfect work of art.\r\nBy this it means, roughly speaking, that all the different\r\nparts fit together and co-operate, and are what they are\r\nbecause of their place in the whole. This belief is sometimes\r\nadvanced dogmatically, while at other times it is\r\ndefended by certain logical arguments. If it is true,\r\nevery part of the universe is a microcosm, a miniature\r\nreflection of the whole. If we knew ourselves thoroughly,\r\naccording to this doctrine, we should know everything.\r\nCommon sense would naturally object that there are\r\npeople—say in China—with whom our relations are so\r\nindirect and trivial that we cannot infer anything important\r\nas to them from any fact about ourselves. If there are\r\nliving beings in Mars or in more distant parts of the\r\nuniverse, the same argument becomes even stronger.\r\nBut further, perhaps the whole contents of the space and\r\ntime in which we live form only one of many universes,\r\neach seeming to itself complete. And thus the conception\r\nof the necessary unity of all that is resolves itself into\r\nthe poverty of imagination, and a freer logic emancipates\r\nus from the strait-waistcoated benevolent institution\r\nwhich idealism palms off as the totality of being.\u003c/p\u003e\r\n\u003cp\u003eAnother very important doctrine held by most, though\r\nnot all, of the school we are examining is the doctrine\r\nthat all reality is what is called “mental” or “spiritual,”\r\nor that, at any rate, all reality is dependent for its existence\r\nupon what is mental. This view is often particularised\r\ninto the form which states that the relation of knower and\r\nknown is fundamental, and that nothing can exist unless\r\nit either knows or is known. Here again the same\r\nlegislative function is ascribed to \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e argumentation:\r\n\u003ca class=\"pagenum\" title=\"10\" id=\"Page_10\"\u003e \u003c/a\u003e\r\nit is thought that there are contradictions in an unknown\r\nreality. Again, if I am not mistaken, the argument is\r\nfallacious, and a better logic will show that no limits can\r\nbe set to the extent and nature of the unknown. And\r\nwhen I speak of the unknown, I do not mean merely\r\nwhat we personally do not know, but what is not known\r\nto any mind. Here as elsewhere, while the older logic\r\nshut out possibilities and imprisoned imagination within\r\nthe walls of the familiar, the newer logic shows rather\r\nwhat may happen, and refuses to decide as to what \u003cem\u003emust\u003c/em\u003e\r\nhappen.\u003c/p\u003e\r\n\u003cp\u003eThe classical tradition in philosophy is the last surviving\r\nchild of two very diverse parents: the Greek belief in\r\nreason, and the mediæval belief in the tidiness of the\r\nuniverse. To the schoolmen, who lived amid wars,\r\nmassacres, and pestilences, nothing appeared so delightful\r\nas safety and order. In their idealising dreams, it was\r\nsafety and order that they sought: the universe of\r\nThomas Aquinas or Dante is as small and neat as a\r\nDutch interior. To us, to whom safety has become\r\nmonotony, to whom the primeval savageries of nature\r\nare so remote as to become a mere pleasing condiment\r\nto our ordered routine, the world of dreams is very\r\ndifferent from what it was amid the wars of Guelf and\r\nGhibelline. Hence William James\u0027s protest against what\r\nhe calls the “block universe” of the classical tradition;\r\nhence Nietzsche\u0027s worship of force; hence the verbal\r\nbloodthirstiness of many quiet literary men. The barbaric\r\nsubstratum of human nature, unsatisfied in action, finds\r\nan outlet in imagination. In philosophy, as elsewhere,\r\nthis tendency is visible; and it is this, rather than formal\r\nargument, that has thrust aside the classical tradition\r\nfor a philosophy which fancies itself more virile and\r\nmore vital.\u003c/p\u003e\r\n\u003ch3\u003e\u003ca class=\"pagenum\" title=\"11\" id=\"Page_11\"\u003e \u003c/a\u003eB. Evolutionism\u003c/h3\u003e\r\n\u003cp\u003eEvolutionism, in one form or another, is the prevailing\r\ncreed of our time. It dominates our politics, our\r\nliterature, and not least our philosophy. Nietzsche,\r\npragmatism, Bergson, are phases in its philosophic\r\ndevelopment, and their popularity far beyond the circles\r\nof professional philosophers shows its consonance with\r\nthe spirit of the age. It believes itself firmly based on\r\nscience, a liberator of hopes, an inspirer of an invigorating\r\nfaith in human power, a sure antidote to the ratiocinative\r\nauthority of the Greeks and the dogmatic authority\r\nof mediæval systems. Against so fashionable and so\r\nagreeable a creed it may seem useless to raise a protest;\r\nand with much of its spirit every modern man must be in\r\nsympathy. But I think that, in the intoxication of a\r\nquick success, much that is important and vital to a true\r\nunderstanding of the universe has been forgotten. Something\r\nof Hellenism must be combined with the new spirit\r\nbefore it can emerge from the ardour of youth into the\r\nwisdom of manhood. And it is time to remember that\r\nbiology is neither the only science, nor yet the model\r\nto which all other sciences must adapt themselves.\r\nEvolutionism, as I shall try to show, is not a truly\r\nscientific philosophy, either in its method or in the\r\nproblems which it considers. The true scientific philosophy\r\nis something more arduous and more aloof, appealing\r\nto less mundane hopes, and requiring a severer\r\ndiscipline for its successful practice.\u003c/p\u003e\r\n\u003cp\u003eDarwin\u0027s \u003ccite\u003eOrigin of Species\u003c/cite\u003e persuaded the world that the\r\ndifference between different species of animals and plants\r\nis not the fixed, immutable difference that it appears to\r\nbe. The doctrine of natural kinds, which had rendered\r\nclassification easy and definite, which was enshrined in\r\n\u003ca class=\"pagenum\" title=\"12\" id=\"Page_12\"\u003e \u003c/a\u003e\r\nthe Aristotelian tradition, and protected by its supposed\r\nnecessity for orthodox dogma, was suddenly swept away\r\nfor ever out of the biological world. The difference\r\nbetween man and the lower animals, which to our human\r\nconceit appears enormous, was shown to be a gradual\r\nachievement, involving intermediate beings who could\r\nnot with certainty be placed either within or without the\r\nhuman family. The sun and planets had already been\r\nshown by Laplace to be very probably derived from a\r\nprimitive more or less undifferentiated nebula. Thus\r\nthe old fixed landmarks became wavering and indistinct,\r\nand all sharp outlines were blurred. Things and species\r\nlost their boundaries, and none could say where they\r\nbegan or where they ended.\u003c/p\u003e\r\n\u003cp\u003eBut if human conceit was staggered for a moment by\r\nits kinship with the ape, it soon found a way to reassert\r\nitself, and that way is the “philosophy” of evolution.\r\nA process which led from the amœba to man appeared to\r\nthe philosophers to be obviously a progress—though\r\nwhether the amœba would agree with this opinion is not\r\nknown. Hence the cycle of changes which science had\r\nshown to be the probable history of the past was\r\nwelcomed as revealing a law of development towards\r\ngood in the universe—an evolution or unfolding of an\r\nideal slowly embodying itself in the actual. But such a\r\nview, though it might satisfy Spencer and those whom\r\nwe may call Hegelian evolutionists, could not be accepted\r\nas adequate by the more whole-hearted votaries of change.\r\nAn ideal to which the world continuously approaches is, to\r\nthese minds, too dead and static to be inspiring. Not only\r\nthe aspirations, but the ideal too, must change and develop\r\nwith the course of evolution; there must be no fixed goal,\r\nbut a continual fashioning of fresh needs by the impulse\r\nwhich is life and which alone gives unity to the process.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"13\" id=\"Page_13\"\u003e \u003c/a\u003eEver since the seventeenth century, those whom\r\nWilliam James described as the “tender-minded” have\r\nbeen engaged in a desperate struggle with the mechanical\r\nview of the course of nature which physical science seems\r\nto impose. A great part of the attractiveness of the\r\nclassical tradition was due to the partial escape from\r\nmechanism which it provided. But now, with the influence\r\nof biology, the “tender-minded” believe that a\r\nmore radical escape is possible, sweeping aside not merely\r\nthe laws of physics, but the whole apparently immutable\r\napparatus of logic, with its fixed concepts, its general\r\nprinciples, and its reasonings which seem able to compel\r\neven the most unwilling assent. The older kind of\r\nteleology, therefore, which regarded the End as a fixed\r\ngoal, already partially visible, towards which we were\r\ngradually approaching, is rejected by M. Bergson as not\r\nallowing enough for the absolute dominion of change.\r\nAfter explaining why he does not accept mechanism, he\r\nproceeds:\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_5\" id=\"FNanchor_5\"\u003e[5]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003e“But radical finalism is quite as unacceptable, and for\r\nthe same reason. The doctrine of teleology, in its\r\nextreme form, as we find it in Leibniz for example,\r\nimplies that things and beings merely realise a programme\r\npreviously arranged. But if there is nothing unforeseen,\r\nno invention or creation in the universe, time is useless\r\nagain. As in the mechanistic hypothesis, here again it is\r\nsupposed that \u003cem\u003eall is given\u003c/em\u003e. Finalism thus understood is\r\nonly inverted mechanism. It springs from the same\r\npostulate, with this sole difference, that in the movement\r\nof our finite intellects along successive things, whose\r\nsuccessiveness is reduced to a mere appearance, it holds\r\nin front of us the light with which it claims to guide us,\r\ninstead of putting it behind. It substitutes the attraction\r\n\u003ca class=\"pagenum\" title=\"14\" id=\"Page_14\"\u003e \u003c/a\u003e\r\nof the future for the impulsion of the past. But succession\r\nremains none the less a mere appearance, as\r\nindeed does movement itself. In the doctrine of Leibniz,\r\ntime is reduced to a confused perception, relative to\r\nthe human standpoint, a perception which would vanish,\r\nlike a rising mist, for a mind seated at the centre of\r\nthings.\u003c/p\u003e\r\n\u003cp\u003e“Yet finalism is not, like mechanism, a doctrine with\r\nfixed rigid outlines. It admits of as many inflections as\r\nwe like. The mechanistic philosophy is to be taken or\r\nleft: it must be left if the least grain of dust, by straying\r\nfrom the path foreseen by mechanics, should show the\r\nslightest trace of spontaneity. The doctrine of final\r\ncauses, on the contrary, will never be definitively refuted.\r\nIf one form of it be put aside, it will take another. Its\r\nprinciple, which is essentially psychological, is very\r\nflexible. It is so extensible, and thereby so comprehensive,\r\nthat one accepts something of it as soon as\r\none rejects pure mechanism. The theory we shall put\r\nforward in this book will therefore necessarily partake of\r\nfinalism to a certain extent.”\u003c/p\u003e\r\n\u003cp\u003eM. Bergson\u0027s form of finalism depends upon his conception\r\nof life. Life, in his philosophy, is a continuous\r\nstream, in which all divisions are artificial and unreal.\r\nSeparate things, beginnings and endings, are mere convenient\r\nfictions: there is only smooth, unbroken transition.\r\nThe beliefs of to-day may count as true to-day,\r\nif they carry us along the stream; but to-morrow they\r\nwill be false, and must be replaced by new beliefs to\r\nmeet the new situation. All our thinking consists of\r\nconvenient fictions, imaginary congealings of the stream:\r\nreality flows on in spite of all our fictions, and though it\r\ncan be lived, it cannot be conceived in thought. Somehow,\r\nwithout explicit statement, the assurance is slipped\r\n\u003ca class=\"pagenum\" title=\"15\" id=\"Page_15\"\u003e \u003c/a\u003e\r\nin that the future, though we cannot foresee it, will be\r\nbetter than the past or the present: the reader is like\r\nthe child who expects a sweet because it has been told\r\nto open its mouth and shut its eyes. Logic, mathematics,\r\nphysics disappear in this philosophy, because they are\r\ntoo “static”; what is real is an impulse and movement\r\ntowards a goal which, like the rainbow, recedes as we\r\nadvance, and makes every place different when we reach\r\nit from what it appeared to be at a distance.\u003c/p\u003e\r\n\u003cp\u003eNow I do not propose at present to enter upon a\r\ntechnical examination of this philosophy. At present I\r\nwish to make only two criticisms of it—first, that its truth\r\ndoes not follow from what science has rendered probable\r\nconcerning the facts of evolution, and secondly, that the\r\nmotives and interests which inspire it are so exclusively\r\npractical, and the problems with which it deals are so\r\nspecial, that it can hardly be regarded as really touching\r\nany of the questions that to my mind constitute genuine\r\nphilosophy.\u003c/p\u003e\r\n\u003cp\u003e(1) What biology has rendered probable is that the\r\ndiverse species arose by adaptation from a less differentiated\r\nancestry. This fact is in itself exceedingly interesting,\r\nbut it is not the kind of fact from which philosophical\r\nconsequences follow. Philosophy is general, and takes\r\nan impartial interest in all that exists. The changes\r\nsuffered by minute portions of matter on the earth\u0027s\r\nsurface are very important to us as active sentient beings;\r\nbut to us as philosophers they have no greater interest\r\nthan other changes in portions of matter elsewhere. And\r\nif the changes on the earth\u0027s surface during the last few\r\nmillions of years appear to our present ethical notions to\r\nbe in the nature of a progress, that gives no ground for\r\nbelieving that progress is a general law of the universe.\r\nExcept under the influence of desire, no one would\r\n\u003ca class=\"pagenum\" title=\"16\" id=\"Page_16\"\u003e \u003c/a\u003e\r\nadmit for a moment so crude a generalisation from such\r\na tiny selection of facts. What does result, not specially\r\nfrom biology, but from all the sciences which deal with\r\nwhat exists, is that we cannot understand the world unless\r\nwe can understand change and continuity. This is even\r\nmore evident in physics than it is in biology. But the\r\nanalysis of change and continuity is not a problem upon\r\nwhich either physics or biology throws any light: it is a\r\nproblem of a new kind, belonging to a different kind of\r\nstudy. The question whether evolutionism offers a true\r\nor a false answer to this problem is not, therefore, a\r\nquestion to be solved by appeals to particular facts, such\r\nas biology and physics reveal. In assuming dogmatically\r\na certain answer to this question, evolutionism ceases to\r\nbe scientific, yet it is only in touching on this question\r\nthat evolutionism reaches the subject-matter of philosophy.\r\nEvolutionism thus consists of two parts: one not philosophical,\r\nbut only a hasty generalisation of the kind\r\nwhich the special sciences might hereafter confirm or\r\nconfute; the other not scientific, but a mere unsupported\r\ndogma, belonging to philosophy by its subject-matter,\r\nbut in no way deducible from the facts upon which\r\nevolution relies.\u003c/p\u003e\r\n\u003cp\u003e(2) The predominant interest of evolutionism is in the\r\nquestion of human destiny, or at least of the destiny of\r\nLife. It is more interested in morality and happiness\r\nthan in knowledge for its own sake. It must be admitted\r\nthat the same may be said of many other philosophies,\r\nand that a desire for the kind of knowledge which\r\nphilosophy really can give is very rare. But if philosophy\r\nis to become scientific—and it is our object to discover\r\nhow this can be achieved—it is necessary first and foremost\r\nthat philosophers should acquire the disinterested\r\nintellectual curiosity which characterises the genuine man\r\n\u003ca class=\"pagenum\" title=\"17\" id=\"Page_17\"\u003e \u003c/a\u003e\r\nof science. Knowledge concerning the future—which is\r\nthe kind of knowledge that must be sought if we are to\r\nknow about human destiny—is possible within certain\r\nnarrow limits. It is impossible to say how much the\r\nlimits may be enlarged with the progress of science.\r\nBut what is evident is that any proposition about the\r\nfuture belongs by its subject-matter to some particular\r\nscience, and is to be ascertained, if at all, by the methods\r\nof that science. Philosophy is not a short cut to the\r\nsame kind of results as those of the other sciences: if it\r\nis to be a genuine study, it must have a province of its\r\nown, and aim at results which the other sciences can\r\nneither prove nor disprove.\u003c/p\u003e\r\n\u003cp\u003eThe consideration that philosophy, if there is such a\r\nstudy, must consist of propositions which could not occur\r\nin the other sciences, is one which has very far-reaching\r\nconsequences. All the questions which have what is called\r\na human interest—such, for example, as the question of\r\na future life—belong, at least in theory, to special sciences,\r\nand are capable, at least in theory, of being decided by\r\nempirical evidence. Philosophers have too often, in the\r\npast, permitted themselves to pronounce on empirical\r\nquestions, and found themselves, as a result, in disastrous\r\nconflict with well-attested facts. We must, therefore,\r\nrenounce the hope that philosophy can promise satisfaction\r\nto our mundane desires. What it can do, when it is\r\npurified from all practical taint, is to help us to understand\r\nthe general aspects of the world and the logical\r\nanalysis of familiar but complex things. Through this\r\nachievement, by the suggestion of fruitful hypotheses,\r\nit may be indirectly useful in other sciences, notably\r\nmathematics, physics, and psychology. But a genuinely\r\nscientific philosophy cannot hope to appeal to any except\r\nthose who have the wish to understand, to escape from\r\n\u003ca class=\"pagenum\" title=\"18\" id=\"Page_18\"\u003e \u003c/a\u003e\r\nintellectual bewilderment. It offers, in its own domain,\r\nthe kind of satisfaction which the other sciences offer.\r\nBut it does not offer, or attempt to offer, a solution of\r\nthe problem of human destiny, or of the destiny of the\r\nuniverse.\u003c/p\u003e\r\n\u003cp\u003eEvolutionism, if what has been said is true, is to be\r\nregarded as a hasty generalisation from certain rather\r\nspecial facts, accompanied by a dogmatic rejection of all\r\nattempts at analysis, and inspired by interests which are\r\npractical rather than theoretical. In spite, therefore, of\r\nits appeal to detailed results in various sciences, it cannot\r\nbe regarded as any more genuinely scientific than the\r\nclassical tradition which it has replaced. How philosophy\r\nis to be rendered scientific, and what is the true subject-matter\r\nof philosophy, I shall try to show first by examples\r\nof certain achieved results, and then more generally.\r\nWe will begin with the problem of the physical conceptions\r\nof space and time and matter, which, as we have\r\nseen, are challenged by the contentions of the evolutionists.\r\nThat these conceptions stand in need of reconstruction\r\nwill be admitted, and is indeed increasingly urged by\r\nphysicists themselves. It will also be admitted that the\r\nreconstruction must take more account of change and\r\nthe universal flux than is done in the older mechanics\r\nwith its fundamental conception of an indestructible\r\nmatter. But I do not think the reconstruction required\r\nis on Bergsonian lines, nor do I think that his rejection\r\nof logic can be anything but harmful. I shall not, however,\r\nadopt the method of explicit controversy, but rather\r\nthe method of independent inquiry, starting from what,\r\nin a pre-philosophic stage, appear to be facts, and keeping\r\nalways as close to these initial data as the requirements of\r\nconsistency will permit.\u003c/p\u003e\r\n\u003cp\u003eAlthough explicit controversy is almost always fruitless\r\n\u003ca class=\"pagenum\" title=\"19\" id=\"Page_19\"\u003e \u003c/a\u003e\r\nin philosophy, owing to the fact that no two philosophers\r\never understand one another, yet it seems necessary to\r\nsay something at the outset in justification of the scientific\r\nas against the mystical attitude. Metaphysics, from the\r\nfirst, has been developed by the union or the conflict of\r\nthese two attitudes. Among the earliest Greek philosophers,\r\nthe Ionians were more scientific and the Sicilians\r\nmore mystical.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_6\" id=\"FNanchor_6\"\u003e[6]\u003c/a\u003e But among the latter, Pythagoras, for\r\nexample, was in himself a curious mixture of the two\r\ntendencies: the scientific attitude led him to his proposition\r\non right-angled triangles, while his mystic insight\r\nshowed him that it is wicked to eat beans. Naturally\r\nenough, his followers divided into two sects, the lovers\r\nof right-angled triangles and the abhorrers of beans; but\r\nthe former sect died out, leaving, however, a haunting\r\nflavour of mysticism over much Greek mathematical\r\nspeculation, and in particular over Plato\u0027s views on\r\nmathematics. Plato, of course, embodies both the\r\nscientific and the mystical attitudes in a higher form\r\nthan his predecessors, but the mystical attitude is distinctly\r\nthe stronger of the two, and secures ultimate\r\nvictory whenever the conflict is sharp. Plato, moreover,\r\nadopted from the Eleatics the device of using logic to\r\ndefeat common sense, and thus to leave the field clear for\r\nmysticism—a device still employed in our own day by\r\nthe adherents of the classical tradition.\u003c/p\u003e\r\n\u003cp\u003eThe logic used in defence of mysticism seems to me\r\nfaulty as logic, and in a later lecture I shall criticise it on\r\nthis ground. But the more thorough-going mystics do\r\nnot employ logic, which they despise: they appeal instead\r\ndirectly to the immediate deliverance of their insight.\r\nNow, although fully developed mysticism is rare in the\r\nWest, some tincture of it colours the thoughts of many\r\n\u003ca class=\"pagenum\" title=\"20\" id=\"Page_20\"\u003e \u003c/a\u003e\r\npeople, particularly as regards matters on which they have\r\nstrong convictions not based on evidence. In all who\r\nseek passionately for the fugitive and difficult goods,\r\nthe conviction is almost irresistible that there is in the\r\nworld something deeper, more significant, than the multiplicity\r\nof little facts chronicled and classified by science.\r\nBehind the veil of these mundane things, they feel,\r\nsomething quite different obscurely shimmers, shining\r\nforth clearly in the great moments of illumination, which\r\nalone give anything worthy to be called real knowledge\r\nof truth. To seek such moments, therefore, is to them\r\nthe way of wisdom, rather than, like the man of science,\r\nto observe coolly, to analyse without emotion, and to\r\naccept without question the equal reality of the trivial\r\nand the important.\u003c/p\u003e\r\n\u003cp\u003eOf the reality or unreality of the mystic\u0027s world I know\r\nnothing. I have no wish to deny it, nor even to declare\r\nthat the insight which reveals it is not a genuine insight.\r\nWhat I do wish to maintain—and it is here that the\r\nscientific attitude becomes imperative—is that insight,\r\nuntested and unsupported, is an insufficient guarantee of\r\ntruth, in spite of the fact that much of the most important\r\ntruth is first suggested by its means. It is common\r\nto speak of an opposition between instinct and reason;\r\nin the eighteenth century, the opposition was drawn in\r\nfavour of reason, but under the influence of Rousseau\r\nand the romantic movement instinct was given the\r\npreference, first by those who rebelled against artificial\r\nforms of government and thought, and then, as the purely\r\nrationalistic defence of traditional theology became increasingly\r\ndifficult, by all who felt in science a menace\r\nto creeds which they associated with a spiritual outlook\r\non life and the world. Bergson, under the name of\r\n“intuition,” has raised instinct to the position of sole\r\n\u003ca class=\"pagenum\" title=\"21\" id=\"Page_21\"\u003e \u003c/a\u003e\r\narbiter of metaphysical truth. But in fact the opposition\r\nof instinct and reason is mainly illusory. Instinct,\r\nintuition, or insight is what first leads to the beliefs\r\nwhich subsequent reason confirms or confutes; but the\r\nconfirmation, where it is possible, consists, in the last\r\nanalysis, of agreement with other beliefs no less instinctive.\r\nReason is a harmonising, controlling force rather than a\r\ncreative one. Even in the most purely logical realms, it\r\nis insight that first arrives at what is new.\u003c/p\u003e\r\n\u003cp\u003eWhere instinct and reason do sometimes conflict is in\r\nregard to single beliefs, held instinctively, and held with\r\nsuch determination that no degree of inconsistency with\r\nother beliefs leads to their abandonment. Instinct, like\r\nall human faculties, is liable to error. Those in whom\r\nreason is weak are often unwilling to admit this as regards\r\nthemselves, though all admit it in regard to others.\r\nWhere instinct is least liable to error is in practical\r\nmatters as to which right judgment is a help to survival;\r\nfriendship and hostility in others, for instance, are often\r\nfelt with extraordinary discrimination through very careful\r\ndisguises. But even in such matters a wrong\r\nimpression may be given by reserve or flattery; and in\r\nmatters less directly practical, such as philosophy deals\r\nwith, very strong instinctive beliefs may be wholly\r\nmistaken, as we may come to know through their\r\nperceived inconsistency with other equally strong beliefs.\r\nIt is such considerations that necessitate the harmonising\r\nmediation of reason, which tests our beliefs by their\r\nmutual compatibility, and examines, in doubtful cases,\r\nthe possible sources of error on the one side and on\r\nthe other. In this there is no opposition to instinct as\r\na whole, but only to blind reliance upon some one interesting\r\naspect of instinct to the exclusion of other more\r\ncommonplace but not less trustworthy aspects. It is\r\n\u003ca class=\"pagenum\" title=\"22\" id=\"Page_22\"\u003e \u003c/a\u003e\r\nsuch onesidedness, not instinct itself, that reason aims\r\nat correcting.\u003c/p\u003e\r\n\u003cp\u003eThese more or less trite maxims may be illustrated\r\nby application to Bergson\u0027s advocacy of “intuition” as\r\nagainst “intellect.” There are, he says, “two profoundly\r\ndifferent ways of knowing a thing. The first implies\r\nthat we move round the object; the second that we\r\nenter into it. The first depends on the point of view at\r\nwhich we are placed and on the symbols by which we\r\nexpress ourselves. The second neither depends on a\r\npoint of view nor relies on any symbol. The first kind\r\nof knowledge may be said to stop at the \u003cem\u003erelative\u003c/em\u003e; the\r\nsecond, in those cases where it is possible, to attain the\r\n\u003cem\u003eabsolute\u003c/em\u003e.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_7\" id=\"FNanchor_7\"\u003e[7]\u003c/a\u003e The second of these, which is intuition, is, he\r\nsays, “the kind of intellectual \u003cem\u003esympathy\u003c/em\u003e by which one\r\nplaces oneself within an object in order to coincide with\r\nwhat is unique in it and therefore inexpressible” (p. 6).\r\nIn illustration, he mentions self-knowledge: “there is\r\none reality, at least, which we all seize from within, by\r\nintuition and not by simple analysis. It is our own\r\npersonality in its flowing through time—our self which\r\nendures” (p. 8). The rest of Bergson\u0027s philosophy\r\nconsists in reporting, through the imperfect medium of\r\nwords, the knowledge gained by intuition, and the\r\nconsequent complete condemnation of all the pretended\r\nknowledge derived from science and common sense.\u003c/p\u003e\r\n\u003cp\u003eThis procedure, since it takes sides in a conflict of\r\ninstinctive beliefs, stands in need of justification by\r\nproving the greater trustworthiness of the beliefs on one\r\nside than of those on the other. Bergson attempts this\r\njustification in two ways—first, by explaining that intellect\r\nis a purely practical faculty designed to secure biological\r\nsuccess; secondly, by mentioning remarkable feats of\r\n\u003ca class=\"pagenum\" title=\"23\" id=\"Page_23\"\u003e \u003c/a\u003e\r\ninstinct in animals, and by pointing out characteristics of\r\nthe world which, though intuition can apprehend them,\r\nare baffling to intellect as he interprets it.\u003c/p\u003e\r\n\u003cp\u003eOf Bergson\u0027s theory that intellect is a purely practical\r\nfaculty developed in the struggle for survival, and not a\r\nsource of true beliefs, we may say, first, that it is only\r\nthrough intellect that we know of the struggle for survival\r\nand of the biological ancestry of man: if the intellect is\r\nmisleading, the whole of this merely inferred history\r\nis presumably untrue. If, on the other hand, we agree\r\nwith M. Bergson in thinking that evolution took place\r\nas Darwin believed, then it is not only intellect, but all\r\nour faculties, that have been developed under the stress\r\nof practical utility. Intuition is seen at its best where it\r\nis directly useful—for example, in regard to other people\u0027s\r\ncharacters and dispositions. Bergson apparently holds that\r\ncapacity for this kind of knowledge is less explicable by\r\nthe struggle for existence than, for example, capacity for\r\npure mathematics. Yet the savage deceived by false\r\nfriendship is likely to pay for his mistake with his life;\r\nwhereas even in the most civilised societies men are not\r\nput to death for mathematical incompetence. All the\r\nmost striking of his instances of intuition in animals have\r\na very direct survival value. The fact is, of course, that\r\nboth intuition and intellect have been developed because\r\nthey are useful, and that, speaking broadly, they are useful\r\nwhen they give truth and become harmful when they\r\ngive falsehood. Intellect, in civilised man, like artistic\r\ncapacity, has occasionally been developed beyond the\r\npoint where it is useful to the individual; intuition, on\r\nthe other hand, seems on the whole to diminish as civilisation\r\nincreases. Speaking broadly, it is greater in children\r\nthan in adults, in the uneducated than in the educated.\r\nProbably in dogs it exceeds anything to be found in\r\n\u003ca class=\"pagenum\" title=\"24\" id=\"Page_24\"\u003e \u003c/a\u003e\r\nhuman beings. But those who find in these facts a\r\nrecommendation of intuition ought to return to running\r\nwild in the woods, dyeing themselves with woad and\r\nliving on hips and haws.\u003c/p\u003e\r\n\u003cp\u003eLet us next examine whether intuition possesses any\r\nsuch infallibility as Bergson claims for it. The best\r\ninstance of it, according to him, is our acquaintance with\r\nourselves; yet self-knowledge is proverbially rare and\r\ndifficult. Most men, for example, have in their nature\r\nmeannesses, vanities, and envies of which they are quite\r\nunconscious, though even their best friends can perceive\r\nthem without any difficulty. It is true that intuition has\r\na convincingness which is lacking to intellect: while it is\r\npresent, it is almost impossible to doubt its truth. But\r\nif it should appear, on examination, to be at least as\r\nfallible as intellect, its greater subjective certainty becomes\r\na demerit, making it only the more irresistibly deceptive.\r\nApart from self-knowledge, one of the most notable\r\nexamples of intuition is the knowledge people believe themselves\r\nto possess of those with whom they are in love:\r\nthe wall between different personalities seems to become\r\ntransparent, and people think they see into another soul\r\nas into their own. Yet deception in such cases is constantly\r\npractised with success; and even where there is no\r\nintentional deception, experience gradually proves, as a\r\nrule, that the supposed insight was illusory, and that the\r\nslower, more groping methods of the intellect are in the\r\nlong run more reliable.\u003c/p\u003e\r\n\u003cp\u003eBergson maintains that intellect can only deal with\r\nthings in so far as they resemble what has been experienced\r\nin the past, while intuition has the power of apprehending\r\nthe uniqueness and novelty that always belong to each\r\nfresh moment. That there is something unique and new\r\nat every moment, is certainly true; it is also true that\r\n\u003ca class=\"pagenum\" title=\"25\" id=\"Page_25\"\u003e \u003c/a\u003e\r\nthis cannot be fully expressed by means of intellectual\r\nconcepts. Only direct acquaintance can give knowledge\r\nof what is unique and new. But direct acquaintance of\r\nthis kind is given fully in sensation, and does not require,\r\nso far as I can see, any special faculty of intuition for its\r\napprehension. It is neither intellect nor intuition, but\r\nsensation, that supplies new data; but when the data\r\nare new in any remarkable manner, intellect is much more\r\ncapable of dealing with them than intuition would be.\r\nThe hen with a brood of ducklings no doubt has intuitions\r\nwhich seem to place her inside them, and not merely to\r\nknow them analytically; but when the ducklings take\r\nto the water, the whole apparent intuition is seen to be\r\nillusory, and the hen is left helpless on the shore.\r\nIntuition, in fact, is an aspect and development of instinct,\r\nand, like all instinct, is admirable in those customary\r\nsurroundings which have moulded the habits of the\r\nanimal in question, but totally incompetent as soon as\r\nthe surroundings are changed in a way which demands\r\nsome non-habitual mode of action.\u003c/p\u003e\r\n\u003cp\u003eThe theoretical understanding of the world, which is\r\nthe aim of philosophy, is not a matter of great practical\r\nimportance to animals, or to savages, or even to most\r\ncivilised men. It is hardly to be supposed, therefore,\r\nthat the rapid, rough and ready methods of instinct or\r\nintuition will find in this field a favourable ground for\r\ntheir application. It is the older kinds of activity, which\r\nbring out our kinship with remote generations of animal\r\nand semi-human ancestors, that show intuition at its best.\r\nIn such matters as self-preservation and love, intuition\r\nwill act sometimes (though not always) with a swiftness\r\nand precision which are astonishing to the critical intellect.\r\nBut philosophy is not one of the pursuits which illustrate\r\nour affinity with the past: it is a highly refined, highly\r\n\u003ca class=\"pagenum\" title=\"26\" id=\"Page_26\"\u003e \u003c/a\u003e\r\ncivilised pursuit, demanding, for its success, a certain\r\nliberation from the life of instinct, and even, at times, a\r\ncertain aloofness from all mundane hopes and fears. It\r\nis not in philosophy, therefore, that we can hope to see\r\nintuition at its best. On the contrary, since the true\r\nobjects of philosophy, and the habits of thought demanded\r\nfor their apprehension, are strange, unusual,\r\nand remote, it is here, more almost than anywhere else,\r\nthat intellect proves superior to intuition, and that quick\r\nunanalysed convictions are least deserving of uncritical\r\nacceptance.\u003c/p\u003e\r\n\u003cp\u003eBefore embarking upon the somewhat difficult and\r\nabstract discussions which lie before us, it will be well\r\nto take a survey of the hopes we may retain and the\r\nhopes we must abandon. The hope of satisfaction to\r\nour more human desires—the hope of demonstrating\r\nthat the world has this or that desirable ethical characteristic—is\r\nnot one which, so far as I can see, philosophy\r\ncan do anything whatever to satisfy. The difference\r\nbetween a good world and a bad one is a difference in\r\nthe particular characteristics of the particular things that\r\nexist in these worlds: it is not a sufficiently abstract\r\ndifference to come within the province of philosophy.\r\nLove and hate, for example, are ethical opposites, but to\r\nphilosophy they are closely analogous attitudes towards\r\nobjects. The general form and structure of those\r\nattitudes towards objects which constitute mental phenomena\r\nis a problem for philosophy; but the difference\r\nbetween love and hate is not a difference of form or\r\nstructure, and therefore belongs rather to the special\r\nscience of psychology than to philosophy. Thus the\r\nethical interests which have often inspired philosophers\r\nmust remain in the background: some kind of ethical\r\ninterest may inspire the whole study, but none must\r\n\u003ca class=\"pagenum\" title=\"27\" id=\"Page_27\"\u003e \u003c/a\u003e\r\nobtrude in the detail or be expected in the special results\r\nwhich are sought.\u003c/p\u003e\r\n\u003cp\u003eIf this view seems at first sight disappointing, we may\r\nremind ourselves that a similar change has been found\r\nnecessary in all the other sciences. The physicist or\r\nchemist is not now required to prove the ethical importance\r\nof his ions or atoms; the biologist is not expected\r\nto prove the utility of the plants or animals which he\r\ndissects. In pre-scientific ages this was not the case.\r\nAstronomy, for example, was studied because men believed\r\nin astrology: it was thought that the movements\r\nof the planets had the most direct and important bearing\r\nupon the lives of human beings. Presumably, when this\r\nbelief decayed and the disinterested study of astronomy\r\nbegan, many who had found astrology absorbingly interesting\r\ndecided that astronomy had too little human\r\ninterest to be worthy of study. Physics, as it appears in\r\nPlato\u0027s \u003ccite\u003eTimæus\u003c/cite\u003e for example, is full of ethical notions: it\r\nis an essential part of its purpose to show that the earth\r\nis worthy of admiration. The modern physicist, on the\r\ncontrary, though he has no wish to deny that the earth\r\nis admirable, is not concerned, as physicist, with its\r\nethical attributes: he is merely concerned to find out\r\nfacts, not to consider whether they are good or bad. In\r\npsychology, the scientific attitude is even more recent and\r\nmore difficult than in the physical sciences: it is natural\r\nto consider that human nature is either good or bad,\r\nand to suppose that the difference between good and bad,\r\nso all-important in practice, must be important in theory\r\nalso. It is only during the last century that an ethically\r\nneutral science of psychology has grown up; and here\r\ntoo ethical neutrality has been essential to scientific\r\nsuccess.\u003c/p\u003e\r\n\u003cp\u003eIn philosophy, hitherto, ethical neutrality has been\r\n\u003ca class=\"pagenum\" title=\"28\" id=\"Page_28\"\u003e \u003c/a\u003e\r\nseldom sought and hardly ever achieved. Men have\r\nremembered their wishes, and have judged philosophies\r\nin relation to their wishes. Driven from the particular\r\nsciences, the belief that the notions of good and evil must\r\nafford a key to the understanding of the world has sought\r\na refuge in philosophy. But even from this last refuge,\r\nif philosophy is not to remain a set of pleasing dreams,\r\nthis belief must be driven forth. It is a commonplace\r\nthat happiness is not best achieved by those who seek\r\nit directly; and it would seem that the same is true of\r\nthe good. In thought, at any rate, those who forget\r\ngood and evil and seek only to know the facts are more\r\nlikely to achieve good than those who view the world\r\nthrough the distorting medium of their own desires.\u003c/p\u003e\r\n\u003cp\u003eThe immense extension of our knowledge of facts in\r\nrecent times has had, as it had in the Renaissance, two\r\neffects upon the general intellectual outlook. On the one\r\nhand, it has made men distrustful of the truth of wide,\r\nambitious systems: theories come and go swiftly, each\r\nserving, for a moment, to classify known facts and promote\r\nthe search for new ones, but each in turn proving\r\ninadequate to deal with the new facts when they have\r\nbeen found. Even those who invent the theories do not,\r\nin science, regard them as anything but a temporary\r\nmakeshift. The ideal of an all-embracing synthesis, such\r\nas the Middle Ages believed themselves to have attained,\r\nrecedes further and further beyond the limits of what\r\nseems feasible. In such a world, as in the world of\r\nMontaigne, nothing seems worth while except the discovery\r\nof more and more facts, each in turn the deathblow\r\nto some cherished theory; the ordering intellect\r\ngrows weary, and becomes slovenly through despair.\u003c/p\u003e\r\n\u003cp\u003eOn the other hand, the new facts have brought new\r\npowers; man\u0027s physical control over natural forces has\r\n\u003ca class=\"pagenum\" title=\"29\" id=\"Page_29\"\u003e \u003c/a\u003e\r\nbeen increasing with unexampled rapidity, and promises\r\nto increase in the future beyond all easily assignable\r\nlimits. Thus alongside of despair as regards ultimate\r\ntheory there is an immense optimism as regards practice:\r\nwhat man can \u003cem\u003edo\u003c/em\u003e seems almost boundless. The old\r\nfixed limits of human power, such as death, or the dependence\r\nof the race on an equilibrium of cosmic forces,\r\nare forgotten, and no hard facts are allowed to break in\r\nupon the dream of omnipotence. No philosophy is\r\ntolerated which sets bounds to man\u0027s capacity of gratifying\r\nhis wishes; and thus the very despair of theory is\r\ninvoked to silence every whisper of doubt as regards the\r\npossibilities of practical achievement.\u003c/p\u003e\r\n\u003cp\u003eIn the welcoming of new fact, and in the suspicion of\r\ndogmatism as regards the universe at large, the modern\r\nspirit should, I think, be accepted as wholly an advance.\r\nBut both in its practical pretensions and in its theoretical\r\ndespair it seems to me to go too far. Most of what is\r\ngreatest in man is called forth in response to the thwarting\r\nof his hopes by immutable natural obstacles; by the\r\npretence of omnipotence, he becomes trivial and a little\r\nabsurd. And on the theoretical side, ultimate metaphysical\r\ntruth, though less all-embracing and harder of attainment\r\nthan it appeared to some philosophers in the past,\r\ncan, I believe, be discovered by those who are willing to\r\ncombine the hopefulness, patience, and open-mindedness\r\nof science with something of the Greek feeling for beauty\r\nin the abstract world of logic and for the ultimate intrinsic\r\nvalue in the contemplation of truth.\u003c/p\u003e\r\n\u003cp\u003eThe philosophy, therefore, which is to be genuinely\r\ninspired by the scientific spirit, must deal with somewhat\r\ndry and abstract matters, and must not hope to find an\r\nanswer to the practical problems of life. To those who\r\nwish to understand much of what has in the past been\r\n\u003ca class=\"pagenum\" title=\"30\" id=\"Page_30\"\u003e \u003c/a\u003e\r\nmost difficult and obscure in the constitution of the\r\nuniverse, it has great rewards to offer—triumphs as noteworthy\r\nas those of Newton and Darwin, and as important\r\nin the long run, for the moulding of our mental habits.\r\nAnd it brings with it—as a new and powerful method of\r\ninvestigation always does—a sense of power and a hope\r\nof progress more reliable and better grounded than any\r\nthat rests on hasty and fallacious generalisation as to the\r\nnature of the universe at large. Many hopes which inspired\r\nphilosophers in the past it cannot claim to fulfil;\r\nbut other hopes, more purely intellectual, it can satisfy\r\nmore fully than former ages could have deemed possible\r\nfor human minds.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"31\" id=\"Page_31\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE II\u003c/small\u003e\u003cbr\u003e\r\nLOGIC AS THE ESSENCE OF\r\nPHILOSOPHY\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"33\" id=\"Page_33\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_2\"\u003eLECTURE II\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nLOGIC AS THE ESSENCE OF PHILOSOPHY\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eThe\u003c/span\u003e topics we discussed in our \u003ca href=\"#Lecture_1\" class=\"pginternal\"\u003efirst lecture\u003c/a\u003e, and the\r\ntopics we shall discuss later, all reduce themselves, in\r\nso far as they are genuinely philosophical, to problems\r\nof logic. This is not due to any accident, but to the\r\nfact that every philosophical problem, when it is subjected\r\nto the necessary analysis and purification, is found either\r\nto be not really philosophical at all, or else to be, in the\r\nsense in which we are using the word, logical. But as\r\nthe word “logic” is never used in the same sense by two\r\ndifferent philosophers, some explanation of what I mean\r\nby the word is indispensable at the outset.\u003c/p\u003e\r\n\u003cp\u003eLogic, in the Middle Ages, and down to the present\r\nday in teaching, meant no more than a scholastic collection\r\nof technical terms and rules of syllogistic inference.\r\nAristotle had spoken, and it was the part of humbler\r\nmen merely to repeat the lesson after him. The trivial\r\nnonsense embodied in this tradition is still set in examinations,\r\nand defended by eminent authorities as an excellent\r\n“propædeutic,” \u003ci\u003ei.e.\u003c/i\u003e a training in those habits of solemn\r\nhumbug which are so great a help in later life. But it is\r\nnot this that I mean to praise in saying that all philosophy\r\nis logic. Ever since the beginning of the seventeenth\r\ncentury, all vigorous minds that have concerned themselves\r\nwith inference have abandoned the mediæval\r\n\u003ca class=\"pagenum\" title=\"34\" id=\"Page_34\"\u003e \u003c/a\u003e\r\ntradition, and in one way or other have widened the\r\nscope of logic.\u003c/p\u003e\r\n\u003cp\u003eThe first extension was the introduction of the\r\ninductive method by Bacon and Galileo—by the former\r\nin a theoretical and largely mistaken form, by the latter\r\nin actual use in establishing the foundations of modern\r\nphysics and astronomy. This is probably the only\r\nextension of the old logic which has become familiar\r\nto the general educated public. But induction, important\r\nas it is when regarded as a method of investigation, does\r\nnot seem to remain when its work is done: in the final\r\nform of a perfected science, it would seem that everything\r\nought to be deductive. If induction remains at all,\r\nwhich is a difficult question, it will remain merely as one\r\nof the principles according to which deductions are\r\neffected. Thus the ultimate result of the introduction\r\nof the inductive method seems not the creation of a new\r\nkind of non-deductive reasoning, but rather the widening\r\nof the scope of deduction by pointing out a way of\r\ndeducing which is certainly not syllogistic, and does not\r\nfit into the mediæval scheme.\u003c/p\u003e\r\n\u003cp\u003eThe question of the scope and validity of induction\r\nis of great difficulty, and of great importance to our\r\nknowledge. Take such a question as, “Will the sun rise\r\nto-morrow?” Our first instinctive feeling is that we\r\nhave abundant reason for saying that it will, because it\r\nhas risen on so many previous mornings. Now, I do not\r\nmyself know whether this does afford a ground or not,\r\nbut I am willing to suppose that it does. The question\r\nwhich then arises is: What is the principle of inference\r\nby which we pass from past sunrises to future ones?\r\nThe answer given by Mill is that the inference depends\r\nupon the law of causation. Let us suppose this to be\r\ntrue; then what is the reason for believing in the law of\r\n\u003ca class=\"pagenum\" title=\"35\" id=\"Page_35\"\u003e \u003c/a\u003e\r\ncausation? There are broadly three possible answers:\r\n(1) that it is itself known \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e; (2) that it is a\r\npostulate; (3) that it is an empirical generalisation from\r\npast instances in which it has been found to hold. The\r\ntheory that causation is known \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e cannot be definitely\r\nrefuted, but it can be rendered very unplausible by\r\nthe mere process of formulating the law exactly, and\r\nthereby showing that it is immensely more complicated\r\nand less obvious than is generally supposed. The theory\r\nthat causation is a postulate, \u003ci\u003ei.e.\u003c/i\u003e that it is something which\r\nwe choose to assert although we know that it is very\r\nlikely false, is also incapable of refutation; but it is\r\nplainly also incapable of justifying any use of the law\r\nin inference. We are thus brought to the theory that\r\nthe law is an empirical generalisation, which is the view\r\nheld by Mill.\u003c/p\u003e\r\n\u003cp\u003eBut if so, how are empirical generalisations to be\r\njustified? The evidence in their favour cannot be\r\nempirical, since we wish to argue from what has been\r\nobserved to what has not been observed, which can only\r\nbe done by means of some known relation of the observed\r\nand the unobserved; but the unobserved, by definition,\r\nis not known empirically, and therefore its relation to\r\nthe observed, if known at all, must be known independently\r\nof empirical evidence. Let us see what Mill says\r\non this subject.\u003c/p\u003e\r\n\u003cp\u003eAccording to Mill, the law of causation is proved by\r\nan admittedly fallible process called “induction by simple\r\nenumeration.” This process, he says, “consists in\r\nascribing the nature of general truths to all propositions\r\nwhich are true in every instance that we happen to know\r\nof.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_8\" id=\"FNanchor_8\"\u003e[8]\u003c/a\u003e As regards its fallibility, he asserts that “the\r\nprecariousness of the method of simple enumeration is in\r\n\u003ca class=\"pagenum\" title=\"36\" id=\"Page_36\"\u003e \u003c/a\u003e\r\nan inverse ratio to the largeness of the generalisation.\r\nThe process is delusive and insufficient, exactly in\r\nproportion as the subject-matter of the observation is\r\nspecial and limited in extent. As the sphere widens, this\r\nunscientific method becomes less and less liable to\r\nmislead; and the most universal class of truths, the law\r\nof causation for instance, and the principles of number\r\nand of geometry, are duly and satisfactorily proved by\r\nthat method alone, nor are they susceptible of any other\r\nproof.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_9\" id=\"FNanchor_9\"\u003e[9]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eIn the above statement, there are two obvious lacunæ:\r\n(1) How is the method of simple enumeration itself\r\njustified? (2) What logical principle, if any, covers the\r\nsame ground as this method, without being liable to its\r\nfailures? Let us take the second question first.\u003c/p\u003e\r\n\u003cp\u003eA method of proof which, when used as directed, gives\r\nsometimes truth and sometimes falsehood—as the method\r\nof simple enumeration does—is obviously not a valid\r\nmethod, for validity demands invariable truth. Thus, if\r\nsimple enumeration is to be rendered valid, it must not\r\nbe stated as Mill states it. We shall have to say, at\r\nmost, that the data render the result \u003cem\u003eprobable\u003c/em\u003e. Causation\r\nholds, we shall say, in every instance we have been able\r\nto test; therefore it \u003cem\u003eprobably\u003c/em\u003e holds in untested instances.\r\nThere are terrible difficulties in the notion of probability,\r\nbut we may ignore them at present. We thus have what\r\nat least \u003cem\u003emay\u003c/em\u003e be a logical principle, since it is without\r\nexception. If a proposition is true in every instance that\r\nwe happen to know of, and if the instances are very\r\nnumerous, then, we shall say, it becomes very probable,\r\non the data, that it will be true in any further instance.\r\nThis is not refuted by the fact that what we declare to be\r\nprobable does not always happen, for an event may be\r\n\u003ca class=\"pagenum\" title=\"37\" id=\"Page_37\"\u003e \u003c/a\u003e\r\nprobable on the data and yet not occur. It is, however,\r\nobviously capable of further analysis, and of more exact\r\nstatement. We shall have to say something like this:\r\nthat every instance of a proposition\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_10\" id=\"FNanchor_10\"\u003e[10]\u003c/a\u003e being true increases\r\nthe probability of its being true in a fresh instance, and\r\nthat a sufficient number of favourable instances will, in\r\nthe absence of instances to the contrary, make the probability\r\nof the truth of a fresh instance approach indefinitely\r\nnear to certainty. Some such principle as this is required\r\nif the method of simple enumeration is to be valid.\u003c/p\u003e\r\n\u003cp\u003eBut this brings us to our other question, namely, how\r\nis our principle known to be true? Obviously, since it\r\nis required to justify induction, it cannot be proved by\r\ninduction; since it goes beyond the empirical data, it\r\ncannot be proved by them alone; since it is required to\r\njustify all inferences from empirical data to what goes\r\nbeyond them, it cannot itself be even rendered in any\r\ndegree probable by such data. Hence, \u003cem\u003eif\u003c/em\u003e it is known,\r\nit is not known by experience, but independently of\r\nexperience. I do not say that any such principle is\r\nknown: I only say that it is required to justify the\r\ninferences from experience which empiricists allow, and\r\nthat it cannot itself be justified empirically.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_11\" id=\"FNanchor_11\"\u003e[11]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eA similar conclusion can be proved by similar\r\narguments concerning any other logical principle. Thus\r\nlogical knowledge is not derivable from experience alone,\r\nand the empiricist\u0027s philosophy can therefore not be\r\naccepted in its entirety, in spite of its excellence in many\r\nmatters which lie outside logic.\u003c/p\u003e\r\n\u003cp\u003eHegel and his followers widened the scope of logic in\r\nquite a different way—a way which I believe to be\r\n\u003ca class=\"pagenum\" title=\"38\" id=\"Page_38\"\u003e \u003c/a\u003e\r\nfallacious, but which requires discussion if only to show\r\nhow their conception of logic differs from the conception\r\nwhich I wish to advocate. In their writings, logic is\r\npractically identical with metaphysics. In broad outline,\r\nthe way this came about is as follows. Hegel believed\r\nthat, by means of \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e reasoning, it could be shown\r\nthat the world \u003cem\u003emust\u003c/em\u003e have various important and interesting\r\ncharacteristics, since any world without these characteristics\r\nwould be impossible and self-contradictory. Thus\r\nwhat he calls “logic” is an investigation of the nature of\r\nthe universe, in so far as this can be inferred merely\r\nfrom the principle that the universe must be logically\r\nself-consistent. I do not myself believe that from this\r\nprinciple alone anything of importance can be inferred as\r\nregards the existing universe. But, however that may\r\nbe, I should not regard Hegel\u0027s reasoning, even if it\r\nwere valid, as properly belonging to logic: it would\r\nrather be an application of logic to the actual world.\r\nLogic itself would be concerned rather with such\r\nquestions as what self-consistency is, which Hegel, so far\r\nas I know, does not discuss. And though he criticises\r\nthe traditional logic, and professes to replace it by an\r\nimproved logic of his own, there is some sense in which\r\nthe traditional logic, with all its faults, is uncritically and\r\nunconsciously assumed throughout his reasoning. It is\r\nnot in the direction advocated by him, it seems to me,\r\nthat the reform of logic is to be sought, but by a more\r\nfundamental, more patient, and less ambitious investigation\r\ninto the presuppositions which his system shares\r\nwith those of most other philosophers.\u003c/p\u003e\r\n\u003cp\u003eThe way in which, as it seems to me, Hegel\u0027s system\r\nassumes the ordinary logic which it subsequently criticises,\r\nis exemplified by the general conception of “categories”\r\nwith which he operates throughout. This conception is,\r\n\u003ca class=\"pagenum\" title=\"39\" id=\"Page_39\"\u003e \u003c/a\u003e\r\nI think, essentially a product of logical confusion, but it\r\nseems in some way to stand for the conception of\r\n“qualities of Reality as a whole.” Mr Bradley has\r\nworked out a theory according to which, in all judgment,\r\nwe are ascribing a predicate to Reality as a whole; and\r\nthis theory is derived from Hegel. Now the traditional\r\nlogic holds that every proposition ascribes a predicate to a\r\nsubject, and from this it easily follows that there can be\r\nonly one subject, the Absolute, for if there were two, the\r\nproposition that there were two would not ascribe a\r\npredicate to either. Thus Hegel\u0027s doctrine, that philosophical\r\npropositions must be of the form, “the Absolute\r\nis such-and-such,” depends upon the traditional belief in\r\nthe universality of the subject-predicate form. This\r\nbelief, being traditional, scarcely self-conscious, and not\r\nsupposed to be important, operates underground, and\r\nis assumed in arguments which, like the refutation of\r\nrelations, appear at first sight such as to establish its\r\ntruth. This is the most important respect in which\r\nHegel uncritically assumes the traditional logic. Other\r\nless important respects—though important enough to be\r\nthe source of such essentially Hegelian conceptions as\r\nthe “concrete universal” and the “union of identity in\r\ndifference”—will be found where he explicitly deals with\r\nformal logic.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_12\" id=\"FNanchor_12\"\u003e[12]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eThere is quite another direction in which a large\r\n\u003ca class=\"pagenum\" title=\"40\" id=\"Page_40\"\u003e \u003c/a\u003e\r\ntechnical development of logic has taken place: I mean\r\nthe direction of what is called logistic or mathematical\r\nlogic. This kind of logic is mathematical in two different\r\nsenses: it is itself a branch of mathematics, and it is the\r\nlogic which is specially applicable to other more traditional\r\nbranches of mathematics. Historically, it began as \u003cem\u003emerely\u003c/em\u003e\r\na branch of mathematics: its special applicability to other\r\nbranches is a more recent development. In both respects,\r\nit is the fulfilment of a hope which Leibniz cherished\r\nthroughout his life, and pursued with all the ardour of\r\nhis amazing intellectual energy. Much of his work on\r\nthis subject has been published recently, since his discoveries\r\nhave been remade by others; but none was\r\npublished by him, because his results persisted in contradicting\r\ncertain points in the traditional doctrine of the\r\nsyllogism. We now know that on these points the\r\ntraditional doctrine is wrong, but respect for Aristotle\r\nprevented Leibniz from realising that this was possible.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_13\" id=\"FNanchor_13\"\u003e[13]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eThe modern development of mathematical logic dates\r\nfrom Boole\u0027s \u003ccite\u003eLaws of Thought\u003c/cite\u003e (1854). But in him and\r\nhis successors, before Peano and Frege, the only thing\r\nreally achieved, apart from certain details, was the invention\r\nof a mathematical symbolism for deducing\r\nconsequences from the premisses which the newer\r\nmethods shared with those of Aristotle. This subject\r\nhas considerable interest as an independent branch of\r\nmathematics, but it has very little to do with real logic.\r\nThe first serious advance in real logic since the time of\r\n\u003ca class=\"pagenum\" title=\"41\" id=\"Page_41\"\u003e \u003c/a\u003e\r\nthe Greeks was made independently by Peano and Frege—both\r\nmathematicians. They both arrived at their\r\nlogical results by an analysis of mathematics. Traditional\r\nlogic regarded the two propositions, “Socrates is mortal”\r\nand “All men are mortal,” as being of the same form;\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_14\" id=\"FNanchor_14\"\u003e[14]\u003c/a\u003e\r\nPeano and Frege showed that they are utterly different in\r\nform. The philosophical importance of logic may be\r\nillustrated by the fact that this confusion—which is still\r\ncommitted by most writers—obscured not only the whole\r\nstudy of the forms of judgment and inference, but also\r\nthe relations of things to their qualities, of concrete\r\nexistence to abstract concepts, and of the world of sense\r\nto the world of Platonic ideas. Peano and Frege, who\r\npointed out the error, did so for technical reasons, and\r\napplied their logic mainly to technical developments; but\r\nthe philosophical importance of the advance which they\r\nmade is impossible to exaggerate.\u003c/p\u003e\r\n\u003cp\u003eMathematical logic, even in its most modern form, is\r\nnot \u003cem\u003edirectly\u003c/em\u003e of philosophical importance except in its\r\nbeginnings. After the beginnings, it belongs rather to\r\nmathematics than to philosophy. Of its beginnings,\r\nwhich are the only part of it that can properly be called\r\n\u003cem\u003ephilosophical\u003c/em\u003e logic, I shall speak shortly. But even the\r\nlater developments, though not directly philosophical, will\r\nbe found of great indirect use in philosophising. They\r\nenable us to deal easily with more abstract conceptions\r\nthan merely verbal reasoning can enumerate; they\r\nsuggest fruitful hypotheses which otherwise could hardly\r\nbe thought of; and they enable us to see quickly what is\r\nthe smallest store of materials with which a given logical\r\nor scientific edifice can be constructed. Not only Frege\u0027s\r\n\u003ca class=\"pagenum\" title=\"42\" id=\"Page_42\"\u003e \u003c/a\u003e\r\ntheory of number, which we shall deal with in \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003eLecture VII.\u003c/a\u003e,\r\nbut the whole theory of physical concepts which\r\nwill be outlined in our next two lectures, is inspired by\r\nmathematical logic, and could never have been imagined\r\nwithout it.\u003c/p\u003e\r\n\u003cp\u003eIn both these cases, and in many others, we shall\r\nappeal to a certain principle called “the principle of\r\nabstraction.” This principle, which might equally well\r\nbe called “the principle which dispenses with abstraction,”\r\nand is one which clears away incredible accumulations of\r\nmetaphysical lumber, was directly suggested by mathematical\r\nlogic, and could hardly have been proved or\r\npractically used without its help. The principle will be\r\nexplained in our \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003efourth lecture\u003c/a\u003e, but its use may be briefly\r\nindicated in advance. When a group of objects have\r\nthat kind of similarity which we are inclined to attribute\r\nto possession of a common quality, the principle in\r\nquestion shows that membership of the group will serve\r\nall the purposes of the supposed common quality, and\r\nthat therefore, unless some common quality is actually\r\nknown, the group or class of similar objects may be used\r\nto replace the common quality, which need not be assumed\r\nto exist. In this and other ways, the indirect uses of\r\neven the later parts of mathematical logic are very great;\r\nbut it is now time to turn our attention to its philosophical\r\nfoundations.\u003c/p\u003e\r\n\u003cp\u003eIn every proposition and in every inference there is,\r\nbesides the particular subject-matter concerned, a certain\r\n\u003cem\u003eform\u003c/em\u003e, a way in which the constituents of the proposition or\r\ninference are put together. If I say, “Socrates is mortal,”\r\n“Jones is angry,” “The sun is hot,” there is something\r\nin common in these three cases, something indicated\r\nby the word “is.” What is in common is the \u003cem\u003eform\u003c/em\u003e of the\r\nproposition, not an actual constituent. If I say a number\r\n\u003ca class=\"pagenum\" title=\"43\" id=\"Page_43\"\u003e \u003c/a\u003e\r\nof things about Socrates—that he was an Athenian, that\r\nhe married Xantippe, that he drank the hemlock—there\r\nis a common constituent, namely Socrates, in all the propositions\r\nI enunciate, but they have diverse forms. If,\r\non the other hand, I take any one of these propositions\r\nand replace its constituents, one at a time, by other constituents,\r\nthe form remains constant, but no constituent\r\nremains. Take (say) the series of propositions, “Socrates\r\ndrank the hemlock,” “Coleridge drank the hemlock,”\r\n“Coleridge drank opium,” “Coleridge ate opium.” The\r\nform remains unchanged throughout this series, but all\r\nthe constituents are altered. Thus form is not another\r\nconstituent, but is the way the constituents are put\r\ntogether. It is forms, in this sense, that are the proper\r\nobject of philosophical logic.\u003c/p\u003e\r\n\u003cp\u003eIt is obvious that the knowledge of logical forms is\r\nsomething quite different from knowledge of existing\r\nthings. The form of “Socrates drank the hemlock” is\r\nnot an existing thing like Socrates or the hemlock, nor\r\ndoes it even have that close relation to existing things\r\nthat drinking has. It is something altogether more\r\nabstract and remote. We might understand all the\r\nseparate words of a sentence without understanding the\r\nsentence: if a sentence is long and complicated, this is\r\napt to happen. In such a case we have knowledge of the\r\nconstituents, but not of the form. We may also have\r\nknowledge of the form without having knowledge of the\r\nconstituents. If I say, “Rorarius drank the hemlock,”\r\nthose among you who have never heard of Rorarius (supposing\r\nthere are any) will understand the form, without\r\nhaving knowledge of all the constituents. In order to\r\nunderstand a sentence, it is necessary to have knowledge\r\nboth of the constituents and of the particular instance of\r\nthe form. It is in this way that a sentence conveys\r\n\u003ca class=\"pagenum\" title=\"44\" id=\"Page_44\"\u003e \u003c/a\u003e\r\ninformation, since it tells us that certain known objects\r\nare related according to a certain known form. Thus\r\nsome kind of knowledge of logical forms, though with\r\nmost people it is not explicit, is involved in all understanding\r\nof discourse. It is the business of philosophical\r\nlogic to extract this knowledge from its concrete integuments,\r\nand to render it explicit and pure.\u003c/p\u003e\r\n\u003cp\u003eIn all inference, form alone is essential: the particular\r\nsubject-matter is irrelevant except as securing the truth\r\nof the premisses. This is one reason for the great importance\r\nof logical form. When I say, “Socrates was a\r\nman, all men are mortal, therefore Socrates was mortal,”\r\nthe connection of premisses and conclusion does not in\r\nany way depend upon its being Socrates and man and\r\nmortality that I am mentioning. The general form of\r\nthe inference may be expressed in some such words as,\r\n“If a thing has a certain property, and whatever has this\r\nproperty has a certain other property, then the thing in\r\nquestion also has that other property.” Here no particular\r\nthings or properties are mentioned: the proposition\r\nis absolutely general. All inferences, when stated fully,\r\nare instances of propositions having this kind of generality.\r\nIf they seem to depend upon the subject-matter otherwise\r\nthan as regards the truth of the premisses, that is because\r\nthe premisses have not been all explicitly stated. In\r\nlogic, it is a waste of time to deal with inferences concerning\r\nparticular cases: we deal throughout with completely\r\ngeneral and purely formal implications, leaving it to\r\nother sciences to discover when the hypotheses are\r\nverified and when they are not.\u003c/p\u003e\r\n\u003cp\u003eBut the forms of propositions giving rise to inferences\r\nare not the simplest forms: they are always hypothetical,\r\nstating that if one proposition is true, then so is another.\r\nBefore considering inference, therefore, logic must consider\r\n\u003ca class=\"pagenum\" title=\"45\" id=\"Page_45\"\u003e \u003c/a\u003e\r\nthose simpler forms which inference presupposes.\r\nHere the traditional logic failed completely: it believed\r\nthat there was only one form of simple proposition (\u003ci\u003ei.e.\u003c/i\u003e\r\nof proposition not stating a relation between two or more\r\nother propositions), namely, the form which ascribes a\r\npredicate to a subject. This is the appropriate form in\r\nassigning the qualities of a given thing—we may say\r\n“this thing is round, and red, and so on.” Grammar\r\nfavours this form, but philosophically it is so far from\r\nuniversal that it is not even very common. If we say\r\n“this thing is bigger than that,” we are not assigning a\r\nmere quality of “this,” but a relation of “this” and “that.”\r\nWe might express the same fact by saying “that thing\r\nis smaller than this,” where grammatically the subject is\r\nchanged. Thus propositions stating that two things have\r\na certain relation have a different form from subject-predicate\r\npropositions, and the failure to perceive this\r\ndifference or to allow for it has been the source of many\r\nerrors in traditional metaphysics.\u003c/p\u003e\r\n\u003cp\u003eThe belief or unconscious conviction that all propositions\r\nare of the subject-predicate form—in other words,\r\nthat every fact consists in some thing having some quality—has\r\nrendered most philosophers incapable of giving any\r\naccount of the world of science and daily life. If they\r\nhad been honestly anxious to give such an account, they\r\nwould probably have discovered their error very quickly;\r\nbut most of them were less anxious to understand the\r\nworld of science and daily life, than to convict it of\r\nunreality in the interests of a super-sensible “real”\r\nworld. Belief in the unreality of the world of sense\r\narises with irresistible force in certain moods—moods\r\nwhich, I imagine, have some simple physiological basis,\r\nbut are none the less powerfully persuasive. The conviction\r\nborn of these moods is the source of most\r\n\u003ca class=\"pagenum\" title=\"46\" id=\"Page_46\"\u003e \u003c/a\u003e\r\nmysticism and of most metaphysics. When the emotional\r\nintensity of such a mood subsides, a man who is\r\nin the habit of reasoning will search for logical reasons\r\nin favour of the belief which he finds in himself. But\r\nsince the belief already exists, he will be very hospitable\r\nto any reason that suggests itself. The paradoxes apparently\r\nproved by his logic are really the paradoxes of\r\nmysticism, and are the goal which he feels his logic must\r\nreach if it is to be in accordance with insight. It is in\r\nthis way that logic has been pursued by those of the\r\ngreat philosophers who were mystics—notably Plato,\r\nSpinoza, and Hegel. But since they usually took for\r\ngranted the supposed insight of the mystic emotion, their\r\nlogical doctrines were presented with a certain dryness,\r\nand were believed by their disciples to be quite independent\r\nof the sudden illumination from which they\r\nsprang. Nevertheless their origin clung to them, and they\r\nremained—to borrow a useful word from Mr Santayana—“malicious”\r\nin regard to the world of science and\r\ncommon sense. It is only so that we can account for\r\nthe complacency with which philosophers have accepted\r\nthe inconsistency of their doctrines with all the common\r\nand scientific facts which seem best established and most\r\nworthy of belief.\u003c/p\u003e\r\n\u003cp\u003eThe logic of mysticism shows, as is natural, the defects\r\nwhich are inherent in anything malicious. While the\r\nmystic mood is dominant, the need of logic is not felt;\r\nas the mood fades, the impulse to logic reasserts itself,\r\nbut with a desire to retain the vanishing insight, or at\r\nleast to prove that it \u003cem\u003ewas\u003c/em\u003e insight, and that what seems to\r\ncontradict it is illusion. The logic which thus arises is\r\nnot quite disinterested or candid, and is inspired by a\r\ncertain hatred of the daily world to which it is to be\r\napplied. Such an attitude naturally does not tend to\r\n\u003ca class=\"pagenum\" title=\"47\" id=\"Page_47\"\u003e \u003c/a\u003e\r\nthe best results. Everyone knows that to read an author\r\nsimply in order to refute him is not the way to understand\r\nhim; and to read the book of Nature with a conviction\r\nthat it is all illusion is just as unlikely to lead to understanding.\r\nIf our logic is to find the common world\r\nintelligible, it must not be hostile, but must be inspired\r\nby a genuine acceptance such as is not usually to be\r\nfound among metaphysicians.\u003c/p\u003e\r\n\u003cp\u003eTraditional logic, since it holds that all propositions\r\nhave the subject-predicate form, is unable to admit the\r\nreality of relations: all relations, it maintains, must be\r\nreduced to properties of the apparently related terms.\r\nThere are many ways of refuting this opinion; one of\r\nthe easiest is derived from the consideration of what are\r\ncalled “asymmetrical” relations. In order to explain\r\nthis, I will first explain two independent ways of classifying\r\nrelations.\u003c/p\u003e\r\n\u003cp\u003eSome relations, when they hold between A and B, also\r\nhold between B and A. Such, for example, is the relation\r\n“brother or sister.” If A is a brother or sister of B,\r\nthen B is a brother or sister of A. Such again is any\r\nkind of similarity, say similarity of colour. Any kind of\r\ndissimilarity is also of this kind: if the colour of A is\r\nunlike the colour of B, then the colour of B is unlike\r\nthe colour of A. Relations of this sort are called \u003cem\u003esymmetrical\u003c/em\u003e.\r\nThus a relation is symmetrical if, whenever it\r\nholds between A and B, it also holds between B and A.\u003c/p\u003e\r\n\u003cp\u003eAll relations that are not symmetrical are called \u003cem\u003enon-symmetrical\u003c/em\u003e.\r\nThus “brother” is non-symmetrical, because,\r\nif A is a brother of B, it may happen that B is a \u003cem\u003esister\u003c/em\u003e\r\nof A.\u003c/p\u003e\r\n\u003cp\u003eA relation is called \u003cem\u003easymmetrical\u003c/em\u003e when, if it holds\r\nbetween A and B, it \u003cem\u003enever\u003c/em\u003e holds between B and A.\r\nThus husband, father, grandfather, etc., are asymmetrical\r\n\u003ca class=\"pagenum\" title=\"48\" id=\"Page_48\"\u003e \u003c/a\u003e\r\nrelations. So are \u003cem\u003ebefore\u003c/em\u003e, \u003cem\u003eafter\u003c/em\u003e, \u003cem\u003egreater\u003c/em\u003e, \u003cem\u003eabove\u003c/em\u003e, \u003cem\u003eto the right\r\nof\u003c/em\u003e, etc. All the relations that give rise to series are of\r\nthis kind.\u003c/p\u003e\r\n\u003cp\u003eClassification into symmetrical, asymmetrical, and merely\r\nnon-symmetrical relations is the first of the two classifications\r\nwe had to consider. The second is into transitive,\r\nintransitive, and merely non-transitive relations, which\r\nare defined as follows.\u003c/p\u003e\r\n\u003cp\u003eA relation is said to be \u003cem\u003etransitive\u003c/em\u003e, if, whenever it holds\r\nbetween A and B and also between B and C, it holds\r\nbetween A and C. Thus \u003cem\u003ebefore\u003c/em\u003e, \u003cem\u003eafter\u003c/em\u003e, \u003cem\u003egreater\u003c/em\u003e, \u003cem\u003eabove\u003c/em\u003e are\r\ntransitive. All relations giving rise to series are transitive,\r\nbut so are many others. The transitive relations\r\njust mentioned were asymmetrical, but many transitive\r\nrelations are symmetrical—for instance, equality in any\r\nrespect, exact identity of colour, being equally numerous\r\n(as applied to collections), and so on.\u003c/p\u003e\r\n\u003cp\u003eA relation is said to be \u003cem\u003enon-transitive\u003c/em\u003e whenever it is not\r\ntransitive. Thus “brother” is non-transitive, because\r\na brother of one\u0027s brother may be oneself. All kinds of\r\ndissimilarity are non-transitive.\u003c/p\u003e\r\n\u003cp\u003eA relation is said to be \u003cem\u003eintransitive\u003c/em\u003e when, if A has the\r\nrelation to B, and B to C, A never has it to C. Thus\r\n“father” is intransitive. So is such a relation as “one\r\ninch taller” or “one year later.”\u003c/p\u003e\r\n\u003cp\u003eLet us now, in the light of this classification, return\r\nto the question whether all relations can be reduced to\r\npredications.\u003c/p\u003e\r\n\u003cp\u003eIn the case of symmetrical relations—\u003ci\u003ei.e.\u003c/i\u003e relations which,\r\nif they hold between A and B, also hold between B and\r\nA—some kind of plausibility can be given to this doctrine.\r\nA symmetrical relation which is transitive, such as equality,\r\ncan be regarded as expressing possession of some common\r\nproperty, while one which is not transitive, such as\r\n\u003ca class=\"pagenum\" title=\"49\" id=\"Page_49\"\u003e \u003c/a\u003e\r\ninequality, can be regarded as expressing possession of\r\ndifferent properties. But when we come to asymmetrical\r\nrelations, such as before and after, greater and less, etc.,\r\nthe attempt to reduce them to properties becomes obviously\r\nimpossible. When, for example, two things are\r\nmerely known to be unequal, without our knowing which\r\nis greater, we may say that the inequality results from\r\ntheir having different magnitudes, because inequality is\r\na symmetrical relation; but to say that when one thing is\r\n\u003cem\u003egreater\u003c/em\u003e than another, and not merely unequal to it, that\r\nmeans that they have different magnitudes, is formally\r\nincapable of explaining the facts. For if the other thing\r\nhad been greater than the one, the magnitudes would\r\nalso have been different, though the fact to be explained\r\nwould not have been the same. Thus mere \u003cem\u003edifference\u003c/em\u003e of\r\nmagnitude is not \u003cem\u003eall\u003c/em\u003e that is involved, since, if it were,\r\nthere would be no difference between one thing being\r\ngreater than another, and the other being greater than\r\nthe one. We shall have to say that the one magnitude is\r\n\u003cem\u003egreater\u003c/em\u003e than the other, and thus we shall have failed to get\r\nrid of the relation “greater.” In short, both possession\r\nof the same property and possession of different properties\r\nare \u003cem\u003esymmetrical\u003c/em\u003e relations, and therefore cannot account for\r\nthe existence of \u003cem\u003easymmetrical\u003c/em\u003e relations.\u003c/p\u003e\r\n\u003cp\u003eAsymmetrical relations are involved in all series—in\r\nspace and time, greater and less, whole and part, and\r\nmany others of the most important characteristics of the\r\nactual world. All these aspects, therefore, the logic which\r\nreduces everything to subjects and predicates is compelled\r\nto condemn as error and mere appearance. To those\r\nwhose logic is not malicious, such a wholesale condemnation\r\nappears impossible. And in fact there is no reason\r\nexcept prejudice, so far as I can discover, for denying the\r\nreality of relations. When once their reality is admitted,\r\n\u003ca class=\"pagenum\" title=\"50\" id=\"Page_50\"\u003e \u003c/a\u003e\r\nall \u003cem\u003elogical\u003c/em\u003e grounds for supposing the world of sense to be\r\nillusory disappear. If this is to be supposed, it must be\r\nfrankly and simply on the ground of mystic insight\r\nunsupported by argument. It is impossible to argue\r\nagainst what professes to be insight, so long as it does not\r\nargue in its own favour. As logicians, therefore, we may\r\nadmit the possibility of the mystic\u0027s world, while yet, so\r\nlong as we do not have his insight, we must continue to\r\nstudy the everyday world with which we are familiar.\r\nBut when he contends that our world is impossible, then\r\nour logic is ready to repel his attack. And the first\r\nstep in creating the logic which is to perform this service\r\nis the recognition of the reality of relations.\u003c/p\u003e\r\n\u003cp\u003eRelations which have two terms are only one kind of\r\nrelations. A relation may have three terms, or four, or\r\nany number. Relations of two terms, being the simplest,\r\nhave received more attention than the others, and have\r\ngenerally been alone considered by philosophers, both\r\nthose who accepted and those who denied the reality of\r\nrelations. But other relations have their importance, and\r\nare indispensable in the solution of certain problems.\r\nJealousy, for example, is a relation between three people.\r\nProfessor Royce mentions the relation “giving”: when\r\nA gives B to C, that is a relation of three terms.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_15\" id=\"FNanchor_15\"\u003e[15]\u003c/a\u003e When\r\na man says to his wife: “My dear, I wish you could\r\ninduce Angelina to accept Edwin,” his wish constitutes\r\na relation between four people, himself, his wife, Angelina,\r\nand Edwin. Thus such relations are by no means\r\nrecondite or rare. But in order to explain exactly how\r\nthey differ from relations of two terms, we must embark\r\nupon a classification of the logical forms of facts, which is\r\nthe first business of logic, and the business in which the\r\ntraditional logic has been most deficient.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"51\" id=\"Page_51\"\u003e \u003c/a\u003eThe existing world consists of many things with many\r\nqualities and relations. A complete description of the\r\nexisting world would require not only a catalogue of the\r\nthings, but also a mention of all their qualities and relations.\r\nWe should have to know not only this, that, and\r\nthe other thing, but also which was red, which yellow,\r\nwhich was earlier than which, which was between which\r\ntwo others, and so on. When I speak of a “fact,” I do\r\nnot mean one of the simple things in the world; I mean\r\nthat a certain thing has a certain quality, or that certain\r\nthings have a certain relation. Thus, for example, I should\r\nnot call Napoleon a fact, but I should call it a fact that\r\nhe was ambitious, or that he married Josephine. Now a\r\nfact, in this sense, is never simple, but always has two or\r\nmore constituents. When it simply assigns a quality to\r\na thing, it has only two constituents, the thing and the\r\nquality. When it consists of a relation between two\r\nthings, it has three constituents, the things and the relation.\r\nWhen it consists of a relation between three things,\r\nit has four constituents, and so on. The constituents of\r\nfacts, in the sense in which we are using the word “fact,”\r\nare not other facts, but are things and qualities or relations.\r\nWhen we say that there are relations of more than two\r\nterms, we mean that there are single facts consisting of a\r\nsingle relation and more than two things. I do not mean\r\nthat one relation of two terms may hold between A and\r\nB, and also between A and C, as, for example, a man is\r\nthe son of his father and also the son of his mother.\r\nThis constitutes two distinct facts: if we choose to treat\r\nit as one fact, it is a fact which has facts for its constituents.\r\nBut the facts I am speaking of have no facts among\r\ntheir constituents, but only things and relations. For\r\nexample, when A is jealous of B on account of C, there\r\nis only one fact, involving three people; there are not\r\n\u003ca class=\"pagenum\" title=\"52\" id=\"Page_52\"\u003e \u003c/a\u003e\r\ntwo instances of jealousy, but only one. It is in such\r\ncases that I speak of a relation of three terms, where the\r\nsimplest possible fact in which the relation occurs is one\r\ninvolving three things in addition to the relation. And\r\nthe same applies to relations of four terms or five or any\r\nother number. All such relations must be admitted in\r\nour inventory of the logical forms of facts: two facts\r\ninvolving the same number of things have the same form,\r\nand two which involve different numbers of things have\r\ndifferent forms.\u003c/p\u003e\r\n\u003cp\u003eGiven any fact, there is an assertion which expresses\r\nthe fact. The fact itself is objective, and independent of\r\nour thought or opinion about it; but the assertion is\r\nsomething which involves thought, and may be either\r\ntrue or false. An assertion may be positive or negative:\r\nwe may assert that Charles I. was executed, or that he\r\ndid \u003cem\u003enot\u003c/em\u003e die in his bed. A negative assertion may be said\r\nto be a \u003cem\u003edenial\u003c/em\u003e. Given a form of words which must be\r\neither true or false, such as “Charles I. died in his bed,”\r\nwe may either assert or deny this form of words: in the\r\none case we have a positive assertion, in the other a\r\nnegative one. A form of words which must be either\r\ntrue or false I shall call a \u003cem\u003eproposition\u003c/em\u003e. Thus a proposition\r\nis the same as what may be significantly asserted or\r\ndenied. A proposition which expresses what we have\r\ncalled a fact, \u003ci\u003ei.e.\u003c/i\u003e which, when asserted, asserts that a\r\ncertain thing has a certain quality, or that certain things\r\nhave a certain relation, will be called an atomic proposition,\r\nbecause, as we shall see immediately, there are\r\nother propositions into which atomic propositions enter\r\nin a way analogous to that in which atoms enter into\r\nmolecules. Atomic propositions, although, like facts,\r\nthey may have any one of an infinite number of forms,\r\nare only one kind of propositions. All other kinds are\r\n\u003ca class=\"pagenum\" title=\"53\" id=\"Page_53\"\u003e \u003c/a\u003e\r\nmore complicated. In order to preserve the parallelism\r\nin language as regards facts and propositions, we shall\r\ngive the name “atomic facts” to the facts we have\r\nhitherto been considering. Thus atomic facts are what\r\ndetermine whether atomic propositions are to be asserted\r\nor denied.\u003c/p\u003e\r\n\u003cp\u003eWhether an atomic proposition, such as “this is red,”\r\nor “this is before that,” is to be asserted or denied can\r\nonly be known empirically. Perhaps one atomic fact\r\nmay sometimes be capable of being inferred from another,\r\nthough this seems very doubtful; but in any case it\r\ncannot be inferred from premisses no one of which is an\r\natomic fact. It follows that, if atomic facts are to be\r\nknown at all, some at least must be known without\r\ninference. The atomic facts which we come to know in\r\nthis way are the facts of sense-perception; at any rate,\r\nthe facts of sense-perception are those which we most\r\nobviously and certainly come to know in this way. If\r\nwe knew all atomic facts, and also knew that there were\r\nnone except those we knew, we should, theoretically, be\r\nable to infer all truths of whatever form.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_16\" id=\"FNanchor_16\"\u003e[16]\u003c/a\u003e Thus logic\r\nwould then supply us with the whole of the apparatus\r\nrequired. But in the first acquisition of knowledge\r\nconcerning atomic facts, logic is useless. In pure logic,\r\nno atomic fact is ever mentioned: we confine ourselves\r\nwholly to forms, without asking ourselves what objects\r\ncan fill the forms. Thus pure logic is independent of\r\natomic facts; but conversely, they are, in a sense,\r\nindependent of logic. Pure logic and atomic facts are\r\nthe two poles, the wholly \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e and the wholly\r\n\u003ca class=\"pagenum\" title=\"54\" id=\"Page_54\"\u003e \u003c/a\u003e\r\nempirical. But between the two lies a vast intermediate\r\nregion, which we must now briefly explore.\u003c/p\u003e\r\n\u003cp\u003e“Molecular” propositions are such as contain conjunctions—\u003cem\u003eif\u003c/em\u003e,\r\n\u003cem\u003eor\u003c/em\u003e, \u003cem\u003eand\u003c/em\u003e, \u003cem\u003eunless\u003c/em\u003e, etc.—and such words are the marks\r\nof a molecular proposition. Consider such an assertion\r\nas, “If it rains, I shall bring my umbrella.” This\r\nassertion is just as capable of truth or falsehood as the\r\nassertion of an atomic proposition, but it is obvious that\r\neither the corresponding fact, or the nature of the correspondence\r\nwith fact, must be quite different from what\r\nit is in the case of an atomic proposition. Whether it\r\nrains, and whether I bring my umbrella, are each severally\r\nmatters of atomic fact, ascertainable by observation. But\r\nthe connection of the two involved in saying that \u003cem\u003eif\u003c/em\u003e the\r\none happens, \u003cem\u003ethen\u003c/em\u003e the other will happen, is something\r\nradically different from either of the two separately.\r\nIt does not require for its truth that it should actually\r\nrain, or that I should actually bring my umbrella; even\r\nif the weather is cloudless, it may still be true that I\r\nshould have brought my umbrella if the weather had\r\nbeen different. Thus we have here a connection of two\r\npropositions, which does not depend upon whether they\r\nare to be asserted or denied, but only upon the second\r\nbeing inferable from the first. Such propositions, therefore,\r\nhave a form which is different from that of any\r\natomic proposition.\u003c/p\u003e\r\n\u003cp\u003eSuch propositions are important to logic, because all\r\ninference depends upon them. If I have told you that\r\nif it rains I shall bring my umbrella, and if you see that\r\nthere is a steady downpour, you can infer that I shall\r\nbring my umbrella. There can be no inference except\r\nwhere propositions are connected in some such way, so\r\nthat from the truth or falsehood of the one something\r\nfollows as to the truth or falsehood of the other. It\r\n\u003ca class=\"pagenum\" title=\"55\" id=\"Page_55\"\u003e \u003c/a\u003e\r\nseems to be the case that we can sometimes know\r\nmolecular propositions, as in the above instance of the\r\numbrella, when we do not know whether the component\r\natomic propositions are true or false. The \u003cem\u003epractical\u003c/em\u003e\r\nutility of inference rests upon this fact.\u003c/p\u003e\r\n\u003cp\u003eThe next kind of propositions we have to consider are\r\n\u003cem\u003egeneral\u003c/em\u003e propositions, such as “all men are mortal,” “all\r\nequilateral triangles are equiangular.” And with these\r\nbelong propositions in which the word “some” occurs,\r\nsuch as “some men are philosophers” or “some philosophers\r\nare not wise.” These are the denials of general\r\npropositions, namely (in the above instances), of “all men\r\nare non-philosophers” and “all philosophers are wise.”\r\nWe will call propositions containing the word “some”\r\n\u003cem\u003enegative\u003c/em\u003e general propositions, and those containing the\r\nword “all” \u003cem\u003epositive\u003c/em\u003e general propositions. These propositions,\r\nit will be seen, begin to have the appearance\r\nof the propositions in logical text-books. But their\r\npeculiarity and complexity are not known to the text-books,\r\nand the problems which they raise are only\r\ndiscussed in the most superficial manner.\u003c/p\u003e\r\n\u003cp\u003eWhen we were discussing atomic facts, we saw that we\r\nshould be able, theoretically, to infer all other truths by\r\nlogic if we knew all atomic facts and also knew that there\r\nwere no other atomic facts besides those we knew. The\r\nknowledge that there are no other atomic facts is positive\r\ngeneral knowledge; it is the knowledge that “all atomic\r\nfacts are known to me,” or at least “all atomic facts are\r\nin this collection”—however the collection may be given.\r\nIt is easy to see that general propositions, such as “all\r\nmen are mortal,” cannot be known by inference from\r\natomic facts alone. If we could know each individual\r\nman, and know that he was mortal, that would not enable\r\nus to know that all men are mortal, unless we \u003cem\u003eknew\u003c/em\u003e that\r\n\u003ca class=\"pagenum\" title=\"56\" id=\"Page_56\"\u003e \u003c/a\u003e\r\nthose were all the men there are, which is a general proposition.\r\nIf we knew every other existing thing throughout\r\nthe universe, and knew that each separate thing was\r\nnot an immortal man, that would not give us our result\r\nunless we \u003cem\u003eknew\u003c/em\u003e that we had explored the whole universe,\r\n\u003ci\u003ei.e.\u003c/i\u003e unless we knew “all things belong to this collection\r\nof things I have examined.” Thus general truths cannot\r\nbe inferred from particular truths alone, but must, if they\r\nare to be known, be either self-evident, or inferred from\r\npremisses of which at least one is a general truth. But\r\nall \u003cem\u003eempirical\u003c/em\u003e evidence is of \u003cem\u003eparticular\u003c/em\u003e truths. Hence, if\r\nthere is any knowledge of general truths at all, there must\r\nbe \u003cem\u003esome\u003c/em\u003e knowledge of general truths which is independent\r\nof empirical evidence, \u003ci\u003ei.e.\u003c/i\u003e does not depend upon the data\r\nof sense.\u003c/p\u003e\r\n\u003cp\u003eThe above conclusion, of which we had an instance in\r\nthe case of the inductive principle, is important, since it\r\naffords a refutation of the older empiricists. They\r\nbelieved that all our knowledge is derived from the\r\nsenses and dependent upon them. We see that, if this\r\nview is to be maintained, we must refuse to admit that\r\nwe know any general propositions. It is perfectly possible\r\nlogically that this should be the case, but it does not\r\nappear to be so in fact, and indeed no one would dream\r\nof maintaining such a view except a theorist at the last\r\nextremity. We must therefore admit that there is general\r\nknowledge not derived from sense, and that some of this\r\nknowledge is not obtained by inference but is primitive.\u003c/p\u003e\r\n\u003cp\u003eSuch general knowledge is to be found in logic.\r\nWhether there is any such knowledge not derived from\r\nlogic, I do not know; but in logic, at any rate, we have\r\nsuch knowledge. It will be remembered that we excluded\r\nfrom pure logic such propositions as, “Socrates is a man,\r\nall men are mortal, therefore Socrates is mortal,” because\r\n\u003ca class=\"pagenum\" title=\"57\" id=\"Page_57\"\u003e \u003c/a\u003e\r\nSocrates and \u003cem\u003eman\u003c/em\u003e and \u003cem\u003emortal\u003c/em\u003e are empirical terms, only\r\nto be understood through particular experience. The\r\ncorresponding proposition in pure logic is: “If anything\r\nhas a certain property, and whatever has this property\r\nhas a certain other property, then the thing in question\r\nhas the other property.” This proposition is absolutely\r\ngeneral: it applies to all things and all properties. And\r\nit is quite self-evident. Thus in such propositions of\r\npure logic we have the self-evident general propositions\r\nof which we were in search.\u003c/p\u003e\r\n\u003cp\u003eA proposition such as, “If Socrates is a man, and all\r\nmen are mortal, then Socrates is mortal,” is true in virtue\r\nof its \u003cem\u003eform\u003c/em\u003e alone. Its truth, in this hypothetical form,\r\ndoes not depend upon whether Socrates actually is a man,\r\nnor upon whether in fact all men are mortal; thus it is\r\nequally true when we substitute other terms for Socrates\r\nand \u003cem\u003eman\u003c/em\u003e and \u003cem\u003emortal\u003c/em\u003e. The general truth of which it is an\r\ninstance is purely formal, and belongs to logic. Since it\r\ndoes not mention any particular thing, or even any\r\nparticular quality or relation, it is wholly independent of\r\nthe accidental facts of the existent world, and can be\r\nknown, theoretically, without any experience of particular\r\nthings or their qualities and relations.\u003c/p\u003e\r\n\u003cp\u003eLogic, we may say, consists of two parts. The first\r\npart investigates what propositions are and what forms\r\nthey may have; this part enumerates the different kinds of\r\natomic propositions, of molecular propositions, of general\r\npropositions, and so on. The second part consists of\r\ncertain supremely general propositions, which assert the\r\ntruth of all propositions of certain forms. This second\r\npart merges into pure mathematics, whose propositions\r\nall turn out, on analysis, to be such general formal truths.\r\nThe first part, which merely enumerates forms, is the\r\nmore difficult, and philosophically the more important;\r\n\u003ca class=\"pagenum\" title=\"58\" id=\"Page_58\"\u003e \u003c/a\u003e\r\nand it is the recent progress in this first part, more than\r\nanything else, that has rendered a truly scientific discussion\r\nof many philosophical problems possible.\u003c/p\u003e\r\n\u003cp\u003eThe problem of the nature of judgment or belief may\r\nbe taken as an example of a problem whose solution\r\ndepends upon an adequate inventory of logical forms.\r\nWe have already seen how the supposed universality of\r\nthe subject-predicate form made it impossible to give a\r\nright analysis of serial order, and therefore made space\r\nand time unintelligible. But in this case it was only\r\nnecessary to admit relations of two terms. The case of\r\njudgment demands the admission of more complicated\r\nforms. If all judgments were true, we might suppose that\r\na judgment consisted in apprehension of a \u003cem\u003efact\u003c/em\u003e, and that\r\nthe apprehension was a relation of a mind to the fact.\r\nFrom poverty in the logical inventory, this view has often\r\nbeen held. But it leads to absolutely insoluble difficulties\r\nin the case of error. Suppose I believe that Charles I.\r\ndied in his bed. There is no objective fact “Charles I.\u0027s\r\ndeath in his bed” to which I can have a relation of apprehension.\r\nCharles I. and death and his bed are objective,\r\nbut they are not, except in my thought, put together as\r\nmy false belief supposes. It is therefore necessary, in\r\nanalysing a belief, to look for some other logical form\r\nthan a two-term relation. Failure to realise this necessity\r\nhas, in my opinion, vitiated almost everything that has\r\nhitherto been written on the theory of knowledge, making\r\nthe problem of error insoluble and the difference between\r\nbelief and perception inexplicable.\u003c/p\u003e\r\n\u003cp\u003eModern logic, as I hope is now evident, has the effect\r\nof enlarging our abstract imagination, and providing an\r\ninfinite number of possible hypotheses to be applied in\r\nthe analysis of any complex fact. In this respect it is the\r\nexact opposite of the logic practised by the classical\r\n\u003ca class=\"pagenum\" title=\"59\" id=\"Page_59\"\u003e \u003c/a\u003e\r\ntradition. In that logic, hypotheses which seem \u003ci lang=\"la\"\u003eprimâ\r\nfacie\u003c/i\u003e possible are professedly proved impossible, and it is\r\ndecreed in advance that reality must have a certain\r\nspecial character. In modern logic, on the contrary,\r\nwhile the \u003ci lang=\"la\"\u003eprimâ facie\u003c/i\u003e hypotheses as a rule remain admissible,\r\nothers, which only logic would have suggested, are\r\nadded to our stock, and are very often found to be\r\nindispensable if a right analysis of the facts is to be\r\nobtained. The old logic put thought in fetters, while the\r\nnew logic gives it wings. It has, in my opinion, introduced\r\nthe same kind of advance into philosophy as Galileo\r\nintroduced into physics, making it possible at last to see\r\nwhat kinds of problems may be capable of solution, and\r\nwhat kinds must be abandoned as beyond human powers.\r\nAnd where a solution appears possible, the new logic\r\nprovides a method which enables us to obtain results\r\nthat do not merely embody personal idiosyncrasies, but\r\nmust command the assent of all who are competent to\r\nform an opinion.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"61\" id=\"Page_61\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE III\u003c/small\u003e\u003cbr\u003e\r\nON OUR KNOWLEDGE OF THE\r\nEXTERNAL WORLD\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"63\" id=\"Page_63\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_3\"\u003eLECTURE III\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nON OUR KNOWLEDGE OF THE EXTERNAL WORLD\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003ePhilosophy\u003c/span\u003e may be approached by many roads, but\r\none of the oldest and most travelled is the road which\r\nleads through doubt as to the reality of the world of\r\nsense. In Indian mysticism, in Greek and modern\r\nmonistic philosophy from Parmenides onward, in Berkeley,\r\nin modern physics, we find sensible appearance criticised\r\nand condemned for a bewildering variety of motives.\r\nThe mystic condemns it on the ground of immediate\r\nknowledge of a more real and significant world behind\r\nthe veil; Parmenides and Plato condemn it because its\r\ncontinual flux is thought inconsistent with the unchanging\r\nnature of the abstract entities revealed by logical\r\nanalysis; Berkeley brings several weapons, but his chief\r\nis the subjectivity of sense-data, their dependence upon\r\nthe organisation and point of view of the spectator;\r\nwhile modern physics, on the basis of sensible evidence\r\nitself, maintains a mad dance of electrons which has,\r\nsuperficially at least, very little resemblance to the\r\nimmediate objects of sight or touch.\u003c/p\u003e\r\n\u003cp\u003eEvery one of these lines of attack raises vital and\r\ninteresting problems.\u003c/p\u003e\r\n\u003cp\u003eThe mystic, so long as he merely reports a positive\r\nrevelation, cannot be refuted; but when he \u003cem\u003edenies\u003c/em\u003e\r\nreality to objects of sense, he may be questioned as to\r\n\u003ca class=\"pagenum\" title=\"64\" id=\"Page_64\"\u003e \u003c/a\u003e\r\nwhat he means by “reality,” and may be asked how\r\ntheir unreality follows from the supposed reality of his\r\nsuper-sensible world. In answering these questions, he\r\nis led to a logic which merges into that of Parmenides\r\nand Plato and the idealist tradition.\u003c/p\u003e\r\n\u003cp\u003eThe logic of the idealist tradition has gradually grown\r\nvery complex and very abstruse, as may be seen from the\r\nBradleian sample considered in our \u003ca href=\"#Lecture_1\" class=\"pginternal\"\u003efirst lecture\u003c/a\u003e. If we\r\nattempted to deal fully with this logic, we should not\r\nhave time to reach any other aspect of our subject; we\r\nwill therefore, while acknowledging that it deserves a\r\nlong discussion, pass by its central doctrines with only\r\nsuch occasional criticism as may serve to exemplify other\r\ntopics, and concentrate our attention on such matters as\r\nits objections to the continuity of motion and the\r\ninfinity of space and time—objections which have been\r\nfully answered by modern mathematicians in a manner\r\nconstituting an abiding triumph for the method of\r\nlogical analysis in philosophy. These objections and the\r\nmodern answers to them will occupy our \u003ca href=\"#Lecture_5\" class=\"pginternal\"\u003efifth\u003c/a\u003e, \u003ca href=\"#Lecture_6\" class=\"pginternal\"\u003esixth\u003c/a\u003e, and\r\n\u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003eseventh\u003c/a\u003e lectures.\u003c/p\u003e\r\n\u003cp\u003eBerkeley\u0027s attack, as reinforced by the physiology of\r\nthe sense-organs and nerves and brain, is very powerful.\r\nI think it must be admitted as probable that the\r\nimmediate objects of sense depend for their existence\r\nupon physiological conditions in ourselves, and that, for\r\nexample, the coloured surfaces which we see cease to exist\r\nwhen we shut our eyes. But it would be a mistake to\r\ninfer that they are dependent upon mind, not real while\r\nwe see them, or not the sole basis for our knowledge\r\nof the external world. This line of argument will be\r\ndeveloped in the present lecture.\u003c/p\u003e\r\n\u003cp\u003eThe discrepancy between the world of physics and\r\nthe world of sense, which we shall consider in our \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003efourth lecture\u003c/a\u003e,\r\n\u003ca class=\"pagenum\" title=\"65\" id=\"Page_65\"\u003e \u003c/a\u003e\r\nwill be found to be more apparent than real,\r\nand it will be shown that whatever there is reason to\r\nbelieve in physics can probably be interpreted in terms\r\nof sense.\u003c/p\u003e\r\n\u003cp\u003eThe instrument of discovery throughout is modern\r\nlogic, a very different science from the logic of the\r\ntext-books and also from the logic of idealism. Our\r\n\u003ca href=\"#Lecture_2\" class=\"pginternal\"\u003esecond lecture\u003c/a\u003e has given a short account of modern logic\r\nand of its points of divergence from the various traditional\r\nkinds of logic.\u003c/p\u003e\r\n\u003cp\u003eIn our \u003ca href=\"#Lecture_8\" class=\"pginternal\"\u003elast lecture\u003c/a\u003e, after a discussion of causality and\r\nfree will, we shall try to reach a general account of the\r\nlogical-analytic method of scientific philosophy, and a\r\ntentative estimate of the hopes of philosophical progress\r\nwhich it allows us to entertain.\u003c/p\u003e\r\n\u003cp\u003eIn this lecture, I wish to apply the logical-analytic\r\nmethod to one of the oldest problems of philosophy,\r\nnamely, the problem of our knowledge of the external\r\nworld. What I have to say on this problem does not\r\namount to an answer of a definite and dogmatic kind;\r\nit amounts only to an analysis and statement of the\r\nquestions involved, with an indication of the directions\r\nin which evidence may be sought. But although not\r\nyet a definite solution, what can be said at present seems\r\nto me to throw a completely new light on the problem,\r\nand to be indispensable, not only in seeking the answer,\r\nbut also in the preliminary question as to what parts of\r\nour problem may possibly have an ascertainable answer.\u003c/p\u003e\r\n\u003cp\u003eIn every philosophical problem, our investigation\r\nstarts from what may be called “data,” by which I\r\nmean matters of common knowledge, vague, complex,\r\ninexact, as common knowledge always is, but yet somehow\r\ncommanding our assent as on the whole and in some\r\ninterpretation pretty certainly true. In the case of our\r\n\u003ca class=\"pagenum\" title=\"66\" id=\"Page_66\"\u003e \u003c/a\u003e\r\npresent problem, the common knowledge involved is of\r\nvarious kinds. There is first our acquaintance with\r\nparticular objects of daily life—furniture, houses, towns,\r\nother people, and so on. Then there is the extension\r\nof such particular knowledge to particular things outside\r\nour personal experience, through history and geography,\r\nnewspapers, etc. And lastly, there is the systematisation\r\nof all this knowledge of particulars by means of physical\r\nscience, which derives immense persuasive force from\r\nits astonishing power of foretelling the future. We are\r\nquite willing to admit that there may be errors of detail\r\nin this knowledge, but we believe them to be discoverable\r\nand corrigible by the methods which have given rise to\r\nour beliefs, and we do not, as practical men, entertain\r\nfor a moment the hypothesis that the whole edifice may\r\nbe built on insecure foundations. In the main, therefore,\r\nand without absolute dogmatism as to this or that special\r\nportion, we may accept this mass of common knowledge\r\nas affording data for our philosophical analysis.\u003c/p\u003e\r\n\u003cp\u003eIt may be said—and this is an objection which must\r\nbe met at the outset—that it is the duty of the philosopher\r\nto call in question the admittedly fallible beliefs of daily\r\nlife, and to replace them by something more solid and\r\nirrefragable. In a sense this is true, and in a sense it is\r\neffected in the course of analysis. But in another sense,\r\nand a very important one, it is quite impossible. While\r\nadmitting that doubt is possible with regard to all our\r\ncommon knowledge, we must nevertheless accept that\r\nknowledge in the main if philosophy is to be possible at\r\nall. There is not any superfine brand of knowledge,\r\nobtainable by the philosopher, which can give us a standpoint\r\nfrom which to criticise the whole of the knowledge\r\nof daily life. The most that can be done is to examine\r\nand purify our common knowledge by an internal scrutiny,\r\n\u003ca class=\"pagenum\" title=\"67\" id=\"Page_67\"\u003e \u003c/a\u003e\r\nassuming the canons by which it has been obtained, and\r\napplying them with more care and with more precision.\r\nPhilosophy cannot boast of having achieved such a degree\r\nof certainty that it can have authority to condemn the\r\nfacts of experience and the laws of science. The philosophic\r\nscrutiny, therefore, though sceptical in regard to\r\nevery detail, is not sceptical as regards the whole. That\r\nis to say, its criticism of details will only be based upon\r\ntheir relation to other details, not upon some external\r\ncriterion which can be applied to all the details equally.\r\nThe reason for this abstention from a universal criticism\r\nis not any dogmatic confidence, but its exact opposite;\r\nit is not that common knowledge \u003cem\u003emust\u003c/em\u003e be true, but that\r\nwe possess no radically different kind of knowledge\r\nderived from some other source. Universal scepticism,\r\nthough logically irrefutable, is practically barren; it can\r\nonly, therefore, give a certain flavour of hesitancy to our\r\nbeliefs, and cannot be used to substitute other beliefs\r\nfor them.\u003c/p\u003e\r\n\u003cp\u003eAlthough data can only be criticised by other data,\r\nnot by an outside standard, yet we may distinguish\r\ndifferent grades of certainty in the different kinds of\r\ncommon knowledge which we enumerated just now.\r\nWhat does not go beyond our own personal sensible\r\nacquaintance must be for us the most certain: the\r\n“evidence of the senses” is proverbially the least open\r\nto question. What depends on testimony, like the facts\r\nof history and geography which are learnt from books,\r\nhas varying degrees of certainty according to the nature\r\nand extent of the testimony. Doubts as to the existence\r\nof Napoleon can only be maintained for a joke, whereas\r\nthe historicity of Agamemnon is a legitimate subject of\r\ndebate. In science, again, we find all grades of certainty\r\nshort of the highest. The law of gravitation, at least\r\n\u003ca class=\"pagenum\" title=\"68\" id=\"Page_68\"\u003e \u003c/a\u003e\r\nas an approximate truth, has acquired by this time the\r\nsame kind of certainty as the existence of Napoleon,\r\nwhereas the latest speculations concerning the constitution\r\nof matter would be universally acknowledged to\r\nhave as yet only a rather slight probability in their favour.\r\nThese varying degrees of certainty attaching to different\r\ndata may be regarded as themselves forming part of our\r\ndata; they, along with the other data, lie within the\r\nvague, complex, inexact body of knowledge which it is\r\nthe business of the philosopher to analyse.\u003c/p\u003e\r\n\u003cp\u003eThe first thing that appears when we begin to analyse\r\nour common knowledge is that some of it is derivative,\r\nwhile some is primitive; that is to say, there is some\r\nthat we only believe because of something else from\r\nwhich it has been inferred in some sense, though not\r\nnecessarily in a strict logical sense, while other parts are\r\nbelieved on their own account, without the support of any\r\noutside evidence. It is obvious that the senses give knowledge\r\nof the latter kind: the immediate facts perceived by\r\nsight or touch or hearing do not need to be proved by\r\nargument, but are completely self-evident. Psychologists,\r\nhowever, have made us aware that what is actually given\r\nin sense is much less than most people would naturally\r\nsuppose, and that much of what at first sight seems to be\r\ngiven is really inferred. This applies especially in regard\r\nto our space-perceptions. For instance, we instinctively\r\ninfer the “real” size and shape of a visible object from\r\nits apparent size and shape, according to its distance and\r\nour point of view. When we hear a person speaking,\r\nour actual sensations usually miss a great deal of what he\r\nsays, and we supply its place by unconscious inference;\r\nin a foreign language, where this process is more difficult,\r\nwe find ourselves apparently grown deaf, requiring, for\r\nexample, to be much nearer the stage at a theatre than\r\n\u003ca class=\"pagenum\" title=\"69\" id=\"Page_69\"\u003e \u003c/a\u003e\r\nwould be necessary in our own country. Thus the first\r\nstep in the analysis of data, namely, the discovery of what\r\nis really given in sense, is full of difficulty. We will,\r\nhowever, not linger on this point; so long as its existence\r\nis realised, the exact outcome does not make any very\r\ngreat difference in our main problem.\u003c/p\u003e\r\n\u003cp\u003eThe next step in our analysis must be the consideration\r\nof how the derivative parts of our common knowledge\r\narise. Here we become involved in a somewhat puzzling\r\nentanglement of logic and psychology. Psychologically,\r\na belief may be called derivative whenever it is caused\r\nby one or more other beliefs, or by some fact of sense\r\nwhich is not simply what the belief asserts. Derivative\r\nbeliefs in this sense constantly arise without any process\r\nof logical inference, merely by association of ideas or\r\nsome equally extra-logical process. From the expression\r\nof a man\u0027s face we judge as to what he is feeling: we\r\nsay we \u003cem\u003esee\u003c/em\u003e that he is angry, when in fact we only see a\r\nfrown. We do not judge as to his state of mind by any\r\nlogical process: the judgment grows up, often without\r\nour being able to say what physical mark of emotion we\r\nactually saw. In such a case, the knowledge is derivative\r\npsychologically; but logically it is in a sense primitive,\r\nsince it is not the result of any logical deduction. There\r\nmay or may not be a possible deduction leading to the\r\nsame result, but whether there is or not, we certainly do\r\nnot employ it. If we call a belief “logically primitive”\r\nwhen it is not actually arrived at by a logical inference,\r\nthen innumerable beliefs are logically primitive which\r\npsychologically are derivative. The separation of these\r\ntwo kinds of primitiveness is vitally important to our\r\npresent discussion.\u003c/p\u003e\r\n\u003cp\u003eWhen we reflect upon the beliefs which are logically\r\nbut not psychologically primitive, we find that, unless\r\n\u003ca class=\"pagenum\" title=\"70\" id=\"Page_70\"\u003e \u003c/a\u003e\r\nthey can on reflection be deduced by a logical process\r\nfrom beliefs which are also psychologically primitive, our\r\nconfidence in their truth tends to diminish the more we\r\nthink about them. We naturally believe, for example,\r\nthat tables and chairs, trees and mountains, are still there\r\nwhen we turn our backs upon them. I do not wish for\r\na moment to maintain that this is certainly not the case,\r\nbut I do maintain that the question whether it is the case\r\nis not to be settled off-hand on any supposed ground of\r\nobviousness. The belief that they persist is, in all men\r\nexcept a few philosophers, logically primitive, but it is\r\nnot psychologically primitive; psychologically, it arises\r\nonly through our having seen those tables and chairs,\r\ntrees and mountains. As soon as the question is seriously\r\nraised whether, because we have seen them, we have a\r\nright to suppose that they are there still, we feel that\r\nsome kind of argument must be produced, and that if\r\nnone is forthcoming, our belief can be no more than a\r\npious opinion. We do not feel this as regards the\r\nimmediate objects of sense: there they are, and as far\r\nas their momentary existence is concerned, no further\r\nargument is required. There is accordingly more need\r\nof justifying our psychologically derivative beliefs than\r\nof justifying those that are primitive.\u003c/p\u003e\r\n\u003cp\u003eWe are thus led to a somewhat vague distinction\r\nbetween what we may call “hard” data and “soft”\r\ndata. This distinction is a matter of degree, and must\r\nnot be pressed; but if not taken too seriously it may help\r\nto make the situation clear. I mean by “hard” data those\r\nwhich resist the solvent influence of critical reflection,\r\nand by “soft” data those which, under the operation\r\nof this process, become to our minds more or less\r\ndoubtful. The hardest of hard data are of two sorts:\r\nthe particular facts of sense, and the general truths of\r\n\u003ca class=\"pagenum\" title=\"71\" id=\"Page_71\"\u003e \u003c/a\u003e\r\nlogic. The more we reflect upon these, the more we\r\nrealise exactly what they are, and exactly what a doubt\r\nconcerning them really means, the more luminously\r\ncertain do they become. \u003cem\u003eVerbal\u003c/em\u003e doubt concerning even\r\nthese is possible, but verbal doubt may occur when what\r\nis nominally being doubted is not really in our thoughts,\r\nand only words are actually present to our minds. Real\r\ndoubt, in these two cases, would, I think, be pathological.\r\nAt any rate, to me they seem quite certain, and I shall\r\nassume that you agree with me in this. Without this\r\nassumption, we are in danger of falling into that universal\r\nscepticism which, as we saw, is as barren as it is irrefutable.\r\nIf we are to continue philosophising, we must make our\r\nbow to the sceptical hypothesis, and, while admitting the\r\nelegant terseness of its philosophy, proceed to the consideration\r\nof other hypotheses which, though perhaps not\r\ncertain, have at least as good a right to our respect as the\r\nhypothesis of the sceptic.\u003c/p\u003e\r\n\u003cp\u003eApplying our distinction of “hard” and “soft” data\r\nto psychologically derivative but logically primitive beliefs,\r\nwe shall find that most, if not all, are to be classed as soft\r\ndata. They may be found, on reflection, to be capable\r\nof logical proof, and they then again become believed,\r\nbut no longer as data. As data, though entitled to a\r\ncertain limited respect, they cannot be placed on a level\r\nwith the facts of sense or the laws of logic. The kind of\r\nrespect which they deserve seems to me such as to warrant\r\nus in hoping, though not too confidently, that the hard\r\ndata may prove them to be at least probable. Also, if\r\nthe hard data are found to throw no light whatever upon\r\ntheir truth or falsehood, we are justified, I think, in\r\ngiving rather more weight to the hypothesis of their\r\ntruth than to the hypothesis of their falsehood. For the\r\npresent, however, let us confine ourselves to the hard\r\n\u003ca class=\"pagenum\" title=\"72\" id=\"Page_72\"\u003e \u003c/a\u003e\r\ndata, with a view to discovering what sort of world can be\r\nconstructed by their means alone.\u003c/p\u003e\r\n\u003cp\u003eOur data now are primarily the facts of sense (\u003ci\u003ei.e.\u003c/i\u003e of\r\n\u003cem\u003eour own\u003c/em\u003e sense-data) and the laws of logic. But even the\r\nseverest scrutiny will allow some additions to this slender\r\nstock. Some facts of memory—especially of recent\r\nmemory—seem to have the highest degree of certainty.\r\nSome introspective facts are as certain as any facts of\r\nsense. And facts of sense themselves must, for our\r\npresent purposes, be interpreted with a certain latitude.\r\nSpatial and temporal relations must sometimes be included,\r\nfor example in the case of a swift motion falling wholly\r\nwithin the specious present. And some facts of comparison,\r\nsuch as the likeness or unlikeness of two shades\r\nof colour, are certainly to be included among hard data.\r\nAlso we must remember that the distinction of hard and\r\nsoft data is psychological and subjective, so that, if there\r\nare other minds than our own—which at our present\r\nstage must be held doubtful—the catalogue of hard data\r\nmay be different for them from what it is for us.\u003c/p\u003e\r\n\u003cp\u003eCertain common beliefs are undoubtedly excluded\r\nfrom hard data. Such is the belief which led us to\r\nintroduce the distinction, namely, that sensible objects\r\nin general persist when we are not perceiving them.\r\nSuch also is the belief in other people\u0027s minds: this belief\r\nis obviously derivative from our perception of their\r\nbodies, and is felt to demand logical justification as soon\r\nas we become aware of its derivativeness. Belief in\r\nwhat is reported by the testimony of others, including\r\nall that we learn from books, is of course involved in the\r\ndoubt as to whether other people have minds at all.\r\nThus the world from which our reconstruction is to\r\nbegin is very fragmentary. The best we can say for it\r\nis that it is slightly more extensive than the world at\r\n\u003ca class=\"pagenum\" title=\"73\" id=\"Page_73\"\u003e \u003c/a\u003e\r\nwhich Descartes arrived by a similar process, since\r\nthat world contained nothing except himself and his\r\nthoughts.\u003c/p\u003e\r\n\u003cp\u003eWe are now in a position to understand and state the\r\nproblem of our knowledge of the external world, and to\r\nremove various misunderstandings which have obscured\r\nthe meaning of the problem. The problem really is:\r\nCan the existence of anything other than our own hard\r\ndata be inferred from the existence of those data? But\r\nbefore considering this problem, let us briefly consider\r\nwhat the problem is \u003cem\u003enot\u003c/em\u003e.\u003c/p\u003e\r\n\u003cp\u003eWhen we speak of the “external” world in this discussion,\r\nwe must not mean “spatially external,” unless\r\n“space” is interpreted in a peculiar and recondite\r\nmanner. The immediate objects of sight, the coloured\r\nsurfaces which make up the visible world, are spatially\r\nexternal in the natural meaning of this phrase. We feel\r\nthem to be “there” as opposed to “here”; without\r\nmaking any assumption of an existence other than hard\r\ndata, we can more or less estimate the distance of a\r\ncoloured surface. It seems probable that distances,\r\nprovided they are not too great, are actually given more\r\nor less roughly in sight; but whether this is the case or\r\nnot, ordinary distances can certainly be estimated approximately\r\nby means of the data of sense alone. The\r\nimmediately given world is spatial, and is further not\r\nwholly contained within our own bodies. Thus our\r\nknowledge of what is external in this sense is not open\r\nto doubt.\u003c/p\u003e\r\n\u003cp\u003eAnother form in which the question is often put is:\r\n“Can we know of the existence of any reality which is\r\nindependent of ourselves?” This form of the question\r\nsuffers from the ambiguity of the two words “independent”\r\nand “self.” To take the Self first: the\r\n\u003ca class=\"pagenum\" title=\"74\" id=\"Page_74\"\u003e \u003c/a\u003e\r\nquestion as to what is to be reckoned part of the Self and\r\nwhat is not, is a very difficult one. Among many other\r\nthings which we may mean by the Self, two may be\r\nselected as specially important, namely, (1) the bare\r\nsubject which thinks and is aware of objects, (2) the\r\nwhole assemblage of things that would necessarily cease\r\nto exist if our lives came to an end. The bare subject,\r\nif it exists at all, is an inference, and is not part of the\r\ndata; therefore this meaning of Self may be ignored in\r\nour present inquiry. The second meaning is difficult to\r\nmake precise, since we hardly know what things depend\r\nupon our lives for their existence. And in this form,\r\nthe definition of Self introduces the word “depend,”\r\nwhich raises the same questions as are raised by the word\r\n“independent.” Let us therefore take up the word\r\n“independent,” and return to the Self later.\u003c/p\u003e\r\n\u003cp\u003eWhen we say that one thing is “independent” of\r\nanother, we may mean either that it is logically possible\r\nfor the one to exist without the other, or that there is no\r\ncausal relation between the two such that the one only\r\noccurs as the effect of the other. The only way, so far\r\nas I know, in which one thing can be \u003cem\u003elogically\u003c/em\u003e dependent\r\nupon another is when the other is \u003cem\u003epart\u003c/em\u003e of the one. The\r\nexistence of a book, for example, is logically dependent\r\nupon that of its pages: without the pages there would\r\nbe no book. Thus in this sense the question, “Can we\r\nknow of the existence of any reality which is independent\r\nof ourselves?” reduces to the question, “Can we know\r\nof the existence of any reality of which our Self is not\r\npart?” In this form, the question brings us back to the\r\nproblem of defining the Self; but I think, however the\r\nSelf may be defined, even when it is taken as the bare\r\nsubject, it cannot be supposed to be part of the immediate\r\nobject of sense; thus in this form of the question we\r\n\u003ca class=\"pagenum\" title=\"75\" id=\"Page_75\"\u003e \u003c/a\u003e\r\nmust admit that we can know of the existence of realities\r\nindependent of ourselves.\u003c/p\u003e\r\n\u003cp\u003eThe question of causal dependence is much more\r\ndifficult. To know that one kind of thing is causally independent\r\nof another, we must know that it actually occurs\r\nwithout the other. Now it is fairly obvious that, whatever\r\nlegitimate meaning we give to the Self, our thoughts and\r\nfeelings are causally dependent upon ourselves, \u003ci\u003ei.e.\u003c/i\u003e do\r\nnot occur when there is no Self for them to belong to.\r\nBut in the case of objects of sense this is not obvious;\r\nindeed, as we saw, the common-sense view is that such\r\nobjects persist in the absence of any percipient. If this\r\nis the case, then they are causally independent of ourselves;\r\nif not, not. Thus in this form the question\r\nreduces to the question whether we can know that objects\r\nof sense, or any other objects not our own thoughts and\r\nfeelings, exist at times when we are not perceiving them.\r\nThis form, in which the difficult word “independent”\r\nno longer occurs, is the form in which we stated the\r\nproblem a minute ago.\u003c/p\u003e\r\n\u003cp\u003eOur question in the above form raises two distinct\r\nproblems, which it is important to keep separate. First,\r\ncan we know that objects of sense, or very similar\r\nobjects, exist at times when we are not perceiving them?\r\nSecondly, if this cannot be known, can we know that\r\nother objects, inferable from objects of sense but not\r\nnecessarily resembling them, exist either when we are\r\nperceiving the objects of sense or at any other time?\r\nThis latter problem arises in philosophy as the problem\r\nof the “thing in itself,” and in science as the problem of\r\nmatter as assumed in physics. We will consider this\r\nlatter problem first.\u003c/p\u003e\r\n\u003cp\u003eOwing to the fact that we feel passive in sensation, we\r\nnaturally suppose that our sensations have outside causes.\r\n\u003ca class=\"pagenum\" title=\"76\" id=\"Page_76\"\u003e \u003c/a\u003e\r\nNow it is necessary here first of all to distinguish between\r\n(1) our sensation, which is a mental event consisting in\r\nour being aware of a sensible object, and (2) the sensible\r\nobject of which we are aware in sensation. When I\r\nspeak of the sensible object, it must be understood that\r\nI do not mean such a thing as a table, which is both\r\nvisible and tangible, can be seen by many people at once,\r\nand is more or less permanent. What I mean is just\r\nthat patch of colour which is momentarily seen when we\r\nlook at the table, or just that particular hardness which\r\nis felt when we press it, or just that particular sound\r\nwhich is heard when we rap it. Each of these I call a\r\nsensible object, and our awareness of it I call a sensation.\r\nNow our sense of passivity, if it really afforded any\r\nargument, would only tend to show that the \u003cem\u003esensation\u003c/em\u003e\r\nhas an outside cause; this cause we should naturally\r\nseek in the sensible object. Thus there is no good\r\nreason, so far, for supposing that sensible objects must\r\nhave outside causes. But both the thing-in-itself of\r\nphilosophy and the matter of physics present themselves\r\nas outside causes of the sensible object as much as of the\r\nsensation. What are the grounds for this common\r\nopinion?\u003c/p\u003e\r\n\u003cp\u003eIn each case, I think, the opinion has resulted from the\r\ncombination of a belief that \u003cem\u003esomething\u003c/em\u003e which can persist\r\nindependently of our consciousness makes itself known\r\nin sensation, with the fact that our sensations often change\r\nin ways which seem to depend upon us rather than upon\r\nanything which would be supposed to persist independently\r\nof us. At first, we believe unreflectingly that\r\neverything is as it seems to be, and that, if we shut our\r\neyes, the objects we had been seeing remain as they were\r\nthough we no longer see them. But there are arguments\r\nagainst this view, which have generally been thought\r\n\u003ca class=\"pagenum\" title=\"77\" id=\"Page_77\"\u003e \u003c/a\u003e\r\nconclusive. It is extraordinarily difficult to see just\r\nwhat the arguments prove; but if we are to make any\r\nprogress with the problem of the external world, we\r\nmust try to make up our minds as to these arguments.\u003c/p\u003e\r\n\u003cp\u003eA table viewed from one place presents a different\r\nappearance from that which it presents from another\r\nplace. This is the language of common sense, but this\r\nlanguage already assumes that there is a real table of\r\nwhich we see the appearances. Let us try to state what\r\nis known in terms of sensible objects alone, without any\r\nelement of hypothesis. We find that as we walk round\r\nthe table, we perceive a series of gradually changing\r\nvisible objects. But in speaking of “walking round the\r\ntable,” we have still retained the hypothesis that there is\r\na single table connected with all the appearances. What\r\nwe ought to say is that, while we have those muscular\r\nand other sensations which make us say we are walking,\r\nour visual sensations change in a continuous way, so that,\r\nfor example, a striking patch of colour is not suddenly\r\nreplaced by something wholly different, but is replaced\r\nby an insensible gradation of slightly different colours\r\nwith slightly different shapes. This is what we really\r\nknow by experience, when we have freed our minds from\r\nthe assumption of permanent “things” with changing\r\nappearances. What is really known is a correlation of\r\nmuscular and other bodily sensations with changes in\r\nvisual sensations.\u003c/p\u003e\r\n\u003cp\u003eBut walking round the table is not the only way of\r\naltering its appearance. We can shut one eye, or put on\r\nblue spectacles, or look through a microscope. All these\r\noperations, in various ways, alter the visual appearance\r\nwhich we call that of the table. More distant objects\r\nwill also alter their appearance if (as we say) the state of\r\nthe atmosphere changes—if there is fog or rain or sunshine.\r\n\u003ca class=\"pagenum\" title=\"78\" id=\"Page_78\"\u003e \u003c/a\u003e\r\nPhysiological changes also alter the appearances\r\nof things. If we assume the world of common sense,\r\nall these changes, including those attributed to physiological\r\ncauses, are changes in the intervening medium.\r\nIt is not quite so easy as in the former case to reduce this\r\nset of facts to a form in which nothing is assumed beyond\r\nsensible objects. Anything intervening between ourselves\r\nand what we see must be invisible: our view in every\r\ndirection is bounded by the nearest visible object. It\r\nmight be objected that a dirty pane of glass, for example,\r\nis visible although we can see things through it. But in\r\nthis case we really see a spotted patchwork: the dirtier\r\nspecks in the glass are visible, while the cleaner parts are\r\ninvisible and allow us to see what is beyond. Thus the\r\ndiscovery that the intervening medium affects the appearances\r\nof things cannot be made by means of the sense of\r\nsight alone.\u003c/p\u003e\r\n\u003cp\u003eLet us take the case of the blue spectacles, which is\r\nthe simplest, but may serve as a type for the others.\r\nThe frame of the spectacles is of course visible, but the\r\nblue glass, if it is clean, is not visible. The blueness,\r\nwhich we say is in the glass, appears as being in the\r\nobjects seen through the glass. The glass itself is known\r\nby means of the sense of touch. In order to know that\r\nit is between us and the objects seen through it, we must\r\nknow how to correlate the space of touch with the space\r\nof sight. This correlation itself, when stated in terms\r\nof the data of sense alone, is by no means a simple\r\nmatter. But it presents no difficulties of principle, and\r\nmay therefore be supposed accomplished. When it has\r\nbeen accomplished, it becomes possible to attach a meaning\r\nto the statement that the blue glass, which we can\r\ntouch, is between us and the object seen, as we say,\r\n“through” it.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"79\" id=\"Page_79\"\u003e \u003c/a\u003eBut we have still not reduced our statement completely\r\nto what is actually given in sense. We have fallen into\r\nthe assumption that the object of which we are conscious\r\nwhen we touch the blue spectacles still exists after we have\r\nceased to touch them. So long as we are touching them,\r\nnothing except our finger can be seen through the part\r\ntouched, which is the only part where we immediately\r\nknow that there is something. If we are to account for\r\nthe blue appearance of objects other than the spectacles,\r\nwhen seen through them, it might seem as if we must\r\nassume that the spectacles still exist when we are not\r\ntouching them; and if this assumption really is necessary,\r\nour main problem is answered: we have means of\r\nknowing of the present existence of objects not given in\r\nsense, though of the same kind as objects formerly given\r\nin sense.\u003c/p\u003e\r\n\u003cp\u003eIt may be questioned, however, whether this assumption\r\nis actually unavoidable, though it is unquestionably\r\nthe most natural one to make. We may say that the\r\nobject of which we become aware when we touch the\r\nspectacles continues to have effects afterwards, though\r\nperhaps it no longer exists. In this view, the supposed\r\ncontinued existence of sensible objects after they have\r\nceased to be sensible will be a fallacious inference from\r\nthe fact that they still have effects. It is often supposed\r\nthat nothing which has ceased to exist can continue to\r\nhave effects, but this is a mere prejudice, due to a wrong\r\nconception of causality. We cannot, therefore, dismiss\r\nour present hypothesis on the ground of \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e impossibility,\r\nbut must examine further whether it can really\r\naccount for the facts.\u003c/p\u003e\r\n\u003cp\u003eIt may be said that our hypothesis is useless in the\r\ncase when the blue glass is never touched at all. How,\r\nin that case, are we to account for the blue appearance of\r\n\u003ca class=\"pagenum\" title=\"80\" id=\"Page_80\"\u003e \u003c/a\u003e\r\nobjects? And more generally, what are we to make of\r\nthe hypothetical sensations of touch which we associate\r\nwith untouched visible objects, which we know would be\r\nverified if we chose, though in fact we do not verify\r\nthem? Must not these be attributed to permanent\r\npossession, by the objects, of the properties which touch\r\nwould reveal?\u003c/p\u003e\r\n\u003cp\u003eLet us consider the more general question first.\r\nExperience has taught us that where we see certain kinds\r\nof coloured surfaces we can, by touch, obtain certain\r\nexpected sensations of hardness or softness, tactile shape,\r\nand so on. This leads us to believe that what is seen is\r\nusually tangible, and that it has, whether we touch it or\r\nnot, the hardness or softness which we should expect to\r\nfeel if we touched it. But the mere fact that we are able\r\nto infer what our tactile sensations would be shows that\r\nit is not logically necessary to assume tactile qualities\r\nbefore they are felt. All that is really known is that the\r\nvisual appearance in question, together with touch, will\r\nlead to certain sensations, which can necessarily be\r\ndetermined in terms of the visual appearance, since\r\notherwise they could not be inferred from it.\u003c/p\u003e\r\n\u003cp\u003eWe can now give a statement of the experienced facts\r\nconcerning the blue spectacles, which will supply an\r\ninterpretation of common-sense beliefs without assuming\r\nanything beyond the existence of sensible objects at the\r\ntimes when they are sensible. By experience of the\r\ncorrelation of touch and sight sensations, we become able\r\nto associate a certain place in touch-space with a certain\r\ncorresponding place in sight-space. Sometimes, namely\r\nin the case of transparent things, we find that there is a\r\ntangible object in a touch-place without there being any\r\nvisible object in the corresponding sight-place. But in\r\nsuch a case as that of the blue spectacles, we find that\r\n\u003ca class=\"pagenum\" title=\"81\" id=\"Page_81\"\u003e \u003c/a\u003e\r\nwhatever object is visible beyond the empty sight-place\r\nin the same line of sight has a different colour from what\r\nit has when there is no tangible object in the intervening\r\ntouch-place; and as we move the tangible object in\r\ntouch-space, the blue patch moves in sight-space. If\r\nnow we find a blue patch moving in this way in sight-space,\r\nwhen we have no sensible experience of an\r\nintervening tangible object, we nevertheless infer that, if\r\nwe put our hand at a certain place in touch-space, we\r\nshould experience a certain touch-sensation. If we are\r\nto avoid non-sensible objects, this must be taken as the\r\nwhole of our meaning when we say that the blue\r\nspectacles are in a certain place, though we have not\r\ntouched them, and have only seen other things rendered\r\nblue by their interposition.\u003c/p\u003e\r\n\u003cp\u003eI think it may be laid down quite generally that, \u003cem\u003ein so\r\nfar\u003c/em\u003e as physics or common sense is verifiable, it must be\r\ncapable of interpretation in terms of actual sense-data\r\nalone. The reason for this is simple. Verification consists\r\nalways in the occurrence of an expected sense-datum.\r\nAstronomers tell us there will be an eclipse of the moon:\r\nwe look at the moon, and find the earth\u0027s shadow biting\r\ninto it, that is to say, we see an appearance quite different\r\nfrom that of the usual full moon. Now if an expected\r\nsense-datum constitutes a verification, what was asserted\r\nmust have been about sense-data; or, at any rate, if part\r\nof what was asserted was not about sense-data, then only\r\nthe other part has been verified. There is in fact a\r\ncertain regularity or conformity to law about the\r\noccurrence of sense-data, but the sense-data that occur\r\nat one time are often causally connected with those that\r\noccur at quite other times, and not, or at least not very\r\nclosely, with those that occur at neighbouring times. If\r\nI look at the moon and immediately afterwards hear a\r\n\u003ca class=\"pagenum\" title=\"82\" id=\"Page_82\"\u003e \u003c/a\u003e\r\ntrain coming, there is no very close causal connection\r\nbetween my two sense-data; but if I look at the moon\r\non two nights a week apart, there is a very close causal\r\nconnection between the two sense-data. The simplest,\r\nor at least the easiest, statement of the connection is\r\nobtained by imagining a “real” moon which goes on\r\nwhether I look at it or not, providing a series of \u003cem\u003epossible\u003c/em\u003e\r\nsense-data of which only those are actual which belong to\r\nmoments when I choose to look at the moon.\u003c/p\u003e\r\n\u003cp\u003eBut the degree of verification obtainable in this way is\r\nvery small. It must be remembered that, at our present\r\nlevel of doubt, we are not at liberty to accept testimony.\r\nWhen we hear certain noises, which are those we should\r\nutter if we wished to express a certain thought, we\r\nassume that that thought, or one very like it, has been in\r\nanother mind, and has given rise to the expression which\r\nwe hear. If at the same time we see a body resembling\r\nour own, moving its lips as we move ours when we\r\nspeak, we cannot resist the belief that it is alive, and that\r\nthe feelings inside it continue when we are not looking at\r\nit. When we see our friend drop a weight upon his\r\ntoe, and hear him say—what we should say in similar\r\ncircumstances, the phenomena \u003cem\u003ecan\u003c/em\u003e no doubt be explained\r\nwithout assuming that he is anything but a series of\r\nshapes and noises seen and heard by us, but practically\r\nno man is so infected with philosophy as not to be quite\r\ncertain that his friend has felt the same kind of pain as\r\nhe himself would feel. We will consider the legitimacy\r\nof this belief presently; for the moment, I only wish to\r\npoint out that it needs the same kind of justification as\r\nour belief that the moon exists when we do not see it,\r\nand that, without it, testimony heard or read is reduced\r\nto noises and shapes, and cannot be regarded as evidence\r\nof the facts which it reports. The verification of physics\r\n\u003ca class=\"pagenum\" title=\"83\" id=\"Page_83\"\u003e \u003c/a\u003e\r\nwhich is possible at our present level is, therefore, only\r\nthat degree of verification which is possible by one man\u0027s\r\nunaided observations, which will not carry us very far\r\ntowards the establishment of a whole science.\u003c/p\u003e\r\n\u003cp\u003eBefore proceeding further, let us summarise the argument\r\nso far as it has gone. The problem is: “Can the\r\nexistence of anything other than our own hard data be\r\ninferred from these data?” It is a mistake to state the\r\nproblem in the form: “Can we know of the existence of\r\nanything other than ourselves and our states?” or: “Can\r\nwe know of the existence of anything independent of\r\nourselves?” because of the extreme difficulty of defining\r\n“self” and “independent” precisely. The felt passivity\r\nof sensation is irrelevant, since, even if it proved anything,\r\nit could only prove that sensations are caused by sensible\r\nobjects. The natural \u003ci lang=\"fr\"\u003enaïve\u003c/i\u003e belief is that things seen\r\npersist, when unseen, exactly or approximately as they\r\nappeared when seen; but this belief tends to be dispelled\r\nby the fact that what common sense regards as the appearance\r\nof one object changes with what common sense\r\nregards as changes in the point of view and in the intervening\r\nmedium, including in the latter our own sense-organs\r\nand nerves and brain. This fact, as just stated,\r\nassumes, however, the common-sense world of stable\r\nobjects which it professes to call in question; hence,\r\nbefore we can discover its precise bearing on our problem,\r\nwe must find a way of stating it which does not involve\r\nany of the assumptions which it is designed to render\r\ndoubtful. What we then find, as the bare outcome of\r\nexperience, is that gradual changes in certain sense-data\r\nare correlated with gradual changes in certain others, or\r\n(in the case of bodily motions) with the other sense-data\r\nthemselves.\u003c/p\u003e\r\n\u003cp\u003eThe assumption that sensible objects persist after they\r\n\u003ca class=\"pagenum\" title=\"84\" id=\"Page_84\"\u003e \u003c/a\u003e\r\nhave ceased to be sensible—for example, that the hardness\r\nof a visible body, which has been discovered by\r\ntouch, continues when the body is no longer touched—may\r\nbe replaced by the statement that the \u003cem\u003eeffects\u003c/em\u003e of\r\nsensible objects persist, \u003ci\u003ei.e.\u003c/i\u003e that what happens now can\r\nonly be accounted for, in many cases, by taking account\r\nof what happened at an earlier time. Everything that\r\none man, by his own personal experience, can verify in\r\nthe account of the world given by common sense and\r\nphysics, will be explicable by some such means, since\r\nverification consists merely in the occurrence of an expected\r\nsense-datum. But what depends upon testimony,\r\nwhether heard or read, cannot be explained in this way,\r\nsince testimony depends upon the existence of minds\r\nother than our own, and thus requires a knowledge of\r\nsomething not given in sense. But before examining\r\nthe question of our knowledge of other minds, let us\r\nreturn to the question of the thing-in-itself, namely, to\r\nthe theory that what exists at times when we are not\r\nperceiving a given sensible object is something quite\r\nunlike that object, something which, together with us\r\nand our sense-organs, causes our sensations, but is never\r\nitself given in sensation.\u003c/p\u003e\r\n\u003cp\u003eThe thing-in-itself, when we start from common-sense\r\nassumptions, is a fairly natural outcome of the difficulties\r\ndue to the changing appearances of what is supposed to\r\nbe one object. It is supposed that the table (for example)\r\ncauses our sense-data of sight and touch, but must, since\r\nthese are altered by the point of view and the intervening\r\nmedium, be quite different from the sense-data to which\r\nit gives rise. There is, in this theory, a tendency to a\r\nconfusion from which it derives some of its plausibility,\r\nnamely, the confusion between a sensation as a psychical\r\noccurrence and its object. A patch of colour, even if it\r\n\u003ca class=\"pagenum\" title=\"85\" id=\"Page_85\"\u003e \u003c/a\u003e\r\nonly exists when it is seen, is still something quite\r\ndifferent from the seeing of it: the seeing of it is mental,\r\nbut the patch of colour is not. This confusion, however,\r\ncan be avoided without our necessarily abandoning the\r\ntheory we are examining. The objection to it, I think,\r\nlies in its failure to realise the radical nature of the reconstruction\r\ndemanded by the difficulties to which it points.\r\nWe cannot speak legitimately of changes in the point of\r\nview and the intervening medium until we have already\r\nconstructed some world more stable than that of momentary\r\nsensation. Our discussion of the blue spectacles and\r\nthe walk round the table has, I hope, made this clear.\r\nBut what remains far from clear is the nature of the\r\nreconstruction required.\u003c/p\u003e\r\n\u003cp\u003eAlthough we cannot rest content with the above theory,\r\nin the terms in which it is stated, we must nevertheless\r\ntreat it with a certain respect, for it is in outline the\r\ntheory upon which physical science and physiology are\r\nbuilt, and it must, therefore, be susceptible of a true\r\ninterpretation. Let us see how this is to be done.\u003c/p\u003e\r\n\u003cp\u003eThe first thing to realise is that there are no such\r\nthings as “illusions of sense.” Objects of sense, even\r\nwhen they occur in dreams, are the most indubitably real\r\nobjects known to us. What, then, makes us call them\r\nunreal in dreams? Merely the unusual nature of their\r\nconnection with other objects of sense. I dream that I\r\nam in America, but I wake up and find myself in England\r\nwithout those intervening days on the Atlantic which,\r\nalas! are inseparably connected with a “real” visit to\r\nAmerica. Objects of sense are called “real” when they\r\nhave the kind of connection with other objects of sense\r\nwhich experience has led us to regard as normal; when\r\nthey fail in this, they are called “illusions.” But what\r\nis illusory is only the inferences to which they give rise;\r\n\u003ca class=\"pagenum\" title=\"86\" id=\"Page_86\"\u003e \u003c/a\u003e\r\nin themselves, they are every bit as real as the objects\r\nof waking life. And conversely, the sensible objects of\r\nwaking life must not be expected to have any more\r\nintrinsic reality than those of dreams. Dreams and\r\nwaking life, in our first efforts at construction, must be\r\ntreated with equal respect; it is only by some reality not\r\n\u003cem\u003emerely\u003c/em\u003e sensible that dreams can be condemned.\u003c/p\u003e\r\n\u003cp\u003eAccepting the indubitable momentary reality of objects\r\nof sense, the next thing to notice is the confusion underlying\r\nobjections derived from their changeableness. As\r\nwe walk round the table, its aspect changes; but it is\r\nthought impossible to maintain either that the table\r\nchanges, or that its various aspects can all “really” exist\r\nin the same place. If we press one eyeball, we shall see\r\ntwo tables; but it is thought preposterous to maintain\r\nthat there are “really” two tables. Such arguments,\r\nhowever, seem to involve the assumption that there can\r\nbe something more real than objects of sense. If we\r\nsee two tables, then there \u003cem\u003eare\u003c/em\u003e two visual tables. It is\r\nperfectly true that, at the same moment, we may discover\r\nby touch that there is only one tactile table. This makes\r\nus declare the two visual tables an illusion, because\r\nusually one visual object corresponds to one tactile object.\r\nBut all that we are warranted in saying is that, in this\r\ncase, the manner of correlation of touch and sight is\r\nunusual. Again, when the aspect of the table changes as\r\nwe walk round it, and we are told there cannot be so\r\nmany different aspects in the same place, the answer is\r\nsimple: what does the critic of the table mean by “the\r\nsame place”? The use of such a phrase presupposes that\r\nall our difficulties have been solved; as yet, we have no\r\nright to speak of a “place” except with reference to one\r\ngiven set of momentary sense-data. When all are\r\nchanged by a bodily movement, no place remains the\r\n\u003ca class=\"pagenum\" title=\"87\" id=\"Page_87\"\u003e \u003c/a\u003e\r\nsame as it was. Thus the difficulty, if it exists, has at\r\nleast not been rightly stated.\u003c/p\u003e\r\n\u003cp\u003eWe will now make a new start, adopting a different\r\nmethod. Instead of inquiring what is the minimum of\r\nassumption by which we can explain the world of sense,\r\nwe will, in order to have a model hypothesis as a help for\r\nthe imagination, construct one possible (not necessary)\r\nexplanation of the facts. It may perhaps then be possible\r\nto pare away what is superfluous in our hypothesis,\r\nleaving a residue which may be regarded as the abstract\r\nanswer to our problem.\u003c/p\u003e\r\n\u003cp\u003eLet us imagine that each mind looks out upon the\r\nworld, as in Leibniz\u0027s monadology, from a point of view\r\npeculiar to itself; and for the sake of simplicity let us\r\nconfine ourselves to the sense of sight, ignoring minds\r\nwhich are devoid of this sense. Each mind sees at each\r\nmoment an immensely complex three-dimensional world;\r\nbut there is absolutely nothing which is seen by two\r\nminds simultaneously. When we say that two people\r\nsee the same thing, we always find that, owing to\r\ndifference of point of view, there are differences, however\r\nslight, between their immediate sensible objects. (I\r\nam here assuming the validity of testimony, but as we\r\nare only constructing a \u003cem\u003epossible\u003c/em\u003e theory, that is a legitimate\r\nassumption.) The three-dimensional world seen by one\r\nmind therefore contains no place in common with that\r\nseen by another, for places can only be constituted by the\r\nthings in or around them. Hence we may suppose, in\r\nspite of the differences between the different worlds, that\r\neach exists entire exactly as it is perceived, and might be\r\nexactly as it is even if it were not perceived. We may\r\nfurther suppose that there are an infinite number of such\r\nworlds which are in fact unperceived. If two men are\r\nsitting in a room, two somewhat similar worlds are\r\n\u003ca class=\"pagenum\" title=\"88\" id=\"Page_88\"\u003e \u003c/a\u003e\r\nperceived by them; if a third man enters and sits\r\nbetween them, a third world, intermediate between the\r\ntwo previous worlds, begins to be perceived. It is true\r\nthat we cannot reasonably suppose just this world to have\r\nexisted before, because it is conditioned by the sense-organs,\r\nnerves, and brain of the newly arrived man; but\r\nwe can reasonably suppose that \u003cem\u003esome\u003c/em\u003e aspect of the universe\r\nexisted from that point of view, though no one was\r\nperceiving it. The system consisting of all views of the\r\nuniverse perceived and unperceived, I shall call the\r\nsystem of “perspectives”; I shall confine the expression\r\n“private worlds” to such views of the universe as are\r\nactually perceived. Thus a “private world” is a\r\nperceived “perspective”; but there may be any number\r\nof unperceived perspectives.\u003c/p\u003e\r\n\u003cp\u003eTwo men are sometimes found to perceive very\r\nsimilar perspectives, so similar that they can use the same\r\nwords to describe them. They say they see the same\r\ntable, because the differences between the two tables they\r\nsee are slight and not practically important. Thus it is\r\npossible, sometimes, to establish a correlation by similarity\r\nbetween a great many of the things of one perspective,\r\nand a great many of the things of another. In case the\r\nsimilarity is very great, we say the points of view of the\r\ntwo perspectives are near together in space; but this\r\nspace in which they are near together is totally different\r\nfrom the spaces inside the two perspectives. It is a\r\nrelation between the perspectives, and is not in either of\r\nthem; no one can perceive it, and if it is to be known it\r\ncan be only by inference. Between two perceived perspectives\r\nwhich are similar, we can imagine a whole series\r\nof other perspectives, some at least unperceived, and such\r\nthat between any two, however similar, there are others\r\nstill more similar. In this way the space which consists\r\n\u003ca class=\"pagenum\" title=\"89\" id=\"Page_89\"\u003e \u003c/a\u003e\r\nof relations between perspectives can be rendered continuous,\r\nand (if we choose) three-dimensional.\u003c/p\u003e\r\n\u003cp\u003eWe can now define the momentary common-sense\r\n“thing,” as opposed to its momentary appearances. By\r\nthe similarity of neighbouring perspectives, many objects\r\nin the one can be correlated with objects in the other,\r\nnamely, with the similar objects. Given an object in one\r\nperspective, form the system of all the objects correlated\r\nwith it in all the perspectives; that system may be\r\nidentified with the momentary common-sense “thing.”\r\nThus an aspect of a “thing” is a member of the system\r\nof aspects which \u003cem\u003eis\u003c/em\u003e the “thing” at that moment. (The\r\ncorrelation of the times of different perspectives raises\r\ncertain complications, of the kind considered in the theory\r\nof relativity; but we may ignore these at present.) All\r\nthe aspects of a thing are real, whereas the thing is a\r\nmere logical construction. It has, however, the merit of\r\nbeing neutral as between different points of view, and of\r\nbeing visible to more than one person, in the only sense\r\nin which it can ever be visible, namely, in the sense that\r\neach sees one of its aspects.\u003c/p\u003e\r\n\u003cp\u003eIt will be observed that, while each perspective contains\r\nits own space, there is only one space in which the\r\nperspectives themselves are the elements. There are as\r\nmany private spaces as there are perspectives; there are\r\ntherefore at least as many as there are percipients, and\r\nthere may be any number of others which have a merely\r\nmaterial existence and are not seen by anyone. But there\r\nis only one perspective-space, whose elements are single\r\nperspectives, each with its own private space. We have\r\nnow to explain how the private space of a single perspective\r\nis correlated with part of the one all-embracing perspective\r\nspace.\u003c/p\u003e\r\n\u003cp\u003ePerspective space is the system of “points of view”\r\n\u003ca class=\"pagenum\" title=\"90\" id=\"Page_90\"\u003e \u003c/a\u003e\r\nof private spaces (perspectives), or, since “points of\r\nview” have not been defined, we may say it is the\r\nsystem of the private spaces themselves. These private\r\nspaces will each count as one point, or at any rate as\r\none element, in perspective space. They are ordered by\r\nmeans of their similarities. Suppose, for example, that\r\nwe start from one which contains the appearance of a\r\ncircular disc, such as would be called a penny, and suppose\r\nthis appearance, in the perspective in question, is circular,\r\nnot elliptic. We can then form a whole series of\r\nperspectives containing a graduated series of circular\r\naspects of varying sizes: for this purpose we only have\r\nto move (as we say) towards the penny or away from it.\r\nThe perspectives in which the penny looks circular will\r\nbe said to lie on a straight line in perspective space, and\r\ntheir order on this line will be that of the sizes of the\r\ncircular aspects. Moreover—though this statement must\r\nbe noticed and subsequently examined—the perspectives in\r\nwhich the penny looks big will be said to be nearer to the\r\npenny than those in which it looks small. It is to be remarked\r\nalso that any other “thing” than our penny might\r\nhave been chosen to define the relations of our perspectives\r\nin perspective space, and that experience shows that the\r\nsame spatial order of perspectives would have resulted.\u003c/p\u003e\r\n\u003cp\u003eIn order to explain the correlation of private spaces\r\nwith perspective space, we have first to explain what is\r\nmeant by “the place (in perspective space) where a thing\r\nis.” For this purpose, let us again consider the penny\r\nwhich appears in many perspectives. We formed a\r\nstraight line of perspectives in which the penny looked\r\ncircular, and we agreed that those in which it looked\r\nlarger were to be considered as nearer to the penny. We\r\ncan form another straight line of perspectives in which\r\nthe penny is seen end-on and looks like a straight line of\r\n\u003ca class=\"pagenum\" title=\"91\" id=\"Page_91\"\u003e \u003c/a\u003e\r\na certain thickness. These two lines will meet in a certain\r\nplace in perspective space, \u003ci\u003ei.e.\u003c/i\u003e in a certain perspective,\r\nwhich may be defined as “the place (in perspective space)\r\nwhere the penny is.” It is true that, in order to prolong\r\nour lines until they reach this place, we shall have to\r\nmake use of other things besides the penny, because, so\r\nfar as experience goes, the penny ceases to present any\r\nappearance after we have come so near to it that it touches\r\nthe eye. But this raises no real difficulty, because the\r\nspatial order of perspectives is found empirically to be\r\nindependent of the particular “things” chosen for defining\r\nthe order. We can, for example, remove our penny and\r\nprolong each of our two straight lines up to their intersection\r\nby placing other pennies further off in such a way\r\nthat the aspects of the one are circular where those of our\r\noriginal penny were circular, and the aspects of the other\r\nare straight where those of our original penny were\r\nstraight. There will then be just one perspective in\r\nwhich one of the new pennies looks circular and the other\r\nstraight. This will be, by definition, the place where the\r\noriginal penny was in perspective space.\u003c/p\u003e\r\n\u003cp\u003eThe above is, of course, only a first rough sketch of the\r\nway in which our definition is to be reached. It neglects\r\nthe size of the penny, and it assumes that we can remove\r\nthe penny without being disturbed by any simultaneous\r\nchanges in the positions of other things. But it is plain\r\nthat such niceties cannot affect the principle, and can only\r\nintroduce complications in its application.\u003c/p\u003e\r\n\u003cp\u003eHaving now defined the perspective which is the place\r\nwhere a given thing is, we can understand what is meant\r\nby saying that the perspectives in which a thing looks\r\nlarge are nearer to the thing than those in which it looks\r\nsmall: they are, in fact, nearer to the perspective which\r\nis the place where the thing is.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"92\" id=\"Page_92\"\u003e \u003c/a\u003eWe can now also explain the correlation between a\r\nprivate space and parts of perspective space. If there is\r\nan aspect of a given thing in a certain private space, then\r\nwe correlate the place where this aspect is in the private\r\nspace with the place where the thing is in perspective\r\nspace.\u003c/p\u003e\r\n\u003cp\u003eWe may define “here” as the place, in perspective\r\nspace, which is occupied by our private world. Thus we\r\ncan now understand what is meant by speaking of a thing\r\nas near to or far from “here.” A thing is near to\r\n“here” if the place where it is is near to my private\r\nworld. We can also understand what is meant by saying\r\nthat our private world is inside our head; for our private\r\nworld is a place in perspective space, and may be part of\r\nthe place where our head is.\u003c/p\u003e\r\n\u003cp\u003eIt will be observed that \u003cem\u003etwo\u003c/em\u003e places in perspective space\r\nare associated with every aspect of a thing: namely, the\r\nplace where the thing is, and the place which is the\r\nperspective of which the aspect in question forms part.\r\nEvery aspect of a thing is a member of two different\r\nclasses of aspects, namely: (1) the various aspects of the\r\nthing, of which at most one appears in any given perspective;\r\n(2) the perspective of which the given aspect\r\nis a member, \u003ci\u003ei.e.\u003c/i\u003e that in which the thing has the given\r\naspect. The physicist naturally classifies aspects in the\r\nfirst way, the psychologist in the second. The two places\r\nassociated with a single aspect correspond to the two\r\nways of classifying it. We may distinguish the two\r\nplaces as that \u003cem\u003eat\u003c/em\u003e which, and that \u003cem\u003efrom\u003c/em\u003e which, the aspect\r\nappears. The “place at which” is the place of the thing\r\nto which the aspect belongs; the “place from which” is\r\nthe place of the perspective to which the aspect belongs.\u003c/p\u003e\r\n\u003cp\u003eLet us now endeavour to state the fact that the aspect\r\nwhich a thing presents at a given place is affected by the\r\n\u003ca class=\"pagenum\" title=\"93\" id=\"Page_93\"\u003e \u003c/a\u003e\r\nintervening medium. The aspects of a thing in different\r\nperspectives are to be conceived as spreading outwards\r\nfrom the place where the thing is, and undergoing various\r\nchanges as they get further away from this place. The\r\nlaws according to which they change cannot be stated if\r\nwe only take account of the aspects that are near the\r\nthing, but require that we should also take account of the\r\nthings that are at the places from which these aspects\r\nappear. This empirical fact can, therefore, be interpreted\r\nin terms of our construction.\u003c/p\u003e\r\n\u003cp\u003eWe have now constructed a largely hypothetical picture\r\nof the world, which contains and places the experienced\r\nfacts, including those derived from testimony. The\r\nworld we have constructed can, with a certain amount of\r\ntrouble, be used to interpret the crude facts of sense, the\r\nfacts of physics, and the facts of physiology. It is therefore\r\na world which \u003cem\u003emay\u003c/em\u003e be actual. It fits the facts, and\r\nthere is no empirical evidence against it; it also is free\r\nfrom logical impossibilities. But have we any good\r\nreason to suppose that it is real? This brings us back\r\nto our original problem, as to the grounds for believing\r\nin the existence of anything outside my private world.\r\nWhat we have derived from our hypothetical construction\r\nis that there are no grounds \u003cem\u003eagainst\u003c/em\u003e the truth of this\r\nbelief, but we have not derived any positive grounds in\r\nits favour. We will resume this inquiry by taking up\r\nagain the question of testimony and the evidence for the\r\nexistence of other minds.\u003c/p\u003e\r\n\u003cp\u003eIt must be conceded to begin with that the argument\r\nin favour of the existence of other people\u0027s minds cannot\r\nbe conclusive. A phantasm of our dreams will appear\r\nto have a mind—a mind to be annoying, as a rule. It\r\nwill give unexpected answers, refuse to conform to our\r\ndesires, and show all those other signs of intelligence to\r\n\u003ca class=\"pagenum\" title=\"94\" id=\"Page_94\"\u003e \u003c/a\u003e\r\nwhich we are accustomed in the acquaintances of our\r\nwaking hours. And yet, when we are awake, we do not\r\nbelieve that the phantasm was, like the appearances of\r\npeople in waking life, representative of a private world\r\nto which we have no direct access. If we are to believe\r\nthis of the people we meet when we are awake, it must\r\nbe on some ground short of demonstration, since it is\r\nobviously possible that what we call waking life may be\r\nonly an unusually persistent and recurrent nightmare.\r\nIt may be that our imagination brings forth all that other\r\npeople seem to say to us, all that we read in books, all\r\nthe daily, weekly, monthly, and quarterly journals that\r\ndistract our thoughts, all the advertisements of soap and\r\nall the speeches of politicians. This \u003cem\u003emay\u003c/em\u003e be true, since\r\nit cannot be shown to be false, yet no one can really\r\nbelieve it. Is there any \u003cem\u003elogical\u003c/em\u003e ground for regarding this\r\npossibility as improbable? Or is there nothing beyond\r\nhabit and prejudice?\u003c/p\u003e\r\n\u003cp\u003eThe minds of other people are among our data, in the\r\nvery wide sense in which we used the word at first.\r\nThat is to say, when we first begin to reflect, we find\r\nourselves already believing in them, not because of any\r\nargument, but because the belief is natural to us. It is,\r\nhowever, a psychologically derivative belief, since it results\r\nfrom observation of people\u0027s bodies; and along with other\r\nsuch beliefs, it does not belong to the hardest of hard\r\ndata, but becomes, under the influence of philosophic\r\nreflection, just sufficiently questionable to make us desire\r\nsome argument connecting it with the facts of sense.\u003c/p\u003e\r\n\u003cp\u003eThe obvious argument is, of course, derived from\r\nanalogy. Other people\u0027s bodies behave as ours do when\r\nwe have certain thoughts and feelings; hence, by analogy,\r\nit is natural to suppose that such behaviour is connected\r\nwith thoughts and feelings like our own. Someone says,\r\n\u003ca class=\"pagenum\" title=\"95\" id=\"Page_95\"\u003e \u003c/a\u003e\r\n“Look out!” and we find we are on the point of being\r\nkilled by a motor-car; we therefore attribute the words\r\nwe heard to the person in question having seen the motor-car\r\nfirst, in which case there are existing things of which\r\nwe are not directly conscious. But this whole scene, with\r\nour inference, may occur in a dream, in which case the\r\ninference is generally considered to be mistaken. Is\r\nthere anything to make the argument from analogy more\r\ncogent when we are (as we think) awake?\u003c/p\u003e\r\n\u003cp\u003eThe analogy in waking life is only to be preferred to\r\nthat in dreams on the ground of its greater extent and\r\nconsistency. If a man were to dream every night about\r\na set of people whom he never met by day, who had\r\nconsistent characters and grew older with the lapse of\r\nyears, he might, like the man in Calderon\u0027s play, find it\r\ndifficult to decide which was the dream-world and which\r\nwas the so-called “real” world. It is only the failure of\r\nour dreams to form a consistent whole either with each\r\nother or with waking life that makes us condemn them.\r\nCertain uniformities are observed in waking life, while\r\ndreams seem quite erratic. The natural hypothesis would\r\nbe that demons and the spirits of the dead visit us while\r\nwe sleep; but the modern mind, as a rule, refuses to\r\nentertain this view, though it is hard to see what could\r\nbe said against it. On the other hand, the mystic, in\r\nmoments of illumination, seems to awaken from a sleep\r\nwhich has filled all his mundane life: the whole world\r\nof sense becomes phantasmal, and he sees, with the clarity\r\nand convincingness that belongs to our morning realisation\r\nafter dreams, a world utterly different from that of\r\nour daily cares and troubles. Who shall condemn him?\r\nWho shall justify him? Or who shall justify the seeming\r\nsolidity of the common objects among which we suppose\r\nourselves to live?\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"96\" id=\"Page_96\"\u003e \u003c/a\u003eThe hypothesis that other people have minds must, I\r\nthink, be allowed to be not susceptible of any very strong\r\nsupport from the analogical argument. At the same time,\r\nit is a hypothesis which systematises a vast body of facts\r\nand never leads to any consequences which there is\r\nreason to think false. There is therefore nothing to be\r\nsaid against its truth, and good reason to use it as a\r\nworking hypothesis. When once it is admitted, it enables\r\nus to extend our knowledge of the sensible world\r\nby testimony, and thus leads to the system of private\r\nworlds which we assumed in our hypothetical construction.\r\nIn actual fact, whatever we may try to think as\r\nphilosophers, we cannot help believing in the minds of\r\nother people, so that the question whether our belief is\r\njustified has a merely speculative interest. And if it is\r\njustified, then there is no further difficulty of principle\r\nin that vast extension of our knowledge, beyond our own\r\nprivate data, which we find in science and common sense.\u003c/p\u003e\r\n\u003cp\u003eThis somewhat meagre conclusion must not be regarded\r\nas the whole outcome of our long discussion. The\r\nproblem of the connection of sense with objective reality\r\nhas commonly been dealt with from a standpoint which\r\ndid not carry initial doubt so far as we have carried it;\r\nmost writers, consciously or unconsciously, have assumed\r\nthat the testimony of others is to be admitted, and therefore\r\n(at least by implication) that others have minds.\r\nTheir difficulties have arisen after this admission, from\r\nthe differences in the appearance which one physical\r\nobject presents to two people at the same time, or to one\r\nperson at two times between which it cannot be supposed\r\nto have changed. Such difficulties have made people\r\ndoubtful how far objective reality could be known by\r\nsense at all, and have made them suppose that there were\r\npositive arguments against the view that it can be so\r\n\u003ca class=\"pagenum\" title=\"97\" id=\"Page_97\"\u003e \u003c/a\u003e\r\nknown. Our hypothetical construction meets these\r\narguments, and shows that the account of the world\r\ngiven by common sense and physical science can be\r\ninterpreted in a way which is logically unobjectionable,\r\nand finds a place for all the data, both hard and soft.\r\nIt is this hypothetical construction, with its reconciliation\r\nof psychology and physics, which is the chief outcome\r\nof our discussion. Probably the construction is only in\r\npart necessary as an initial assumption, and can be obtained\r\nfrom more slender materials by the logical methods\r\nof which we shall have an example in the definitions of\r\npoints, instants, and particles; but I do not yet know to\r\nwhat lengths this diminution in our initial assumptions\r\ncan be carried.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"99\" id=\"Page_99\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE IV\u003c/small\u003e\u003cbr\u003e\r\nTHE WORLD OF PHYSICS AND\r\nTHE WORLD OF SENSE\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"101\" id=\"Page_101\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_4\"\u003eLECTURE IV\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nTHE WORLD OF PHYSICS AND THE WORLD OF SENSE\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eAmong\u003c/span\u003e the objections to the reality of objects of sense,\r\nthere is one which is derived from the apparent difference\r\nbetween matter as it appears in physics and things as they\r\nappear in sensation. Men of science, for the most part,\r\nare willing to condemn immediate data as “merely subjective,”\r\nwhile yet maintaining the truth of the physics\r\ninferred from those data. But such an attitude, though\r\nit may be \u003cem\u003ecapable\u003c/em\u003e of justification, obviously stands in need\r\nof it; and the only justification possible must be one\r\nwhich exhibits matter as a logical construction from sense-data—unless,\r\nindeed, there were some wholly \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e\r\nprinciple by which unknown entities could be inferred\r\nfrom such as are known. It is therefore necessary to\r\nfind some way of bridging the gulf between the world\r\nof physics and the world of sense, and it is this problem\r\nwhich will occupy us in the present lecture. Physicists\r\nappear to be unconscious of the gulf, while psychologists,\r\nwho are conscious of it, have not the mathematical knowledge\r\nrequired for spanning it. The problem is difficult,\r\nand I do not know its solution in detail. All that I can\r\nhope to do is to make the problem felt, and to indicate\r\nthe kind of methods by which a solution is to be sought.\u003c/p\u003e\r\n\u003cp\u003eLet us begin by a brief description of the two contrasted\r\nworlds. We will take first the world of physics,\r\n\u003ca class=\"pagenum\" title=\"102\" id=\"Page_102\"\u003e \u003c/a\u003e\r\nfor, though the other world is given while the physical\r\nworld is inferred, to us now the world of physics is the\r\nmore familiar, the world of pure sense having become\r\nstrange and difficult to rediscover. Physics started from\r\nthe common-sense belief in fairly permanent and fairly\r\nrigid bodies—tables and chairs, stones, mountains, the\r\nearth and moon and sun. This common-sense belief,\r\nit should be noticed, is a piece of audacious metaphysical\r\ntheorising; objects are not continually present to sensation,\r\nand it may be doubted whether they are there when\r\nthey are not seen or felt. This problem, which has been\r\nacute since the time of Berkeley, is ignored by common\r\nsense, and has therefore hitherto been ignored by\r\nphysicists. We have thus here a first departure from\r\nthe immediate data of sensation, though it is a departure\r\nmerely by way of extension, and was probably made by\r\nour savage ancestors in some very remote prehistoric\r\nepoch.\u003c/p\u003e\r\n\u003cp\u003eBut tables and chairs, stones and mountains, are not\r\n\u003cem\u003equite\u003c/em\u003e permanent or \u003cem\u003equite\u003c/em\u003e rigid. Tables and chairs lose\r\ntheir legs, stones are split by frost, and mountains are\r\ncleft by earthquakes and eruptions. Then there are\r\nother things, which seem material, and yet present almost\r\nno permanence or rigidity. Breath, smoke, clouds, are\r\nexamples of such things—so, in a lesser degree, are ice\r\nand snow; and rivers and seas, though fairly permanent,\r\nare not in any degree rigid. Breath, smoke, clouds, and\r\ngenerally things that can be seen but not touched, were\r\nthought to be hardly real; to this day the usual mark\r\nof a ghost is that it can be seen but not touched. Such\r\nobjects were peculiar in the fact that they seemed to disappear\r\ncompletely, not merely to be transformed into\r\nsomething else. Ice and snow, when they disappear, are\r\nreplaced by water; and it required no great theoretical\r\n\u003ca class=\"pagenum\" title=\"103\" id=\"Page_103\"\u003e \u003c/a\u003e\r\neffort to invent the hypothesis that the water was the\r\nsame thing as the ice and snow, but in a new form.\r\nSolid bodies, when they break, break into parts which\r\nare practically the same in shape and size as they were\r\nbefore. A stone can be hammered into a powder, but the\r\npowder consists of grains which retain the character they\r\nhad before the pounding. Thus the ideal of absolutely\r\nrigid and absolutely permanent bodies, which early physicists\r\npursued throughout the changing appearances, seemed\r\nattainable by supposing ordinary bodies to be composed\r\nof a vast number of tiny atoms. This billiard-ball view\r\nof matter dominated the imagination of physicists until\r\nquite modern times, until, in fact, it was replaced by the\r\nelectromagnetic theory, which in its turn is developing into\r\na new atomism. Apart from the special form of the\r\natomic theory which was invented for the needs of chemistry,\r\nsome kind of atomism dominated the whole of\r\ntraditional dynamics, and was implied in every statement\r\nof its laws and axioms.\u003c/p\u003e\r\n\u003cp\u003eThe pictorial accounts which physicists give of the\r\nmaterial world as they conceive it undergo violent changes\r\nunder the influence of modifications in theory which are\r\nmuch slighter than the layman might suppose from the\r\nalterations of the description. Certain features, however,\r\nhave remained fairly stable. It is always assumed that\r\nthere is \u003cem\u003esomething\u003c/em\u003e indestructible which is capable of motion\r\nin space; what is indestructible is always very small, but\r\ndoes not always occupy a mere point in space. There is\r\nsupposed to be one all-embracing space in which the\r\nmotion takes place, and until lately we might have assumed\r\none all-embracing time also. But the principle of relativity\r\nhas given prominence to the conception of “local\r\ntime,” and has somewhat diminished men\u0027s confidence in\r\nthe one even-flowing stream of time. Without dogmatising\r\n\u003ca class=\"pagenum\" title=\"104\" id=\"Page_104\"\u003e \u003c/a\u003e\r\nas to the ultimate outcome of the principle of\r\nrelativity, however, we may safely say, I think, that it\r\ndoes not destroy the possibility of correlating different\r\nlocal times, and does not therefore have such far-reaching\r\nphilosophical consequences as is sometimes supposed. In\r\nfact, in spite of difficulties as to measurement, the one\r\nall-embracing time still, I think, underlies all that physics\r\nhas to say about motion. We thus have still in physics,\r\nas we had in Newton\u0027s time, a set of indestructible entities\r\nwhich may be called particles, moving relatively to each\r\nother in a single space and a single time.\u003c/p\u003e\r\n\u003cp\u003eThe world of immediate data is quite different from\r\nthis. Nothing is permanent; even the things that we\r\nthink are fairly permanent, such as mountains, only become\r\ndata when we see them, and are not immediately\r\ngiven as existing at other moments. So far from one\r\nall-embracing space being given, there are several spaces\r\nfor each person, according to the different senses which\r\ngive relations that may be called spatial. Experience\r\nteaches us to obtain one space from these by correlation,\r\nand experience, together with instinctive theorising, teaches\r\nus to correlate our spaces with those which we believe to\r\nexist in the sensible worlds of other people. The construction\r\nof a single time offers less difficulty so long as\r\nwe confine ourselves to one person\u0027s private world, but the\r\ncorrelation of one private time with another is a matter\r\nof great difficulty. Thus, apart from any of the fluctuating\r\nhypotheses of physics, three main problems arise in\r\nconnecting the world of physics with the world of sense,\r\nnamely (1) the construction of permanent “things,” (2)\r\nthe construction of a single space, and (3) the construction\r\nof a single time. We will consider these three\r\nproblems in succession.\u003c/p\u003e\r\n\u003cp\u003e(1) The belief in indestructible “things” very early\r\n\u003ca class=\"pagenum\" title=\"105\" id=\"Page_105\"\u003e \u003c/a\u003e\r\ntook the form of atomism. The underlying motive in\r\natomism was not, I think, any empirical success in interpreting\r\nphenomena, but rather an instinctive belief that\r\nbeneath all the changes of the sensible world there must\r\nbe something permanent and unchanging. This belief\r\nwas, no doubt, fostered and nourished by its practical\r\nsuccesses, culminating in the conservation of mass; but\r\nit was not produced by these successes. On the contrary,\r\nthey were produced by it. Philosophical writers\r\non physics sometimes speak as though the conservation\r\nof something or other were essential to the possibility\r\nof science, but this, I believe, is an entirely erroneous\r\nopinion. If the \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e belief in permanence had not\r\nexisted, the same laws which are now formulated in terms\r\nof this belief might just as well have been formulated\r\nwithout it. Why should we suppose that, when ice\r\nmelts, the water which replaces it is the same thing in a\r\nnew form? Merely because this supposition enables us\r\nto state the phenomena in a way which is consonant with\r\nour prejudices. What we really know is that, under\r\ncertain conditions of temperature, the appearance we call\r\nice is replaced by the appearance we call water. We can\r\ngive laws according to which the one appearance will be\r\nsucceeded by the other, but there is no reason except\r\nprejudice for regarding both as appearances of the same\r\nsubstance.\u003c/p\u003e\r\n\u003cp\u003eOne task, if what has just been said is correct, which\r\nconfronts us in trying to connect the world of sense with\r\nthe world of physics, is the task of reconstructing the\r\nconception of matter without the \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e beliefs which\r\nhistorically gave rise to it. In spite of the revolutionary\r\nresults of modern physics, the empirical successes of the\r\nconception of matter show that there must be some legitimate\r\nconception which fulfils roughly the same functions.\r\n\u003ca class=\"pagenum\" title=\"106\" id=\"Page_106\"\u003e \u003c/a\u003e\r\nThe time has hardly come when we can state precisely\r\nwhat this legitimate conception is, but we can see in a\r\ngeneral way what it must be like. For this purpose, it\r\nis only necessary to take our ordinary common-sense\r\nstatements and reword them without the assumption of\r\npermanent substance. We say, for example, that things\r\nchange gradually—sometimes very quickly, but not without\r\npassing through a continuous series of intermediate\r\nstates. What this means is that, given any sensible\r\nappearance, there will usually be, \u003cem\u003eif we watch\u003c/em\u003e, a continuous\r\nseries of appearances connected with the given\r\none, leading on by imperceptible gradations to the new\r\nappearances which common-sense regards as those of the\r\nsame thing. Thus a thing may be defined as a certain\r\nseries of appearances, connected with each other by\r\ncontinuity and by certain causal laws. In the case of\r\nslowly changing things, this is easily seen. Consider,\r\nsay, a wall-paper which fades in the course of years. It\r\nis an effort not to conceive of it as one “thing” whose\r\ncolour is slightly different at one time from what it is at\r\nanother. But what do we really \u003cem\u003eknow\u003c/em\u003e about it? We\r\nknow that under suitable circumstances—\u003ci\u003ei.e.\u003c/i\u003e when we are,\r\nas is said, “in the room”—we perceive certain colours\r\nin a certain pattern: not always precisely the same\r\ncolours, but sufficiently similar to feel familiar. If we\r\ncan state the laws according to which the colour varies,\r\nwe can state all that is empirically verifiable; the assumption\r\nthat there is a constant entity, the wall-paper, which\r\n“has” these various colours at various times, is a piece\r\nof gratuitous metaphysics. We may, if we like, \u003cem\u003edefine\u003c/em\u003e\r\nthe wall-paper as the series of its aspects. These are\r\ncollected together by the same motives which led us to\r\nregard the wall-paper as one thing, namely a combination\r\nof sensible continuity and causal connection. More\r\n\u003ca class=\"pagenum\" title=\"107\" id=\"Page_107\"\u003e \u003c/a\u003e\r\ngenerally, a “thing” will be defined as a certain series\r\nof aspects, namely those which would commonly be said\r\nto be \u003cem\u003eof\u003c/em\u003e the thing. To say that a certain aspect is an\r\naspect \u003cem\u003eof\u003c/em\u003e a certain thing will merely mean that it is one\r\nof those which, taken serially, \u003cem\u003eare\u003c/em\u003e the thing. Everything\r\nwill then proceed as before: whatever was verifiable is\r\nunchanged, but our language is so interpreted as to avoid\r\nan unnecessary metaphysical assumption of permanence.\u003c/p\u003e\r\n\u003cp\u003eThe above extrusion of permanent things affords an\r\nexample of the maxim which inspires all scientific philosophising,\r\nnamely “Occam\u0027s razor”: \u003cem\u003eEntities are not to\r\nbe multiplied without necessity.\u003c/em\u003e In other words, in dealing\r\nwith any subject-matter, find out what entities are\r\nundeniably involved, and state everything in terms of\r\nthese entities. Very often the resulting statement is\r\nmore complicated and difficult than one which, like\r\ncommon sense and most philosophy, assumes hypothetical\r\nentities whose existence there is no good reason to believe\r\nin. We find it easier to imagine a wall-paper with\r\nchanging colours than to think merely of the series of\r\ncolours; but it is a mistake to suppose that what is easy\r\nand natural in thought is what is most free from unwarrantable\r\nassumptions, as the case of “things” very\r\naptly illustrates.\u003c/p\u003e\r\n\u003cp\u003eThe above summary account of the genesis of “things,”\r\nthough it may be correct in outline, has omitted some\r\nserious difficulties which it is necessary briefly to consider.\r\nStarting from a world of helter-skelter sense-data, we\r\nwish to collect them into series, each of which can be\r\nregarded as consisting of the successive appearances of\r\none “thing.” There is, to begin with, some conflict\r\nbetween what common sense regards as one thing, and\r\nwhat physics regards an unchanging collection of particles.\r\nTo common sense, a human body is one thing, but to\r\n\u003ca class=\"pagenum\" title=\"108\" id=\"Page_108\"\u003e \u003c/a\u003e\r\nscience the matter composing it is continually changing.\r\nThis conflict, however, is not very serious, and may, for\r\nour rough preliminary purpose, be largely ignored. The\r\nproblem is: by what principles shall we select certain\r\ndata from the chaos, and call them all appearances of the\r\nsame thing?\u003c/p\u003e\r\n\u003cp\u003eA rough and approximate answer to this question is\r\nnot very difficult. There are certain fairly stable collections\r\nof appearances, such as landscapes, the furniture of\r\nrooms, the faces of acquaintances. In these cases, we\r\nhave little hesitation in regarding them on successive\r\noccasions as appearances of one thing or collection of\r\nthings. But, as the \u003ccite\u003eComedy of Errors\u003c/cite\u003e illustrates, we may\r\nbe led astray if we judge by mere resemblance. This\r\nshows that something more is involved, for two different\r\nthings may have any degree of likeness up to exact\r\nsimilarity.\u003c/p\u003e\r\n\u003cp\u003eAnother insufficient criterion of one thing is \u003cem\u003econtinuity\u003c/em\u003e.\r\nAs we have already seen, if we watch what we regard as\r\none changing thing, we usually find its changes to be continuous\r\nso far as our senses can perceive. We are thus\r\nled to assume that, if we see two finitely different appearances\r\nat two different times, and if we have reason to\r\nregard them as belonging to the same thing, then there\r\nwas a continuous series of intermediate states of that\r\nthing during the time when we were not observing it.\r\nAnd so it comes to be thought that continuity of change\r\nis necessary and sufficient to constitute one thing. But\r\nin fact it is neither. It is not \u003cem\u003enecessary\u003c/em\u003e, because the\r\nunobserved states, in the case where our attention has\r\nnot been concentrated on the thing throughout, are\r\npurely hypothetical, and cannot possibly be our ground\r\nfor supposing the earlier and later appearances to belong\r\nto the same thing; on the contrary, it is because we suppose\r\n\u003ca class=\"pagenum\" title=\"109\" id=\"Page_109\"\u003e \u003c/a\u003e\r\nthis that we assume intermediate unobserved states.\r\nContinuity is also not sufficient, since we can, for example,\r\npass by sensibly continuous gradations from any one drop\r\nof the sea to any other drop. The utmost we can say is\r\nthat discontinuity during uninterrupted observation is as\r\na rule a mark of difference between things, though even\r\nthis cannot be said in such cases as sudden explosions.\u003c/p\u003e\r\n\u003cp\u003eThe assumption of continuity is, however, successfully\r\nmade in physics. This proves something, though not\r\nanything of very obvious utility to our present problem:\r\nit proves that nothing in the known world is inconsistent\r\nwith the hypothesis that all changes are really continuous,\r\nthough from too great rapidity or from our lack of\r\nobservation they may not always appear continuous. In\r\nthis hypothetical sense, continuity may be allowed to be a\r\n\u003cem\u003enecessary\u003c/em\u003e condition if two appearances are to be classed as\r\nappearances of the same thing. But it is not a \u003cem\u003esufficient\u003c/em\u003e\r\ncondition, as appears from the instance of the drops in\r\nthe sea. Thus something more must be sought before\r\nwe can give even the roughest definition of a “thing.”\u003c/p\u003e\r\n\u003cp\u003eWhat is wanted further seems to be something in the\r\nnature of fulfilment of causal laws. This statement, as it\r\nstands, is very vague, but we will endeavour to give it\r\nprecision. When I speak of “causal laws,” I mean any\r\nlaws which connect events at different times, or even, as\r\na limiting case, events at the same time provided the\r\nconnection is not logically demonstrable. In this very\r\ngeneral sense, the laws of dynamics are causal laws, and\r\nso are the laws correlating the simultaneous appearances\r\nof one “thing” to different senses. The question is:\r\nHow do such laws help in the definition of a “thing”?\u003c/p\u003e\r\n\u003cp\u003eTo answer this question, we must consider what it is\r\nthat is proved by the empirical success of physics.\r\nWhat is proved is that its hypotheses, though unverifiable\r\n\u003ca class=\"pagenum\" title=\"110\" id=\"Page_110\"\u003e \u003c/a\u003e\r\nwhere they go beyond sense-data, are at no point in\r\ncontradiction with sense-data, but, on the contrary, are\r\nideally such as to render all sense-data calculable from a\r\nsufficient collection of data all belonging to a given\r\nperiod of time. Now physics has found it empirically\r\npossible to collect sense-data into series, each series being\r\nregarded as belonging to one “thing,” and behaving,\r\nwith regard to the laws of physics, in a way in which\r\nseries not belonging to one thing would in general not\r\nbehave. If it is to be unambiguous whether two\r\nappearances belong to the same thing or not, there must\r\nbe only one way of grouping appearances so that the\r\nresulting things obey the laws of physics. It would be\r\nvery difficult to prove that this is the case, but for our\r\npresent purposes we may let this point pass, and assume\r\nthat there is only one way. We must include in our\r\ndefinition of a “thing” those of its aspects, if any, which\r\nare not observed. Thus we may lay down the following\r\ndefinition: \u003cem\u003eThings are those series of aspects which obey the\r\nlaws of physics.\u003c/em\u003e That such series exist is an empirical\r\nfact, which constitutes the verifiability of physics.\u003c/p\u003e\r\n\u003cp\u003eIt may still be objected that the “matter” of physics is\r\nsomething other than series of sense-data. Sense-data,\r\nit may be said, belong to psychology and are, at any\r\nrate in some sense, subjective, whereas physics is quite\r\nindependent of psychological considerations, and does not\r\nassume that its matter only exists when it is perceived.\u003c/p\u003e\r\n\u003cp\u003eTo this objection there are two answers, both of some\r\nimportance.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ea\u003c/i\u003e) We have been considering, in the above account,\r\nthe question of the \u003cem\u003everifiability\u003c/em\u003e of physics. Now verifiability\r\nis by no means the same thing as truth; it is, in\r\nfact, something far more subjective and psychological.\r\nFor a proposition to be verifiable, it is not enough that\r\n\u003ca class=\"pagenum\" title=\"111\" id=\"Page_111\"\u003e \u003c/a\u003e\r\nit should be true, but it must also be such as we can\r\n\u003cem\u003ediscover\u003c/em\u003e to be true. Thus verifiability depends upon our\r\ncapacity for acquiring knowledge, and not only upon\r\nthe objective truth. In physics, as ordinarily set forth,\r\nthere is much that is unverifiable: there are hypotheses\r\nas to (\u003cspan class=\"greek\" title=\"a\" lang=\"grc\"\u003eα\u003c/span\u003e) how things would appear to a spectator in a\r\nplace where, as it happens, there is no spectator; (\u003cspan class=\"greek\" title=\"b\" lang=\"grc\"\u003eβ\u003c/span\u003e) how\r\nthings would appear at times when, in fact, they are not\r\nappearing to anyone; (\u003cspan class=\"greek\" title=\"g\" lang=\"grc\"\u003eγ\u003c/span\u003e) things which never appear at\r\nall. All these are introduced to simplify the statement of\r\nthe causal laws, but none of them form an integral part\r\nof what is \u003cem\u003eknown\u003c/em\u003e to be true in physics. This brings us\r\nto our second answer.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003eb\u003c/i\u003e) If physics is to consist wholly of propositions\r\nknown to be true, or at least capable of being proved or\r\ndisproved, the three kinds of hypothetical entities we\r\nhave just enumerated must all be capable of being\r\nexhibited as logical functions of sense-data. In order to\r\nshow how this might possibly be done, let us recall the\r\nhypothetical Leibnizian universe of \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003eLecture III\u003c/a\u003e. In that\r\nuniverse, we had a number of perspectives, two of which\r\nnever had any entity in common, but often contained\r\nentities which could be sufficiently correlated to be\r\nregarded as belonging to the same thing. We will call\r\none of these an “actual” private world when there is an\r\nactual spectator to which it appears, and “ideal” when\r\nit is merely constructed on principles of continuity. A\r\nphysical thing consists, at each instant, of the whole set\r\nof its aspects at that instant, in all the different worlds;\r\nthus a momentary state of a thing is a whole set of\r\naspects. An “ideal” appearance will be an aspect\r\nmerely calculated, but not actually perceived by any\r\nspectator. An “ideal” state of a thing will be a state at\r\na moment when all its appearances are ideal. An ideal\r\n\u003ca class=\"pagenum\" title=\"112\" id=\"Page_112\"\u003e \u003c/a\u003e\r\nthing will be one whose states at all times are ideal.\r\nIdeal appearances, states, and things, since they are\r\ncalculated, must be functions of actual appearances, states,\r\nand things; in fact, ultimately, they must be functions of\r\nactual appearances. Thus it is unnecessary, for the\r\nenunciation of the laws of physics, to assign any reality to\r\nideal elements: it is enough to accept them as logical\r\nconstructions, provided we have means of knowing how\r\nto determine when they become actual. This, in fact, we\r\nhave with some degree of approximation; the starry\r\nheaven, for instance, becomes actual whenever we choose\r\nto look at it. It is open to us to believe that the ideal\r\nelements exist, and there can be no reason for \u003cem\u003edis\u003c/em\u003ebelieving\r\nthis; but unless in virtue of some \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e law we cannot\r\n\u003cem\u003eknow\u003c/em\u003e it, for empirical knowledge is confined to what we\r\nactually observe.\u003c/p\u003e\r\n\u003cp\u003e(2) The three main conceptions of physics are space,\r\ntime, and matter. Some of the problems raised by the\r\nconception of matter have been indicated in the above\r\ndiscussion of “things.” But space and time also raise\r\ndifficult problems of much the same kind, namely,\r\ndifficulties in reducing the haphazard untidy world of\r\nimmediate sensation to the smooth orderly world of\r\ngeometry and kinematics. Let us begin with the consideration\r\nof space.\u003c/p\u003e\r\n\u003cp\u003ePeople who have never read any psychology seldom\r\nrealise how much mental labour has gone into the construction\r\nof the one all-embracing space into which all\r\nsensible objects are supposed to fit. Kant, who was\r\nunusually ignorant of psychology, described space as “an\r\ninfinite given whole,” whereas a moment\u0027s psychological\r\nreflection shows that a space which is infinite is not given,\r\nwhile a space which can be called given is not infinite.\r\nWhat the nature of “given” space really is, is a difficult\r\n\u003ca class=\"pagenum\" title=\"113\" id=\"Page_113\"\u003e \u003c/a\u003e\r\nquestion, upon which psychologists are by no means\r\nagreed. But some general remarks may be made, which\r\nwill suffice to show the problems, without taking sides\r\non any psychological issue still in debate.\u003c/p\u003e\r\n\u003cp\u003eThe first thing to notice is that different senses have\r\ndifferent spaces. The space of sight is quite different\r\nfrom the space of touch: it is only by experience in\r\ninfancy that we learn to correlate them. In later life,\r\nwhen we see an object within reach, we know how to\r\ntouch it, and more or less what it will feel like; if we\r\ntouch an object with our eyes shut, we know where we\r\nshould have to look for it, and more or less what it\r\nwould look like. But this knowledge is derived from\r\nearly experience of the correlation of certain kinds of\r\ntouch-sensations with certain kinds of sight-sensations.\r\nThe one space into which both kinds of sensations fit\r\nis an intellectual construction, not a datum. And\r\nbesides touch and sight, there are other kinds of sensation\r\nwhich give other, though less important spaces:\r\nthese also have to be fitted into the one space by means\r\nof experienced correlations. And as in the case of things,\r\nso here: the one all-embracing space, though convenient\r\nas a way of speaking, need not be supposed really to\r\nexist. All that experience makes certain is the several\r\nspaces of the several senses, correlated by empirically discovered\r\nlaws. The one space may turn out to be valid\r\nas a logical construction, compounded of the several\r\nspaces, but there is no good reason to assume its independent\r\nmetaphysical reality.\u003c/p\u003e\r\n\u003cp\u003eAnother respect in which the spaces of immediate\r\nexperience differ from the space of geometry and physics\r\nis in regard to \u003cem\u003epoints\u003c/em\u003e. The space of geometry and physics\r\nconsists of an infinite number of points, but no one has\r\never seen or touched a point. If there are points in a\r\n\u003ca class=\"pagenum\" title=\"114\" id=\"Page_114\"\u003e \u003c/a\u003e\r\nsensible space, they must be an inference. It is not easy\r\nto see any way in which, as independent entities, they\r\ncould be validly inferred from the data; thus here again,\r\nwe shall have, if possible, to find some logical construction,\r\nsome complex assemblage of immediately given\r\nobjects, which will have the geometrical properties\r\nrequired of points. It is customary to think of points\r\nas simple and infinitely small, but geometry in no way\r\ndemands that we should think of them in this way. All\r\nthat is necessary for geometry is that they should have\r\nmutual relations possessing certain enumerated abstract\r\nproperties, and it may be that an assemblage of data of\r\nsensation will serve this purpose. Exactly how this is to\r\nbe done, I do not yet know, but it seems fairly certain\r\nthat it can be done.\u003c/p\u003e\r\n\u003cp\u003eThe following illustrative method, simplified so as to\r\nbe easily manipulated, has been invented by Dr Whitehead\r\nfor the purpose of showing how points might be manufactured\r\nfrom sense-data. We have first of all to observe\r\nthat there are no infinitesimal sense-data: any surface we\r\ncan see, for example, must be of some finite extent.\r\nBut what at first appears as one undivided whole is often\r\nfound, under the influence of attention, to split up into\r\nparts contained within the whole. Thus one spatial\r\nobject may be contained within another, and entirely\r\nenclosed by the other. This relation of enclosure, by\r\nthe help of some very natural hypotheses, will enable us\r\nto define a “point” as a certain class of spatial objects,\r\nnamely all those (as it will turn out in the end) which\r\nwould naturally be said to contain the point. In order\r\nto obtain a definition of a “point” in this way, we\r\nproceed as follows:\u003c/p\u003e\r\n\u003cp\u003eGiven any set of volumes or surfaces, they will not in\r\ngeneral converge into one point. But if they get smaller\r\n\u003ca class=\"pagenum\" title=\"115\" id=\"Page_115\"\u003e \u003c/a\u003e\r\nand smaller, while of any two of the set there is always\r\none that encloses the other, then we begin to have the\r\nkind of conditions which would enable us to treat them\r\nas having a point for their limit. The hypotheses required\r\nfor the relation of enclosure are that (1) it must be transitive;\r\n(2) of two \u003cem\u003edifferent\u003c/em\u003e spatial objects, it is impossible\r\nfor each to enclose the other, but a single spatial object\r\nalways encloses itself; (3) any set of spatial objects such\r\nthat there is at least one spatial object enclosed by them\r\nall has a lower limit or minimum, \u003ci\u003ei.e.\u003c/i\u003e an object enclosed\r\nby all of them and enclosing all objects which are\r\nenclosed by all of them; (4) to prevent trivial exceptions,\r\nwe must add that there are to be instances of enclosure,\r\n\u003ci\u003ei.e.\u003c/i\u003e there are really to be objects of which one encloses\r\nthe other. When an enclosure-relation has these properties,\r\nwe will call it a “point-producer.” Given any\r\nrelation of enclosure, we will call a set of objects an\r\n“enclosure-series” if, of any two of them, one is\r\ncontained in the other. We require a condition which\r\nshall secure that an enclosure-series converges to a point,\r\nand this is obtained as follows: Let our enclosure-series\r\nbe such that, given any other enclosure-series of which\r\nthere are members enclosed in any arbitrarily chosen\r\nmember of our first series, then there are members of\r\nour first series enclosed in any arbitrarily chosen member\r\nof our second series. In this case, our first enclosure-series\r\nmay be called a “punctual enclosure-series.” Then\r\na “point” is all the objects which enclose members of a\r\ngiven punctual enclosure-series. In order to ensure infinite\r\ndivisibility, we require one further property to be added\r\nto those defining point-producers, namely that any object\r\nwhich encloses itself also encloses an object other than itself.\r\nThe “points” generated by point-producers with this property\r\nwill be found to be such as geometry requires.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"116\" id=\"Page_116\"\u003e \u003c/a\u003e(3) The question of time, so long as we confine ourselves\r\nto one private world, is rather less complicated\r\nthan that of space, and we can see pretty clearly how it\r\nmight be dealt with by such methods as we have been\r\nconsidering. Events of which we are conscious do not\r\nlast merely for a mathematical instant, but always for\r\nsome finite time, however short. Even if there be a\r\nphysical world such as the mathematical theory of motion\r\nsupposes, impressions on our sense-organs produce sensations\r\nwhich are not merely and strictly instantaneous, and\r\ntherefore the objects of sense of which we are immediately\r\nconscious are not strictly instantaneous. Instants, therefore,\r\nare not among the data of experience, and, if\r\nlegitimate, must be either inferred or constructed. It is\r\ndifficult to see how they can be validly inferred; thus we\r\nare left with the alternative that they must be constructed.\r\nHow is this to be done?\u003c/p\u003e\r\n\u003cp\u003eImmediate experience provides us with two time-relations\r\namong events: they may be simultaneous, or\r\none may be earlier and the other later. These two are\r\nboth part of the crude data; it is not the case that only\r\nthe events are given, and their time-order is added by\r\nour subjective activity. The time-order, within certain\r\nlimits, is as much given as the events. In any story of\r\nadventure you will find such passages as the following:\r\n“With a cynical smile he pointed the revolver at the\r\nbreast of the dauntless youth. ‘At the word \u003cem\u003ethree\u003c/em\u003e I\r\nshall fire,’ he said. The words one and two had already\r\nbeen spoken with a cool and deliberate distinctness. The\r\nword \u003cem\u003ethree\u003c/em\u003e was forming on his lips. At this moment a\r\nblinding flash of lightning rent the air.” Here we have\r\nsimultaneity—not due, as Kant would have us believe, to\r\nthe subjective mental apparatus of the dauntless youth,\r\nbut given as objectively as the revolver and the lightning.\r\n\u003ca class=\"pagenum\" title=\"117\" id=\"Page_117\"\u003e \u003c/a\u003e\r\nAnd it is equally given in immediate experience that the\r\nwords \u003cem\u003eone\u003c/em\u003e and \u003cem\u003etwo\u003c/em\u003e come earlier than the flash. These\r\ntime-relations hold between events which are not strictly\r\ninstantaneous. Thus one event may begin sooner than\r\nanother, and therefore be before it, but may continue\r\nafter the other has begun, and therefore be also simultaneous\r\nwith it. If it persists after the other is over, it\r\nwill also be later than the other. Earlier, simultaneous,\r\nand later, are not inconsistent with each other when we\r\nare concerned with events which last for a finite time,\r\nhowever short; they only become inconsistent when we\r\nare dealing with something instantaneous.\u003c/p\u003e\r\n\u003cp\u003eIt is to be observed that we cannot give what may be\r\ncalled \u003cem\u003eabsolute\u003c/em\u003e dates, but only dates determined by events.\r\nWe cannot point to a time itself, but only to some event\r\noccurring at that time. There is therefore no reason in\r\nexperience to suppose that there are times as opposed to\r\nevents: the events, ordered by the relations of simultaneity\r\nand succession, are all that experience provides.\r\nHence, unless we are to introduce superfluous metaphysical\r\nentities, we must, in defining what mathematical\r\nphysics can regard as an instant, proceed by means of\r\nsome construction which assumes nothing beyond events\r\nand their temporal relations.\u003c/p\u003e\r\n\u003cp\u003eIf we wish to assign a date exactly by means of events,\r\nhow shall we proceed? If we take any one event, we\r\ncannot assign our date exactly, because the event is not\r\ninstantaneous, that is to say, it may be simultaneous with\r\ntwo events which are not simultaneous with each other.\r\nIn order to assign a date exactly, we must be able,\r\ntheoretically, to determine whether any given event is\r\nbefore, at, or after this date, and we must know that any\r\nother date is either before or after this date, but not\r\nsimultaneous with it. Suppose, now, instead of taking\r\n\u003ca class=\"pagenum\" title=\"118\" id=\"Page_118\"\u003e \u003c/a\u003e\r\none event A, we take two events A and B, and suppose\r\nA and B partly overlap, but B ends before A ends.\r\nThen an event which is simultaneous with both A and B\r\nmust exist during the time when A and B overlap; thus\r\nwe have come rather nearer to a precise date than when\r\nwe considered A and B alone. Let C be an event which\r\nis simultaneous with both A and B, but which ends before\r\neither A or B has ended. Then an event which is simultaneous\r\nwith A and B and C must exist during the time\r\nwhen all three\r\noverlap, which is\r\na still shorter\r\ntime. Proceeding\r\nin this way,\r\nby taking more\r\nand more events,\r\na new event which\r\nis dated as simultaneous with all of them becomes gradually\r\nmore and more accurately dated. This suggests a\r\nway by which a completely accurate date can be defined.\u003c/p\u003e\r\n\u003cdiv class=\"figcenter\" style=\"width: 300px;\"\u003e\r\n\u003cimg alt=\"\" height=\"146\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-our-knowledge-of-the-external-world-bertrand-russell-p0118-image.png\" width=\"300\" id=\"img_images_p0118-image.png\"\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003eLet us take a group of events of which any two overlap,\r\nso that there is some time, however short, when they all\r\nexist. If there is any other event which is simultaneous\r\nwith all of these, let us add it to the group; let us go\r\non until we have constructed a group such that no event\r\noutside the group is simultaneous with all of them, but\r\nall the events inside the group are simultaneous with\r\neach other. Let us define this whole group as an instant\r\nof time. It remains to show that it has the properties we\r\nexpect of an instant.\u003c/p\u003e\r\n\u003cp\u003eWhat are the properties we expect of instants? First,\r\nthey must form a series: of any two, one must be before\r\nthe other, and the other must be not before the one; if\r\none is before another, and the other before a third, the\r\n\u003ca class=\"pagenum\" title=\"119\" id=\"Page_119\"\u003e \u003c/a\u003e\r\nfirst must be before the third. Secondly, every event\r\nmust be at a certain number of instants; two events are\r\nsimultaneous if they are at the same instant, and one is\r\nbefore the other if there is an instant, at which the one\r\nis, which is earlier than some instant at which the other\r\nis. Thirdly, if we assume that there is always some\r\nchange going on somewhere during the time when any\r\ngiven event persists, the series of instants ought to be\r\ncompact, \u003ci\u003ei.e.\u003c/i\u003e given any two instants, there ought to be\r\nother instants between them. Do instants, as we have\r\ndefined them, have these properties?\u003c/p\u003e\r\n\u003cp\u003eWe shall say that an event is “at” an instant when it\r\nis a member of the group by which the instant is constituted;\r\nand we shall say that one instant is before\r\nanother if the group which is the one instant contains an\r\nevent which is earlier than, but not simultaneous with,\r\nsome event in the group which is the other instant.\r\nWhen one event is earlier than, but not simultaneous\r\nwith another, we shall say that it “wholly precedes” the\r\nother. Now we know that of two events which are not\r\nsimultaneous, there must be one which wholly precedes\r\nthe other, and in that case the other cannot also wholly\r\nprecede the one; we also know that, if one event wholly\r\nprecedes another, and the other wholly precedes a third,\r\nthen the first wholly precedes the third. From these\r\nfacts it is easy to deduce that the instants as we have\r\ndefined them form a series.\u003c/p\u003e\r\n\u003cp\u003eWe have next to show that every event is “at” at least\r\none instant, \u003ci\u003ei.e.\u003c/i\u003e that, given any event, there is at least\r\none class, such as we used in defining instants, of which\r\nit is a member. For this purpose, consider all the events\r\nwhich are simultaneous with a given event, and do not\r\nbegin later, \u003ci\u003ei.e.\u003c/i\u003e are not wholly after anything simultaneous\r\nwith it. We will call these the “initial \u003cins title=\"contemporaries\"\u003econtemporaries”\u003c/ins\u003e\r\n\u003ca class=\"pagenum\" title=\"120\" id=\"Page_120\"\u003e \u003c/a\u003e\r\nof the given event. It will be found that this class of events\r\nis the first instant at which the given event exists, provided\r\nevery event wholly after some contemporary of the given\r\nevent is wholly after some \u003cem\u003einitial\u003c/em\u003e contemporary of it.\u003c/p\u003e\r\n\u003cp\u003eFinally, the series of instants will be compact if, given\r\nany two events of which one wholly precedes the other,\r\nthere are events wholly after the one and simultaneous\r\nwith something wholly before the other. Whether this\r\nis the case or not, is an empirical question; but if it is not,\r\nthere is no reason to expect the time-series to be compact.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_17\" id=\"FNanchor_17\"\u003e[17]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eThus our definition of instants secures all that mathematics\r\nrequires, without having to assume the existence\r\nof any disputable metaphysical entities.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"121\" id=\"Page_121\"\u003e \u003c/a\u003eInstants may also be defined by means of the enclosure-relation,\r\nexactly as was done in the case of points. One\r\nobject will be temporally enclosed by another when it is\r\nsimultaneous with the other, but not before or after it.\r\nWhatever encloses temporally or is enclosed temporally\r\nwe shall call an “event.” In order that the relation\r\nof temporal enclosure may be a “point-producer,” we\r\nrequire (1) that it should be transitive, \u003ci\u003ei.e.\u003c/i\u003e that if one\r\nevent encloses another, and the other a third, then the\r\nfirst encloses the third; (2) that every event encloses\r\nitself, but if one event encloses another different event,\r\nthen the other does not enclose the one; (3) that given\r\nany set of events such that there is at least one event\r\nenclosed by all of them, then there is an event enclosing\r\nall that they all enclose, and itself enclosed by all of them;\r\n(4) that there is at least one event. To ensure infinite\r\ndivisibility, we require also that every event should enclose\r\nevents other than itself. Assuming these characteristics,\r\ntemporal enclosure is an infinitely divisible point-producer.\r\nWe can now form an “enclosure-series” of events, by\r\nchoosing a group of events such that of any two there is\r\none which encloses the other; this will be a “punctual\r\nenclosure-series” if, given any other enclosure-series such\r\nthat every member of our first series encloses some\r\nmember of our second, then every member of our second\r\nseries encloses some member of our first. Then an\r\n“instant” is the class of all events which enclose members\r\nof a given punctual enclosure-series.\u003c/p\u003e\r\n\u003cp\u003eThe correlation of the times of different private worlds\r\nso as to produce the one all-embracing time of physics is\r\na more difficult matter. We saw, in \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003eLecture III.\u003c/a\u003e, that\r\ndifferent private worlds often contain correlated appearances,\r\nsuch as common sense would regard as appearances\r\nof the same “thing.” When two appearances in different\r\n\u003ca class=\"pagenum\" title=\"122\" id=\"Page_122\"\u003e \u003c/a\u003e\r\nworlds are so correlated as to belong to one momentary\r\n“state” of a thing, it would be natural to regard them as\r\nsimultaneous, and as thus affording a simple means of\r\ncorrelating different private times. But this can only be\r\nregarded as a first approximation. What we call one\r\nsound will be heard sooner by people near the source of\r\nthe sound than by people further from it, and the same\r\napplies, though in a less degree, to light. Thus two correlated\r\nappearances in different worlds are not necessarily\r\nto be regarded as occurring at the same date in physical\r\ntime, though they will be parts of one momentary state\r\nof a thing. The correlation of different private times is\r\nregulated by the desire to secure the simplest possible\r\nstatement of the laws of physics, and thus raises rather\r\ncomplicated technical problems; but from the point of\r\nview of philosophical theory, there is no very serious\r\ndifficulty of principle involved.\u003c/p\u003e\r\n\u003cp\u003eThe above brief outline must not be regarded as more\r\nthan tentative and suggestive. It is intended merely to\r\nshow the kind of way in which, given a world with the\r\nkind of properties that psychologists find in the world\r\nof sense, it may be possible, by means of purely logical\r\nconstructions, to make it amenable to mathematical treatment\r\nby defining series or classes of sense-data which can\r\nbe called respectively particles, points, and instants. If\r\nsuch constructions are possible, then mathematical physics\r\nis applicable to the real world, in spite of the fact that\r\nits particles, points, and instants are not to be found\r\namong actually existing entities.\u003c/p\u003e\r\n\u003cp\u003eThe problem which the above considerations are intended\r\nto elucidate is one whose importance and even\r\nexistence has been concealed by the unfortunate separation\r\nof different studies which prevails throughout the\r\ncivilised world. Physicists, ignorant and contemptuous\r\n\u003ca class=\"pagenum\" title=\"123\" id=\"Page_123\"\u003e \u003c/a\u003e\r\nof philosophy, have been content to assume their\r\nparticles, points, and instants in practice, while conceding,\r\nwith ironical politeness, that their concepts laid no claim\r\nto metaphysical validity. Metaphysicians, obsessed by\r\nthe idealistic opinion that only mind is real, and the\r\nParmenidean belief that the real is unchanging, repeated\r\none after another the supposed contradictions in the\r\nnotions of matter, space, and time, and therefore naturally\r\nmade no endeavour to invent a tenable theory of particles,\r\npoints, and instants. Psychologists, who have done invaluable\r\nwork in bringing to light the chaotic nature of\r\nthe crude materials supplied by unmanipulated sensation,\r\nhave been ignorant of mathematics and modern logic,\r\nand have therefore been content to say that matter, space,\r\nand time are “intellectual constructions,” without making\r\nany attempt to show in detail either how the intellect\r\ncan construct them, or what secures the practical validity\r\nwhich physics shows them to possess. Philosophers, it\r\nis to be hoped, will come to recognise that they cannot\r\nachieve any solid success in such problems without some\r\nslight knowledge of logic, mathematics, and physics; meanwhile,\r\nfor want of students with the necessary equipment,\r\nthis vital problem remains unattempted and unknown.\u003c/p\u003e\r\n\u003cp\u003eThere are, it is true, two authors, both physicists, who\r\nhave done something, though not much, to bring about\r\na recognition of the problem as one demanding study.\r\nThese two authors are Poincaré and Mach, Poincaré\r\nespecially in his \u003ccite\u003eScience and Hypothesis\u003c/cite\u003e, Mach especially\r\nin his \u003ccite\u003eAnalysis of Sensations\u003c/cite\u003e. Both of them, however,\r\nadmirable as their work is, seem to me to suffer from a\r\ngeneral philosophical bias. Poincaré is Kantian, while\r\nMach is ultra-empiricist; with Poincaré almost all the\r\nmathematical part of physics is merely conventional, while\r\nwith Mach the sensation as a mental event is identified\r\n\u003ca class=\"pagenum\" title=\"124\" id=\"Page_124\"\u003e \u003c/a\u003e\r\nwith its object as a part of the physical world. Nevertheless,\r\nboth these authors, and especially Mach, deserve\r\nmention as having made serious contributions to the consideration\r\nof our problem.\u003c/p\u003e\r\n\u003cp\u003eWhen a point or an instant is defined as a class of\r\nsensible qualities, the first impression produced is likely\r\nto be one of wild and wilful paradox. Certain considerations\r\napply here, however, which will again be relevant\r\nwhen we come to the definition of numbers. There is\r\na whole type of problems which can be solved by such\r\ndefinitions, and almost always there will be at first an\r\neffect of paradox. Given a set of objects any two of\r\nwhich have a relation of the sort called “symmetrical and\r\ntransitive,” it is almost certain that we shall come to\r\nregard them as all having some common quality, or as\r\nall having the same relation to some one object outside\r\nthe set. This kind of case is important, and I shall\r\ntherefore try to make it clear even at the cost of some\r\nrepetition of previous definitions.\u003c/p\u003e\r\n\u003cp\u003eA relation is said to be “symmetrical” when, if one\r\nterm has this relation to another, then the other also has\r\nit to the one. Thus “brother or sister” is a “symmetrical”\r\nrelation: if one person is a brother or a sister of\r\nanother, then the other is a brother or sister of the one.\r\nSimultaneity, again, is a symmetrical relation; so is\r\nequality in size. A relation is said to be “transitive”\r\nwhen, if one term has this relation to another, and the\r\nother to a third, then the one has it to the third. The\r\nsymmetrical relations mentioned just now are also transitive—provided,\r\nin the case of “brother or sister,” we\r\nallow a person to be counted as his or her own brother\r\nor sister, and provided, in the case of simultaneity, we\r\nmean complete simultaneity, \u003ci\u003ei.e.\u003c/i\u003e beginning and ending\r\ntogether.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"125\" id=\"Page_125\"\u003e \u003c/a\u003eBut many relations are transitive without being symmetrical—for\r\ninstance, such relations as “greater,”\r\n“earlier,” “to the right of,” “ancestor of,” in fact all\r\nsuch relations as give rise to series. Other relations\r\nare symmetrical without being transitive—for example,\r\ndifference in any respect. If A is of a different age from\r\nB, and B of a different age from C, it does not follow that\r\nA is of a different age from C. Simultaneity, again, in\r\nthe case of events which last for a finite time, will not\r\nnecessarily be transitive if it only means that the times of\r\nthe two events overlap. If A ends just after B has\r\nbegun, and B ends just after C has begun, A and B will\r\nbe simultaneous in this sense, and so will B and C, but\r\nA and C may well not be simultaneous.\u003c/p\u003e\r\n\u003cp\u003eAll the relations which can naturally be represented as\r\nequality in any respect, or as possession of a common\r\nproperty, are transitive and symmetrical—this applies, for\r\nexample, to such relations as being of the same height or\r\nweight or colour. Owing to the fact that possession of a\r\ncommon property gives rise to a transitive symmetrical\r\nrelation, we come to imagine that wherever such a relation\r\noccurs it must be due to a common property. “Being\r\nequally numerous” is a transitive symmetrical relation of\r\ntwo collections; hence we imagine that both have a\r\ncommon property, called their number. “Existing at a\r\ngiven instant” (in the sense in which we defined an\r\ninstant) is a transitive symmetrical relation; hence we\r\ncome to think that there really is an instant which confers\r\na common property on all the things existing at that\r\ninstant. “Being states of a given thing” is a transitive\r\nsymmetrical relation; hence we come to imagine that\r\nthere really is a thing, other than the series of states,\r\nwhich accounts for the transitive symmetrical relation.\r\nIn all such cases, the class of terms that have the given\r\n\u003ca class=\"pagenum\" title=\"126\" id=\"Page_126\"\u003e \u003c/a\u003e\r\ntransitive symmetrical relation to a given term will fulfil\r\nall the formal requisites of a common property of all the\r\nmembers of the class. Since there certainly is the class,\r\nwhile any other common property may be illusory, it is\r\nprudent, in order to avoid needless assumptions, to\r\nsubstitute the class for the common property which would\r\nbe ordinarily assumed. This is the reason for the\r\ndefinitions we have adopted, and this is the source of the\r\napparent paradoxes. No harm is done if there are such\r\ncommon properties as language assumes, since we do not\r\ndeny them, but merely abstain from asserting them.\r\nBut if there are not such common properties in any given\r\ncase, then our method has secured us against error. In\r\nthe absence of special knowledge, therefore, the method\r\nwe have adopted is the only one which is safe, and which\r\navoids the risk of introducing fictitious metaphysical\r\nentities.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"127\" id=\"Page_127\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE V\u003c/small\u003e\u003cbr\u003e\r\nTHE THEORY OF CONTINUITY\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"129\" id=\"Page_129\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_5\"\u003eLECTURE V\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nTHE THEORY OF CONTINUITY\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eThe\u003c/span\u003e theory of continuity, with which we shall be occupied\r\nin the present lecture, is, in most of its refinements\r\nand developments, a purely mathematical subject—very\r\nbeautiful, very important, and very delightful, but not,\r\nstrictly speaking, a part of philosophy. The logical basis\r\nof the theory alone belongs to philosophy, and alone\r\nwill occupy us to-night. The way the problem of continuity\r\nenters into philosophy is, broadly speaking, the\r\nfollowing: Space and time are treated by mathematicians\r\nas consisting of points and instants, but they also have a\r\nproperty, easier to feel than to define, which is called\r\ncontinuity, and is thought by many philosophers to be\r\ndestroyed when they are resolved into points and instants.\r\nZeno, as we shall see, proved that analysis into points and\r\ninstants was impossible if we adhered to the view that the\r\nnumber of points or instants in a finite space or time must\r\nbe finite. Later philosophers, believing infinite number\r\nto be self-contradictory, have found here an antinomy:\r\nSpaces and times could not consist of a \u003cem\u003efinite\u003c/em\u003e number of\r\npoints and instants, for such reasons as Zeno\u0027s; they\r\ncould not consist of an \u003cem\u003einfinite\u003c/em\u003e number of points and\r\ninstants, because infinite numbers were supposed to be\r\nself-contradictory. Therefore spaces and times, if real at\r\nall, must not be regarded as composed of points and\r\ninstants.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"130\" id=\"Page_130\"\u003e \u003c/a\u003eBut even when points and instants, as independent\r\nentities, are discarded, as they were by the theory\r\nadvocated in our \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003elast lecture\u003c/a\u003e, the problems of continuity,\r\nas I shall try to show presently, remain, in a practically\r\nunchanged form. Let us therefore, to begin with, admit\r\npoints and instants, and consider the problems in connection\r\nwith this simpler or at least more familiar\r\nhypothesis.\u003c/p\u003e\r\n\u003cp\u003eThe argument against continuity, in so far as it rests\r\nupon the supposed difficulties of infinite numbers, has\r\nbeen disposed of by the positive theory of the infinite,\r\nwhich will be considered in \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003eLecture VII\u003c/a\u003e. But there\r\nremains a feeling—of the kind that led Zeno to the\r\ncontention that the arrow in its flight is at rest—which\r\nsuggests that points and instants, even if they are\r\ninfinitely numerous, can only give a jerky motion, a\r\nsuccession of different immobilities, not the smooth\r\ntransitions with which the senses have made us familiar.\r\nThis feeling is due, I believe, to a failure to realise\r\nimaginatively, as well as abstractly, the nature of continuous\r\nseries as they appear in mathematics. When a\r\ntheory has been apprehended logically, there is often a\r\nlong and serious labour still required in order to \u003cem\u003efeel\u003c/em\u003e it:\r\nit is necessary to dwell upon it, to thrust out from the\r\nmind, one by one, the misleading suggestions of false but\r\nmore familiar theories, to acquire the kind of intimacy\r\nwhich, in the case of a foreign language, would enable\r\nus to think and dream in it, not merely to construct laborious\r\nsentences by the help of grammar and dictionary.\r\nIt is, I believe, the absence of this kind of intimacy\r\nwhich makes many philosophers regard the mathematical\r\ndoctrine of continuity as an inadequate explanation of the\r\ncontinuity which we experience in the world of sense.\u003c/p\u003e\r\n\u003cp\u003eIn the present lecture, I shall first try to explain in\r\n\u003ca class=\"pagenum\" title=\"131\" id=\"Page_131\"\u003e \u003c/a\u003e\r\noutline what the mathematical theory of continuity is in\r\nits philosophically important essentials. The application\r\nto actual space and time will not be in question to begin\r\nwith. I do not see any reason to suppose that the points\r\nand instants which mathematicians introduce in dealing\r\nwith space and time are actual physically existing entities,\r\nbut I do see reason to suppose that the continuity of\r\nactual space and time may be more or less analogous to\r\nmathematical continuity. The theory of mathematical\r\ncontinuity is an abstract logical theory, not dependent for\r\nits validity upon any properties of actual space and time.\r\nWhat is claimed for it is that, when it is understood,\r\ncertain characteristics of space and time, previously very\r\nhard to analyse, are found not to present any logical\r\ndifficulty. What we know empirically about space and\r\ntime is insufficient to enable us to decide between various\r\nmathematically possible alternatives, but these alternatives\r\nare all fully intelligible and fully adequate to the observed\r\nfacts. For the present, however, it will be well to forget\r\nspace and time and the continuity of sensible change, in\r\norder to return to these topics equipped with the weapons\r\nprovided by the abstract theory of continuity.\u003c/p\u003e\r\n\u003cp\u003eContinuity, in mathematics, is a property only possible\r\nto a \u003cem\u003eseries\u003c/em\u003e of terms, \u003ci\u003ei.e.\u003c/i\u003e to terms arranged in an order, so\r\nthat we can say of any two that one comes \u003cem\u003ebefore\u003c/em\u003e the\r\nother. Numbers in order of magnitude, the points on a\r\nline from left to right, the moments of time from earlier\r\nto later, are instances of series. The notion of order,\r\nwhich is here introduced, is one which is not required in\r\nthe theory of cardinal number. It is possible to know\r\nthat two classes have the same number of terms without\r\nknowing any order in which they are to be taken. We\r\nhave an instance of this in such a case as English husbands\r\nand English wives: we can see that there must be the\r\n\u003ca class=\"pagenum\" title=\"132\" id=\"Page_132\"\u003e \u003c/a\u003e\r\nsame number of husbands as of wives, without having to\r\narrange them in a series. But continuity, which we are\r\nnow to consider, is essentially a property of an order: it\r\ndoes not belong to a set of terms in themselves, but only\r\nto a set in a certain order. A set of terms which can be\r\narranged in one order can always also be arranged in\r\nother orders, and a set of terms which can be arranged\r\nin a continuous order can always also be arranged in\r\norders which are not continuous. Thus the essence\r\nof continuity must not be sought in the nature of the\r\nset of terms, but in the nature of their arrangement\r\nin a series.\u003c/p\u003e\r\n\u003cp\u003eMathematicians have distinguished different degrees\r\nof continuity, and have confined the word “continuous,”\r\nfor technical purposes, to series having a certain high\r\ndegree of continuity. But for philosophical purposes, all\r\nthat is important in continuity is introduced by the lowest\r\ndegree of continuity, which is called “compactness.” A\r\nseries is called “compact” when no two terms are\r\nconsecutive, but between any two there are others. One\r\nof the simplest examples of a compact series is the series\r\nof fractions in order of magnitude. Given any two fractions,\r\nhowever near together, there are other fractions\r\ngreater than the one and smaller than the other, and\r\ntherefore no two fractions are consecutive. There is no\r\nfraction, for example, which is next after \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e: if we choose\r\nsome fraction which is very little greater than \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e, say \u003csup\u003e51\u003c/sup\u003e⁄\u003csub\u003e100\u003c/sub\u003e\r\nwe can find others, such as \u003csup\u003e101\u003c/sup\u003e⁄\u003csub\u003e200\u003c/sub\u003e, which are nearer to \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e.\r\nThus between any two fractions, however little they\r\ndiffer, there are an infinite number of other fractions.\r\nMathematical space and time also have this property of\r\ncompactness, though whether actual space and time have\r\nit is a further question, dependent upon empirical evidence,\r\nand probably incapable of being answered with certainty.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"133\" id=\"Page_133\"\u003e \u003c/a\u003eIn the case of abstract objects such as fractions, it is\r\nperhaps not very difficult to realise the logical possibility\r\nof their forming a compact series. The difficulties that\r\nmight be felt are those of infinity, for in a compact series\r\nthe number of terms between any two given terms must\r\nbe infinite. But when these difficulties have been solved,\r\nthe mere compactness in itself offers no great obstacle to\r\nthe imagination. In more concrete cases, however, such\r\nas motion, compactness becomes much more repugnant\r\nto our habits of thought. It will therefore be desirable\r\nto consider explicitly the mathematical account of motion,\r\nwith a view to making its logical possibility felt. The\r\nmathematical account of motion is perhaps artificially\r\nsimplified when regarded as describing what actually\r\noccurs in the physical world; but what actually occurs\r\nmust be capable, by a certain amount of logical manipulation,\r\nof being brought within the scope of the mathematical\r\naccount, and must, in its analysis, raise just such problems\r\nas are raised in their simplest form by this account.\r\nNeglecting, therefore, for the present, the question of\r\nits physical adequacy, let us devote ourselves merely to\r\nconsidering its possibility as a formal statement of the\r\nnature of motion.\u003c/p\u003e\r\n\u003cp\u003eIn order to simplify our problem as much as possible,\r\nlet us imagine a tiny speck of light moving along a scale.\r\nWhat do we mean by saying that the motion is continuous?\r\nIt is not necessary for our purposes to consider the whole\r\nof what the mathematician means by this statement: only\r\npart of what he means is philosophically important. One\r\npart of what he means is that, if we consider any two\r\npositions of the speck occupied at any two instants, there\r\nwill be other intermediate positions occupied at intermediate\r\ninstants. However near together we take the\r\ntwo positions, the speck will not jump suddenly from\r\n\u003ca class=\"pagenum\" title=\"134\" id=\"Page_134\"\u003e \u003c/a\u003e\r\nthe one to the other, but will pass through an infinite\r\nnumber of other positions on the way. Every distance,\r\nhowever small, is traversed by passing through all the\r\ninfinite series of positions between the two ends of the\r\ndistance.\u003c/p\u003e\r\n\u003cp\u003eBut at this point imagination suggests that we may\r\ndescribe the continuity of motion by saying that the speck\r\nalways passes from one position at one instant to \u003cem\u003ethe next\u003c/em\u003e\r\nposition at \u003cem\u003ethe next\u003c/em\u003e instant. As soon as we say this or\r\nimagine it, we fall into error, because there is no \u003cem\u003enext\u003c/em\u003e\r\npoint or \u003cem\u003enext\u003c/em\u003e instant. If there were, we should find\r\nZeno\u0027s paradoxes, in some form, unavoidable, as will\r\nappear in our \u003ca href=\"#Lecture_6\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e. One simple paradox may\r\nserve as an illustration. If our speck is in motion along\r\nthe scale throughout the whole of a certain time, it cannot\r\nbe at the same point at two consecutive instants. But it\r\ncannot, from one instant to the next, travel further than\r\nfrom one point to the next, for if it did, there would be\r\nno instant at which it was in the positions intermediate\r\nbetween that at the first instant and that at the next, and\r\nwe agreed that the continuity of motion excludes the\r\npossibility of such sudden jumps. It follows that our\r\nspeck must, so long as it moves, pass from one point at\r\none instant to the next point at the next instant. Thus\r\nthere will be just one perfectly definite velocity with\r\nwhich all motions must take place: no motion can be\r\nfaster than this, and no motion can be slower. Since this\r\nconclusion is false, we must reject the hypothesis upon\r\nwhich it is based, namely that there are consecutive points\r\nand instants.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_18\" id=\"FNanchor_18\"\u003e[18]\u003c/a\u003e Hence the continuity of motion must not\r\nbe supposed to consist in a body\u0027s occupying consecutive\r\npositions at consecutive times.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"135\" id=\"Page_135\"\u003e \u003c/a\u003eThe difficulty to imagination lies chiefly, I think, in\r\nkeeping out the suggestion of \u003cem\u003einfinitesimal\u003c/em\u003e distances and\r\ntimes. Suppose we halve a given distance, and then\r\nhalve the half, and so on, we can continue the process as\r\nlong as we please, and the longer we continue it, the\r\nsmaller the resulting distance becomes. This infinite\r\ndivisibility seems, at first sight, to imply that there are\r\ninfinitesimal distances, \u003ci\u003ei.e.\u003c/i\u003e distances so small that any\r\nfinite fraction of an inch would be greater. This, however,\r\nis an error. The continued bisection of our distance,\r\nthough it gives us continually smaller distances, gives us\r\nalways \u003cem\u003efinite\u003c/em\u003e distances. If our original distance was an\r\ninch, we reach successively half an inch, a quarter of an\r\ninch, an eighth, a sixteenth, and so on; but every one\r\nof this infinite series of diminishing distances is finite.\r\n“But,” it may be said, “\u003cem\u003ein the end\u003c/em\u003e the distance will grow\r\ninfinitesimal.” No, because there is no end. The\r\nprocess of bisection is one which can, theoretically, be\r\ncarried on for ever, without any last term being attained.\r\nThus infinite divisibility of distances, which must be\r\nadmitted, does not imply that there are distances so small\r\nthat any finite distance would be larger.\u003c/p\u003e\r\n\u003cp\u003eIt is easy, in this kind of question, to fall into an\r\nelementary logical blunder. Given any finite distance,\r\nwe can find a smaller distance; this may be expressed in\r\nthe ambiguous form “there is a distance smaller than any\r\nfinite distance.” But if this is then interpreted as meaning\r\n“there is a distance such that, whatever finite distance\r\nmay be chosen, the distance in question is smaller,” then\r\nthe statement is false. Common language is ill adapted\r\nto expressing matters of this kind, and philosophers who\r\nhave been dependent on it have frequently been misled\r\nby it.\u003c/p\u003e\r\n\u003cp\u003eIn a continuous motion, then, we shall say that at any\r\n\u003ca class=\"pagenum\" title=\"136\" id=\"Page_136\"\u003e \u003c/a\u003e\r\ngiven instant the moving body occupies a certain position,\r\nand at other instants it occupies other positions; the\r\ninterval between any two instants and between any two\r\npositions is always finite, but the continuity of the motion\r\nis shown in the fact that, however near together we take\r\nthe two positions and the two instants, there are an\r\ninfinite number of positions still nearer together, which\r\nare occupied at instants that are also still nearer together.\r\nThe moving body never jumps from one position to\r\nanother, but always passes by a gradual transition through\r\nan infinite number of intermediaries. At a given instant,\r\nit is where it is, like Zeno\u0027s arrow;\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_19\" id=\"FNanchor_19\"\u003e[19]\u003c/a\u003e but we cannot say\r\nthat it is at rest at the instant, since the instant does not\r\nlast for a finite time, and there is not a beginning and\r\nend of the instant with an interval between them. Rest\r\nconsists in being in the same position at all the instants\r\nthroughout a certain finite period, however short; it does\r\nnot consist simply in a body\u0027s being where it is at a given\r\ninstant. This whole theory, as is obvious, depends upon\r\nthe nature of compact series, and demands, for its full\r\ncomprehension, that compact series should have become\r\nfamiliar and easy to the imagination as well as to deliberate\r\nthought.\u003c/p\u003e\r\n\u003cp\u003eWhat is required may be expressed in mathematical\r\nlanguage by saying that the position of a moving body\r\nmust be a continuous function of the time. To define\r\naccurately what this means, we proceed as follows.\r\nConsider a particle which, at the moment \u003ci\u003et\u003c/i\u003e, is at the\r\npoint P. Choose now any\r\nsmall portion P\u003csub\u003e1\u003c/sub\u003eP\u003csub\u003e2\u003c/sub\u003e of the\r\npath of the particle, this\r\nportion being one which contains P. We say then\r\nthat, if the motion of the particle is continuous at the\r\n\u003ca class=\"pagenum\" title=\"137\" id=\"Page_137\"\u003e \u003c/a\u003e\r\ntime \u003ci\u003et\u003c/i\u003e, it must be possible to find two instants \u003ci\u003et\u003c/i\u003e\u003csub\u003e1\u003c/sub\u003e, \u003ci\u003et\u003c/i\u003e\u003csub\u003e2\u003c/sub\u003e, one\r\nearlier than \u003ci\u003et\u003c/i\u003e and one later, such that throughout the\r\nwhole time from \u003ci\u003et\u003c/i\u003e\u003csub\u003e1\u003c/sub\u003e to \u003ci\u003et\u003c/i\u003e\u003csub\u003e2\u003c/sub\u003e (both included), the particle lies\r\nbetween P\u003csub\u003e1\u003c/sub\u003e and P\u003csub\u003e2\u003c/sub\u003e. And we say that this must still\r\nhold however small we make the portion P\u003csub\u003e1\u003c/sub\u003eP\u003csub\u003e2\u003c/sub\u003e. When\r\nthis is the case, we say that the motion is continuous at\r\nthe time \u003ci\u003et\u003c/i\u003e; and when the motion is continuous at all\r\ntimes, we say that the motion as a whole is continuous.\r\nIt is obvious that if the particle were to jump suddenly\r\nfrom P to some other point Q, our definition would fail for\r\nall intervals P\u003csub\u003e1\u003c/sub\u003eP\u003csub\u003e2\u003c/sub\u003e which were too small to include Q.\r\nThus our definition affords an analysis of the continuity\r\nof motion, while admitting points and instants and\r\ndenying infinitesimal distances in space or periods in\r\ntime.\u003c/p\u003e\r\n\u003cdiv class=\"figcenter\" style=\"width: 300px;\"\u003e\r\n\u003cimg alt=\"\" height=\"49\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-our-knowledge-of-the-external-world-bertrand-russell-p0136-image.png\" width=\"300\" id=\"img_images_p0136-image.png\"\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003ePhilosophers, mostly in ignorance of the mathematician\u0027s\r\nanalysis, have adopted other and more heroic\r\nmethods of dealing with the \u003ci lang=\"la\"\u003eprimâ facie\u003c/i\u003e difficulties of\r\ncontinuous motion. A typical and recent example of\r\nphilosophic theories of motion is afforded by Bergson,\r\nwhose views on this subject I have examined elsewhere.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_20\" id=\"FNanchor_20\"\u003e[20]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eApart from definite arguments, there are certain\r\nfeelings, rather than reasons, which stand in the way of\r\nan acceptance of the mathematical account of motion.\r\nTo begin with, if a body is moving at all fast, we \u003cem\u003esee\u003c/em\u003e its\r\nmotion just as we see its colour. A \u003cem\u003eslow\u003c/em\u003e motion, like that\r\nof the hour-hand of a watch, is only known in the way\r\nwhich mathematics would lead us to expect, namely by\r\nobserving a change of position after a lapse of time; but,\r\nwhen we observe the motion of the second-hand, we do\r\nnot merely see first one position and then another—we\r\nsee something as directly sensible as colour. What is\r\nthis something that we see, and that we call visible motion?\r\n\u003ca class=\"pagenum\" title=\"138\" id=\"Page_138\"\u003e \u003c/a\u003e\r\nWhatever it is, it is \u003cem\u003enot\u003c/em\u003e the successive occupation\r\nof successive positions: something beyond the mathematical\r\ntheory of motion is required to account for it.\r\nOpponents of the mathematical theory emphasise this fact.\r\n“Your theory,” they say, “may be very logical, and\r\nmight apply admirably to some other world; but in this\r\nactual world, actual motions are quite different from what\r\nyour theory would declare them to be, and require, therefore,\r\nsome different philosophy from yours for their\r\nadequate explanation.”\u003c/p\u003e\r\n\u003cp\u003eThe objection thus raised is one which I have no wish\r\nto underrate, but I believe it can be fully answered without\r\ndeparting from the methods and the outlook which\r\nhave led to the mathematical theory of motion. Let us,\r\nhowever, first try to state the objection more fully.\u003c/p\u003e\r\n\u003cp\u003eIf the mathematical theory is adequate, nothing happens\r\nwhen a body moves except that it is in different places\r\nat different times. But in this sense the hour-hand\r\nand the second-hand are equally in motion, yet in the\r\nsecond-hand there is something perceptible to our senses\r\nwhich is absent in the hour-hand. We can see, at each\r\nmoment, that the second-hand \u003cem\u003eis moving\u003c/em\u003e, which is different\r\nfrom seeing it first in one place and then in another.\r\nThis seems to involve our seeing it simultaneously in a\r\nnumber of places, although it must also involve our\r\nseeing that it is in some of these places earlier than in\r\nothers. If, for example, I move my hand quickly from\r\nleft to right, you seem to see the whole movement at\r\nonce, in spite of the fact that you know it begins at the\r\nleft and ends at the right. It is this kind of consideration,\r\nI think, which leads Bergson and many others to\r\nregard a movement as really one indivisible whole, not\r\nthe series of separate states imagined by the mathematician.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"139\" id=\"Page_139\"\u003e \u003c/a\u003eTo this objection there are three supplementary\r\nanswers, physiological, psychological, and logical. We\r\nwill consider them successively.\u003c/p\u003e\r\n\u003cp\u003e(1) The physiological answer merely shows that, if the\r\nphysical world is what the mathematician supposes, its\r\nsensible appearance may nevertheless be expected to be\r\nwhat it is. The aim of this answer is thus the modest\r\none of showing that the mathematical account is not impossible\r\nas applied to the physical world; it does not\r\neven attempt to show that this account is necessary, or\r\nthat an analogous account applies in psychology.\u003c/p\u003e\r\n\u003cp\u003eWhen any nerve is stimulated, so as to cause a sensation,\r\nthe sensation does not cease instantaneously with\r\nthe cessation of the stimulus, but dies away in a short\r\nfinite time. A flash of lightning, brief as it is to our\r\nsight, is briefer still as a physical phenomenon: we\r\ncontinue to see it for a few moments after the light-waves\r\nhave ceased to strike the eye. Thus in the case of a\r\nphysical motion, if it is sufficiently swift, we shall actually\r\nat one instant see the moving body throughout a finite\r\nportion of its course, and not only at the exact spot where\r\nit is at that instant. Sensations, however, as they die\r\naway, grow gradually fainter; thus the sensation due to\r\na stimulus which is recently past is not exactly like the\r\nsensation due to a present stimulus. It follows from\r\nthis that, when we see a rapid motion, we shall not only\r\nsee a number of positions of the moving body simultaneously,\r\nbut we shall see them with different degrees of\r\nintensity—the present position most vividly, and the\r\nothers with diminishing vividness, until sensation fades\r\naway into immediate memory. This state of things\r\naccounts fully for the perception of motion. A motion\r\nis \u003cem\u003eperceived\u003c/em\u003e, not merely \u003cem\u003einferred\u003c/em\u003e, when it is sufficiently\r\nswift for many positions to be sensible at one time; and\r\n\u003ca class=\"pagenum\" title=\"140\" id=\"Page_140\"\u003e \u003c/a\u003e\r\nthe earlier and later parts of one perceived motion are\r\ndistinguished by the less and greater vividness of the\r\nsensations.\u003c/p\u003e\r\n\u003cp\u003eThis answer shows that physiology can account for our\r\nperception of motion. But physiology, in speaking of\r\nstimulus and sense-organs and a physical motion distinct\r\nfrom the immediate object of sense, is assuming the truth\r\nof physics, and is thus only capable of showing the\r\nphysical account to be possible, not of showing it to be\r\n\u003cem\u003enecessary\u003c/em\u003e. This consideration brings us to the psychological\r\nanswer.\u003c/p\u003e\r\n\u003cp\u003e(2) The psychological answer to our difficulty about\r\nmotion is part of a vast theory, not yet worked out, and\r\nonly capable, at present, of being vaguely outlined. We\r\nconsidered this theory in the \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003ethird\u003c/a\u003e and \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003efourth\u003c/a\u003e lectures;\r\nfor the present, a mere sketch of its application to our\r\npresent problem must suffice. The world of physics,\r\nwhich was assumed in the physiological answer, is obviously\r\ninferred from what is given in sensation; yet as\r\nsoon as we seriously consider what is actually given in\r\nsensation, we find it apparently very different from the\r\nworld of physics. The question is thus forced upon us:\r\nIs the inference from sense to physics a valid one?\r\nI believe the answer to be affirmative, for reasons which I\r\nsuggested in the \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003ethird\u003c/a\u003e and \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003efourth\u003c/a\u003e lectures; but the answer\r\ncannot be either short or easy. It consists, broadly speaking,\r\nin showing that, although the particles, points, and\r\ninstants with which physics operates are not themselves\r\ngiven in experience, and are very likely not actually existing\r\nthings, yet, out of the materials provided in sensation,\r\nit is possible to make logical constructions having the\r\nmathematical properties which physics assigns to particles,\r\npoints, and instants. If this can be done, then all the\r\npropositions of physics can be translated, by a sort of\r\n\u003ca class=\"pagenum\" title=\"141\" id=\"Page_141\"\u003e \u003c/a\u003e\r\ndictionary, into propositions about the kinds of objects\r\nwhich are given in sensation.\u003c/p\u003e\r\n\u003cp\u003eApplying these general considerations to the case of\r\nmotion, we find that, even within the sphere of immediate\r\nsense-data, it is necessary, or at any rate more consonant\r\nwith the facts than any other equally simple view, to\r\ndistinguish instantaneous states of objects, and to regard\r\nsuch states as forming a compact series. Let us consider\r\na body which is moving swiftly enough for its motion to\r\nbe perceptible, and long enough for its motion to be not\r\nwholly comprised in one sensation. Then, in spite of the\r\nfact that we see a finite extent of the motion at one\r\ninstant, the extent which we see at one instant is different\r\nfrom that which we see at another. Thus we are brought\r\nback, after all, to a series of momentary views of the\r\nmoving body, and this series will be compact, like the\r\nformer physical series of points. In fact, though the\r\n\u003cem\u003eterms\u003c/em\u003e of the series seem different, the mathematical character\r\nof the series is unchanged, and the whole mathematical\r\ntheory of motion will apply to it \u003cem\u003everbatim\u003c/em\u003e.\u003c/p\u003e\r\n\u003cp\u003eWhen we are considering the actual data of sensation\r\nin this connection, it is important to realise that two\r\nsense-data may be, and \u003cem\u003emust\u003c/em\u003e sometimes be, really different\r\nwhen we cannot perceive any difference between them.\r\nAn old but conclusive reason for believing this was\r\nemphasised by Poincaré.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_21\" id=\"FNanchor_21\"\u003e[21]\u003c/a\u003e In all cases of sense-data\r\ncapable of gradual change, we may find one sense-datum\r\nindistinguishable from another, and that other indistinguishable\r\nfrom a third, while yet the first and third\r\nare quite easily distinguishable. Suppose, for example, a\r\nperson with his eyes shut is holding a weight in his hand,\r\nand someone noiselessly adds a small extra weight. If\r\n\u003ca class=\"pagenum\" title=\"142\" id=\"Page_142\"\u003e \u003c/a\u003e\r\nthe extra weight is small enough, no difference will be\r\nperceived in the sensation. After a time, another small\r\nextra weight may be added, and still no change will be\r\nperceived; but if both extra weights had been added at\r\nonce, it may be that the change would be quite easily\r\nperceptible. Or, again, take shades of colour. It would\r\nbe easy to find three stuffs of such closely similar shades\r\nthat no difference could be perceived between the first\r\nand second, nor yet between the second and third, while\r\nyet the first and third would be distinguishable. In such\r\na case, the second shade cannot be the same as the first,\r\nor it would be distinguishable from the third; nor the\r\nsame as the third, or it would be distinguishable from the\r\nfirst. It must, therefore, though indistinguishable from\r\nboth, be really intermediate between them.\u003c/p\u003e\r\n\u003cp\u003eSuch considerations as the above show that, although\r\nwe cannot distinguish sense-data unless they differ by\r\nmore than a certain amount, it is perfectly reasonable to\r\nsuppose that sense-data of a given kind, such as weights\r\nor colours, really form a compact series. The objections\r\nwhich may be brought from a psychological point of view\r\nagainst the mathematical theory of motion are not, therefore,\r\nobjections to this theory properly understood, but\r\nonly to a quite unnecessary assumption of simplicity in\r\nthe momentary object of sense. Of the immediate object\r\nof sense, in the case of a visible motion, we may say that\r\nat each instant it is in all the positions which remain\r\nsensible at that instant; but this set of positions changes\r\ncontinuously from moment to moment, and is amenable\r\nto exactly the same mathematical treatment as if it were\r\na mere point. When we assert that some mathematical\r\naccount of phenomena is correct, all that we primarily\r\nassert is that \u003cem\u003esomething\u003c/em\u003e definable in terms of the crude\r\nphenomena satisfies our formulæ; and in this sense the\r\n\u003ca class=\"pagenum\" title=\"143\" id=\"Page_143\"\u003e \u003c/a\u003e\r\nmathematical theory of motion is applicable to the data\r\nof sensation as well as to the supposed particles of abstract\r\nphysics.\u003c/p\u003e\r\n\u003cp\u003eThere are a number of distinct questions which are apt\r\nto be confused when the mathematical continuum is said\r\nto be inadequate to the facts of sense. We may state\r\nthese, in order of diminishing generality, as follows:—\u003c/p\u003e\r\n\u003cblockquote\u003e\u003cdiv\u003e\r\n\u003cp\u003e(\u003ci\u003ea\u003c/i\u003e) Are series possessing mathematical continuity\r\nlogically possible?\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003eb\u003c/i\u003e) Assuming that they are possible logically, are\r\nthey not impossible as applied to actual sense-data,\r\nbecause, among actual sense-data, there are no such\r\nfixed mutually external terms as are to be found, \u003ci\u003ee.g.\u003c/i\u003e,\r\nin the series of fractions?\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ec\u003c/i\u003e) Does not the assumption of points and instants\r\nmake the whole mathematical account\r\nfictitious?\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ed\u003c/i\u003e) Finally, assuming that all these objections\r\nhave been answered, is there, in actual empirical fact,\r\nany sufficient reason to believe the world of sense\r\ncontinuous?\u003c/p\u003e\r\n\u003c/div\u003e\u003c/blockquote\u003e\r\n\u003cp\u003eLet us consider these questions in succession.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ea\u003c/i\u003e) The question of the logical possibility of the\r\nmathematical continuum turns partly on the elementary\r\nmisunderstandings we considered at the beginning of the\r\npresent lecture, partly on the possibility of the mathematical\r\ninfinite, which will occupy our next two lectures,\r\nand partly on the logical form of the answer to the\r\nBergsonian objection which we stated a few minutes ago.\r\nI shall say no more on this topic at present, since it is\r\ndesirable first to complete the psychological answer.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003eb\u003c/i\u003e) The question whether sense-data are composed of\r\nmutually external units is not one which can be decided\r\nby empirical evidence. It is often urged that, as a\r\n\u003ca class=\"pagenum\" title=\"144\" id=\"Page_144\"\u003e \u003c/a\u003e\r\nmatter of immediate experience, the sensible flux is\r\ndevoid of divisions, and is falsified by the dissections of\r\nthe intellect. Now I have no wish to argue that this\r\nview is \u003cem\u003econtrary\u003c/em\u003e to immediate experience: I wish only to\r\nmaintain that it is essentially incapable of being \u003cem\u003eproved\u003c/em\u003e by\r\nimmediate experience. As we saw, there must be among\r\nsense-data differences so slight as to be imperceptible:\r\nthe fact that sense-data are immediately given does not\r\nmean that their differences also \u003cem\u003emust\u003c/em\u003e be immediately\r\ngiven (though they \u003cem\u003emay\u003c/em\u003e be). Suppose, for example, a\r\ncoloured surface on which the colour changes gradually—so\r\ngradually that the difference of colour in two\r\nvery neighbouring portions is imperceptible, while the\r\ndifference between more widely separated portions is\r\nquite noticeable. The effect produced, in such a case,\r\nwill be precisely that of “interpenetration,” of transition\r\nwhich is not a matter of discrete units. And since it\r\ntends to be supposed that the colours, being immediate\r\ndata, must \u003cem\u003eappear\u003c/em\u003e different if they \u003cem\u003eare\u003c/em\u003e different, it seems\r\neasily to follow that “interpenetration” must be the\r\nultimately right account. But this does not follow.\r\nIt is unconsciously assumed, as a premiss for a\r\n\u003ci lang=\"la\"\u003ereductio ad absurdum\u003c/i\u003e of the analytic view, that, if A and\r\nB are immediate data, and A differs from B, then the\r\nfact that they differ must also be an immediate datum.\r\nIt is difficult to say how this assumption arose, but I\r\nthink it is to be connected with the confusion between\r\n“acquaintance” and “knowledge about.” Acquaintance,\r\nwhich is what we derive from sense, does not, theoretically\r\nat least, imply even the smallest “knowledge about,” \u003ci\u003ei.e.\u003c/i\u003e\r\nit does not imply knowledge of any proposition concerning\r\nthe object with which we are acquainted. It is a\r\nmistake to speak as if acquaintance had degrees: there\r\nis merely acquaintance and non-acquaintance. When we\r\n\u003ca class=\"pagenum\" title=\"145\" id=\"Page_145\"\u003e \u003c/a\u003e\r\nspeak of becoming “better acquainted,” as for instance\r\nwith a person, what we must mean is, becoming\r\nacquainted with more parts of a certain whole; but the\r\nacquaintance with each part is either complete or nonexistent.\r\nThus it is a mistake to say that if we were\r\nperfectly acquainted with an object we should know all\r\nabout it. “Knowledge about” is knowledge of propositions,\r\nwhich is not involved necessarily in acquaintance\r\nwith the constituents of the propositions. To know that\r\ntwo shades of colour are different is knowledge about\r\nthem; hence acquaintance with the two shades does not\r\nin any way necessitate the knowledge that they are\r\ndifferent.\u003c/p\u003e\r\n\u003cp\u003eFrom what has just been said it follows that the nature\r\nof sense-data cannot be validly used to prove that they\r\nare not composed of mutually external units. It may be\r\nadmitted, on the other hand, that nothing in their\r\nempirical character specially necessitates the view that\r\nthey are composed of mutually external units. This\r\nview, if it is held, must be held on logical, not on\r\nempirical, grounds. I believe that the logical grounds\r\nare adequate to the conclusion. They rest, at bottom,\r\nupon the impossibility of explaining complexity without\r\nassuming constituents. It is undeniable that the visual\r\nfield, for example, is complex; and so far as I can see,\r\nthere is always self-contradiction in the theories which,\r\nwhile admitting this complexity, attempt to deny that it\r\nresults from a combination of mutually external units.\r\nBut to pursue this topic would lead us too far from our\r\ntheme, and I shall therefore say no more about it at\r\npresent.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ec\u003c/i\u003e) It is sometimes urged that the mathematical\r\naccount of motion is rendered fictitious by its assumption\r\nof points and instants. Now there are here two different\r\n\u003ca class=\"pagenum\" title=\"146\" id=\"Page_146\"\u003e \u003c/a\u003e\r\nquestions to be distinguished. There is the question of\r\nabsolute or relative space and time, and there is the\r\nquestion whether what occupies space and time must\r\nbe composed of elements which have no extension or\r\nduration. And each of these questions in turn may\r\ntake two forms, namely: (\u003cspan class=\"greek\" title=\"a\" lang=\"grc\"\u003eα\u003c/span\u003e) is the hypothesis \u003cem\u003econsistent\u003c/em\u003e\r\nwith the facts and with logic? (\u003cspan class=\"greek\" title=\"b\" lang=\"grc\"\u003eβ\u003c/span\u003e) is it \u003cem\u003enecessitated\u003c/em\u003e by the\r\nfacts or by logic? I wish to answer, in each case, yes to\r\nthe first form of the question, and no to the second.\r\nBut in any case the mathematical account of motion will\r\nnot be fictitious, provided a right interpretation is given\r\nto the words “point” and “instant.” A few words on\r\neach alternative will serve to make this clear.\u003c/p\u003e\r\n\u003cp\u003eFormally, mathematics adopts an absolute theory of\r\nspace and time, \u003ci\u003ei.e.\u003c/i\u003e it assumes that, besides the things\r\nwhich are in space and time, there are also entities, called\r\n“points” and “instants,” which are occupied by things.\r\nThis view, however, though advocated by Newton, has\r\nlong been regarded by mathematicians as merely a\r\nconvenient fiction. There is, so far as I can see, no\r\nconceivable evidence either for or against it. It is\r\nlogically possible, and it is consistent with the facts. But\r\nthe facts are also consistent with the denial of spatial and\r\ntemporal entities over and above things with spatial and\r\ntemporal relations. Hence, in accordance with Occam\u0027s\r\nrazor, we shall do well to abstain from either assuming\r\nor denying points and instants. This means, so far as\r\npractical working out is concerned, that we adopt the\r\nrelational theory; for in practice the refusal to assume\r\npoints and instants has the same effect as the denial of\r\nthem. But in strict theory the two are quite different,\r\nsince the denial introduces an element of unverifiable\r\ndogma which is wholly absent when we merely refrain\r\nfrom the assertion. Thus, although we shall derive\r\n\u003ca class=\"pagenum\" title=\"147\" id=\"Page_147\"\u003e \u003c/a\u003e\r\npoints and instants from things, we shall leave the bare\r\npossibility open that they may also have an independent\r\nexistence as simple entities.\u003c/p\u003e\r\n\u003cp\u003eWe come now to the question whether the things in\r\nspace and time are to be conceived as composed of\r\nelements without extension or duration, \u003ci\u003ei.e.\u003c/i\u003e of elements\r\nwhich only occupy a point and an instant. Physics,\r\nformally, assumes in its differential equations that things\r\nconsist of elements which occupy only a point at each\r\ninstant, but persist throughout time. For reasons explained\r\nin \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003eLecture IV.\u003c/a\u003e, the persistence of things through\r\ntime is to be regarded as the formal result of a logical\r\nconstruction, not as necessarily implying any actual persistence.\r\nThe same motives, in fact, which lead to the\r\ndivision of things into point-particles, ought presumably\r\nto lead to their division into instant-particles, so that the\r\nultimate \u003cem\u003eformal\u003c/em\u003e constituent of the matter in physics will be\r\na point-instant-particle. But such objects, as well as the\r\nparticles of physics, are not data. The same economy of\r\nhypothesis, which dictates the practical adoption of a\r\nrelative rather than an absolute space and time, also\r\ndictates the practical adoption of material elements which\r\nhave a finite extension and duration. Since, as we saw\r\nin \u003ca href=\"#Lecture_4\" class=\"pginternal\"\u003eLecture IV.\u003c/a\u003e, points and instants can be constructed as\r\nlogical functions of such elements, the mathematical\r\naccount of motion, in which a particle passes continuously\r\nthrough a continuous series of points, can be interpreted\r\nin a form which assumes only elements which agree with\r\nour actual data in having a finite extension and duration.\r\nThus, so far as the use of points and instants is concerned,\r\nthe mathematical account of motion can be freed from\r\nthe charge of employing fictions.\u003c/p\u003e\r\n\u003cp\u003e(\u003ci\u003ed\u003c/i\u003e) But we must now face the question: Is there, in\r\nactual empirical fact, any sufficient reason to believe the\r\n\u003ca class=\"pagenum\" title=\"148\" id=\"Page_148\"\u003e \u003c/a\u003e\r\nworld of sense continuous? The answer here must, I\r\nthink, be in the negative. We may say that the\r\nhypothesis of continuity is perfectly consistent with the\r\nfacts and with logic, and that it is technically simpler\r\nthan any other tenable hypothesis. But since our powers\r\nof discrimination among very similar sensible objects are\r\nnot infinitely precise, it is quite impossible to decide\r\nbetween different theories which only differ in regard to\r\nwhat is below the margin of discrimination. If, for\r\nexample, a coloured surface which we see consists of a\r\nfinite number of very small surfaces, and if a motion\r\nwhich we see consists, like a cinematograph, of a large\r\nfinite number of successive positions, there will be\r\nnothing empirically discoverable to show that objects of\r\nsense are not continuous. In what is called \u003cem\u003eexperienced\u003c/em\u003e\r\ncontinuity, such as is said to be given in sense, there is a\r\nlarge negative element: absence of perception of difference\r\noccurs in cases which are \u003cem\u003ethought\u003c/em\u003e to give perception of\r\nabsence of difference. When, for example, we cannot\r\ndistinguish a colour A from a colour B, nor a colour B\r\nfrom a colour C, but can distinguish A from C, the\r\nindistinguishability is a purely negative fact, namely, that\r\nwe do not \u003cem\u003eperceive\u003c/em\u003e a difference. Even in regard to\r\nimmediate data, this is no reason for denying that there\r\nis a difference. Thus, if we see a coloured surface whose\r\ncolour changes gradually, its sensible appearance if the\r\nchange is continuous will be indistinguishable from what\r\nit would be if the change were by small finite jumps. If\r\nthis is true, as it seems to be, it follows that there can\r\nnever be any empirical evidence to demonstrate that the\r\nsensible world is continuous, and not a collection of a\r\nvery large finite number of elements of which each differs\r\nfrom its neighbour in a finite though very small degree.\r\nThe continuity of space and time, the infinite number of\r\n\u003ca class=\"pagenum\" title=\"149\" id=\"Page_149\"\u003e \u003c/a\u003e\r\ndifferent shades in the spectrum, and so on, are all in\r\nthe nature of unverifiable hypotheses—perfectly possible\r\nlogically, perfectly consistent with the known facts, and\r\nsimpler technically than any other tenable hypotheses,\r\nbut not the sole hypotheses which are logically and empirically\r\nadequate.\u003c/p\u003e\r\n\u003cp\u003eIf a relational theory of instants is constructed, in\r\nwhich an “instant” is defined as a group of events\r\nsimultaneous with each other and not all simultaneous\r\nwith any event outside the group, then if our resulting\r\nseries of instants is to be compact, it must be possible,\r\nif \u003ci\u003ex\u003c/i\u003e wholly precedes \u003ci\u003ey\u003c/i\u003e, to find an event \u003ci\u003ez\u003c/i\u003e, simultaneous\r\nwith part of \u003ci\u003ex\u003c/i\u003e, which wholly precedes some event which\r\nwholly precedes \u003ci\u003ey\u003c/i\u003e. Now this requires that the number\r\nof events concerned should be infinite in any finite\r\nperiod of time. If this is to be the case in the world\r\nof one man\u0027s sense-data, and if each sense-datum is to\r\nhave not less than a certain finite temporal extension, it\r\nwill be necessary to assume that we always have an\r\ninfinite number of sense-data simultaneous with any given\r\nsense-datum. Applying similar considerations to space,\r\nand assuming that sense-data are to have not less than a\r\ncertain spatial extension, it will be necessary to suppose\r\nthat an infinite number of sense-data overlap spatially\r\nwith any given sense-datum. This hypothesis is possible,\r\nif we suppose a single sense-datum, \u003ci\u003ee.g.\u003c/i\u003e in sight, to be a\r\nfinite surface, enclosing other surfaces which are also\r\nsingle sense-data. But there are difficulties in such a\r\nhypothesis, and I do not know whether these difficulties\r\ncould be successfully met. If they cannot, we must do\r\none of two things: either declare that the world of one\r\nman\u0027s sense-data is not continuous, or else refuse to\r\nadmit that there is any lower limit to the duration and\r\nextension of a single sense-datum. I do not know what\r\n\u003ca class=\"pagenum\" title=\"150\" id=\"Page_150\"\u003e \u003c/a\u003e\r\nis the right course to adopt as regards these alternatives.\r\nThe logical analysis we have been considering provides\r\nthe apparatus for dealing with the various hypotheses,\r\nand the empirical decision between them is a problem for\r\nthe psychologist.\u003c/p\u003e\r\n\u003cp\u003e(3) We have now to consider the \u003cem\u003elogical\u003c/em\u003e answer to the\r\nalleged difficulties of the mathematical theory of motion,\r\nor rather to the positive theory which is urged on the\r\nother side. The view urged explicitly by Bergson, and\r\nimplied in the doctrines of many philosophers, is, that a\r\nmotion is something indivisible, not validly analysable\r\ninto a series of states. This is part of a much more\r\ngeneral doctrine, which holds that analysis always falsifies,\r\nbecause the parts of a complex whole are different, as\r\ncombined in that whole, from what they would otherwise\r\nbe. It is very difficult to state this doctrine in any form\r\nwhich has a precise meaning. Often arguments are used\r\nwhich have no bearing whatever upon the question. It\r\nis urged, for example, that when a man becomes a father,\r\nhis nature is altered by the new relation in which he finds\r\nhimself, so that he is not strictly identical with the man\r\nwho was previously not a father. This may be true, but\r\nit is a causal psychological fact, not a logical fact. The\r\ndoctrine would require that a man who is a father cannot\r\nbe strictly identical with a man who is a son, because he\r\nis modified in one way by the relation of fatherhood and\r\nin another by that of sonship. In fact, we may give a\r\nprecise statement of the doctrine we are combating in the\r\nform: \u003cem\u003eThere can never be two facts concerning the same thing.\u003c/em\u003e\r\nA fact concerning a thing always is or involves a relation\r\nto one or more entities; thus two facts concerning the\r\nsame thing would involve two relations of the same\r\nthing. But the doctrine in question holds that a thing\r\nis so modified by its relations that it cannot be the same\r\n\u003ca class=\"pagenum\" title=\"151\" id=\"Page_151\"\u003e \u003c/a\u003e\r\nin one relation as in another. Hence, if this doctrine is\r\ntrue, there can never be more than one fact concerning\r\nany one thing. I do not think the philosophers in\r\nquestion have realised that this is the precise statement\r\nof the view they advocate, because in this form the view\r\nis so contrary to plain truth that its falsehood is evident\r\nas soon as it is stated. The discussion of this question,\r\nhowever, involves so many logical subtleties, and is so\r\nbeset with difficulties, that I shall not pursue it further\r\nat present.\u003c/p\u003e\r\n\u003cp\u003eWhen once the above general doctrine is rejected, it\r\nis obvious that, where there is change, there must be\r\na succession of states. There cannot be change—and\r\nmotion is only a particular case of change—unless there\r\nis something different at one time from what there is\r\nat some other time. Change, therefore, must involve\r\nrelations and complexity, and must demand analysis. So\r\nlong as our analysis has only gone as far as other smaller\r\nchanges, it is not complete; if it is to be complete, it must\r\nend with terms that are not changes, but are related by a\r\nrelation of earlier and later. In the case of changes which\r\nappear continuous, such as motions, it seems to be impossible\r\nto find anything other than change so long as we deal\r\nwith finite periods of time, however short. We are thus\r\ndriven back, by the logical necessities of the case, to the\r\nconception of instants without duration, or at any rate\r\nwithout any duration which even the most delicate\r\ninstruments can reveal. This conception, though it can\r\nbe made to seem difficult, is really easier than any other\r\nthat the facts allow. It is a kind of logical framework\r\ninto which any tenable theory must fit—not necessarily\r\nitself the statement of the crude facts, but a form in\r\nwhich statements which are true of the crude facts can\r\nbe made by a suitable interpretation. The direct consideration\r\n\u003ca class=\"pagenum\" title=\"152\" id=\"Page_152\"\u003e \u003c/a\u003e\r\nof the crude facts of the physical world has\r\nbeen undertaken in earlier lectures; in the present\r\nlecture, we have only been concerned to show that\r\nnothing in the crude facts is inconsistent with the mathematical\r\ndoctrine of continuity, or demands a continuity\r\nof a radically different kind from that of mathematical\r\nmotion.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"153\" id=\"Page_153\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE VI\u003c/small\u003e\u003cbr\u003e\r\nTHE PROBLEM OF INFINITY\r\nCONSIDERED HISTORICALLY\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"155\" id=\"Page_155\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_6\"\u003eLECTURE VI\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nTHE PROBLEM OF INFINITY CONSIDERED HISTORICALLY\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eIt\u003c/span\u003e will be remembered that, when we enumerated the\r\ngrounds upon which the reality of the sensible world\r\nhas been questioned, one of those mentioned was the\r\nsupposed impossibility of infinity and continuity. In\r\nview of our earlier discussion of physics, it would seem\r\nthat no \u003cem\u003econclusive\u003c/em\u003e empirical evidence exists in favour of\r\ninfinity or continuity in objects of sense or in matter.\r\nNevertheless, the explanation which assumes infinity and\r\ncontinuity remains incomparably easier and more natural,\r\nfrom a scientific point of view, than any other, and since\r\nGeorg Cantor has shown that the supposed contradictions\r\nare illusory, there is no longer any reason to struggle\r\nafter a finitist explanation of the world.\u003c/p\u003e\r\n\u003cp\u003eThe supposed difficulties of continuity all have their\r\nsource in the fact that a continuous series must have an\r\ninfinite number of terms, and are in fact difficulties concerning\r\ninfinity. Hence, in freeing the infinite from\r\ncontradiction, we are at the same time showing the logical\r\npossibility of continuity as assumed in science.\u003c/p\u003e\r\n\u003cp\u003eThe kind of way in which infinity has been used to\r\ndiscredit the world of sense may be illustrated by Kant\u0027s\r\nfirst two antinomies. In the first, the thesis states:\r\n“The world has a beginning in time, and as regards\r\nspace is enclosed within limits”; the antithesis states:\r\n\u003ca class=\"pagenum\" title=\"156\" id=\"Page_156\"\u003e \u003c/a\u003e\r\n“The world has no beginning and no limits in space,\r\nbut is infinite in respect of both time and space.” Kant\r\nprofesses to prove both these propositions, whereas, if\r\nwhat we have said on modern logic has any truth, it must\r\nbe impossible to prove either. In order, however, to\r\nrescue the world of sense, it is enough to destroy the\r\nproof of \u003cem\u003eone\u003c/em\u003e of the two. For our present purpose, it is\r\nthe proof that the world is \u003cem\u003efinite\u003c/em\u003e that interests us. Kant\u0027s\r\nargument as regards space here rests upon his argument\r\nas regards time. We need therefore only examine the\r\nargument as regards time. What he says is as follows:\u003c/p\u003e\r\n\u003cp\u003e“For let us assume that the world has no beginning\r\nas regards time, so that up to every given instant an\r\neternity has elapsed, and therefore an infinite series of\r\nsuccessive states of the things in the world has passed by.\r\nBut the infinity of a series consists just in this, that it can\r\nnever be completed by successive synthesis. Therefore\r\nan infinite past world-series is impossible, and accordingly\r\na beginning of the world is a necessary condition of its\r\nexistence; which was the first thing to be proved.”\u003c/p\u003e\r\n\u003cp\u003eMany different criticisms might be passed on this\r\nargument, but we will content ourselves with a bare\r\nminimum. To begin with, it is a mistake to define the\r\ninfinity of a series as “impossibility of completion by\r\nsuccessive synthesis.” The notion of infinity, as we\r\nshall see in the \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e, is primarily a property of\r\n\u003cem\u003eclasses\u003c/em\u003e, and only derivatively applicable to series; classes\r\nwhich are infinite are given all at once by the defining\r\nproperty of their members, so that there is no question\r\nof “completion” or of “successive synthesis.” And the\r\nword “synthesis,” by suggesting the mental activity of\r\nsynthesising, introduces, more or less surreptitiously,\r\nthat reference to mind by which all Kant\u0027s philosophy\r\nwas infected. In the second place, when Kant says that\r\n\u003ca class=\"pagenum\" title=\"157\" id=\"Page_157\"\u003e \u003c/a\u003e\r\nan infinite series can “never” be completed by successive\r\nsynthesis, all that he has even conceivably a right to say\r\nis that it cannot be completed \u003cem\u003ein a finite time\u003c/em\u003e. Thus what\r\nhe really proves is, at most, that if the world had no\r\nbeginning, it must have already existed for an infinite\r\ntime. This, however, is a very poor conclusion, by no\r\nmeans suitable for his purposes. And with this result\r\nwe might, if we chose, take leave of the first antinomy.\u003c/p\u003e\r\n\u003cp\u003eIt is worth while, however, to consider how Kant came\r\nto make such an elementary blunder. What happened\r\nin his imagination was obviously something like this:\r\nStarting from the present and going backwards in time,\r\nwe have, if the world had no beginning, an infinite series\r\nof events. As we see from the word “synthesis,” he\r\nimagined a mind trying to grasp these successively, \u003cem\u003ein the\r\nreverse order\u003c/em\u003e to that in which they had occurred, \u003ci\u003ei.e.\u003c/i\u003e going\r\nfrom the present backwards. \u003cem\u003eThis\u003c/em\u003e series is obviously one\r\nwhich has no end. But the series of events up to the\r\npresent has an end, since it ends with the present.\r\nOwing to the inveterate subjectivism of his mental habits,\r\nhe failed to notice that he had reversed the sense of the\r\nseries by substituting backward synthesis for forward\r\nhappening, and thus he supposed that it was necessary to\r\nidentify the mental series, which had no end, with the\r\nphysical series, which had an end but no beginning. It\r\nwas this mistake, I think, which, operating unconsciously,\r\nled him to attribute validity to a singularly flimsy piece of\r\nfallacious reasoning.\u003c/p\u003e\r\n\u003cp\u003eThe second antinomy illustrates the dependence of the\r\nproblem of continuity upon that of infinity. The thesis\r\nstates: “Every complex substance in the world consists\r\nof simple parts, and there exists everywhere nothing but\r\nthe simple or what is composed of it.” The antithesis\r\nstates: “No complex thing in the world consists of\r\n\u003ca class=\"pagenum\" title=\"158\" id=\"Page_158\"\u003e \u003c/a\u003e\r\nsimple parts, and everywhere in it there exists nothing\r\nsimple.” Here, as before, the proofs of both thesis and\r\nantithesis are open to criticism, but for the purpose of\r\nvindicating physics and the world of sense it is enough\r\nto find a fallacy in \u003cem\u003eone\u003c/em\u003e of the proofs. We will choose for\r\nthis purpose the proof of the antithesis, which begins as\r\nfollows:\u003c/p\u003e\r\n\u003cp\u003e“Assume that a complex thing (as substance) consists\r\nof simple parts. Since all external relation, and therefore\r\nall composition out of substances, is only possible in\r\nspace, the space occupied by a complex thing must\r\nconsist of as many parts as the thing consists of. Now\r\nspace does not consist of simple parts, but of spaces.”\u003c/p\u003e\r\n\u003cp\u003eThe rest of his argument need not concern us, for the\r\nnerve of the proof lies in the one statement: “Space\r\ndoes not consist of simple parts, but of spaces.” This\r\nis like Bergson\u0027s objection to “the absurd proposition\r\nthat motion is made up of immobilities.” Kant does\r\nnot tell us why he holds that a space must consist of\r\nspaces rather than of simple parts. Geometry regards\r\nspace as made up of points, which are simple; and\r\nalthough, as we have seen, this view is not scientifically\r\nor logically \u003cem\u003enecessary\u003c/em\u003e, it remains \u003ci lang=\"la\"\u003eprimâ facie\u003c/i\u003e possible, and\r\nits mere possibility is enough to vitiate Kant\u0027s argument.\r\nFor, if his proof of the thesis of the antinomy\r\nwere valid, and if the antithesis could only be avoided\r\nby assuming points, then the antinomy itself would\r\nafford a conclusive reason in favour of points. Why,\r\nthen, did Kant think it impossible that space should\r\nbe composed of points?\u003c/p\u003e\r\n\u003cp\u003eI think two considerations probably influenced him.\r\nIn the first place, the essential thing about space is spatial\r\norder, and mere points, by themselves, will not account\r\nfor spatial order. It is obvious that his argument\r\n\u003ca class=\"pagenum\" title=\"159\" id=\"Page_159\"\u003e \u003c/a\u003e\r\nassumes absolute space; but it is spatial \u003cem\u003erelations\u003c/em\u003e that are\r\nalone important, and they cannot be reduced to points.\r\nThis ground for his view depends, therefore, upon his\r\nignorance of the logical theory of order and his oscillations\r\nbetween absolute and relative space. But there is\r\nalso another ground for his opinion, which is more\r\nrelevant to our present topic. This is the ground\r\nderived from infinite divisibility. A space may be halved,\r\nand then halved again, and so on \u003ci lang=\"la\"\u003ead infinitum\u003c/i\u003e, and at\r\nevery stage of the process the parts are still spaces, not\r\npoints. In order to reach points by such a method, it\r\nwould be necessary to come to the end of an unending\r\nprocess, which is impossible. But just as an infinite class\r\ncan be given all at once by its defining concept, though\r\nit cannot be reached by successive enumeration, so an\r\ninfinite set of points can be given all at once as making\r\nup a line or area or volume, though they can never be\r\nreached by the process of successive division. Thus the\r\ninfinite divisibility of space gives no ground for denying\r\nthat space is composed of points. Kant does not give his\r\ngrounds for this denial, and we can therefore only conjecture\r\nwhat they were. But the above two grounds,\r\nwhich we have seen to be fallacious, seem sufficient to\r\naccount for his opinion, and we may therefore conclude\r\nthat the antithesis of the second antinomy is unproved.\u003c/p\u003e\r\n\u003cp\u003eThe above illustration of Kant\u0027s antinomies has only been\r\nintroduced in order to show the relevance of the problem\r\nof infinity to the problem of the reality of objects of sense.\r\nIn the remainder of the present lecture, I wish to state and\r\nexplain the problem of infinity, to show how it arose, and\r\nto show the irrelevance of all the solutions proposed by\r\nphilosophers. In the \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003efollowing lecture\u003c/a\u003e, I shall try to\r\nexplain the true solution, which has been discovered by\r\nthe mathematicians, but nevertheless belongs essentially\r\n\u003ca class=\"pagenum\" title=\"160\" id=\"Page_160\"\u003e \u003c/a\u003e\r\nto philosophy. The solution is definitive, in the sense\r\nthat it entirely satisfies and convinces all who study it\r\ncarefully. For over two thousand years the human\r\nintellect was baffled by the problem; its many failures\r\nand its ultimate success make this problem peculiarly apt\r\nfor the illustration of method.\u003c/p\u003e\r\n\u003cp\u003eThe problem appears to have first arisen in some such\r\nway as the following.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_22\" id=\"FNanchor_22\"\u003e[22]\u003c/a\u003e Pythagoras and his followers,\r\nwho were interested, like Descartes, in the application of\r\nnumber to geometry, adopted in that science more\r\narithmetical methods than those with which Euclid has\r\nmade us familiar. They, or their contemporaries the\r\natomists, believed, apparently, that space is composed of\r\nindivisible points, while time is composed of indivisible\r\ninstants.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_23\" id=\"FNanchor_23\"\u003e[23]\u003c/a\u003e This belief would not, by itself, have raised\r\nthe difficulties which they encountered, but it was presumably\r\naccompanied by another belief, that the number\r\nof points in any finite area or of instants in any finite\r\nperiod must be finite. I do not suppose that this latter\r\nbelief was a conscious one, because probably no other\r\npossibility had occurred to them. But the belief nevertheless\r\noperated, and very soon brought them into\r\nconflict with facts which they themselves discovered.\r\nBefore explaining how this occurred, however, it is\r\nnecessary to say one word in explanation of the phrase\r\n“finite number.” The \u003cem\u003eexact\u003c/em\u003e explanation is a matter for\r\nour \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e; for the present, it must suffice to say\r\nthat I mean 0 and 1 and 2 and 3 and so on, for ever—in\r\nother words, any number that can be obtained by\r\n\u003ca class=\"pagenum\" title=\"161\" id=\"Page_161\"\u003e \u003c/a\u003e\r\nsuccessively adding ones. This includes all the numbers\r\nthat can be expressed by means of our ordinary numerals,\r\nand since such numbers can be made greater and greater,\r\nwithout ever reaching an unsurpassable maximum, it is\r\neasy to suppose that there are no other numbers. But\r\nthis supposition, natural as it is, is mistaken.\u003c/p\u003e\r\n\u003cp\u003eWhether the Pythagoreans themselves believed space\r\nand time to be composed of indivisible points and instants\r\nis a debatable question.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_24\" id=\"FNanchor_24\"\u003e[24]\u003c/a\u003e It would seem that the\r\ndistinction between space and matter had not yet been\r\nclearly made, and that therefore, when an atomistic view\r\nis expressed, it is difficult to decide whether particles of\r\nmatter or points of space are intended. There is an\r\ninteresting passage\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_25\" id=\"FNanchor_25\"\u003e[25]\u003c/a\u003e in Aristotle\u0027s \u003ccite\u003ePhysics\u003c/cite\u003e,\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_26\" id=\"FNanchor_26\"\u003e[26]\u003c/a\u003e where he\r\nsays:\u003c/p\u003e\r\n\u003cp\u003e“The Pythagoreans all maintained the existence of the\r\nvoid, and said that it enters into the heaven itself from\r\nthe boundless breath, inasmuch as the heaven breathes\r\nin the void also; and the void differentiates natures, as\r\n\u003ca class=\"pagenum\" title=\"162\" id=\"Page_162\"\u003e \u003c/a\u003e\r\nif it were a sort of separation of consecutives, and as if\r\nit were their differentiation; and that this also is what\r\nis first in numbers, for it is the void which differentiates\r\nthem.”\u003c/p\u003e\r\n\u003cp\u003eThis seems to imply that they regarded matter as consisting\r\nof atoms with empty space in between. But if\r\nso, they must have thought space could be studied by\r\nonly paying attention to the atoms, for otherwise it would\r\nbe hard to account for their arithmetical methods in\r\ngeometry, or for their statement that “things are\r\nnumbers.”\u003c/p\u003e\r\n\u003cp\u003eThe difficulty which beset the Pythagoreans in their\r\nattempts to apply numbers arose through their discovery\r\nof incommensurables, and this, in turn, arose as follows.\r\nPythagoras, as we all learnt in youth, discovered the\r\nproposition that the sum of the squares on the sides of\r\na right-angled triangle is equal to the square on the\r\nhypotenuse. It is said that he sacrificed an ox when he\r\ndiscovered this theorem; if so, the ox was the first\r\nmartyr to science. But the theorem, though it has\r\nremained his chief claim to immortality, was soon found\r\nto have a consequence fatal to his whole philosophy.\r\nConsider the case of a right-angled triangle whose two\r\nsides are equal, such a triangle as is formed by two sides\r\nof a square and a diagonal. Here, in virtue of the\r\ntheorem, the square on the diagonal is double of the\r\nsquare on either of the sides. But Pythagoras or his\r\nearly followers easily proved that the square of one whole\r\nnumber cannot be double of the square of another.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_27\" id=\"FNanchor_27\"\u003e[27]\u003c/a\u003e\r\n\u003ca class=\"pagenum\" title=\"163\" id=\"Page_163\"\u003e \u003c/a\u003e\r\nThus the length of the side and the length of the\r\ndiagonal are incommensurable; that is to say, however\r\nsmall a unit of length you take, if it is contained an exact\r\nnumber of times in the side, it is not contained any exact\r\nnumber of times in the diagonal, and \u003ci lang=\"la\"\u003evice versa\u003c/i\u003e.\u003c/p\u003e\r\n\u003cp\u003eNow this fact might have been assimilated by some\r\nphilosophies without any great difficulty, but to the\r\nphilosophy of Pythagoras it was absolutely fatal. Pythagoras\r\nheld that number is the constitutive essence of\r\nall things, yet no two numbers could express the ratio\r\nof the side of a square to the diagonal. It would seem\r\nprobable that we may expand his difficulty, without\r\ndeparting from his thought, by assuming that he regarded\r\nthe length of a line as determined by the number of atoms\r\ncontained in it—a line two inches long would contain\r\ntwice as many atoms as a line one inch long, and so on.\r\nBut if this were the truth, then there must be a definite\r\nnumerical ratio between any two finite lengths, because\r\nit was supposed that the number of atoms in each, however\r\nlarge, must be finite. Here there was an insoluble\r\ncontradiction. The Pythagoreans, it is said, resolved to\r\nkeep the existence of incommensurables a profound\r\nsecret, revealed only to a few of the supreme heads\r\nof the sect; and one of their number, Hippasos of\r\nMetapontion, is even said to have been shipwrecked at\r\nsea for impiously disclosing the terrible discovery to their\r\nenemies. It must be remembered that Pythagoras was the\r\nfounder of a new religion as well as the teacher of a new\r\nscience: if the science came to be doubted, the disciples\r\nmight fall into sin, and perhaps even eat beans, which\r\naccording to Pythagoras is as bad as eating parents\u0027 bones.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"164\" id=\"Page_164\"\u003e \u003c/a\u003eThe problem first raised by the discovery of incommensurables\r\nproved, as time went on, to be one of the\r\nmost severe and at the same time most far-reaching\r\nproblems that have confronted the human intellect in its\r\nendeavour to understand the world. It showed at once\r\nthat numerical measurement of lengths, if it was to be\r\nmade accurate, must require an arithmetic more advanced\r\nand more difficult than any that the ancients possessed.\r\nThey therefore set to work to reconstruct geometry on\r\na basis which did not assume the universal possibility of\r\nnumerical measurement—a reconstruction which, as may\r\nbe seen in Euclid, they effected with extraordinary skill\r\nand with great logical acumen. The moderns, under\r\nthe influence of Cartesian geometry, have reasserted the\r\nuniversal possibility of numerical measurement, extending\r\narithmetic, partly for that purpose, so as to include what\r\nare called “irrational” numbers, which give the ratios\r\nof incommensurable lengths. But although irrational\r\nnumbers have long been used without a qualm, it is only\r\nin quite recent years that logically satisfactory definitions\r\nof them have been given. With these definitions, the\r\nfirst and most obvious form of the difficulty which confronted\r\nthe Pythagoreans has been solved; but other\r\nforms of the difficulty remain to be considered, and it is\r\nthese that introduce us to the problem of infinity in its\r\npure form.\u003c/p\u003e\r\n\u003cp\u003eWe saw that, accepting the view that a length is composed\r\nof points, the existence of incommensurables proves\r\nthat every finite length must contain an infinite number\r\nof points. In other words, if we were to take away points\r\none by one, we should never have taken away all the\r\npoints, however long we continued the process. The\r\nnumber of points, therefore, cannot be \u003cem\u003ecounted\u003c/em\u003e, for counting\r\nis a process which enumerates things one by one. The\r\n\u003ca class=\"pagenum\" title=\"165\" id=\"Page_165\"\u003e \u003c/a\u003e\r\nproperty of being unable to be counted is characteristic\r\nof infinite collections, and is a source of many of their\r\nparadoxical qualities. So paradoxical are these qualities\r\nthat until our own day they were thought to constitute\r\nlogical contradictions. A long line of philosophers, from\r\nZeno\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_28\" id=\"FNanchor_28\"\u003e[28]\u003c/a\u003e to M. Bergson, have based much of their metaphysics\r\nupon the supposed impossibility of infinite collections.\r\nBroadly speaking, the difficulties were stated by Zeno,\r\nand nothing material was added until we reach Bolzano\u0027s\r\n\u003ccite lang=\"de\"\u003eParadoxien des Unendlichen\u003c/cite\u003e, a little work written in 1847–8,\r\nand published posthumously in 1851. Intervening attempts\r\nto deal with the problem are futile and negligible.\r\nThe definitive solution of the difficulties is due, not to\r\nBolzano, but to Georg Cantor, whose work on this subject\r\nfirst appeared in 1882.\u003c/p\u003e\r\n\u003cp\u003eIn order to understand Zeno, and to realise how little\r\nmodern orthodox metaphysics has added to the achievements\r\nof the Greeks, we must consider for a moment his\r\nmaster Parmenides, in whose interest the paradoxes were\r\ninvented.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_29\" id=\"FNanchor_29\"\u003e[29]\u003c/a\u003e Parmenides expounded his views in a poem\r\ndivided into two parts, called “the way of truth” and\r\n“the way of opinion”—like Mr Bradley\u0027s “Appearance”\r\nand “Reality,” except that Parmenides tells us first about\r\nreality and then about appearance. “The way of opinion,”\r\nin his philosophy, is, broadly speaking, Pythagoreanism;\r\nit begins with a warning: “Here I shall close my trustworthy\r\nspeech and thought about the truth. Henceforward\r\nlearn the opinions of mortals, giving ear to the\r\ndeceptive ordering of my words.” What has gone before\r\nhas been revealed by a goddess, who tells him what\r\n\u003ca class=\"pagenum\" title=\"166\" id=\"Page_166\"\u003e \u003c/a\u003e\r\nreally \u003cem\u003eis\u003c/em\u003e. Reality, she says, is uncreated, indestructible,\r\nunchanging, indivisible; it is “immovable in the bonds\r\nof mighty chains, without beginning and without end;\r\nsince coming into being and passing away have been driven\r\nafar, and true belief has cast them away.” The fundamental\r\nprinciple of his inquiry is stated in a sentence\r\nwhich would not be out of place in Hegel:\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_30\" id=\"FNanchor_30\"\u003e[30]\u003c/a\u003e “Thou\r\ncanst not know what is not—that is impossible—nor\r\nutter it; for it is the same thing that can be thought and\r\nthat can be.” And again: “It needs must be that what\r\ncan be thought and spoken of is; for it is possible for it\r\nto be, and it is not possible for what is nothing to be.”\r\nThe impossibility of change follows from this principle;\r\nfor what is past can be spoken of, and therefore, by the\r\nprinciple, still is.\u003c/p\u003e\r\n\u003cp\u003eThe great conception of a reality behind the passing\r\nillusions of sense, a reality one, indivisible, and unchanging,\r\nwas thus introduced into Western philosophy by\r\nParmenides, not, it would seem, for mystical or religious\r\nreasons, but on the basis of a logical argument as to the\r\nimpossibility of not-being. All the great metaphysical\r\nsystems—notably those of Plato, Spinoza, and Hegel—are\r\nthe outcome of this fundamental idea. It is difficult\r\nto disentangle the truth and the error in this view. The\r\ncontention that time is unreal and that the world of sense\r\nis illusory must, I think, be regarded as based upon\r\nfallacious reasoning. Nevertheless, there is some sense—easier\r\nto feel than to state—in which time is an unimportant\r\nand superficial characteristic of reality. Past and\r\nfuture must be acknowledged to be as real as the present,\r\nand a certain emancipation from slavery to time is\r\nessential to philosophic thought. The importance of\r\n\u003ca class=\"pagenum\" title=\"167\" id=\"Page_167\"\u003e \u003c/a\u003e\r\ntime is rather practical than theoretical, rather in relation\r\nto our desires than in relation to truth. A truer image\r\nof the world, I think, is obtained by picturing things as\r\nentering into the stream of time from an eternal world\r\noutside, than from a view which regards time as the\r\ndevouring tyrant of all that is. Both in thought and\r\nin feeling, to realise the unimportance of time is the\r\ngate of wisdom. But unimportance is not unreality;\r\nand therefore what we shall have to say about Zeno\u0027s\r\narguments in support of Parmenides must be mainly\r\ncritical.\u003c/p\u003e\r\n\u003cp\u003eThe relation of Zeno to Parmenides is explained by\r\nPlato\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_31\" id=\"FNanchor_31\"\u003e[31]\u003c/a\u003e in the dialogue in which Socrates, as a young\r\nman, learns logical acumen and philosophic disinterestedness\r\nfrom their dialectic. I quote from Jowett\u0027s\r\ntranslation:\u003c/p\u003e\r\n\u003cp\u003e“I see, Parmenides, said Socrates, that Zeno is your\r\nsecond self in his writings too; he puts what you say in\r\nanother way, and would fain deceive us into believing\r\nthat he is telling us what is new. For you, in your\r\npoems, say All is one, and of this you adduce excellent\r\nproofs; and he on the other hand says There is no\r\nMany; and on behalf of this he offers overwhelming\r\nevidence. To deceive the world, as you have done, by\r\nsaying the same thing in different ways, one of you\r\naffirming the one, and the other denying the many, is a\r\nstrain of art beyond the reach of most of us.\u003c/p\u003e\r\n\u003cp\u003e“Yes, Socrates, said Zeno. But although you are as\r\nkeen as a Spartan hound in pursuing the track, you do\r\nnot quite apprehend the true motive of the composition,\r\nwhich is not really such an ambitious work as you\r\nimagine; for what you speak of was an accident; I had\r\nno serious intention of deceiving the world. The truth\r\n\u003ca class=\"pagenum\" title=\"168\" id=\"Page_168\"\u003e \u003c/a\u003e\r\nis, that these writings of mine were meant to protect the\r\narguments of Parmenides against those who scoff at him\r\nand show the many ridiculous and contradictory results\r\nwhich they suppose to follow from the affirmation of the\r\none. My answer is an address to the partisans of the\r\nmany, whose attack I return with interest by retorting\r\nupon them that their hypothesis of the being of the many\r\nif carried out appears in a still more ridiculous light than\r\nthe hypothesis of the being of the one.”\u003c/p\u003e\r\n\u003cp\u003eZeno\u0027s four arguments against motion were intended\r\nto exhibit the contradictions that result from supposing\r\nthat there is such a thing as change, and thus to support\r\nthe Parmenidean doctrine that reality is unchanging.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_32\" id=\"FNanchor_32\"\u003e[32]\u003c/a\u003e\r\nUnfortunately, we only know his arguments through\r\nAristotle,\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_33\" id=\"FNanchor_33\"\u003e[33]\u003c/a\u003e who stated them in order to refute them.\r\nThose philosophers in the present day who have had their\r\ndoctrines stated by opponents will realise that a just or\r\nadequate presentation of Zeno\u0027s position is hardly to be\r\nexpected from Aristotle; but by some care in interpretation\r\nit seems possible to reconstruct the so-called\r\n“sophisms” which have been “refuted” by every tyro\r\nfrom that day to this.\u003c/p\u003e\r\n\u003cp\u003eZeno\u0027s arguments would seem to be “ad hominem”;\r\nthat is to say, they seem to assume premisses granted by\r\nhis opponents, and to show that, granting these premisses,\r\nit is possible to deduce consequences which his opponents\r\nmust deny. In order to decide whether they are valid\r\narguments or “sophisms,” it is necessary to guess at the\r\ntacit premisses, and to decide who was the “homo” at\r\nwhom they were aimed. Some maintain that they were\r\n\u003ca class=\"pagenum\" title=\"169\" id=\"Page_169\"\u003e \u003c/a\u003e\r\naimed at the Pythagoreans,\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_34\" id=\"FNanchor_34\"\u003e[34]\u003c/a\u003e while others have held that\r\nthey were intended to refute the atomists.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_35\" id=\"FNanchor_35\"\u003e[35]\u003c/a\u003e M. Evellin,\r\non the contrary, holds that they constitute a refutation of\r\ninfinite divisibility,\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_36\" id=\"FNanchor_36\"\u003e[36]\u003c/a\u003e while M. G. Noël, in the interests of\r\nHegel, maintains that the first two arguments refute\r\ninfinite divisibility, while the next two refute indivisibles.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_37\" id=\"FNanchor_37\"\u003e[37]\u003c/a\u003e\r\nAmid such a bewildering variety of interpretations, we\r\ncan at least not complain of any restrictions on our\r\nliberty of choice.\u003c/p\u003e\r\n\u003cp\u003eThe historical questions raised by the above-mentioned\r\ndiscussions are no doubt largely insoluble, owing to the\r\nvery scanty material from which our evidence is derived.\r\nThe points which seem fairly clear are the following:\r\n(1) That, in spite of MM. Milhaud and Paul Tannery,\r\nZeno is anxious to prove that motion is really impossible,\r\nand that he desires to prove this because he follows\r\nParmenides in denying plurality;\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_38\" id=\"FNanchor_38\"\u003e[38]\u003c/a\u003e (2) that the third and\r\nfourth arguments proceed on the hypothesis of indivisibles,\r\na hypothesis which, whether adopted by the\r\nPythagoreans or not, was certainly much advocated, as\r\nmay be seen from the treatise \u003ccite\u003eOn Indivisible Lines\u003c/cite\u003e attributed\r\nto Aristotle. As regards the first two arguments,\r\nthey would seem to be valid on the hypothesis of indivisibles,\r\nand also, without this hypothesis, to be such as\r\n\u003ca class=\"pagenum\" title=\"170\" id=\"Page_170\"\u003e \u003c/a\u003e\r\nwould be valid if the traditional contradictions in infinite\r\nnumbers were insoluble, which they are not.\u003c/p\u003e\r\n\u003cp\u003eWe may conclude, therefore, that Zeno\u0027s polemic is\r\ndirected against the view that space and time consist of\r\npoints and instants; and that as against the view that a\r\nfinite stretch of space or time consists of a finite number\r\nof points and instants, his arguments are not sophisms,\r\nbut perfectly valid.\u003c/p\u003e\r\n\u003cp\u003eThe conclusion which Zeno wishes us to draw is that\r\nplurality is a delusion, and spaces and times are really\r\nindivisible. The other conclusion which is possible,\r\nnamely, that the number of points and instants is infinite,\r\nwas not tenable so long as the infinite was infected with\r\ncontradictions. In a fragment which is not one of the\r\nfour famous arguments against motion, Zeno says:\u003c/p\u003e\r\n\u003cp\u003e“If things are a many, they must be just as many as\r\nthey are, and neither more nor less. Now, if they are\r\nas many as they are, they will be finite in number.\u003c/p\u003e\r\n\u003cp\u003e“If things are a many, they will be infinite in number;\r\nfor there will always be other things between them, and\r\nothers again between these. And so things are infinite\r\nin number.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_39\" id=\"FNanchor_39\"\u003e[39]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eThis argument attempts to prove that, if there are\r\nmany things, the number of them must be both finite\r\nand infinite, which is impossible; hence we are to conclude\r\nthat there is only one thing. But the weak point\r\nin the argument is the phrase: “If they are just as\r\nmany as they are, they will be finite in number.” This\r\nphrase is not very clear, but it is plain that it assumes\r\nthe impossibility of definite infinite numbers. Without\r\nthis assumption, which is now known to be false, the\r\narguments of Zeno, though they suffice (on certain very\r\nreasonable assumptions) to dispel the hypothesis of finite\r\n\u003ca class=\"pagenum\" title=\"171\" id=\"Page_171\"\u003e \u003c/a\u003e\r\nindivisibles, do not suffice to prove that motion and\r\nchange and plurality are impossible. They are not, however,\r\non any view, mere foolish quibbles: they are\r\nserious arguments, raising difficulties which it has taken\r\ntwo thousand years to answer, and which even now are\r\nfatal to the teachings of most philosophers.\u003c/p\u003e\r\n\u003cp\u003eThe first of Zeno\u0027s arguments is the argument of the\r\nrace-course, which is paraphrased by Burnet as follows:\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_40\" id=\"FNanchor_40\"\u003e[40]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003e“You cannot get to the end of a race-course. You\r\ncannot traverse an infinite number of points in a finite\r\ntime. You must traverse the half of any given distance\r\nbefore you traverse the whole, and the half of that again\r\nbefore you can traverse it. This goes on \u003ci lang=\"la\"\u003ead infinitum\u003c/i\u003e,\r\nso that there are an infinite number of points in any\r\ngiven space, and you cannot touch an infinite number\r\none by one in a finite time.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_41\" id=\"FNanchor_41\"\u003e[41]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eZeno appeals here, in the first place, to the fact that\r\n\u003ca class=\"pagenum\" title=\"172\" id=\"Page_172\"\u003e \u003c/a\u003e\r\nany distance, however small, can be halved. From this\r\nit follows, of course, that there must be an infinite\r\nnumber of points in a line. But, Aristotle represents\r\nhim as arguing, you cannot touch an infinite number of\r\npoints \u003cem\u003eone by one\u003c/em\u003e in a finite time. The words “one by\r\none” are important. (1) If \u003cem\u003eall\u003c/em\u003e the points touched are\r\nconcerned, then, though you pass through them continuously,\r\nyou do not touch them “one by one.” That is\r\nto say, after touching one, there is not another which you\r\ntouch next: no two points are next each other, but\r\nbetween any two there are always an infinite number of\r\nothers, which cannot be enumerated one by one. (2)\r\nIf, on the other hand, only the successive middle points\r\nare concerned, obtained by always halving what remains\r\nof the course, then the points are reached one by one,\r\nand, though they are infinite in number, they are in fact\r\nall reached in a finite time. His argument to the contrary\r\nmay be supposed to appeal to the view that a finite time\r\nmust consist of a finite number of instants, in which case\r\nwhat he says would be perfectly true on the assumption\r\nthat the possibility of continued dichotomy is undeniable.\r\nIf, on the other hand, we suppose the argument directed\r\nagainst the partisans of infinite divisibility, we must\r\nsuppose it to proceed as follows:\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_42\" id=\"FNanchor_42\"\u003e[42]\u003c/a\u003e “The points given by\r\nsuccessive halving of the distances still to be traversed\r\nare infinite in number, and are reached in succession, each\r\nbeing reached a finite time later than its predecessor;\r\nbut the sum of an infinite number of finite times must be\r\ninfinite, and therefore the process will never be completed.”\r\nIt is very possible that this is historically the right interpretation,\r\nbut in this form the argument is invalid. If\r\nhalf the course takes half a minute, and the next quarter\r\n\u003ca class=\"pagenum\" title=\"173\" id=\"Page_173\"\u003e \u003c/a\u003e\r\ntakes a quarter of a minute, and so on, the whole course\r\nwill take a minute. The apparent force of the argument,\r\non this interpretation, lies solely in the mistaken supposition\r\nthat there cannot be anything beyond the whole of\r\nan infinite series, which can be seen to be false by observing\r\nthat 1 is beyond the whole of the infinite series \u003csup\u003e1\u003c/sup\u003e⁄\u003csub\u003e2\u003c/sub\u003e, \u003csup\u003e3\u003c/sup\u003e⁄\u003csub\u003e4\u003c/sub\u003e,\r\n\u003csup\u003e7\u003c/sup\u003e⁄\u003csub\u003e8\u003c/sub\u003e, \u003csup\u003e15\u003c/sup\u003e⁄\u003csub\u003e16\u003c/sub\u003e, …\u003c/p\u003e\r\n\u003cp\u003eThe second of Zeno\u0027s arguments is the one concerning\r\nAchilles and the tortoise, which has achieved more\r\nnotoriety than the others. It is paraphrased by Burnet\r\nas follows:\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_43\" id=\"FNanchor_43\"\u003e[43]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003e“Achilles will never overtake the tortoise. He must\r\nfirst reach the place from which the tortoise started. By\r\nthat time the tortoise will have got some way ahead.\r\nAchilles must then make up that, and again the tortoise\r\nwill be ahead. He is always coming nearer, but he never\r\nmakes up to it.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_44\" id=\"FNanchor_44\"\u003e[44]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eThis argument is essentially the same as the previous\r\none. It shows that, if Achilles ever overtakes the tortoise,\r\nit must be after an infinite number of instants have elapsed\r\nsince he started. This is in fact true; but the view that\r\nan infinite number of instants make up an infinitely long\r\ntime is not true, and therefore the conclusion that Achilles\r\nwill never overtake the tortoise does not follow.\u003c/p\u003e\r\n\u003cp\u003eThe third argument,\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_45\" id=\"FNanchor_45\"\u003e[45]\u003c/a\u003e that of the arrow, is very interesting.\r\nThe text has been questioned. Burnet accepts the\r\nalterations of Zeller, and paraphrases thus:\u003c/p\u003e\r\n\u003cp\u003e“The arrow in flight is at rest. For, if everything is\r\n\u003ca class=\"pagenum\" title=\"174\" id=\"Page_174\"\u003e \u003c/a\u003e\r\nat rest when it occupies a space equal to itself, and what\r\nis in flight at any given moment always occupies a space\r\nequal to itself, it cannot move.”\u003c/p\u003e\r\n\u003cp\u003eBut according to Prantl, the literal translation of the\r\nunemended text of Aristotle\u0027s statement of the argument\r\nis as follows: “If everything, when it is behaving\r\nin a uniform manner, is continually either moving or\r\nat rest, but what is moving is always in the \u003cem\u003enow\u003c/em\u003e, then\r\nthe moving arrow is motionless.” This form of the\r\nargument brings out its force more clearly than Burnet\u0027s\r\nparaphrase.\u003c/p\u003e\r\n\u003cp\u003eHere, if not in the first two arguments, the view that\r\na finite part of time consists of a finite series of successive\r\ninstants seems to be assumed; at any rate the plausibility\r\nof the argument seems to depend upon supposing that\r\nthere are consecutive instants. Throughout an instant,\r\nit is said, a moving body is where it is: it cannot move\r\nduring the instant, for that would require that the instant\r\nshould have parts. Thus, suppose we consider a period\r\nconsisting of a thousand instants, and suppose the arrow\r\nis in flight throughout this period. At each of the\r\nthousand instants, the arrow is where it is, though at the\r\nnext instant it is somewhere else. It is never moving,\r\nbut in some miraculous way the change of position has to\r\noccur \u003cem\u003ebetween\u003c/em\u003e the instants, that is to say, not at any time\r\nwhatever. This is what M. Bergson calls the cinematographic\r\nrepresentation of reality. The more the difficulty\r\nis meditated, the more real it becomes. The solution\r\nlies in the theory of continuous series: we find it hard\r\nto avoid supposing that, when the arrow is in flight, there\r\nis a \u003cem\u003enext\u003c/em\u003e position occupied at the \u003cem\u003enext\u003c/em\u003e moment; but in\r\nfact there is no next position and no next moment, and\r\nwhen once this is imaginatively realised, the difficulty is\r\nseen to disappear.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"175\" id=\"Page_175\"\u003e \u003c/a\u003eThe fourth and last of Zeno\u0027s arguments is\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_46\" id=\"FNanchor_46\"\u003e[46]\u003c/a\u003e the argument\r\nof the stadium.\u003c/p\u003e\r\n\u003cp\u003eThe argument as stated by Burnet is as follows:\u003c/p\u003e\r\n\u003ctable class=\"zeno-stadium\" id=\"zeno-stadium-1\" data-summary=\"Zeno\u0027s argument of the stadium\"\u003e\r\n\u003ctbody\u003e\u003ctr\u003e\r\n\u003cth colspan=\"5\"\u003eFirst Position.\u003c/th\u003e\r\n\u003cth style=\"width: 1.5em;\"\u003e \u003c/th\u003e\r\n\u003cth colspan=\"7\"\u003eSecond Position.\u003c/th\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eA\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd\u003eA\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eB\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd\u003eB\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd colspan=\"2\"\u003e \u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eC\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd\u003eC\u003c/td\u003e\r\n\u003ctd colspan=\"2\"\u003e \u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003ctd\u003e.\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003c/tbody\u003e\u003c/table\u003e\r\n\u003cp\u003e“Half the time may be equal to double the time.\r\nLet us suppose three rows of bodies, one of which (A)\r\nis at rest while the other two (B, C) are moving with\r\nequal velocity in opposite directions. By the time they\r\nare all in the same part of the course, B will have passed\r\ntwice as many of the bodies in C as in A. Therefore the\r\ntime which it takes to pass C is twice as long as the time\r\nit takes to pass A. But the time which B and C take to\r\nreach the position of A is the same. Therefore double\r\nthe time is equal to the half.”\u003c/p\u003e\r\n\u003cp\u003eGaye\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_47\" id=\"FNanchor_47\"\u003e[47]\u003c/a\u003e devoted an interesting article to the interpretation\r\nof this argument. His translation of Aristotle\u0027s\r\nstatement is as follows:\u003c/p\u003e\r\n\u003cp\u003e“The fourth argument is that concerning the two\r\nrows of bodies, each row being composed of an equal\r\nnumber of bodies of equal size, passing each other on a\r\nrace-course as they proceed with equal velocity in opposite\r\ndirections, the one row originally occupying the space\r\nbetween the goal and the middle point of the course, and\r\nthe other that between the middle point and the starting-post.\r\nThis, he thinks, involves the conclusion that half\r\na given time is equal to double the time. The fallacy of\r\nthe reasoning lies in the assumption that a body occupies\r\nan equal time in passing with equal velocity a body that\r\nis in motion and a body of equal size that is at rest, an\r\n\u003ca class=\"pagenum\" title=\"176\" id=\"Page_176\"\u003e \u003c/a\u003e\r\nassumption which is false. For instance (so runs the\r\nargument), let A A … be the stationary bodies of equal\r\nsize, B B … the bodies, equal in number and in size\r\nto A A …, originally occupying the half of the course\r\nfrom the starting-post to the middle of the A\u0027s, and\r\nC C … those originally occupying the other half from\r\nthe goal to the middle of the A\u0027s, equal in number, size,\r\nand velocity, to B B … Then three consequences follow.\r\nFirst, as the B\u0027s and C\u0027s pass one another, the first B\r\nreaches the last C at the same moment at which the first\r\nC reaches the last B. Secondly, at this moment the first\r\nC has passed all the A\u0027s, whereas the first B has passed\r\nonly half the A\u0027s and has consequently occupied only\r\nhalf the time occupied by the first C, since each of the\r\ntwo occupies an equal time in passing each A. Thirdly,\r\nat the same moment all the B\u0027s have passed all the C\u0027s:\r\nfor the first C and the first B will simultaneously reach\r\nthe opposite ends of the course, since (so says Zeno) the\r\ntime occupied by the first C in passing each of the B\u0027s is\r\nequal to that occupied by it in passing each of the A\u0027s,\r\nbecause an equal time is occupied by both the first B and\r\nthe first C in passing all the A\u0027s. This is the argument:\r\nbut it presupposes the aforesaid fallacious assumption.”\u003c/p\u003e\r\n\u003ctable class=\"zeno-stadium\" id=\"zeno-stadium-2\" data-summary=\"Zeno\u0027s argument of the stadium\"\u003e\r\n\u003ctbody\u003e\u003ctr\u003e\r\n\u003cth colspan=\"3\"\u003eFirst Position.\u003c/th\u003e\r\n\u003cth style=\"width: 1.5em;\"\u003e \u003c/th\u003e\r\n\u003cth colspan=\"5\"\u003eSecond Position.\u003c/th\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eB\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eB′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eB″\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd colspan=\"3\"\u003e \u003c/td\u003e\r\n\u003ctd\u003eB\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eB′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eB″\u003cbr\u003e·\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eA\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eA′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eA″\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd colspan=\"2\"\u003e \u003c/td\u003e\r\n\u003ctd\u003eA\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eA′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eA″\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003eC\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eC′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eC″\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003e \u003c/td\u003e\r\n\u003ctd\u003eC\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eC′\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd\u003eC″\u003cbr\u003e·\u003c/td\u003e\r\n\u003ctd colspan=\"2\"\u003e \u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003c/tbody\u003e\u003c/table\u003e\r\n\u003cp\u003eThis argument is not quite easy to follow, and it is\r\nonly valid as against the assumption that a finite time\r\nconsists of a finite\r\nnumber of instants.\r\nWe may re-state it\r\nin different language.\r\nLet us suppose three\r\ndrill-sergeants, A, A′,\r\nand A″, standing in a\r\nrow, while the two files of soldiers march past them in\r\nopposite directions. At the first moment which we consider,\r\n\u003ca class=\"pagenum\" title=\"177\" id=\"Page_177\"\u003e \u003c/a\u003e\r\nthe three men B, B′, B″ in one row, and the three\r\nmen C, C′, C″ in the other row, are respectively opposite\r\nto A, A′, and A″. At the very next moment, each row\r\nhas moved on, and now B and C″ are opposite A′. Thus\r\nB and C″ are opposite each other. When, then, did B\r\npass C′? It must have been somewhere between the\r\ntwo moments which we supposed consecutive, and therefore\r\nthe two moments cannot really have been consecutive.\r\nIt follows that there must be other moments between any\r\ntwo given moments, and therefore that there must be an\r\ninfinite number of moments in any given interval of time.\u003c/p\u003e\r\n\u003cp\u003eThe above difficulty, that B must have passed C′ at\r\nsome time between two consecutive moments, is a genuine\r\none, but is not precisely the difficulty raised by Zeno.\r\nWhat Zeno professes to prove is that “half of a given\r\ntime is equal to double that time.” The most intelligible\r\nexplanation of the argument known to me is that of\r\nGaye.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_48\" id=\"FNanchor_48\"\u003e[48]\u003c/a\u003e Since, however, his explanation is not easy to\r\nset forth shortly, I will re-state what seems to me to be\r\nthe logical essence of Zeno\u0027s contention. If we suppose\r\nthat time consists of a series of consecutive instants, and\r\nthat motion consists in passing through a series of consecutive\r\npoints, then the fastest possible motion is one\r\nwhich, at each instant, is at a point consecutive to that\r\nat which it was at the previous instant. Any slower\r\nmotion must be one which has intervals of rest interspersed,\r\nand any faster motion must wholly omit some\r\npoints. All this is evident from the fact that we cannot\r\nhave more than one event for each instant. But now, in\r\nthe case of our A\u0027s and B\u0027s and C\u0027s, B is opposite a fresh\r\nA every instant, and therefore the number of A\u0027s passed\r\ngives the number of instants since the beginning of the\r\nmotion. But during the motion B has passed twice as\r\n\u003ca class=\"pagenum\" title=\"178\" id=\"Page_178\"\u003e \u003c/a\u003e\r\nmany C\u0027s, and yet cannot have passed more than one each\r\ninstant. Hence the number of instants since the motion\r\nbegan is twice the number of A\u0027s passed, though we previously\r\nfound it was equal to this number. From this\r\nresult, Zeno\u0027s conclusion follows.\u003c/p\u003e\r\n\u003cp\u003eZeno\u0027s arguments, in some form, have afforded grounds\r\nfor almost all the theories of space and time and infinity\r\nwhich have been constructed from his day to our\r\nown. We have seen that all his arguments are valid\r\n(with certain reasonable hypotheses) on the assumption\r\nthat finite spaces and times consist of a finite number of\r\npoints and instants, and that the third and fourth almost\r\ncertainly in fact proceeded on this assumption, while the\r\nfirst and second, which were perhaps intended to refute\r\nthe opposite assumption, were in that case fallacious.\r\nWe may therefore escape from his paradoxes either by\r\nmaintaining that, though space and time do consist of\r\npoints and instants, the number of them in any finite\r\ninterval is infinite; or by denying that space and time\r\nconsist of points and instants at all; or lastly, by denying\r\nthe reality of space and time altogether. It would seem\r\nthat Zeno himself, as a supporter of Parmenides, drew\r\nthe last of these three possible deductions, at any rate in\r\nregard to time. In this a very large number of philosophers\r\nhave followed him. Many others, like M. Bergson,\r\nhave preferred to deny that space and time consist of\r\npoints and instants. Either of these solutions will meet\r\nthe difficulties in the form in which Zeno raised them.\r\nBut, as we saw, the difficulties can also be met if infinite\r\nnumbers are admissible. And on grounds which are\r\nindependent of space and time, infinite numbers, and\r\nseries in which no two terms are consecutive, must in\r\nany case be admitted. Consider, for example, all the\r\nfractions less than 1, arranged in order of magnitude.\r\n\u003ca class=\"pagenum\" title=\"179\" id=\"Page_179\"\u003e \u003c/a\u003e\r\nBetween any two of them, there are others, for example,\r\nthe arithmetical mean of the two. Thus no two fractions\r\nare consecutive, and the total number of them is infinite.\r\nIt will be found that much of what Zeno says as regards\r\nthe series of points on a line can be equally well applied\r\nto the series of fractions. And we cannot deny that there\r\nare fractions, so that two of the above ways of escape\r\nare closed to us. It follows that, if we are to solve\r\nthe whole class of difficulties derivable from Zeno\u0027s by\r\nanalogy, we must discover some tenable theory of infinite\r\nnumbers. What, then, are the difficulties which, until\r\nthe last thirty years, led philosophers to the belief that\r\ninfinite numbers are impossible?\u003c/p\u003e\r\n\u003cp\u003eThe difficulties of infinity are of two kinds, of which\r\nthe first may be called sham, while the others involve, for\r\ntheir solution, a certain amount of new and not altogether\r\neasy thinking. The sham difficulties are those suggested\r\nby the etymology, and those suggested by confusion of\r\nthe mathematical infinite with what philosophers impertinently\r\ncall the “true” infinite. Etymologically,\r\n“infinite” should mean “having no end.” But in fact\r\nsome infinite series have ends, some have not; while\r\nsome collections are infinite without being serial, and can\r\ntherefore not properly be regarded as either endless or\r\nhaving ends. The series of instants from any earlier one\r\nto any later one (both included) is infinite, but has two\r\nends; the series of instants from the beginning of time\r\nto the present moment has one end, but is infinite.\r\nKant, in his first antinomy, seems to hold that it is harder\r\nfor the past to be infinite than for the future to be so,\r\non the ground that the past is now completed, and that\r\nnothing infinite can be completed. It is very difficult to\r\nsee how he can have imagined that there was any sense in\r\nthis remark; but it seems most probable that he was\r\n\u003ca class=\"pagenum\" title=\"180\" id=\"Page_180\"\u003e \u003c/a\u003e\r\nthinking of the infinite as the “unended.” It is odd that\r\nhe did not see that the future too has one end at the\r\npresent, and is precisely on a level with the past. His\r\nregarding the two as different in this respect illustrates\r\njust that kind of slavery to time which, as we agreed in\r\nspeaking of Parmenides, the true philosopher must learn\r\nto leave behind him.\u003c/p\u003e\r\n\u003cp\u003eThe confusions introduced into the notions of philosophers\r\nby the so-called “true” infinite are curious. They\r\nsee that this notion is not the same as the mathematical\r\ninfinite, but they choose to believe that it is the notion\r\nwhich the mathematicians are vainly trying to reach.\r\nThey therefore inform the mathematicians, kindly but\r\nfirmly, that they are mistaken in adhering to the “false”\r\ninfinite, since plainly the “true” infinite is something\r\nquite different. The reply to this is that what they call the\r\n“true” infinite is a notion totally irrelevant to the problem\r\nof the mathematical infinite, to which it has only a fanciful\r\nand verbal analogy. So remote is it that I do not\r\npropose to confuse the issue by even mentioning what\r\nthe “true” infinite is. It is the “false” infinite that\r\nconcerns us, and we have to show that the epithet “false”\r\nis undeserved.\u003c/p\u003e\r\n\u003cp\u003eThere are, however, certain genuine difficulties in\r\nunderstanding the infinite, certain habits of mind derived\r\nfrom the consideration of finite numbers, and easily\r\nextended to infinite numbers under the mistaken notion\r\nthat they represent logical necessities. For example,\r\nevery number that we are accustomed to, except 0, has\r\nanother number immediately before it, from which it\r\nresults by adding 1; but the first infinite number does\r\nnot have this property. The numbers before it form an\r\ninfinite series, containing all the ordinary finite numbers,\r\nhaving no maximum, no last finite number, after which\r\n\u003ca class=\"pagenum\" title=\"181\" id=\"Page_181\"\u003e \u003c/a\u003e\r\none little step would plunge us into the infinite. If it is\r\nassumed that the first infinite number is reached by a\r\nsuccession of small steps, it is easy to show that it is self-contradictory.\r\nThe first infinite number is, in fact,\r\nbeyond the whole unending series of finite numbers.\r\n“But,” it will be said, “there cannot be anything beyond\r\nthe whole of an unending series.” This, we may point\r\nout, is the very principle upon which Zeno relies in the\r\narguments of the race-course and the Achilles. Take the\r\nrace-course: there is the moment when the runner still\r\nhas half his distance to run, then the moment when he\r\nstill has a quarter, then when he still has an eighth, and\r\nso on in a strictly unending series. Beyond the whole\r\nof this series is the moment when he reaches the goal.\r\nThus there certainly can be something beyond the whole\r\nof an unending series. But it remains to show that this\r\nfact is only what might have been expected.\u003c/p\u003e\r\n\u003cp\u003eThe difficulty, like most of the vaguer difficulties\r\nbesetting the mathematical infinite, is derived, I think,\r\nfrom the more or less unconscious operation of the idea\r\nof \u003cem\u003ecounting\u003c/em\u003e. If you set to work to count the terms in an\r\ninfinite collection, you will never have completed your\r\ntask. Thus, in the case of the runner, if half, three-quarters,\r\nseven-eighths, and so on of the course were\r\nmarked, and the runner was not allowed to pass any of\r\nthe marks until the umpire said “Now,” then Zeno\u0027s\r\nconclusion would be true in practice, and he would never\r\nreach the goal.\u003c/p\u003e\r\n\u003cp\u003eBut it is not essential to the existence of a collection,\r\nor even to knowledge and reasoning concerning it, that we\r\nshould be able to pass its terms in review one by one.\r\nThis may be seen in the case of finite collections; we can\r\nspeak of “mankind” or “the human race,” though many\r\nof the individuals in this collection are not personally\r\n\u003ca class=\"pagenum\" title=\"182\" id=\"Page_182\"\u003e \u003c/a\u003e\r\nknown to us. We can do this because we know of\r\nvarious characteristics which every individual has if he\r\nbelongs to the collection, and not if he does not. And\r\nexactly the same happens in the case of infinite collections:\r\nthey may be known by their characteristics although their\r\nterms cannot be enumerated. In this sense, an unending\r\nseries may nevertheless form a whole, and there may be\r\nnew terms beyond the whole of it.\u003c/p\u003e\r\n\u003cp\u003eSome purely arithmetical peculiarities of infinite\r\nnumbers have also caused perplexity. For instance, an\r\ninfinite number is not increased by adding one to it, or\r\nby doubling it. Such peculiarities have seemed to many\r\nto contradict logic, but in fact they only contradict confirmed\r\nmental habits. The whole difficulty of the subject\r\nlies in the necessity of thinking in an unfamiliar way,\r\nand in realising that many properties which we have\r\nthought inherent in number are in fact peculiar to\r\nfinite numbers. If this is remembered, the positive\r\ntheory of infinity, which will occupy the \u003ca href=\"#Lecture_7\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e,\r\nwill not be found so difficult as it is to those who cling\r\nobstinately to the prejudices instilled by the arithmetic\r\nwhich is learnt in childhood.\r\n\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"183\" id=\"Page_183\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE VII\u003c/small\u003e\u003cbr\u003e\r\nTHE POSITIVE THEORY\r\nOF INFINITY\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"185\" id=\"Page_185\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_7\"\u003eLECTURE VII\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nTHE POSITIVE THEORY OF INFINITY\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eThe\u003c/span\u003e positive theory of infinity, and the general theory\r\nof number to which it has given rise, are among the\r\ntriumphs of scientific method in philosophy, and are\r\ntherefore specially suitable for illustrating the logical-analytic\r\ncharacter of that method. The work in this\r\nsubject has been done by mathematicians, and its results\r\ncan be expressed in mathematical symbolism. Why,\r\nthen, it may be said, should the subject be regarded\r\nas philosophy rather than as mathematics? This raises\r\na difficult question, partly concerned with the use of\r\nwords, but partly also of real importance in understanding\r\nthe function of philosophy. Every subject-matter, it\r\nwould seem, can give rise to philosophical investigations\r\nas well as to the appropriate science, the difference\r\nbetween the two treatments being in the direction of\r\nmovement and in the kind of truths which it is sought\r\nto establish. In the special sciences, when they have\r\nbecome fully developed, the movement is forward and\r\nsynthetic, from the simpler to the more complex. But\r\nin philosophy we follow the inverse direction: from the\r\ncomplex and relatively concrete we proceed towards the\r\nsimple and abstract by means of analysis, seeking, in the\r\nprocess, to eliminate the particularity of the original\r\nsubject-matter, and to confine our attention entirely to\r\nthe logical \u003cem\u003eform\u003c/em\u003e of the facts concerned.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"186\" id=\"Page_186\"\u003e \u003c/a\u003eBetween philosophy and pure mathematics there is a\r\ncertain affinity, in the fact that both are general and \u003ci lang=\"la\"\u003ea\r\npriori\u003c/i\u003e. Neither of them asserts propositions which, like\r\nthose of history and geography, depend upon the actual\r\nconcrete facts being just what they are. We may illustrate\r\nthis characteristic by means of Leibniz\u0027s conception of\r\nmany \u003cem\u003epossible\u003c/em\u003e worlds, of which one only is \u003cem\u003eactual\u003c/em\u003e. In all\r\nthe many possible worlds, philosophy and mathematics\r\nwill be the same; the differences will only be in respect of\r\nthose particular facts which are chronicled by the descriptive\r\nsciences. Any quality, therefore, by which our\r\nactual world is distinguished from other abstractly possible\r\nworlds, must be ignored by mathematics and philosophy\r\nalike. Mathematics and philosophy differ, however, in\r\ntheir manner of treating the general properties in which\r\nall possible worlds agree; for while mathematics, starting\r\nfrom comparatively simple propositions, seeks to build up\r\nmore and more complex results by deductive synthesis,\r\nphilosophy, starting from data which are common knowledge,\r\nseeks to purify and generalise them into the\r\nsimplest statements of abstract form that can be obtained\r\nfrom them by logical analysis.\u003c/p\u003e\r\n\u003cp\u003eThe difference between philosophy and mathematics\r\nmay be illustrated by our present problem, namely, the\r\nnature of number. Both start from certain facts about\r\nnumbers which are evident to inspection. But mathematics\r\nuses these facts to deduce more and more complicated\r\ntheorems, while philosophy seeks, by analysis, to\r\ngo behind these facts to others, simpler, more fundamental,\r\nand inherently more fitted to form the premisses of\r\nthe science of arithmetic. The question, “What is a\r\nnumber?” is the pre-eminent philosophic question in\r\nthis subject, but it is one which the mathematician as such\r\nneed not ask, provided he knows enough of the properties\r\n\u003ca class=\"pagenum\" title=\"187\" id=\"Page_187\"\u003e \u003c/a\u003e\r\nof numbers to enable him to deduce his theorems. We,\r\nsince our object is philosophical, must grapple with the\r\nphilosopher\u0027s question. The answer to the question,\r\n“What is a number?” which we shall reach in this\r\nlecture, will be found to give also, by implication, the\r\nanswer to the difficulties of infinity which we considered\r\nin the \u003ca href=\"#Lecture_6\" class=\"pginternal\"\u003eprevious lecture\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eThe question “What is a number?” is one which,\r\nuntil quite recent times, was never considered in the\r\nkind of way that is capable of yielding a precise answer.\r\nPhilosophers were content with some vague dictum such\r\nas, “Number is unity in plurality.” A typical definition of\r\nthe kind that contented philosophers is the following from\r\nSigwart\u0027s \u003ccite\u003eLogic\u003c/cite\u003e (§ 66, section 3): “Every number is not\r\nmerely a \u003cem\u003eplurality\u003c/em\u003e, but a plurality thought \u003cem\u003eas held together\r\nand closed, and to that extent as a unity\u003c/em\u003e.” Now there is\r\nin such definitions a very elementary blunder, of the\r\nsame kind that would be committed if we said “yellow\r\nis a flower” because some flowers are yellow. Take, for\r\nexample, the number 3. A single collection of three\r\nthings might conceivably be described as “a plurality\r\nthought as held together and closed, and to that extent\r\nas a unity”; but a collection of three things is not the\r\nnumber 3. The number 3 is something which all collections\r\nof three things have in common, but is not itself\r\na collection of three things. The definition, therefore,\r\napart from any other defects, has failed to reach the\r\nnecessary degree of abstraction: the number 3 is something\r\nmore abstract than any collection of three things.\u003c/p\u003e\r\n\u003cp\u003eSuch vague philosophic definitions, however, remained\r\ninoperative because of their very vagueness. What\r\nmost men who thought about numbers really had in\r\nmind was that numbers are the result of \u003cem\u003ecounting\u003c/em\u003e. “On\r\nthe consciousness of the law of counting,” says Sigwart\r\n\u003ca class=\"pagenum\" title=\"188\" id=\"Page_188\"\u003e \u003c/a\u003e\r\nat the beginning of his discussion of number, “rests\r\nthe possibility of spontaneously prolonging the series\r\nof numbers \u003ci lang=\"la\"\u003ead infinitum\u003c/i\u003e.” It is this view of number\r\nas generated by counting which has been the chief\r\npsychological obstacle to the understanding of infinite\r\nnumbers. Counting, because it is familiar, is erroneously\r\nsupposed to be simple, whereas it is in fact a highly\r\ncomplex process, which has no meaning unless the\r\nnumbers reached in counting have some significance\r\nindependent of the process by which they are reached.\r\nAnd infinite numbers cannot be reached at all in this\r\nway. The mistake is of the same kind as if cows were\r\ndefined as what can be bought from a cattle-merchant.\r\nTo a person who knew several cattle-merchants, but had\r\nnever seen a cow, this might seem an admirable definition.\r\nBut if in his travels he came across a herd of wild\r\ncows, he would have to declare that they were not cows\r\nat all, because no cattle-merchant could sell them. So\r\ninfinite numbers were declared not to be numbers at all,\r\nbecause they could not be reached by counting.\u003c/p\u003e\r\n\u003cp\u003eIt will be worth while to consider for a moment what\r\ncounting actually is. We count a set of objects when\r\nwe let our attention pass from one to another, until we\r\nhave attended once to each, saying the names of the\r\nnumbers in order with each successive act of attention.\r\nThe last number named in this process is the number\r\nof the objects, and therefore counting is a method of\r\nfinding out what the number of the objects is. But this\r\noperation is really a very complicated one, and those\r\nwho imagine that it is the logical source of number show\r\nthemselves remarkably incapable of analysis. In the\r\nfirst place, when we say “one, two, three …” as we\r\ncount, we cannot be said to be discovering the number\r\nof the objects counted unless we attach some meaning\r\n\u003ca class=\"pagenum\" title=\"189\" id=\"Page_189\"\u003e \u003c/a\u003e\r\nto the words one, two, three, … A child may learn\r\nto know these words in order, and to repeat them\r\ncorrectly like the letters of the alphabet, without attaching\r\nany meaning to them. Such a child may count\r\ncorrectly from the point of view of a grown-up listener,\r\nwithout having any idea of numbers at all. The\r\noperation of counting, in fact, can only be intelligently\r\nperformed by a person who already has some idea what\r\nthe numbers are; and from this it follows that counting\r\ndoes not give the logical basis of number.\u003c/p\u003e\r\n\u003cp\u003eAgain, how do we know that the last number reached\r\nin the process of counting is the number of the objects\r\ncounted? This is just one of those facts that are too\r\nfamiliar for their significance to be realised; but those\r\nwho wish to be logicians must acquire the habit of\r\ndwelling upon such facts. There are two propositions\r\ninvolved in this fact: first, that the number of numbers\r\nfrom 1 up to any given number is that given number—for\r\ninstance, the number of numbers from 1 to 100 is\r\na hundred; secondly, that if a set of numbers can be\r\nused as names of a set of objects, each number occurring\r\nonly once, then the number of numbers used as names\r\nis the same as the number of objects. The first of\r\nthese propositions is capable of an easy arithmetical\r\nproof so long as finite numbers are concerned; but with\r\ninfinite numbers, after the first, it ceases to be true. The\r\nsecond proposition remains true, and is in fact, as we\r\nshall see, an immediate consequence of the definition\r\nof number. But owing to the falsehood of the first\r\nproposition where infinite numbers are concerned, counting,\r\neven if it were practically possible, would not be a\r\nvalid method of discovering the number of terms in an\r\ninfinite collection, and would in fact give different results\r\naccording to the manner in which it was carried out.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"190\" id=\"Page_190\"\u003e \u003c/a\u003eThere are two respects in which the infinite numbers\r\nthat are known differ from finite numbers: first, infinite\r\nnumbers have, while finite numbers have not, a property\r\nwhich I shall call \u003cem\u003ereflexiveness\u003c/em\u003e; secondly, finite numbers\r\nhave, while infinite numbers have not, a property which\r\nI shall call \u003cem\u003einductiveness\u003c/em\u003e. Let us consider these two\r\nproperties successively.\u003c/p\u003e\r\n\u003cp\u003e(1) \u003cem\u003eReflexiveness.\u003c/em\u003e—A number is said to be \u003cem\u003ereflexive\u003c/em\u003e\r\nwhen it is not increased by adding 1 to it. It follows\r\nat once that any finite number can be added to a reflexive\r\nnumber without increasing it. This property of infinite\r\nnumbers was always thought, until recently, to be self-contradictory;\r\nbut through the work of Georg Cantor\r\nit has come to be recognised that, though at first\r\nastonishing, it is no more self-contradictory than the\r\nfact that people at the antipodes do not tumble off.\r\nIn virtue of this property, given any infinite collection\r\nof objects, any finite number of objects can be added\r\nor taken away without increasing or diminishing the\r\nnumber of the collection. Even an infinite number of\r\nobjects may, under certain conditions, be added or taken\r\naway without altering the number. This may be made\r\nclearer by the help of some examples.\u003c/p\u003e\r\n\u003cp\u003eImagine all the natural numbers 0, 1, 2, 3, … to be\r\nwritten down in a row, and immediately beneath them\r\nwrite down the numbers 1, 2, 3, 4, …, so that 1 is\r\nunder 0, 2 is under 1, and so on. Then every number\r\nin the top row has a number directly under it in\r\nthe bottom row, and no number occurs twice in either\r\nrow. It follows that the number of numbers in the\r\ntwo rows must be the same. But all the numbers\r\nthat occur in the bottom row also occur in the top\r\n\u003ca class=\"pagenum\" title=\"191\" id=\"Page_191\"\u003e \u003c/a\u003e\r\nrow, and one more, namely 0; thus the number\r\nof terms in the top row is obtained by adding one to\r\nthe number of the bottom row. So long, therefore,\r\nas it was supposed that a number must be increased by\r\nadding 1 to it, this state of things constituted a contradiction,\r\nand led to the denial that there are infinite\r\nnumbers.\u003c/p\u003e\r\n\u003ctable id=\"numbers\" data-summary=\"1, 2, 3, 4 below 0, 1, 2, 3\"\u003e\r\n\u003ctbody\u003e\u003ctr\u003e\r\n\u003ctd\u003e0,\u003c/td\u003e\r\n\u003ctd\u003e1,\u003c/td\u003e\r\n\u003ctd\u003e2,\u003c/td\u003e\r\n\u003ctd\u003e3,\u003c/td\u003e\r\n\u003ctd\u003e…\u003c/td\u003e\r\n\u003ctd\u003e\u003ci\u003en\u003c/i\u003e …\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003ctr\u003e\r\n\u003ctd\u003e1,\u003c/td\u003e\r\n\u003ctd\u003e2,\u003c/td\u003e\r\n\u003ctd\u003e3,\u003c/td\u003e\r\n\u003ctd\u003e4,\u003c/td\u003e\r\n\u003ctd\u003e…\u003c/td\u003e\r\n\u003ctd\u003e\u003ci\u003en\u003c/i\u003e + 1 …\u003c/td\u003e\r\n\u003c/tr\u003e\r\n\u003c/tbody\u003e\u003c/table\u003e\r\n\u003cp\u003eThe following example is even more surprising.\r\nWrite the natural numbers 1, 2, 3, 4, … in the top\r\nrow, and the even numbers 2, 4, 6, 8, … in the\r\nbottom row, so that under each number in the top row\r\nstands its double in the bottom row. Then, as before,\r\nthe number of numbers in the two rows is the same, yet\r\nthe second row results from taking away all the odd\r\nnumbers—an infinite collection—from the top row.\r\nThis example is given by Leibniz to prove that there\r\ncan be no infinite numbers. He believed in infinite\r\ncollections, but, since he thought that a number must\r\nalways be increased when it is added to and diminished\r\nwhen it is subtracted from, he maintained that infinite\r\ncollections do not have numbers. “The number of\r\nall numbers,” he says, “implies a contradiction, which\r\nI show thus: To any number there is a corresponding\r\nnumber equal to its double. Therefore the number\r\nof all numbers is not greater than the number of\r\neven numbers, \u003ci\u003ei.e.\u003c/i\u003e the whole is not greater than its\r\npart.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_49\" id=\"FNanchor_49\"\u003e[49]\u003c/a\u003e In dealing with this argument, we ought\r\nto substitute “the number of all finite numbers”\r\nfor “the number of all numbers”; we then obtain\r\nexactly the illustration given by our two rows, one\r\ncontaining all the finite numbers, the other only the even\r\nfinite numbers. It will be seen that Leibniz regards it as\r\nself-contradictory to maintain that the whole is not\r\n\u003ca class=\"pagenum\" title=\"192\" id=\"Page_192\"\u003e \u003c/a\u003e\r\ngreater than its part. But the word “greater” is one\r\nwhich is capable of many meanings; for our purpose, we\r\nmust substitute the less ambiguous phrase “containing a\r\ngreater number of terms.” In this sense, it is not self-contradictory\r\nfor whole and part to be equal; it is the\r\nrealisation of this fact which has made the modern theory\r\nof infinity possible.\u003c/p\u003e\r\n\u003cp\u003eThere is an interesting discussion of the reflexiveness\r\nof infinite wholes in the first of Galileo\u0027s Dialogues on\r\nMotion. I quote from a translation published in 1730.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_50\" id=\"FNanchor_50\"\u003e[50]\u003c/a\u003e\r\nThe personages in the dialogue are Salviati, Sagredo, and\r\nSimplicius, and they reason as follows:\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSimp.\u003c/i\u003e Here already arises a Doubt which I think is\r\nnot to be resolv\u0027d; and that is this: Since \u0027tis plain that\r\none Line is given greater than another, and since both\r\ncontain infinite Points, we must surely necessarily infer,\r\nthat we have found in the same Species something greater\r\nthan Infinite, since the Infinity of Points of the greater\r\nLine exceeds the Infinity of Points of the lesser. But\r\nnow, to assign an Infinite greater than an Infinite, is what\r\nI can\u0027t possibly conceive.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSalv.\u003c/i\u003e These are some of those Difficulties which\r\narise from Discourses which our finite Understanding\r\nmakes about Infinites, by ascribing to them Attributes\r\nwhich we give to Things finite and terminate, which I\r\nthink most improper, because those Attributes of Majority,\r\nMinority, and Equality, agree not with Infinities, of which\r\nwe can\u0027t say that one is greater than, less than, or equal\r\nto another. For Proof whereof I have something come\r\n\u003ca class=\"pagenum\" title=\"193\" id=\"Page_193\"\u003e \u003c/a\u003e\r\ninto my Head, which (that I may be the better understood)\r\nI will propose by way of Interrogatories to\r\n\u003ci\u003eSimplicius\u003c/i\u003e, who started this Difficulty. To begin then: I\r\nsuppose you know which are square Numbers, and which\r\nnot?\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSimp.\u003c/i\u003e I know very well that a square Number is\r\nthat which arises from the Multiplication of any Number\r\ninto itself; thus 4 and 9 are square Numbers, that arising\r\nfrom 2, and this from 3, multiplied by themselves.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSalv.\u003c/i\u003e Very well; And you also know, that as the\r\nProducts are call\u0027d Squares, the Factors are call\u0027d Roots:\r\nAnd that the other Numbers, which proceed not from\r\nNumbers multiplied into themselves, are not Squares.\r\nWhence taking in all Numbers, both Squares and Not\r\nSquares, if I should say, that the Not Squares are more\r\nthan the Squares, should I not be in the right?\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSimp.\u003c/i\u003e Most certainly.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSalv.\u003c/i\u003e If I go on with you then, and ask you, How\r\nmany squar\u0027d Numbers there are? you may truly answer,\r\nThat there are as many as are their proper Roots, since\r\nevery Square has its own Root, and every Root its own\r\nSquare, and since no Square has more than one Root,\r\nnor any Root more than one Square.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSimp.\u003c/i\u003e Very true.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSalv.\u003c/i\u003e But now, if I should ask how many Roots\r\nthere are, you can\u0027t deny but there are as many as there\r\nare Numbers, since there\u0027s no Number but what\u0027s the\r\nRoot to some Square. And this being granted, we may\r\nlikewise affirm, that there are as many square Numbers,\r\nas there are Numbers; for there are as many Squares as\r\nthere are Roots, and as many Roots as Numbers. And\r\nyet in the Beginning of this, we said, there were many\r\nmore Numbers than Squares, the greater Part of\r\nNumbers being not Squares: And tho\u0027 the Number of\r\n\u003ca class=\"pagenum\" title=\"194\" id=\"Page_194\"\u003e \u003c/a\u003e\r\nSquares decreases in a greater proportion, as we go on to\r\nbigger Numbers, for count to an Hundred you\u0027ll find 10\r\nSquares, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which\r\nis the same as to say the 10th Part are Squares; in Ten\r\nthousand only the 100th Part are Squares; in a Million\r\nonly the 1000th: And yet in an infinite Number, if we\r\ncan but comprehend it, we may say the Squares are as\r\nmany as all the Numbers taken together.\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSagr.\u003c/i\u003e What must be determin\u0027d then in this Case?\u003c/p\u003e\r\n\u003cp\u003e“\u003ci\u003eSalv.\u003c/i\u003e I see no other way, but by saying that all\r\nNumbers are infinite; Squares are Infinite, their Roots\r\nInfinite, and that the Number of Squares is not less than\r\nthe Number of Numbers, nor this less than that: and\r\nthen by concluding that the Attributes or Terms of\r\nEquality, Majority, and Minority, have no Place in\r\nInfinites, but are confin\u0027d to terminate Quantities.”\u003c/p\u003e\r\n\u003cp\u003eThe way in which the problem is expounded in the\r\nabove discussion is worthy of Galileo, but the solution\r\nsuggested is not the right one. It is actually the case\r\nthat the number of square (finite) numbers is the same as\r\nthe number of (finite) numbers. The fact that, so long\r\nas we confine ourselves to numbers less than some given\r\nfinite number, the proportion of squares tends towards\r\nzero as the given finite number increases, does not\r\ncontradict the fact that the number of all finite squares\r\nis the same as the number of all finite numbers. This\r\nis only an instance of the fact, now familiar to mathematicians,\r\nthat the \u003cem\u003elimit\u003c/em\u003e of a function as the variable\r\n\u003cem\u003eapproaches\u003c/em\u003e a given point may not be the same as its \u003cem\u003evalue\u003c/em\u003e\r\nwhen the variable actually \u003cem\u003ereaches\u003c/em\u003e the given point. But\r\nalthough the infinite numbers which Galileo discusses are\r\nequal, Cantor has shown that what Simplicius could not\r\nconceive is true, namely, that there are an infinite number\r\nof different infinite numbers, and that the conception of\r\n\u003ca class=\"pagenum\" title=\"195\" id=\"Page_195\"\u003e \u003c/a\u003e\r\n\u003cem\u003egreater\u003c/em\u003e and \u003cem\u003eless\u003c/em\u003e can be perfectly well applied to them.\r\nThe whole of Simplicius\u0027s difficulty comes, as is evident,\r\nfrom his belief that, if \u003cem\u003egreater\u003c/em\u003e and \u003cem\u003eless\u003c/em\u003e can be applied, a\r\npart of an infinite collection must have fewer terms than\r\nthe whole; and when this is denied, all contradictions\r\ndisappear. As regards greater and less lengths of lines,\r\nwhich is the problem from which the above discussion\r\nstarts, that involves a meaning of \u003cem\u003egreater\u003c/em\u003e and \u003cem\u003eless\u003c/em\u003e which is\r\nnot arithmetical. The number of points is the same in\r\na long line and in a short one, being in fact the same as\r\nthe number of points in all space. The \u003cem\u003egreater\u003c/em\u003e and \u003cem\u003eless\u003c/em\u003e\r\nof metrical geometry involves the new metrical conception\r\nof \u003cem\u003econgruence\u003c/em\u003e, which cannot be developed out of\r\narithmetical considerations alone. But this question has\r\nnot the fundamental importance which belongs to the\r\narithmetical theory of infinity.\u003c/p\u003e\r\n\u003cp\u003e(2) \u003cem\u003eNon-inductiveness.\u003c/em\u003e—The second property by which\r\ninfinite numbers are distinguished from finite numbers\r\nis the property of non-inductiveness. This will be best\r\nexplained by defining the positive property of inductiveness\r\nwhich characterises the finite numbers, and which\r\nis named after the method of proof known as “mathematical\r\ninduction.”\u003c/p\u003e\r\n\u003cp\u003eLet us first consider what is meant by calling a property\r\n“hereditary” in a given series. Take such a property\r\nas being named Jones. If a man is named Jones, so is\r\nhis son; we will therefore call the property of being\r\ncalled Jones hereditary with respect to the relation of\r\nfather and son. If a man is called Jones, all his descendants\r\nin the direct male line are called Jones; this\r\nfollows from the fact that the property is hereditary.\r\nNow, instead of the relation of father and son, consider\r\nthe relation of a finite number to its immediate successor,\r\nthat is, the relation which holds between 0 and 1, between\r\n\u003ca class=\"pagenum\" title=\"196\" id=\"Page_196\"\u003e \u003c/a\u003e\r\n1 and 2, between 2 and 3, and so on. If a property of\r\nnumbers is hereditary with respect to this relation, then\r\nif it belongs to (say) 100, it must belong also to all finite\r\nnumbers greater than 100; for, being hereditary, it\r\nbelongs to 101 because it belongs to 100, and it belongs\r\nto 102 because it belongs to 101, and so on—where the\r\n“and so on” will take us, sooner or later, to any finite\r\nnumber greater than 100. Thus, for example, the\r\nproperty of being greater than 99 is hereditary in the\r\nseries of finite numbers; and generally, a property is\r\nhereditary in this series when, given any number that\r\npossesses the property, the next number must always\r\nalso possess it.\u003c/p\u003e\r\n\u003cp\u003eIt will be seen that a hereditary property, though it\r\nmust belong to all the finite numbers greater than a given\r\nnumber possessing the property, need not belong to all\r\nthe numbers less than this number. For example, the\r\nhereditary property of being greater than 99 belongs to\r\n100 and all greater numbers, but not to any smaller\r\nnumber. Similarly, the hereditary property of being\r\ncalled Jones belongs to all the descendants (in the direct\r\nmale line) of those who have this property, but not to\r\nall their ancestors, because we reach at last a first Jones,\r\nbefore whom the ancestors have no surname. It is\r\nobvious, however, that any hereditary property possessed\r\nby Adam must belong to all men; and similarly any\r\nhereditary property possessed by 0 must belong to all\r\nfinite numbers. This is the principle of what is called\r\n“mathematical induction.” It frequently happens, when\r\nwe wish to prove that all finite numbers have some\r\nproperty, that we have first to prove that 0 has the\r\nproperty, and then that the property is hereditary, \u003ci\u003ei.e.\u003c/i\u003e\r\nthat, if it belongs to a given number, then it belongs to\r\nthe next number. Owing to the fact that such proofs\r\n\u003ca class=\"pagenum\" title=\"197\" id=\"Page_197\"\u003e \u003c/a\u003e\r\nare called “inductive,” I shall call the properties to which\r\nthey are applicable “inductive” properties. Thus an\r\ninductive property of numbers is one which is hereditary\r\nand belongs to 0.\u003c/p\u003e\r\n\u003cp\u003eTaking any one of the natural numbers, say 29, it is\r\neasy to see that it must have all inductive properties.\r\nFor since such properties belong to 0 and are hereditary,\r\nthey belong to 1; therefore, since they are hereditary,\r\nthey belong to 2, and so on; by twenty-nine repetitions\r\nof such arguments we show that they belong to 29.\r\nWe may \u003cem\u003edefine\u003c/em\u003e the “inductive” numbers as \u003cem\u003eall those that\r\npossess all inductive properties\u003c/em\u003e; they will be the same as\r\nwhat are called the “natural” numbers, \u003ci\u003ei.e.\u003c/i\u003e the ordinary\r\nfinite whole numbers. To all such numbers, proofs by\r\nmathematical induction can be validly applied. They\r\nare those numbers, we may loosely say, which can be\r\nreached from 0 by successive additions of 1; in other\r\nwords, they are all the numbers that can be reached by\r\ncounting.\u003c/p\u003e\r\n\u003cp\u003eBut beyond all these numbers, there are the infinite\r\nnumbers, and infinite numbers do not have all inductive\r\nproperties. Such numbers, therefore, may be called non-inductive.\r\nAll those properties of numbers which are\r\nproved by an imaginary step-by-step process from one\r\nnumber to the next are liable to fail when we come to\r\ninfinite numbers. The first of the infinite numbers has\r\nno immediate predecessor, because there is no greatest\r\nfinite number; thus no succession of steps from one\r\nnumber to the next will ever reach from a finite number\r\nto an infinite one, and the step-by-step method of proof\r\nfails. This is another reason for the supposed self-contradictions\r\nof infinite \u003cins title=\"number\"\u003enumbers\u003c/ins\u003e. Many of the most\r\nfamiliar properties of numbers, which custom had led\r\npeople to regard as logically necessary, are in fact only\r\n\u003ca class=\"pagenum\" title=\"198\" id=\"Page_198\"\u003e \u003c/a\u003e\r\ndemonstrable by the step-by-step method, and fail to be\r\ntrue of infinite numbers. But so soon as we realise the\r\nnecessity of proving such properties by mathematical\r\ninduction, and the strictly limited scope of this method\r\nof proof, the supposed contradictions are seen to contradict,\r\nnot logic, but only our prejudices and mental habits.\u003c/p\u003e\r\n\u003cp\u003eThe property of being increased by the addition of\r\n1—\u003ci\u003ei.e.\u003c/i\u003e the property of non-reflexiveness—may serve to\r\nillustrate the limitations of mathematical induction. It\r\nis easy to prove that 0 is increased by the addition of 1,\r\nand that, if a given number is increased by the addition of\r\n1, so is the next number, \u003ci\u003ei.e.\u003c/i\u003e the number obtained by the\r\naddition of 1. It follows that each of the natural numbers\r\nis increased by the addition of 1. This follows generally\r\nfrom the general argument, and follows for each particular\r\ncase by a sufficient number of applications of the argument.\r\nWe first prove that 0 is not equal to 1; then,\r\nsince the property of being increased by 1 is hereditary,\r\nit follows that 1 is not equal to 2; hence it follows that\r\n2 is not equal to 3; if we wish to prove that 30,000 is\r\nnot equal to 30,001, we can do so by repeating this\r\nreasoning 30,000 times. But we cannot prove in this\r\nway that \u003cem\u003eall\u003c/em\u003e numbers are increased by the addition of 1;\r\nwe can only prove that this holds of the numbers attainable\r\nby successive additions of 1 starting from 0. The\r\nreflexive numbers, which lie beyond all those attainable\r\nin this way, are as a matter of fact not increased by the\r\naddition of 1.\u003c/p\u003e\r\n\u003cp\u003eThe two properties of reflexiveness and non-inductiveness,\r\nwhich we have considered as characteristics of\r\ninfinite numbers, have not so far been proved to be\r\nalways found together. It is known that all reflexive\r\nnumbers are non-inductive, but it is not known that all\r\nnon-inductive numbers are reflexive. Fallacious proofs\r\n\u003ca class=\"pagenum\" title=\"199\" id=\"Page_199\"\u003e \u003c/a\u003e\r\nof this proposition have been published by many writers,\r\nincluding myself, but up to the present no valid proof\r\nhas been discovered. The infinite numbers actually\r\nknown, however, are all reflexive as well as non-inductive;\r\nthus, in mathematical practice, if not in theory,\r\nthe two properties are always associated. For our purposes,\r\ntherefore, it will be convenient to ignore the bare\r\npossibility that there may be non-inductive non-reflexive\r\nnumbers, since all known numbers are either inductive\r\nor reflexive.\u003c/p\u003e\r\n\u003cp\u003eWhen infinite numbers are first introduced to people,\r\nthey are apt to refuse the name of numbers to them,\r\nbecause their behaviour is so different from that of finite\r\nnumbers that it seems a wilful misuse of terms to call\r\nthem numbers at all. In order to meet this feeling, we\r\nmust now turn to the logical basis of arithmetic, and\r\nconsider the logical definition of numbers.\u003c/p\u003e\r\n\u003cp\u003eThe logical definition of numbers, though it seems an\r\nessential support to the theory of infinite numbers, was\r\nin fact discovered independently and by a different man.\r\nThe theory of infinite numbers—that is to say, the arithmetical\r\nas opposed to the logical part of the theory—was\r\ndiscovered by Georg Cantor, and published by him in\r\n1882–3.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_51\" id=\"FNanchor_51\"\u003e[51]\u003c/a\u003e The definition of number was discovered\r\nabout the same time by a man whose great genius has\r\nnot received the recognition it deserves—I mean Gottlob\r\nFrege of Jena. His first work, \u003ccite lang=\"de\"\u003eBegriffsschrift\u003c/cite\u003e, published\r\nin 1879, contained the very important theory of hereditary\r\nproperties in a series to which I alluded in connection\r\nwith inductiveness. His definition of number is contained\r\nin his second work, published in 1884, and entitled\r\n\u003ccite lang=\"de\"\u003eDie Grundlagen der Arithmetik, eine logisch-mathematische\r\n\u003ca class=\"pagenum\" title=\"200\" id=\"Page_200\"\u003e \u003c/a\u003e\r\nUntersuchung über den Begriff der Zahl\u003c/cite\u003e.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_52\" id=\"FNanchor_52\"\u003e[52]\u003c/a\u003e It is with this\r\nbook that the logical theory of arithmetic begins, and it\r\nwill repay us to consider Frege\u0027s analysis in some detail.\u003c/p\u003e\r\n\u003cp\u003eFrege begins by noting the increased desire for logical\r\nstrictness in mathematical demonstrations which distinguishes\r\nmodern mathematicians from their predecessors,\r\nand points out that this must lead to a critical investigation\r\nof the definition of number. He proceeds to show\r\nthe inadequacy of previous philosophical theories, especially\r\nof the “synthetic \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e” theory of Kant and the\r\nempirical theory of Mill. This brings him to the question:\r\nWhat kind of object is it that number can properly\r\nbe ascribed to? He points out that physical things may\r\nbe regarded as one or many: for example, if a tree has\r\na thousand leaves, they may be taken altogether as constituting\r\nits foliage, which would count as one, not as a\r\nthousand; and \u003cem\u003eone\u003c/em\u003e pair of boots is the same object as\r\n\u003cem\u003etwo\u003c/em\u003e boots. It follows that physical things are not the\r\nsubjects of which number is properly predicated; for\r\nwhen we have discovered the proper subjects, the number\r\nto be ascribed must be unambiguous. This leads to a\r\ndiscussion of the very prevalent view that number is\r\nreally something psychological and subjective, a view\r\nwhich Frege emphatically rejects. “Number,” he says,\r\n“is as little an object of psychology or an outcome of\r\n\u003cins title=\"pscyhical\"\u003epsychical\u003c/ins\u003e processes as the North Sea…. The botanist\r\nwishes to state something which is just as much a fact\r\nwhen he gives the number of petals in a flower as when\r\nhe gives its colour. The one depends as little as the\r\nother upon our caprice. There is therefore a certain\r\n\u003ca class=\"pagenum\" title=\"201\" id=\"Page_201\"\u003e \u003c/a\u003e\r\nsimilarity between number and colour; but this does not\r\nconsist in the fact that both are sensibly perceptible in\r\nexternal things, but in the fact that both are objective”\r\n(p. 34).\u003c/p\u003e\r\n\u003cp\u003e“I distinguish the objective,” he continues, “from the\r\npalpable, the spatial, the actual. The earth\u0027s axis, the\r\ncentre of mass of the solar system, are objective, but I\r\nshould not call them actual, like the earth itself” (p. 35).\r\nHe concludes that number is neither spatial and physical,\r\nnor subjective, but non-sensible and objective. This\r\nconclusion is important, since it applies to all the subject-matter\r\nof mathematics and logic. Most philosophers\r\nhave thought that the physical and the mental between\r\nthem exhausted the world of being. Some have argued\r\nthat the objects of mathematics were obviously not subjective,\r\nand therefore must be physical and empirical;\r\nothers have argued that they were obviously not physical,\r\nand therefore must be subjective and mental. Both sides\r\nwere right in what they denied, and wrong in what they\r\nasserted; Frege has the merit of accepting both denials,\r\nand finding a third assertion by recognising the world\r\nof logic, which is neither mental nor physical.\u003c/p\u003e\r\n\u003cp\u003eThe fact is, as Frege points out, that no number, not\r\neven 1, is applicable to physical things, but only to general\r\nterms or descriptions, such as “man,” “satellite of the\r\nearth,” “satellite of Venus.” The general term “man”\r\nis applicable to a certain number of objects: there are in\r\nthe world so and so many men. The unity which philosophers\r\nrightly feel to be necessary for the assertion of a\r\nnumber is the unity of the general term, and it is the\r\ngeneral term which is the proper subject of number.\r\nAnd this applies equally when there is one object or none\r\nwhich falls under the general term. “Satellite of the\r\nearth” is a term only applicable to one object, namely,\r\n\u003ca class=\"pagenum\" title=\"202\" id=\"Page_202\"\u003e \u003c/a\u003e\r\nthe moon. But “one” is not a property of the moon\r\nitself, which may equally well be regarded as many\r\nmolecules: it is a property of the general term “earth\u0027s\r\nsatellite.” Similarly, 0 is a property of the general term\r\n“satellite of Venus,” because Venus has no satellite.\r\nHere at last we have an intelligible theory of the\r\nnumber 0. This was impossible if numbers applied to\r\nphysical objects, because obviously no physical object\r\ncould have the number 0. Thus, in seeking our definition\r\nof number we have arrived so far at the result that\r\nnumbers are properties of general terms or general descriptions,\r\nnot of physical things or of mental occurrences.\u003c/p\u003e\r\n\u003cp\u003eInstead of speaking of a general term, such as “man,”\r\nas the subject of which a number can be asserted, we may,\r\nwithout making any serious change, take the subject as\r\nthe class or collection of objects—\u003ci\u003ei.e.\u003c/i\u003e “mankind” in the\r\nabove instance—to which the general term in question is\r\napplicable. Two general terms, such as “man” and\r\n“featherless biped,” which are applicable to the same\r\ncollection of objects, will obviously have the same number\r\nof instances; thus the number depends upon the class,\r\nnot upon the selection of this or that general term to\r\ndescribe it, provided several general terms can be found\r\nto describe the same class. But some general term is\r\nalways necessary in order to describe a class. Even when\r\nthe terms are enumerated, as “this and that and the\r\nother,” the collection is constituted by the general property\r\nof being either this, or that, or the other, and only so\r\nacquires the unity which enables us to speak of it as \u003cem\u003eone\u003c/em\u003e\r\ncollection. And in the case of an infinite class, enumeration\r\nis impossible, so that description by a general characteristic\r\ncommon and peculiar to the members of the class\r\nis the only possible description. Here, as we see, the\r\ntheory of number to which Frege was led by purely logical\r\n\u003ca class=\"pagenum\" title=\"203\" id=\"Page_203\"\u003e \u003c/a\u003e\r\nconsiderations becomes of use in showing how infinite\r\nclasses can be amenable to number in spite of being incapable\r\nof enumeration.\u003c/p\u003e\r\n\u003cp\u003eFrege next asks the question: When do two collections\r\nhave the same number of terms? In ordinary life, we\r\ndecide this question by counting; but counting, as we saw,\r\nis impossible in the case of infinite collections, and is not\r\nlogically fundamental with finite collections. We want,\r\ntherefore, a different method of answering our question.\r\nAn illustration may help to make the method clear. I\r\ndo not know how many married men there are in England,\r\nbut I do know that the number is the same as the number\r\nof married women. The reason I know this is that the\r\nrelation of husband and wife relates one man to one\r\nwoman and one woman to one man. A relation of this\r\nsort is called a one-one relation. The relation of father\r\nto son is called a one-many relation, because a man can\r\nhave only one father but may have many sons; conversely,\r\nthe relation of son to father is called a many-one relation.\r\nBut the relation of husband to wife (in Christian countries)\r\nis called one-one, because a man cannot have more than\r\none wife, or a woman more than one husband. Now,\r\nwhenever there is a one-one relation between all the\r\nterms of one collection and all the terms of another\r\nseverally, as in the case of English husbands and English\r\nwives, the number of terms in the one collection is the\r\nsame as the number in the other; but when there is not\r\nsuch a relation, the number is different. This is the\r\nanswer to the question: When do two collections have\r\nthe same number of terms?\u003c/p\u003e\r\n\u003cp\u003eWe can now at last answer the question: What is\r\nmeant by the number of terms in a given collection?\r\nWhen there is a one-one relation between all the terms of\r\none collection and all the terms of another severally, we\r\n\u003ca class=\"pagenum\" title=\"204\" id=\"Page_204\"\u003e \u003c/a\u003e\r\nshall say that the two collections are “similar.” We\r\nhave just seen that two similar collections have the same\r\nnumber of terms. This leads us to define the number\r\nof a given collection as the class of all collections that are\r\nsimilar to it; that is to say, we set up the following\r\nformal definition:\u003c/p\u003e\r\n\u003cp\u003e“The number of terms in a given class” is defined as\r\nmeaning “the class of all classes that are similar to the\r\ngiven class.”\u003c/p\u003e\r\n\u003cp\u003eThis definition, as Frege (expressing it in slightly\r\ndifferent terms) showed, yields the usual arithmetical properties\r\nof numbers. It is applicable equally to finite and\r\ninfinite numbers, and it does not require the admission\r\nof some new and mysterious set of metaphysical entities.\r\nIt shows that it is not physical objects, but classes or the\r\ngeneral terms by which they are defined, of which\r\nnumbers can be asserted; and it applies to 0 and 1 without\r\nany of the difficulties which other theories find in\r\ndealing with these two special cases.\u003c/p\u003e\r\n\u003cp\u003eThe above definition is sure to produce, at first sight,\r\na feeling of oddity, which is liable to cause a certain dissatisfaction.\r\nIt defines the number 2, for instance, as the\r\nclass of all couples, and the number 3 as the class of all\r\ntriads. This does not \u003cem\u003eseem\u003c/em\u003e to be what we have hitherto\r\nbeen meaning when we spoke of 2 and 3, though it\r\nwould be difficult to say \u003cem\u003ewhat\u003c/em\u003e we had been meaning.\r\nThe answer to a feeling cannot be a logical argument,\r\nbut nevertheless the answer in this case is not without\r\nimportance. In the first place, it will be found that when\r\nan idea which has grown familiar as an unanalysed whole\r\nis first resolved accurately into its component parts—which\r\nis what we do when we define it—there is almost\r\nalways a feeling of unfamiliarity produced by the analysis,\r\nwhich tends to cause a protest against the definition. In\r\n\u003ca class=\"pagenum\" title=\"205\" id=\"Page_205\"\u003e \u003c/a\u003e\r\nthe second place, it may be admitted that the definition,\r\nlike all definitions, is to a certain extent arbitrary. In\r\nthe case of the small finite numbers, such as 2 and 3, it\r\nwould be possible to frame definitions more nearly in\r\naccordance with our unanalysed feeling of what we mean;\r\nbut the method of such definitions would lack uniformity,\r\nand would be found to fail sooner or later—at latest when\r\nwe reached infinite numbers.\u003c/p\u003e\r\n\u003cp\u003eIn the third place, the real desideratum about such a\r\ndefinition as that of number is not that it should represent\r\nas nearly as possible the ideas of those who have not\r\ngone through the analysis required in order to reach a\r\ndefinition, but that it should give us objects having the\r\nrequisite properties. Numbers, in fact, must satisfy the\r\nformulæ of arithmetic; any indubitable set of objects\r\nfulfilling this requirement may be called numbers. So\r\nfar, the simplest set known to fulfil this requirement is\r\nthe set introduced by the above definition. In comparison\r\nwith this merit, the question whether the objects to which\r\nthe definition applies are like or unlike the vague ideas\r\nof numbers entertained by those who cannot give a\r\ndefinition, is one of very little importance. All the\r\nimportant requirements are fulfilled by the above definition,\r\nand the sense of oddity which is at first unavoidable\r\nwill be found to wear off very quickly with the growth\r\nof familiarity.\u003c/p\u003e\r\n\u003cp\u003eThere is, however, a certain logical doctrine which may\r\nbe thought to form an objection to the above definition\r\nof numbers as classes of classes—I mean the doctrine\r\nthat there are no such objects as classes at all. It might\r\nbe thought that this doctrine would make havoc of a\r\ntheory which reduces numbers to classes, and of the\r\nmany other theories in which we have made use of classes.\r\nThis, however, would be a mistake: none of these theories\r\n\u003ca class=\"pagenum\" title=\"206\" id=\"Page_206\"\u003e \u003c/a\u003e\r\nare any the worse for the doctrine that classes are fictions.\r\nWhat the doctrine is, and why it is not destructive, I will\r\ntry briefly to explain.\u003c/p\u003e\r\n\u003cp\u003eOn account of certain rather complicated difficulties,\r\nculminating in definite contradictions, I was led to the\r\nview that nothing that can be said significantly about\r\nthings, \u003ci\u003ei.e.\u003c/i\u003e particulars, can be said significantly (\u003ci\u003ei.e.\u003c/i\u003e either\r\ntruly or falsely) about classes of things. That is to say,\r\nif, in any sentence in which a thing is mentioned, you\r\nsubstitute a class for the thing, you no longer have a\r\nsentence that has any meaning: the sentence is no longer\r\neither true or false, but a meaningless collection of words.\r\nAppearances to the contrary can be dispelled by a\r\nmoment\u0027s reflection. For example, in the sentence,\r\n“Adam is fond of apples,” you may substitute \u003cem\u003emankind\u003c/em\u003e,\r\nand say, “Mankind is fond of apples.” But obviously\r\nyou do not mean that there is one individual, called\r\n“mankind,” which munches apples: you mean that the\r\nseparate individuals who compose mankind are each\r\nseverally fond of apples.\u003c/p\u003e\r\n\u003cp\u003eNow, if nothing that can be said significantly about a\r\nthing can be said significantly about a class of things, it\r\nfollows that classes of things cannot have the same kind\r\nof reality as things have; for if they had, a class could\r\nbe substituted for a thing in a proposition predicating\r\nthe kind of reality which would be common to both.\r\nThis view is really consonant to common sense. In the\r\nthird or fourth century \u003cspan class=\"small-caps all-upper\"\u003eB.C.\u003c/span\u003e there lived a Chinese philosopher\r\nnamed Hui Tzŭ, who maintained that “a bay\r\nhorse and a dun cow are three; because taken separately\r\nthey are two, and taken together they are one: two and\r\none make three.”\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_53\" id=\"FNanchor_53\"\u003e[53]\u003c/a\u003e The author from whom I quote says\r\nthat Hui Tzŭ “was particularly fond of the quibbles\r\n\u003ca class=\"pagenum\" title=\"207\" id=\"Page_207\"\u003e \u003c/a\u003e\r\nwhich so delighted the sophists or unsound reasoners of\r\nancient Greece,” and this no doubt represents the judgment\r\nof common sense upon such arguments. Yet if\r\ncollections of things were things, his contention would be\r\nirrefragable. It is only because the bay horse and the\r\ndun cow taken together are not a new thing that we can\r\nescape the conclusion that there are three things wherever\r\nthere are two.\u003c/p\u003e\r\n\u003cp\u003eWhen it is admitted that classes are not things, the\r\nquestion arises: What do we mean by statements which\r\nare nominally about classes? Take such a statement as,\r\n“The class of people interested in mathematical logic is\r\nnot very numerous.” Obviously this reduces itself to,\r\n“Not very many people are interested in mathematical\r\nlogic.” For the sake of definiteness, let us substitute\r\nsome particular number, say 3, for “very many.” Then\r\nour statement is, “Not three people are interested in\r\nmathematical logic.” This may be expressed in the\r\nform: “If \u003ci\u003ex\u003c/i\u003e is interested in mathematical logic, and also\r\n\u003ci\u003ey\u003c/i\u003e is interested, and also \u003ci\u003ez\u003c/i\u003e is interested, then \u003ci\u003ex\u003c/i\u003e is identical\r\nwith \u003ci\u003ey\u003c/i\u003e, or \u003ci\u003ex\u003c/i\u003e is identical with \u003ci\u003ez\u003c/i\u003e, or \u003ci\u003ey\u003c/i\u003e is identical with \u003ci\u003ez\u003c/i\u003e.”\r\nHere there is no longer any reference at all to a “class.”\r\nIn some such way, all statements nominally about a class\r\ncan be reduced to statements about what follows from\r\nthe hypothesis of anything\u0027s having the defining property\r\nof the class. All that is wanted, therefore, in order to\r\nrender the \u003cem\u003everbal\u003c/em\u003e use of classes legitimate, is a uniform\r\nmethod of interpreting propositions in which such a use\r\noccurs, so as to obtain propositions in which there is no\r\nlonger any such use. The definition of such a method\r\nis a technical matter, which Dr Whitehead and I have\r\ndealt with elsewhere, and which we need not enter into\r\non this occasion.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_54\" id=\"FNanchor_54\"\u003e[54]\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"208\" id=\"Page_208\"\u003e \u003c/a\u003eIf the theory that classes are merely symbolic is accepted,\r\nit follows that numbers are not actual entities, but that\r\npropositions in which numbers verbally occur have not\r\nreally any constituents corresponding to numbers, but\r\nonly a certain logical form which is not a part of propositions\r\nhaving this form. This is in fact the case with\r\nall the apparent objects of logic and mathematics. Such\r\nwords as \u003cem\u003eor\u003c/em\u003e, \u003cem\u003enot\u003c/em\u003e, \u003cem\u003eif\u003c/em\u003e, \u003cem\u003ethere is\u003c/em\u003e, \u003cem\u003eidentity\u003c/em\u003e, \u003cem\u003egreater\u003c/em\u003e, \u003cem\u003eplus\u003c/em\u003e, \u003cem\u003enothing\u003c/em\u003e,\r\n\u003cem\u003eeverything\u003c/em\u003e, \u003cem\u003efunction\u003c/em\u003e, and so on, are not names of definite\r\nobjects, like “John” or “Jones,” but are words which\r\nrequire a context in order to have meaning. All of them\r\nare \u003cem\u003eformal\u003c/em\u003e, that is to say, their occurrence indicates a\r\ncertain form of proposition, not a certain constituent.\r\n“Logical constants,” in short, are not entities; the\r\nwords expressing them are not names, and cannot\r\nsignificantly be made into logical subjects except when it\r\nis the words themselves, as opposed to their meanings,\r\nthat are being discussed.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_55\" id=\"FNanchor_55\"\u003e[55]\u003c/a\u003e This fact has a very important\r\nbearing on all logic and philosophy, since it shows how\r\nthey differ from the special sciences. But the questions\r\nraised are so large and so difficult that it is impossible to\r\npursue them further on this occasion.\u003c/p\u003e\r\n\u003cp class=\"chapter-page\"\u003e\u003ca class=\"pagenum\" title=\"209\" id=\"Page_209\"\u003e \u003c/a\u003e\u003csmall\u003eLECTURE VIII\u003c/small\u003e\u003cbr\u003e\r\nON THE NOTION OF CAUSE,\r\nWITH APPLICATIONS TO THE\r\nFREE-WILL PROBLEM\u003c/p\u003e\r\n\u003ch2\u003e\u003csmall\u003e\u003ca class=\"pagenum\" title=\"211\" id=\"Page_211\"\u003e \u003c/a\u003e\u003ca id=\"Lecture_8\"\u003eLECTURE VIII\u003c/a\u003e\u003c/small\u003e\u003cbr\u003e\r\nON THE NOTION OF CAUSE, WITH APPLICATIONS TO THE FREE-WILL PROBLEM\u003c/h2\u003e\r\n\u003cp class=\"no-indent\"\u003e\u003cspan class=\"small-caps\"\u003eThe\u003c/span\u003e nature of philosophic analysis, as illustrated in our\r\nprevious lectures, can now be stated in general terms.\r\nWe start from a body of common knowledge, which\r\nconstitutes our data. On examination, the data are\r\nfound to be complex, rather vague, and largely interdependent\r\nlogically. By analysis we reduce them to propositions\r\nwhich are as nearly as possible simple and\r\nprecise, and we arrange them in deductive chains, in\r\nwhich a certain number of initial propositions form a\r\nlogical guarantee for all the rest. These initial propositions\r\nare \u003cem\u003epremisses\u003c/em\u003e for the body of knowledge in\r\nquestion. Premisses are thus quite different from data—they\r\nare simpler, more precise, and less infected with\r\nlogical redundancy. If the work of analysis has been\r\nperformed completely, they will be wholly free from\r\nlogical redundancy, wholly precise, and as simple as is\r\nlogically compatible with their leading to the given body\r\nof knowledge. The discovery of these premisses belongs\r\nto philosophy; but the work of deducing the body of\r\ncommon knowledge from them belongs to mathematics,\r\nif “mathematics” is interpreted in a somewhat liberal\r\nsense.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"212\" id=\"Page_212\"\u003e \u003c/a\u003eBut besides the logical analysis of the common\r\nknowledge which forms our data, there is the consideration\r\nof its degree of certainty. When we have arrived\r\nat its premisses, we may find that some of them seem\r\nopen to doubt, and we may find further that this doubt\r\nextends to those of our original data which depend upon\r\nthese doubtful premisses. In our \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003ethird lecture\u003c/a\u003e, for\r\nexample, we saw that the part of physics which depends\r\nupon testimony, and thus upon the existence of other\r\nminds than our own, does not seem so certain as the\r\npart which depends exclusively upon our own sense-data\r\nand the laws of logic. Similarly, it used to be felt that\r\nthe parts of geometry which depend upon the axiom of\r\nparallels have less certainty than the parts which are\r\nindependent of this premiss. We may say, generally,\r\nthat what commonly passes as knowledge is not all\r\nequally certain, and that, when analysis into premisses has\r\nbeen effected, the degree of certainty of any consequence\r\nof the premisses will depend upon that of the most\r\ndoubtful premiss employed in proving this consequence.\r\nThus analysis into premisses serves not only a logical\r\npurpose, but also the purpose of facilitating an estimate\r\nas to the degree of certainty to be attached to this or\r\nthat derivative belief. In view of the fallibility of all\r\nhuman beliefs, this service seems at least as important\r\nas the purely logical services rendered by philosophical\r\nanalysis.\u003c/p\u003e\r\n\u003cp\u003eIn the present lecture, I wish to apply the analytic\r\nmethod to the notion of “cause,” and to illustrate the\r\ndiscussion by applying it to the problem of free will.\r\nFor this purpose I shall inquire: I., what is meant by\r\na causal law; II., what is the evidence that causal laws\r\nhave held hitherto; III., what is the evidence that they\r\nwill continue to hold in the future; IV., how the\r\n\u003ca class=\"pagenum\" title=\"213\" id=\"Page_213\"\u003e \u003c/a\u003e\r\ncausality which is used in science differs from that of\r\ncommon sense and traditional philosophy; V., what new\r\nlight is thrown on the question of free will by our\r\nanalysis of the notion of “cause.”\u003c/p\u003e\r\n\u003cp\u003eI. By a “causal law” I mean any general proposition\r\nin virtue of which it is possible to infer the existence of\r\none thing or event from the existence of another or of\r\na number of others. If you hear thunder without having\r\nseen lightning, you infer that there nevertheless was a\r\nflash, because of the general proposition, “All thunder is\r\npreceded by lightning.” When Robinson Crusoe sees a\r\nfootprint, he infers a human being, and he might justify\r\nhis inference by the general proposition, “All marks in\r\nthe ground shaped like a human foot are subsequent to\r\na human being\u0027s standing where the marks are.” When\r\nwe see the sun set, we expect that it will rise again the\r\nnext day. When we hear a man speaking, we infer that\r\nhe has certain thoughts. All these inferences are due to\r\ncausal laws.\u003c/p\u003e\r\n\u003cp\u003eA causal law, we said, allows us to infer the existence\r\nof one \u003cem\u003ething\u003c/em\u003e (or \u003cem\u003eevent\u003c/em\u003e) from the existence of one or more\r\nothers. The word “thing” here is to be understood as\r\nonly applying to particulars, \u003ci\u003ei.e.\u003c/i\u003e as excluding such logical\r\nobjects as numbers or classes or abstract properties and\r\nrelations, and including sense-data, with whatever is\r\nlogically of the same type as sense-data.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_56\" id=\"FNanchor_56\"\u003e[56]\u003c/a\u003e In so far as a\r\ncausal law is directly verifiable, the thing inferred and the\r\nthing from which it is inferred must both be data,\r\nthough they need not both be data at the same time.\r\nIn fact, a causal law which is being used to extend our\r\nknowledge of existence must be applied to what, at the\r\n\u003ca class=\"pagenum\" title=\"214\" id=\"Page_214\"\u003e \u003c/a\u003e\r\nmoment, is not a datum; it is in the possibility of such\r\napplication that the practical utility of a causal law\r\nconsists. The important point, for our present purpose,\r\nhowever, is that what is inferred is a “thing,” a\r\n“particular,” an object having the kind of reality that\r\nbelongs to objects of sense, not an abstract object such\r\nas virtue or the square root of two.\u003c/p\u003e\r\n\u003cp\u003eBut we cannot become acquainted with a particular\r\nexcept by its being actually given. Hence the particular\r\ninferred by a causal law must be only \u003cem\u003edescribed\u003c/em\u003e with more\r\nor less exactness; it cannot be \u003cem\u003enamed\u003c/em\u003e until the inference\r\nis verified. Moreover, since the causal law is \u003cem\u003egeneral\u003c/em\u003e, and\r\ncapable of applying to many cases, the given particular\r\nfrom which we infer must allow the inference in virtue\r\nof some general characteristic, not in virtue of its being\r\njust the particular that it is. This is obvious in all our\r\nprevious instances: we infer the unperceived lightning\r\nfrom the thunder, not in virtue of any peculiarity of the\r\nthunder, but in virtue of its resemblance to other claps\r\nof thunder. Thus a causal law must state that the\r\nexistence of a thing of a certain sort (or of a number of\r\nthings of a number of assigned sorts) implies the existence\r\nof another thing having a relation to the first which\r\nremains invariable so long as the first is of the kind in\r\nquestion.\u003c/p\u003e\r\n\u003cp\u003eIt is to be observed that what is constant in a causal\r\nlaw is not the object or objects given, nor yet the object\r\ninferred, both of which may vary within wide limits, but\r\nthe \u003cem\u003erelation\u003c/em\u003e between what is given and what is inferred.\r\nThe principle, “same cause, same effect,” which is sometimes\r\nsaid to be the principle of causality, is much\r\nnarrower in its scope than the principle which really\r\noccurs in science; indeed, if strictly interpreted, it has\r\nno scope at all, since the “same” cause never recurs\r\n\u003ca class=\"pagenum\" title=\"215\" id=\"Page_215\"\u003e \u003c/a\u003e\r\nexactly. We shall return to this point at a later stage of\r\nthe discussion.\u003c/p\u003e\r\n\u003cp\u003eThe particular which is inferred may be uniquely\r\ndetermined by the causal law, or may be only described\r\nin such general terms that many different particulars\r\nmight satisfy the description. This depends upon\r\nwhether the constant relation affirmed by the causal law\r\nis one which only one term can have to the data, or one\r\nwhich many terms may have. If many terms may have\r\nthe relation in question, science will not be satisfied until\r\nit has found some more stringent law, which will enable\r\nus to determine the inferred things uniquely.\u003c/p\u003e\r\n\u003cp\u003eSince all known things are in time, a causal law must\r\ntake account of temporal relations. It will be part of the\r\ncausal law to state a relation of succession or coexistence\r\nbetween the thing given and the thing inferred. When\r\nwe hear thunder and infer that there was lightning, the\r\nlaw states that the thing inferred is earlier than the thing\r\ngiven. Conversely, when we see lightning and wait\r\nexpectantly for the thunder, the law states that the thing\r\ngiven is earlier than the thing inferred. When we infer a\r\nman\u0027s thoughts from his words, the law states that the two\r\nare (at least approximately) simultaneous.\u003c/p\u003e\r\n\u003cp\u003eIf a causal law is to achieve the precision at which\r\nscience aims, it must not be content with a vague \u003cem\u003eearlier\u003c/em\u003e\r\nor \u003cem\u003elater\u003c/em\u003e, but must state how much earlier or how much\r\nlater. That is to say, the time-relation between the thing\r\ngiven and the thing inferred ought to be capable of exact\r\nstatement; and usually the inference to be drawn is\r\ndifferent according to the length and direction of the\r\ninterval. “A quarter of an hour ago this man was alive;\r\nan hour hence he will be cold.” Such a statement\r\ninvolves two causal laws, one inferring from a datum\r\nsomething which existed a quarter of \u003cins title=\"a\"\u003ean\u003c/ins\u003e hour ago, the\r\n\u003ca class=\"pagenum\" title=\"216\" id=\"Page_216\"\u003e \u003c/a\u003e\r\nother inferring from the same datum something which\r\nwill exist an hour hence.\u003c/p\u003e\r\n\u003cp\u003eOften a causal law involves not one datum, but many,\r\nwhich need not be all simultaneous with each other,\r\nthough their time-relations must be given. The general\r\nscheme of a causal law will be as follows:\u003c/p\u003e\r\n\u003cp\u003e“Whenever things occur in certain relations to each\r\nother (among which their time-relations must be included),\r\nthen a thing having a fixed relation to these things will\r\noccur at a date fixed relatively to their dates.”\u003c/p\u003e\r\n\u003cp\u003eThe things given will not, in practice, be things that\r\nonly exist for an instant, for such things, if there are any,\r\ncan never be data. The things given will each occupy\r\nsome finite time. They may be not static things, but\r\nprocesses, especially motions. We have considered in an\r\nearlier lecture the sense in which a motion may be a\r\ndatum, and need not now recur to this topic.\u003c/p\u003e\r\n\u003cp\u003eIt is not essential to a causal law that the object\r\ninferred should be later than some or all of the data.\r\nIt may equally well be earlier or at the same time. The\r\nonly thing essential is that the law should be such as to\r\nenable us to infer the existence of an object which we can\r\nmore or less accurately describe in terms of the data.\u003c/p\u003e\r\n\u003cp\u003eII. I come now to our second question, namely: What\r\nis the nature of the evidence that causal laws have held\r\nhitherto, at least in the observed portions of the past?\r\nThis question must not be confused with the further\r\nquestion: Does this evidence warrant us in assuming the\r\ntruth of causal laws in the future and in unobserved\r\nportions of the past? For the present, I am only asking\r\nwhat are the grounds which lead to a belief in causal laws,\r\nnot whether these grounds are adequate to support the\r\nbelief in universal causation.\u003c/p\u003e\r\n\u003cp\u003eThe first step is the discovery of approximate unanalysed\r\n\u003ca class=\"pagenum\" title=\"217\" id=\"Page_217\"\u003e \u003c/a\u003e\r\nuniformities of sequence or coexistence. After lightning\r\ncomes thunder, after a blow received comes pain, after\r\napproaching a fire comes warmth; again, there are uniformities\r\nof coexistence, for example between touch and\r\nsight, between certain sensations in the throat and the\r\nsound of one\u0027s own voice, and so on. Every such\r\nuniformity of sequence or coexistence, after it has been\r\nexperienced a certain number of times, is followed by an\r\nexpectation that it will be repeated on future occasions,\r\n\u003ci\u003ei.e.\u003c/i\u003e that where one of the correlated events is found, the\r\nother will be found also. The connection of experienced\r\npast uniformity with expectation as to the future is just\r\none of those uniformities of sequence which we have\r\nobserved to be true hitherto. This affords a psychological\r\naccount of what may be called the animal belief in\r\ncausation, because it is something which can be observed\r\nin horses and dogs, and is rather a habit of acting than\r\na real belief. So far, we have merely repeated Hume,\r\nwho carried the discussion of cause up to this point,\r\nbut did not, apparently, perceive how much remained\r\nto be said.\u003c/p\u003e\r\n\u003cp\u003eIs there, in fact, any characteristic, such as might be\r\ncalled causality or uniformity, which is found to hold\r\nthroughout the observed past? And if so, how is it to\r\nbe stated?\u003c/p\u003e\r\n\u003cp\u003eThe particular uniformities which we mentioned before,\r\nsuch as lightning being followed by thunder, are not\r\nfound to be free from exceptions. We sometimes see\r\nlightning without hearing thunder; and although, in\r\nsuch a case, we suppose that thunder might have been\r\nheard if we had been nearer to the lightning, that is a\r\nsupposition based on theory, and therefore incapable of\r\nbeing invoked to support the theory. What does seem,\r\nhowever, to be shown by scientific experience is this:\r\n\u003ca class=\"pagenum\" title=\"218\" id=\"Page_218\"\u003e \u003c/a\u003e\r\nthat where an observed uniformity fails, some wider\r\nuniformity can be found, embracing more circumstances,\r\nand subsuming both the successes and the failures of the\r\nprevious uniformity. Unsupported bodies in air fall,\r\nunless they are balloons or aeroplanes; but the principles\r\nof mechanics give uniformities which apply to balloons\r\nand aeroplanes just as accurately as to bodies that fall.\r\nThere is much that is hypothetical and more or less\r\nartificial in the uniformities affirmed by mechanics,\r\nbecause, when they cannot otherwise be made applicable,\r\nunobserved bodies are inferred in order to account for\r\nobserved peculiarities. Still, it is an empirical fact that\r\nit is possible to preserve the laws by assuming such\r\nbodies, and that they never have to be assumed in circumstances\r\nin which they ought to be observable. Thus\r\nthe empirical verification of mechanical laws may be\r\nadmitted, although we must also admit that it is less\r\ncomplete and triumphant than is sometimes supposed.\u003c/p\u003e\r\n\u003cp\u003eAssuming now, what must be admitted to be doubtful,\r\nthat the whole of the past has proceeded according to\r\ninvariable laws, what can we say as to the nature of these\r\nlaws? They will not be of the simple type which asserts\r\nthat the same cause always produces the same effect.\r\nWe may take the law of gravitation as a sample of the\r\nkind of law that appears to be verified without exception.\r\nIn order to state this law in a form which observation\r\ncan confirm, we will confine it to the solar system. It\r\nthen states that the motions of planets and their satellites\r\nhave at every instant an acceleration compounded of\r\naccelerations towards all the other bodies in the solar\r\nsystem, proportional to the masses of those bodies and\r\ninversely proportional to the squares of their distances.\r\nIn virtue of this law, given the state of the solar system\r\nthroughout any finite time, however short, its state at all\r\n\u003ca class=\"pagenum\" title=\"219\" id=\"Page_219\"\u003e \u003c/a\u003e\r\nearlier and later times is determinate except in so far as\r\nother forces than gravitation or other bodies than those\r\nin the solar system have to be taken into consideration.\r\nBut other forces, so far as science can discover, appear\r\nto be equally regular, and equally capable of being\r\nsummed up in single causal laws. If the mechanical\r\naccount of matter were complete, the whole physical\r\nhistory of the universe, past and future, could be inferred\r\nfrom a sufficient number of data concerning an assigned\r\nfinite time, however short.\u003c/p\u003e\r\n\u003cp\u003eIn the mental world, the evidence for the universality\r\nof causal laws is less complete than in the physical world.\r\nPsychology cannot boast of any triumph comparable to\r\ngravitational astronomy. Nevertheless, the evidence is\r\nnot very greatly less than in the physical world. The\r\ncrude and approximate causal laws from which science\r\nstarts are just as easy to discover in the mental sphere as\r\nin the physical. In the world of sense, there are to begin\r\nwith the correlations of sight and touch and so on, and\r\nthe facts which lead us to connect various kinds of sensations\r\nwith eyes, ears, nose, tongue, etc. Then there\r\nare such facts as that our body moves in answer to our\r\nvolitions. Exceptions exist, but are capable of being\r\nexplained as easily as the exceptions to the rule that unsupported\r\nbodies in air fall. There is, in fact, just such\r\na degree of evidence for causal laws in psychology as will\r\nwarrant the psychologist in assuming them as a matter\r\nof course, though not such a degree as will suffice to\r\nremove all doubt from the mind of a sceptical inquirer.\r\nIt should be observed that causal laws in which the given\r\nterm is mental and the inferred term physical, or \u003ci lang=\"la\"\u003evice\r\nversa\u003c/i\u003e, are at least as easy to discover as causal laws in\r\nwhich both terms are mental.\u003c/p\u003e\r\n\u003cp\u003eIt will be noticed that, although we have spoken of\r\n\u003ca class=\"pagenum\" title=\"220\" id=\"Page_220\"\u003e \u003c/a\u003e\r\ncausal laws, we have not hitherto introduced the word\r\n“cause.” At this stage, it will be well to say a few words\r\non legitimate and illegitimate uses of this word. The word\r\n“cause,” in the scientific account of the world, belongs\r\nonly to the early stages, in which small preliminary, approximate\r\ngeneralisations are being ascertained with a\r\nview to subsequent larger and more invariable laws. We\r\nmay say, “Arsenic causes death,” so long as we are\r\nignorant of the precise process by which the result is\r\nbrought about. But in a sufficiently advanced science,\r\nthe word “cause” will not occur in any statement of\r\ninvariable laws. There is, however, a somewhat rough\r\nand loose use of the word “cause” which may be preserved.\r\nThe approximate uniformities which lead to its\r\npre-scientific employment may turn out to be true in all\r\nbut very rare and exceptional circumstances, perhaps in\r\nall circumstances that actually occur. In such cases, it is\r\nconvenient to be able to speak of the antecedent event\r\nas the “cause” and the subsequent event as the “effect.”\r\nIn this sense, provided it is realised that the sequence is\r\nnot necessary and may have exceptions, it is still possible\r\nto employ the words “cause” and “effect.” It is in this\r\nsense, and in this sense only, that we shall intend the\r\nwords when we speak of one particular event “causing”\r\nanother particular event, as we must sometimes do if we\r\nare to avoid intolerable circumlocution.\u003c/p\u003e\r\n\u003cp\u003eIII. We come now to our third question, namely:\r\nWhat reason can be given for believing that causal laws\r\nwill hold in future, or that they have held in unobserved\r\nportions of the past?\u003c/p\u003e\r\n\u003cp\u003eWhat we have said so far is that there have been\r\nhitherto certain observed causal laws, and that all the\r\nempirical evidence we possess is compatible with the\r\nview that everything, both mental and physical, so far\r\n\u003ca class=\"pagenum\" title=\"221\" id=\"Page_221\"\u003e \u003c/a\u003e\r\nas our observation has extended, has happened in accordance\r\nwith causal laws. The law of universal causation,\r\nsuggested by these facts, may be enunciated as follows:\u003c/p\u003e\r\n\u003cp\u003e“There are such invariable relations between different\r\nevents at the same or different times that, given the state\r\nof the whole universe throughout any finite time, however\r\nshort, every previous and subsequent event can theoretically\r\nbe determined as a function of the given events\r\nduring that time.”\u003c/p\u003e\r\n\u003cp\u003eHave we any reason to believe this universal law?\r\nOr, to ask a more modest question, have we any reason\r\nto believe that a particular causal law, such as the law of\r\ngravitation, will continue to hold in the future?\u003c/p\u003e\r\n\u003cp\u003eAmong observed causal laws is this, that observation of\r\nuniformities is followed by expectation of their recurrence.\r\nA horse who has been driven always along a certain road\r\nexpects to be driven along that road again; a dog who is\r\nalways fed at a certain hour expects food at that hour\r\nand not at any other. Such expectations, as Hume\r\npointed out, explain only too well the common-sense\r\nbelief in uniformities of sequence, but they afford absolutely\r\nno logical ground for beliefs as to the future,\r\nnot even for the belief that we shall continue to expect\r\nthe continuation of experienced uniformities, for that is\r\nprecisely one of those causal laws for which a ground has\r\nto be sought. If Hume\u0027s account of causation is the\r\nlast word, we have not only no reason to suppose that\r\nthe sun will rise to-morrow, but no reason to suppose\r\nthat five minutes hence we shall still expect it to rise\r\nto-morrow.\u003c/p\u003e\r\n\u003cp\u003eIt may, of course, be said that all inferences as to the\r\nfuture are in fact invalid, and I do not see how such\r\na view could be disproved. But, while admitting the\r\nlegitimacy of such a view, we may nevertheless inquire:\r\n\u003ca class=\"pagenum\" title=\"222\" id=\"Page_222\"\u003e \u003c/a\u003e\r\nIf inferences as to the future \u003cem\u003eare\u003c/em\u003e valid, what principle\r\nmust be involved in making them?\u003c/p\u003e\r\n\u003cp\u003eThe principle involved is the principle of induction,\r\nwhich, if it is true, must be an \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e logical law, not\r\ncapable of being proved or disproved by experience. It\r\nis a difficult question how this principle ought to be\r\nformulated; but if it is to warrant the inferences which\r\nwe wish to make by its means, it must lead to the following\r\nproposition: “If, in a great number of instances, a\r\nthing of a certain kind is associated in a certain way with\r\na thing of a certain other kind, it is probable that a thing\r\nof the one kind is always similarly associated with a thing\r\nof the other kind; and as the number of instances\r\nincreases, the probability approaches indefinitely near to\r\ncertainty.” It may well be questioned whether this\r\nproposition is true; but if we admit it, we can infer that\r\nany characteristic of the whole of the observed past is\r\nlikely to apply to the future and to the unobserved past.\r\nThis proposition, therefore, if it is true, will warrant the\r\ninference that causal laws probably hold at all times, future\r\nas well as past; but without this principle, the observed\r\ncases of the truth of causal laws afford no presumption as\r\nto the unobserved cases, and therefore the existence of a\r\nthing not directly observed can never be validly inferred.\u003c/p\u003e\r\n\u003cp\u003eIt is thus the principle of induction, rather than the\r\nlaw of causality, which is at the bottom of all inferences\r\nas to the existence of things not immediately given.\r\nWith the principle of induction, all that is wanted for\r\nsuch inferences can be proved; without it, all such\r\ninferences are invalid. This principle has not received\r\nthe attention which its great importance deserves. Those\r\nwho were interested in deductive logic naturally enough\r\nignored it, while those who emphasised the scope of\r\ninduction wished to maintain that all logic is empirical,\r\n\u003ca class=\"pagenum\" title=\"223\" id=\"Page_223\"\u003e \u003c/a\u003e\r\nand therefore could not be expected to realise that\r\ninduction itself, their own darling, required a logical\r\nprinciple which obviously could not be proved inductively,\r\nand must therefore be \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e if it could be\r\nknown at all.\u003c/p\u003e\r\n\u003cp\u003eThe view that the law of causality itself is \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e\r\ncannot, I think, be maintained by anyone who realises\r\nwhat a complicated principle it is. In the form which\r\nstates that “every event has a cause” it looks simple;\r\nbut on examination, “cause” is merged in “causal law,”\r\nand the definition of a “causal law” is found to be far\r\nfrom simple. There must necessarily be \u003cem\u003esome\u003c/em\u003e \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e\r\nprinciple involved in inference from the existence of one\r\nthing to that of another, if such inference is ever valid;\r\nbut it would appear from the above analysis that the\r\nprinciple in question is induction, not causality. Whether\r\ninferences from past to future are valid depends wholly,\r\nif our discussion has been sound, upon the inductive\r\nprinciple: if it is true, such inferences are valid, and if\r\nit is false, they are invalid.\u003c/p\u003e\r\n\u003cp\u003eIV. I come now to the question how the conception of\r\ncausal laws which we have arrived at is related to the\r\ntraditional conception of cause as it occurs in philosophy\r\nand common sense.\u003c/p\u003e\r\n\u003cp\u003eHistorically, the notion of cause has been bound up\r\nwith that of human volition. The typical cause would\r\nbe the fiat of a king. The cause is supposed to be\r\n“active,” the effect “passive.” From this it is easy to\r\npass on to the suggestion that a “true” cause must\r\ncontain some prevision of the effect; hence the effect\r\nbecomes the “end” at which the cause aims, and teleology\r\nreplaces causation in the explanation of nature.\r\nBut all such ideas, as applied to physics, are mere\r\nanthropomorphic superstitions. It is as a reaction against\r\n\u003ca class=\"pagenum\" title=\"224\" id=\"Page_224\"\u003e \u003c/a\u003e\r\nthese errors that Mach and others have urged a purely\r\n“descriptive” view of physics: physics, they say, does\r\nnot aim at telling us “why” things happen, but only\r\n“how” they happen. And if the question “why?”\r\nmeans anything more than the search for a general law\r\naccording to which a phenomenon occurs, then it is\r\ncertainly the case that this question cannot be answered\r\nin physics and ought not to be asked. In this sense, the\r\ndescriptive view is indubitably in the right. But in\r\nusing causal laws to support inferences from the observed\r\nto the unobserved, physics ceases to be \u003cem\u003epurely\u003c/em\u003e descriptive,\r\nand it is these laws which give the scientifically useful\r\npart of the traditional notion of “cause.” There is\r\ntherefore \u003cem\u003esomething\u003c/em\u003e to preserve in this notion, though it\r\nis a very tiny part of what is commonly assumed in\r\northodox metaphysics.\u003c/p\u003e\r\n\u003cp\u003eIn order to understand the difference between the kind\r\nof cause which science uses and the kind which we\r\nnaturally imagine, it is necessary to shut out, by an effort,\r\neverything that differentiates between past and future.\r\nThis is an extraordinarily difficult thing to do, because\r\nour mental life is so intimately bound up with difference.\r\nNot only do memory and hope make a difference in our\r\nfeelings as regards past and future, but almost our whole\r\nvocabulary is filled with the idea of activity, of things\r\ndone now for the sake of their future effects. All\r\ntransitive verbs involve the notion of cause as activity,\r\nand would have to be replaced by some cumbrous periphrasis\r\nbefore this notion could be eliminated.\u003c/p\u003e\r\n\u003cp\u003eConsider such a statement as, “Brutus killed Cæsar.”\r\nOn another occasion, Brutus and Cæsar might engage\r\nour attention, but for the present it is the killing that\r\nwe have to study. We may say that to kill a person is\r\nto cause his death intentionally. This means that desire\r\n\u003ca class=\"pagenum\" title=\"225\" id=\"Page_225\"\u003e \u003c/a\u003e\r\nfor a person\u0027s death causes a certain act, because it is\r\nbelieved that that act will cause the person\u0027s death; or\r\nmore accurately, the desire and the belief jointly cause\r\nthe act. Brutus desires that Cæsar should be dead, and\r\nbelieves that he will be dead if he is stabbed; Brutus\r\ntherefore stabs him, and the stab causes Cæsar\u0027s death,\r\nas Brutus expected it would. Every act which realises\r\na purpose involves two causal steps in this way: C is\r\ndesired, and it is believed (truly if the purpose is achieved)\r\nthat B will cause C; the desire and the belief together\r\ncause B, which in turn causes C. Thus we have first A,\r\nwhich is a desire for C and a belief that B (an act) will\r\ncause C; then we have B, the act caused by A, and\r\nbelieved to be a cause of C; then, if the belief was\r\ncorrect, we have C, caused by B, and if the belief was\r\nincorrect we have disappointment. Regarded purely\r\nscientifically, this series A, B, C may equally well be\r\nconsidered in the inverse order, as they would be at a\r\ncoroner\u0027s inquest. But from the point of view of Brutus,\r\nthe desire, which comes at the beginning, is what makes\r\nthe whole series interesting. We feel that if his desires\r\nhad been different, the effects which he in fact produced\r\nwould not have occurred. This is true, and gives him\r\na sense of power and freedom. It is equally true that\r\nif the effects had not occurred, his desires would have\r\nbeen different, since being what they were the effects\r\ndid occur. Thus the desires are determined by their\r\nconsequences just as much as the consequences by the\r\ndesires; but as we cannot (in general) know in advance\r\nthe consequences of our desires without knowing our\r\ndesires, this form of inference is uninteresting as applied\r\nto our own acts, though quite vital as applied to those\r\nof others.\u003c/p\u003e\r\n\u003cp\u003eA cause, considered scientifically, has none of that\r\n\u003ca class=\"pagenum\" title=\"226\" id=\"Page_226\"\u003e \u003c/a\u003e\r\nanalogy with volition which makes us imagine that the\r\neffect is \u003cem\u003ecompelled\u003c/em\u003e by it. A cause is an event or group\r\nof events, of some known general character, and having\r\na known relation to some other event, called the effect;\r\nthe relation being of such a kind that only one event, or\r\nat any rate only one well-defined sort of event, can have\r\nthe relation to a given cause. It is customary only to\r\ngive the name “effect” to an event which is later than\r\nthe cause, but there is no kind of reason for this restriction.\r\nWe shall do better to allow the effect to be before\r\nthe cause or simultaneous with it, because nothing of\r\nany scientific importance depends upon its being after\r\nthe cause.\u003c/p\u003e\r\n\u003cp\u003eIf the inference from cause to effect is to be indubitable,\r\nit seems that the cause can hardly stop short of\r\nthe whole universe. So long as anything is left out,\r\nsomething may be left out which alters the expected\r\nresult. But for practical and scientific purposes, phenomena\r\ncan be collected into groups which are causally\r\nself-contained, or nearly so. In the common notion of\r\ncausation, the cause is a single event—we say the lightning\r\ncauses the thunder, and so on. But it is difficult to\r\nknow what we mean by a single event; and it generally\r\nappears that, in order to have anything approaching\r\ncertainty concerning the effect, it is necessary to include\r\nmany more circumstances in the cause than unscientific\r\ncommon sense would suppose. But often a probable\r\ncausal connection, where the cause is fairly simple, is of\r\nmore practical importance than a more indubitable connection\r\nin which the cause is so complex as to be hard\r\nto ascertain.\u003c/p\u003e\r\n\u003cp\u003eTo sum up: the strict, certain, universal law of causation\r\nwhich philosophers advocate is an ideal, possibly\r\ntrue, but not \u003cem\u003eknown\u003c/em\u003e to be true in virtue of any available\r\n\u003ca class=\"pagenum\" title=\"227\" id=\"Page_227\"\u003e \u003c/a\u003e\r\nevidence. What is actually known, as a matter of\r\nempirical science, is that certain constant relations are\r\nobserved to hold between the members of a group of\r\nevents at certain times, and that when such relations\r\nfail, as they sometimes do, it is usually possible to discover\r\na new, more constant relation by enlarging the\r\ngroup. Any such constant relation between events of\r\nspecified kinds with given intervals of time between them\r\nis a “causal law.” But all causal laws are liable to\r\nexceptions, if the cause is less than the whole state of\r\nthe universe; we believe, on the basis of a good deal\r\nof experience, that such exceptions can be dealt with by\r\nenlarging the group we call the cause, but this belief,\r\nwherever it is still unverified, ought not to be regarded\r\nas certain, but only as suggesting a direction for further\r\ninquiry.\u003c/p\u003e\r\n\u003cp\u003eA very common causal group consists of volitions\r\nand the consequent bodily acts, though exceptions arise\r\n(for example) through sudden paralysis. Another very\r\nfrequent connection (though here the exceptions are\r\nmuch more numerous) is between a bodily act and the\r\nrealisation of the purpose which led to the act. These\r\nconnections are patent, whereas the causes of desires are\r\nmore obscure. Thus it is natural to begin causal series\r\nwith desires, to suppose that all causes are analogous to\r\ndesires, and that desires themselves arise spontaneously.\r\nSuch a view, however, is not one which any serious\r\npsychologist would maintain. But this brings us to the\r\nquestion of the application of our analysis of cause to the\r\nproblem of free will.\u003c/p\u003e\r\n\u003cp\u003eV. The problem of free will is so intimately bound up\r\nwith the analysis of causation that, old as it is, we need\r\nnot despair of obtaining new light on it by the help of\r\nnew views on the notion of cause. The free-will problem\r\n\u003ca class=\"pagenum\" title=\"228\" id=\"Page_228\"\u003e \u003c/a\u003e\r\nhas, at one time or another, stirred men\u0027s passions profoundly,\r\nand the fear that the will might not be free has\r\nbeen to some men a source of great unhappiness. I\r\nbelieve that, under the influence of a cool analysis, the\r\ndoubtful questions involved will be found to have no such\r\nemotional importance as is sometimes thought, since the\r\ndisagreeable consequences supposed to flow from a denial\r\nof free will do not flow from this denial in any form in\r\nwhich there is reason to make it. It is not, however, on\r\nthis account chiefly that I wish to discuss this problem,\r\nbut rather because it affords a good example of the\r\nclarifying effect of analysis and of the interminable controversies\r\nwhich may result from its neglect.\u003c/p\u003e\r\n\u003cp\u003eLet us first try to discover what it is we really desire\r\nwhen we desire free will. Some of our reasons for desiring\r\nfree will are profound, some trivial. To begin with\r\nthe former: we do not wish to feel ourselves in the\r\nhands of fate, so that, however much we may desire to\r\nwill one thing, we may nevertheless be compelled by an\r\noutside force to will another. We do not wish to think\r\nthat, however much we may desire to act well, heredity\r\nand surroundings may force us into acting ill. We wish\r\nto feel that, in cases of doubt, our choice is momentous\r\nand lies within our power. Besides these desires, which\r\nare worthy of all respect, we have, however, others not\r\nso respectable, which equally make us desire free will.\r\nWe do not like to think that other people, if they knew\r\nenough, could predict our actions, though we know that\r\nwe can often predict those of other people, especially if\r\nthey are elderly. Much as we esteem the old gentleman\r\nwho is our neighbour in the country, we know that when\r\ngrouse are mentioned he will tell the story of the grouse\r\nin the gun-room. But we ourselves are not so mechanical:\r\nwe never tell an anecdote to the same person twice,\r\n\u003ca class=\"pagenum\" title=\"229\" id=\"Page_229\"\u003e \u003c/a\u003e\r\nor even once unless he is sure to enjoy it; although we\r\nonce met (say) Bismarck, we are quite capable of hearing\r\nhim mentioned without relating the occasion when we\r\nmet him. In this sense, everybody thinks that he himself\r\nhas free will, though he knows that no one else has.\r\nThe desire for this kind of free will seems to be no better\r\nthan a form of vanity. I do not believe that this desire\r\ncan be gratified with any certainty; but the other, more\r\nrespectable desires are, I believe, not inconsistent with\r\nany tenable form of determinism.\u003c/p\u003e\r\n\u003cp\u003eWe have thus two questions to consider: (1) Are\r\nhuman actions theoretically predictable from a sufficient\r\nnumber of antecedents? (2) Are human actions subject\r\nto an external compulsion? The two questions, as I\r\nshall try to show, are entirely distinct, and we may\r\nanswer the first in the affirmative without therefore being\r\nforced to give an affirmative answer to the second.\u003c/p\u003e\r\n\u003cp\u003e(1) \u003cem\u003eAre human actions theoretically predictable from a\r\nsufficient number of antecedents?\u003c/em\u003e Let us first endeavour\r\nto give precision to this question. We may state the\r\nquestion thus: Is there some constant relation between\r\nan act and a certain number of earlier events, such that,\r\nwhen the earlier events are given, only one act, or at\r\nmost only acts with some well-marked character, can\r\nhave this relation to the earlier events? If this is the\r\ncase, then, as soon as the earlier events are known, it is\r\ntheoretically possible to predict either the precise act,\r\nor at least the character necessary to its fulfilling the\r\nconstant relation.\u003c/p\u003e\r\n\u003cp\u003eTo this question, a negative answer has been given by\r\nBergson, in a form which calls in question the general\r\napplicability of the law of causation. He maintains that\r\nevery event, and more particularly every mental event,\r\nembodies so much of the past that it could not possibly\r\n\u003ca class=\"pagenum\" title=\"230\" id=\"Page_230\"\u003e \u003c/a\u003e\r\nhave occurred at any earlier time, and is therefore necessarily\r\nquite different from all previous and subsequent\r\nevents. If, for example, I read a certain poem many\r\ntimes, my experience on each occasion is modified by the\r\nprevious readings, and my emotions are never repeated\r\nexactly. The principle of causation, according to him,\r\nasserts that the same cause, if repeated, will produce the\r\nsame effect. But owing to memory, he contends, this\r\nprinciple does not apply to mental events. What is\r\napparently the same cause, if repeated, is modified by the\r\nmere fact of repetition, and cannot produce the same\r\neffect. He infers that every mental event is a genuine\r\nnovelty, not predictable from the past, because the past\r\ncontains nothing exactly like it by which we could imagine\r\nit. And on this ground he regards the freedom of the\r\nwill as unassailable.\u003c/p\u003e\r\n\u003cp\u003eBergson\u0027s contention has undoubtedly a great deal of\r\ntruth, and I have no wish to deny its importance. But\r\nI do not think its consequences are quite what he believes\r\nthem to be. It is not necessary for the determinist to\r\nmaintain that he can foresee the whole particularity of\r\nthe act which will be performed. If he could foresee\r\nthat A was going to murder B, his foresight would not\r\nbe invalidated by the fact that he could not know all the\r\ninfinite complexity of A\u0027s state of mind in committing\r\nthe murder, nor whether the murder was to be performed\r\nwith a knife or with a revolver. If the \u003cem\u003ekind\u003c/em\u003e of act which\r\nwill be performed can be foreseen within narrow limits,\r\nit is of little practical interest that there are fine shades\r\nwhich cannot be foreseen. No doubt every time the\r\nstory of the grouse in the gun-room is told, there will\r\nbe slight differences due to increasing habitualness, but\r\nthey do not invalidate the prediction that the story will\r\nbe told. And there is nothing in Bergson\u0027s argument\r\n\u003ca class=\"pagenum\" title=\"231\" id=\"Page_231\"\u003e \u003c/a\u003e\r\nto show that we can never predict what \u003cem\u003ekind\u003c/em\u003e of act will\r\nbe performed.\u003c/p\u003e\r\n\u003cp\u003eAgain, his statement of the law of causation is inadequate.\r\nThe law does not state merely that, if the \u003cem\u003esame\u003c/em\u003e cause is\r\nrepeated, the \u003cem\u003esame\u003c/em\u003e effect will result. It states rather that\r\nthere is a constant relation between causes of certain kinds\r\nand effects of certain kinds. For example, if a body falls\r\nfreely, there is a constant relation between the height\r\nthrough which it falls and the time it takes in falling. It\r\nis not necessary to have a body fall through the \u003cem\u003esame\u003c/em\u003e\r\nheight which has been previously observed, in order to\r\nbe able to foretell the length of time occupied in falling.\r\nIf this were necessary, no prediction would be possible,\r\nsince it would be impossible to make the height exactly\r\nthe same on two occasions. Similarly, the attraction which\r\nthe sun will exert on the earth is not only known at distances\r\nfor which it has been observed, but at all distances,\r\nbecause it is known to vary as the inverse square of the\r\ndistance. In fact, what is found to be repeated is always\r\nthe \u003cem\u003erelation\u003c/em\u003e of cause and effect, not the cause itself; all\r\nthat is necessary as regards the cause is that it should be\r\nof the same \u003cem\u003ekind\u003c/em\u003e (in the relevant respect) as earlier causes\r\nwhose effects have been observed.\u003c/p\u003e\r\n\u003cp\u003eAnother respect in which Bergson\u0027s statement of\r\ncausation is inadequate is in its assumption that the cause\r\nmust be \u003cem\u003eone\u003c/em\u003e event, whereas it may be two or more events,\r\nor even some continuous process. The substantive\r\nquestion at issue is whether mental events are determined\r\nby the past. Now in such a case as the repeated reading\r\nof a poem, it is obvious that our feelings in reading the\r\npoem are most emphatically dependent upon the past,\r\nbut not upon one single event in the past. All our\r\nprevious readings of the poem must be included in the\r\ncause. But we easily perceive a certain law according to\r\n\u003ca class=\"pagenum\" title=\"232\" id=\"Page_232\"\u003e \u003c/a\u003e\r\nwhich the effect varies as the previous readings increase\r\nin number, and in fact Bergson himself tacitly assumes\r\nsuch a law. We decide at last not to read the poem\r\nagain, because we know that this time the effect would\r\nbe boredom. We may not know all the niceties and\r\nshades of the boredom we should feel, but we know\r\nenough to guide our decision, and the prophecy of boredom\r\nis none the less true for being more or less general.\r\nThus the kinds of cases upon which Bergson relies are\r\ninsufficient to show the impossibility of prediction in the\r\nonly sense in which prediction has practical or emotional\r\ninterest. We may therefore leave the consideration of\r\nhis arguments and address ourselves to the problem\r\ndirectly.\u003c/p\u003e\r\n\u003cp\u003eThe law of causation, according to which later events\r\ncan theoretically be predicted by means of earlier events,\r\nhas often been held to be \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e, a necessity of thought,\r\na category without which science would be impossible.\r\nThese claims seem to me excessive. In certain directions\r\nthe law has been verified empirically, and in other directions\r\nthere is no positive evidence against it. But science\r\ncan use it where it has been found to be true, without\r\nbeing forced into any assumption as to its truth in other\r\nfields. We cannot, therefore, feel any \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e certainty\r\nthat causation must apply to human volitions.\u003c/p\u003e\r\n\u003cp\u003eThe question how far human volitions are subject to\r\ncausal laws is a purely empirical one. Empirically it\r\nseems plain that the great majority of our volitions have\r\ncauses, but it cannot, on this account, be held necessarily\r\ncertain that all have causes. There are, however, precisely\r\nthe same kinds of reasons for regarding it as probable\r\nthat they all have causes as there are in the case of\r\nphysical events.\u003c/p\u003e\r\n\u003cp\u003eWe may suppose—though this is doubtful—that there\r\n\u003ca class=\"pagenum\" title=\"233\" id=\"Page_233\"\u003e \u003c/a\u003e\r\nare laws of correlation of the mental and the physical,\r\nin virtue of which, given the state of all the matter in the\r\nworld, and therefore of all the brains and living organisms,\r\nthe state of all the minds in the world could be inferred,\r\nwhile conversely the state of all the matter in the world\r\ncould be inferred if the state of all the minds were given.\r\nIt is obvious that there is \u003cem\u003esome\u003c/em\u003e degree of correlation between\r\nbrain and mind, and it is impossible to say how\r\ncomplete it may be. This, however, is not the point\r\nwhich I wish to elicit. What I wish to urge is that, even\r\nif we admit the most extreme claims of determinism and\r\nof correlation of mind and brain, still the consequences\r\ninimical to what is worth preserving in free will do not\r\nfollow. The belief that they follow results, I think,\r\nentirely from the assimilation of causes to volitions, and\r\nfrom the notion that causes \u003cem\u003ecompel\u003c/em\u003e their effects in some\r\nsense analogous to that in which a human authority can\r\ncompel a man to do what he would rather not do. This\r\nassimilation, as soon as the true nature of scientific causal\r\nlaws is realised, is seen to be a sheer mistake. But this\r\nbrings us to the second of the two questions which we\r\nraised in regard to free will, namely, whether, assuming\r\ndeterminism, our actions can be in any proper sense\r\nregarded as compelled by outside forces.\u003c/p\u003e\r\n\u003cp\u003e(2) \u003cem\u003eAre human actions subject to an external compulsion?\u003c/em\u003e\r\nWe have, in deliberation, a subjective sense of freedom,\r\nwhich is sometimes alleged against the view that volitions\r\nhave causes. This sense of freedom, however, is only a\r\nsense that we can choose which we please of a number\r\nof alternatives: it does not show us that there is no\r\ncausal connection between what we please to choose and\r\nour previous history. The supposed inconsistency of\r\nthese two springs from the habit of conceiving causes as\r\nanalogous to volitions—a habit which often survives unconsciously\r\n\u003ca class=\"pagenum\" title=\"234\" id=\"Page_234\"\u003e \u003c/a\u003e\r\nin those who intend to conceive causes in\r\na more scientific manner. If a cause is analogous to a\r\nvolition, outside causes will be analogous to an alien will,\r\nand acts predictable from outside causes will be subject\r\nto compulsion. But this view of cause is one to which\r\nscience lends no countenance. Causes, we have seen, do\r\nnot \u003cem\u003ecompel\u003c/em\u003e their effects, any more than effects \u003cem\u003ecompel\u003c/em\u003e their\r\ncauses. There is a mutual relation, so that either can be\r\ninferred from the other. When the geologist infers the\r\npast state of the earth from its present state, we should\r\nnot say that the present state \u003cem\u003ecompels\u003c/em\u003e the past state to\r\nhave been what it was; yet it renders it necessary as a\r\nconsequence of the data, in the only sense in which\r\neffects are rendered necessary by their causes. The\r\ndifference which we \u003cem\u003efeel\u003c/em\u003e, in this respect, between causes\r\nand effects is a mere confusion due to the fact that we\r\nremember past events but do not happen to have memory\r\nof the future.\u003c/p\u003e\r\n\u003cp\u003eThe apparent indeterminateness of the future, upon\r\nwhich some advocates of free will rely, is merely a result\r\nof our ignorance. It is plain that no desirable kind of\r\nfree will can be dependent simply upon our ignorance;\r\nfor if that were the case, animals would be more free\r\nthan men, and savages than civilised people. Free will\r\nin any valuable sense must be compatible with the fullest\r\nknowledge. Now, quite apart from any assumption as to\r\ncausality, it is obvious that complete knowledge would\r\nembrace the future as well as the past. Our knowledge\r\nof the past is not wholly based upon causal inferences,\r\nbut is partly derived from memory. It is a mere\r\naccident that we have no memory of the future. We\r\nmight—as in the pretended visions of seers—see future\r\nevents immediately, in the way in which we see past\r\nevents. They certainly will be what they will be, and\r\n\u003ca class=\"pagenum\" title=\"235\" id=\"Page_235\"\u003e \u003c/a\u003e\r\nare in this sense just as determined as the past. If we\r\nsaw future events in the same immediate way in which\r\nwe see past events, what kind of free will would still be\r\npossible? Such a kind would be wholly independent of\r\ndeterminism: it could not be contrary to even the most\r\nentirely universal reign of causality. And such a kind\r\nmust contain whatever is worth having in free will, since\r\nit is impossible to believe that mere ignorance can be the\r\nessential condition of any good thing. Let us therefore\r\nimagine a set of beings who know the whole future\r\nwith absolute certainty, and let us ask ourselves whether\r\nthey could have anything that we should call free will.\u003c/p\u003e\r\n\u003cp\u003eSuch beings as we are imagining would not have to\r\nwait for the event in order to know what decision they\r\nwere going to adopt on some future occasion. They\r\nwould know now what their volitions were going to\r\nbe. But would they have any reason to regret this\r\nknowledge? Surely not, unless the foreseen volitions\r\nwere in themselves regrettable. And it is less likely\r\nthat the foreseen volitions would be regrettable if the\r\nsteps which would lead to them were also foreseen. It\r\nis difficult not to suppose that what is foreseen is fated,\r\nand must happen however much it may be dreaded.\r\nBut human actions are the outcome of desire, and no\r\nforeseeing can be true unless it takes account of desire.\r\nA foreseen volition will have to be one which does not\r\nbecome odious through being foreseen. The beings we\r\nare imagining would easily come to know the causal\r\nconnections of volitions, and therefore their volitions\r\nwould be better calculated to satisfy their desires than\r\nours are. Since volitions are the outcome of desires, a\r\nprevision of volitions contrary to desires could not be a\r\ntrue one. It must be remembered that the supposed\r\nprevision would not create the future any more than\r\n\u003ca class=\"pagenum\" title=\"236\" id=\"Page_236\"\u003e \u003c/a\u003e\r\nmemory creates the past. We do not think we were\r\nnecessarily not free in the past, merely because we can\r\nnow remember our past volitions. Similarly, we might\r\nbe free in the future, even if we could now see what our\r\nfuture volitions were going to be. Freedom, in short, in\r\nany valuable sense, demands only that our volitions shall\r\nbe, as they are, the result of our own desires, not of an\r\noutside force compelling us to will what we would\r\nrather not will. Everything else is confusion of thought,\r\ndue to the feeling that knowledge \u003cem\u003ecompels\u003c/em\u003e the happening\r\nof what it knows when this is future, though it is at once\r\nobvious that knowledge has no such power in regard to\r\nthe past. Free will, therefore, is true in the only form\r\nwhich is important; and the desire for other forms is a\r\nmere effect of insufficient analysis.\u003c/p\u003e\r\n\u003chr class=\"thought-break\"\u003e\r\n\u003cp\u003eWhat has been said on philosophical method in the\r\nforegoing lectures has been rather by means of illustrations\r\nin particular cases than by means of general precepts.\r\nNothing of any value can be said on method except\r\nthrough examples; but now, at the end of our course,\r\nwe may collect certain general maxims which may possibly\r\nbe a help in acquiring a philosophical habit of\r\nmind and a guide in looking for solutions of philosophic\r\nproblems.\u003c/p\u003e\r\n\u003cp\u003ePhilosophy does not become scientific by making use\r\nof other sciences, in the kind of way in which (\u003ci\u003ee.g.\u003c/i\u003e)\r\nHerbert Spencer does. Philosophy aims at what is\r\n\u003cem\u003egeneral\u003c/em\u003e, and the special sciences, however they may \u003cem\u003esuggest\u003c/em\u003e\r\nlarge generalisations, cannot make them certain. And\r\na hasty generalisation, such as Spencer\u0027s generalisation\r\nof evolution, is none the less hasty because what is\r\ngeneralised is the latest scientific theory. Philosophy is\r\na study apart from the other sciences: its results cannot\r\n\u003ca class=\"pagenum\" title=\"237\" id=\"Page_237\"\u003e \u003c/a\u003e\r\nbe established by the other sciences, and conversely must\r\nnot be such as some other science might conceivably contradict.\r\nProphecies as to the future of the universe, for\r\nexample, are not the business of philosophy; whether\r\nthe universe is progressive, retrograde, or stationary, it\r\nis not for the philosopher to say.\u003c/p\u003e\r\n\u003cp\u003eIn order to become a scientific philosopher, a certain\r\npeculiar mental discipline is required. There must be\r\npresent, first of all, the desire to know philosophical\r\ntruth, and this desire must be sufficiently strong to\r\nsurvive through years when there seems no hope of its\r\nfinding any satisfaction. The desire to know philosophical\r\ntruth is very rare—in its purity, it is not often\r\nfound even among philosophers. It is obscured sometimes—particularly\r\nafter long periods of fruitless search—by\r\nthe desire to \u003cem\u003ethink\u003c/em\u003e we know. Some plausible\r\nopinion presents itself, and by turning our attention away\r\nfrom the objections to it, or merely by not making great\r\nefforts to find objections to it, we may obtain the comfort\r\nof believing it, although, if we had resisted the wish for\r\ncomfort, we should have come to see that the opinion\r\nwas false. Again the desire for unadulterated truth is\r\noften obscured, in professional philosophers, by love of\r\nsystem: the one little fact which will not come inside\r\nthe philosopher\u0027s edifice has to be pushed and tortured\r\nuntil it seems to consent. Yet the one little fact is more\r\nlikely to be important for the future than the system\r\nwith which it is inconsistent. Pythagoras invented a\r\nsystem which fitted admirably with all the facts he knew,\r\nexcept the incommensurability of the diagonal of a square\r\nand the side; this one little fact stood out, and remained\r\na fact even after Hippasos of Metapontion was drowned\r\nfor revealing it. To us, the discovery of this fact is the\r\nchief claim of Pythagoras to immortality, while his\r\n\u003ca class=\"pagenum\" title=\"238\" id=\"Page_238\"\u003e \u003c/a\u003e\r\nsystem has become a matter of merely historical curiosity.\u003ca class=\"fnanchor pginternal\" href=\"#Footnote_57\" id=\"FNanchor_57\"\u003e[57]\u003c/a\u003e\r\nLove of system, therefore, and the system-maker\u0027s vanity\r\nwhich becomes associated with it, are among the snares\r\nthat the student of philosophy must guard against.\u003c/p\u003e\r\n\u003cp\u003eThe desire to establish this or that result, or generally\r\nto discover evidence for agreeable results, of whatever\r\nkind, has of course been the chief obstacle to honest\r\nphilosophising. So strangely perverted do men become\r\nby unrecognised passions, that a determination in advance\r\nto arrive at this or that conclusion is generally regarded\r\nas a mark of virtue, and those whose studies lead to\r\nan opposite conclusion are thought to be wicked. No\r\ndoubt it is commoner to wish to arrive at an agreeable\r\nresult than to wish to arrive at a true result. But only\r\nthose in whom the desire to arrive at a \u003cem\u003etrue\u003c/em\u003e result is\r\nparamount can hope to serve any good purpose by the\r\nstudy of philosophy.\u003c/p\u003e\r\n\u003cp\u003eBut even when the desire to know exists in the\r\nrequisite strength, the mental vision by which abstract\r\ntruth is recognised is hard to distinguish from vivid\r\nimaginability and consonance with mental habits. It is\r\nnecessary to practise methodological doubt, like Descartes,\r\nin order to loosen the hold of mental habits; and it is\r\nnecessary to cultivate logical imagination, in order to\r\nhave a number of hypotheses at command, and not to be\r\nthe slave of the one which common sense has rendered\r\neasy to imagine. These two processes, of doubting the\r\nfamiliar and imagining the unfamiliar, are correlative,\r\nand form the chief part of the mental training required\r\nfor a philosopher.\u003c/p\u003e\r\n\u003cp\u003eThe naïve beliefs which we find in ourselves when we\r\nfirst begin the process of philosophic reflection may turn\r\n\u003ca class=\"pagenum\" title=\"239\" id=\"Page_239\"\u003e \u003c/a\u003e\r\nout, in the end, to be almost all capable of a true interpretation;\r\nbut they ought all, before being admitted into\r\nphilosophy, to undergo the ordeal of sceptical criticism.\r\nUntil they have gone through this ordeal, they are mere\r\nblind habits, ways of behaving rather than intellectual\r\nconvictions. And although it may be that a majority\r\nwill pass the test, we may be pretty sure that some will\r\nnot, and that a serious readjustment of our outlook\r\nought to result. In order to break the dominion of\r\nhabit, we must do our best to doubt the senses, reason,\r\nmorals, everything in short. In some directions, doubt\r\nwill be found possible; in others, it will be checked by\r\nthat direct vision of abstract truth upon which the possibility\r\nof philosophical knowledge depends.\u003c/p\u003e\r\n\u003cp\u003eAt the same time, and as an essential aid to the direct\r\nperception of the truth, it is necessary to acquire fertility\r\nin imagining abstract hypotheses. This is, I think, what\r\nhas most of all been lacking hitherto in philosophy. So\r\nmeagre was the logical apparatus that all the hypotheses\r\nphilosophers could imagine were found to be inconsistent\r\nwith the facts. Too often this state of things led to the\r\nadoption of heroic measures, such as a wholesale denial\r\nof the facts, when an imagination better stocked with\r\nlogical tools would have found a key to unlock the\r\nmystery. It is in this way that the study of logic becomes\r\nthe central study in philosophy: it gives the method of\r\nresearch in philosophy, just as mathematics gives the\r\nmethod in physics. And as physics, which, from Plato\r\nto the Renaissance, was as unprogressive, dim, and superstitious\r\nas philosophy, became a science through Galileo\u0027s\r\nfresh observation of facts and subsequent mathematical\r\nmanipulation, so philosophy, in our own day, is becoming\r\nscientific through the simultaneous acquisition of new\r\nfacts and logical methods.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"240\" id=\"Page_240\"\u003e \u003c/a\u003eIn spite, however, of the new possibility of progress in\r\nphilosophy, the first effect, as in the case of physics, is\r\nto diminish very greatly the extent of what is thought to\r\nbe known. Before Galileo, people believed themselves\r\npossessed of immense knowledge on all the most interesting\r\nquestions in physics. He established certain facts as\r\nto the way in which bodies fall, not very interesting on\r\ntheir own account, but of quite immeasurable interest as\r\nexamples of real knowledge and of a new method whose\r\nfuture fruitfulness he himself divined. But his few facts\r\nsufficed to destroy the whole vast system of supposed\r\nknowledge handed down from Aristotle, as even the\r\npalest morning sun suffices to extinguish the stars. So\r\nin philosophy: though some have believed one system,\r\nand others another, almost all have been of opinion that\r\na great deal was known; but all this supposed knowledge\r\nin the traditional systems must be swept away, and a new\r\nbeginning must be made, which we shall esteem fortunate\r\nindeed if it can attain results comparable to Galileo\u0027s law\r\nof falling bodies.\u003c/p\u003e\r\n\u003cp\u003eBy the practice of methodological doubt, if it is\r\ngenuine and prolonged, a certain humility as to our\r\nknowledge is induced: we become glad to know \u003cem\u003eanything\u003c/em\u003e\r\nin philosophy, however seemingly trivial. Philosophy\r\nhas suffered from the lack of this kind of modesty. It\r\nhas made the mistake of attacking the interesting problems\r\nat once, instead of proceeding patiently and slowly,\r\naccumulating whatever solid knowledge was obtainable,\r\nand trusting the great problems to the future. Men of\r\nscience are not ashamed of what is intrinsically trivial, if\r\nits consequences are likely to be important; the \u003cem\u003eimmediate\u003c/em\u003e\r\noutcome of an experiment is hardly ever interesting on\r\nits own account. So in philosophy, it is often desirable\r\nto expend time and care on matters which, judged alone,\r\n\u003ca class=\"pagenum\" title=\"241\" id=\"Page_241\"\u003e \u003c/a\u003e\r\nmight seem frivolous, for it is often only through the\r\nconsideration of such matters that the greater problems\r\ncan be approached.\u003c/p\u003e\r\n\u003cp\u003eWhen our problem has been selected, and the necessary\r\nmental discipline has been acquired, the method to be\r\npursued is fairly uniform. The big problems which\r\nprovoke philosophical inquiry are found, on examination,\r\nto be complex, and to depend upon a number of component\r\nproblems, usually more abstract than those of\r\nwhich they are the components. It will generally be\r\nfound that all our initial data, all the facts that we seem\r\nto know to begin with, suffer from vagueness, confusion,\r\nand complexity. Current philosophical ideas share these\r\ndefects; it is therefore necessary to create an apparatus\r\nof precise conceptions as general and as free from complexity\r\nas possible, before the data can be analysed into\r\nthe kind of premisses which philosophy aims at discovering.\r\nIn this process of analysis, the source of difficulty is\r\ntracked further and further back, growing at each stage\r\nmore abstract, more refined, more difficult to apprehend.\r\nUsually it will be found that a number of these extraordinarily\r\nabstract questions underlie any one of the big\r\nobvious problems. When everything has been done that\r\ncan be done by method, a stage is reached where only\r\ndirect philosophic vision can carry matters further. Here\r\nonly genius will avail. What is wanted, as a rule, is\r\nsome new effort of logical imagination, some glimpse of\r\na possibility never conceived before, and then the direct\r\nperception that this possibility is realised in the case in\r\nquestion. Failure to think of the right possibility leaves\r\ninsoluble difficulties, balanced arguments pro and con,\r\nutter bewilderment and despair. But the right possibility,\r\nas a rule, when once conceived, justifies itself swiftly by\r\nits astonishing power of absorbing apparently conflicting\r\n\u003ca class=\"pagenum\" title=\"242\" id=\"Page_242\"\u003e \u003c/a\u003e\r\nfacts. From this point onward, the work of the philosopher\r\nis synthetic and comparatively easy; it is in the\r\nvery last stage of the analysis that the real difficulty\r\nconsists.\u003c/p\u003e\r\n\u003cp\u003eOf the prospect of progress in philosophy, it would be\r\nrash to speak with confidence. Many of the traditional\r\nproblems of philosophy, perhaps most of those which\r\nhave interested a wider circle than that of technical\r\nstudents, do not appear to be soluble by scientific\r\nmethods. Just as astronomy lost much of its human\r\ninterest when it ceased to be astrology, so philosophy\r\nmust lose in attractiveness as it grows less prodigal of\r\npromises. But to the large and still growing body of\r\nmen engaged in the pursuit of science—men who hitherto,\r\nnot without justification, have turned aside from philosophy\r\nwith a certain contempt—the new method, successful\r\nalready in such time-honoured problems as number,\r\ninfinity, continuity, space and time, should make an\r\nappeal which the older methods have wholly failed to\r\nmake. Physics, with its principle of relativity and its\r\nrevolutionary investigations into the nature of matter, is\r\nfeeling the need for that kind of novelty in fundamental\r\nhypotheses which scientific philosophy aims at facilitating.\r\nThe one and only condition, I believe, which is necessary\r\nin order to secure for philosophy in the near future an\r\nachievement surpassing all that has hitherto been accomplished\r\nby philosophers, is the creation of a school of\r\nmen with scientific training and philosophical interests,\r\nunhampered by the traditions of the past, and not misled\r\nby the literary methods of those who copy the ancients\r\nin all except their merits.\u003c/p\u003e\r\n\u003ch2\u003e\u003ca class=\"pagenum\" title=\"243\" id=\"Page_243\"\u003e \u003c/a\u003eINDEX\u003c/h2\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eAbsolute, \u003ca href=\"#Page_6\" class=\"pginternal\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_39\" class=\"pginternal\"\u003e39\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAbstraction, principle of, \u003ca href=\"#Page_42\" class=\"pginternal\"\u003e42\u003c/a\u003e, \u003ca href=\"#Page_124\" class=\"pginternal\"\u003e124\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eAchilles, Zeno\u0027s argument of, \u003ca href=\"#Page_173\" class=\"pginternal\"\u003e173\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAcquaintance, \u003ca href=\"#Page_25\" class=\"pginternal\"\u003e25\u003c/a\u003e, \u003ca href=\"#Page_144\" class=\"pginternal\"\u003e144\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eActivity, \u003ca href=\"#Page_224\" class=\"pginternal\"\u003e224\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eAllman, \u003ca href=\"#Footnote_24\" class=\"pginternal\"\u003e161 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eAnalysis, \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e, \u003ca href=\"#Page_204\" class=\"pginternal\"\u003e204\u003c/a\u003e, \u003ca href=\"#Page_211\" class=\"pginternal\"\u003e211\u003c/a\u003e, \u003ca href=\"#Page_241\" class=\"pginternal\"\u003e241\u003c/a\u003e.\u003cbr\u003e\r\n legitimacy of, \u003ca href=\"#Page_150\" class=\"pginternal\"\u003e150\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAnaximander, \u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAntinomies, Kant\u0027s, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eAquinas, \u003ca href=\"#Page_10\" class=\"pginternal\"\u003e10\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAristotle, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e, \u003ca href=\"#Footnote_23\" class=\"pginternal\"\u003e160 n.\u003c/a\u003e, \u003ca href=\"#Page_161\" class=\"pginternal\"\u003e161\u003c/a\u003e ff., \u003ca href=\"#Page_240\" class=\"pginternal\"\u003e240\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eArrow, Zeno\u0027s argument of, \u003ca href=\"#Page_173\" class=\"pginternal\"\u003e173\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAssertion, \u003ca href=\"#Page_52\" class=\"pginternal\"\u003e52\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAtomism, logical, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eAtomists, \u003ca href=\"#Page_160\" class=\"pginternal\"\u003e160\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eBelief, \u003ca href=\"#Page_58\" class=\"pginternal\"\u003e58\u003c/a\u003e.\u003cbr\u003e\r\n primitive and derivative, \u003ca href=\"#Page_69\" class=\"pginternal\"\u003e69\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eBergson, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e, \u003ca href=\"#Page_13\" class=\"pginternal\"\u003e13\u003c/a\u003e, \u003ca href=\"#Page_20\" class=\"pginternal\"\u003e20\u003c/a\u003e ff., \u003ca href=\"#Page_137\" class=\"pginternal\"\u003e137\u003c/a\u003e, \u003ca href=\"#Page_138\" class=\"pginternal\"\u003e138\u003c/a\u003e, \u003ca href=\"#Page_150\" class=\"pginternal\"\u003e150\u003c/a\u003e, \u003ca href=\"#Page_158\" class=\"pginternal\"\u003e158\u003c/a\u003e, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e, \u003ca href=\"#Page_174\" class=\"pginternal\"\u003e174\u003c/a\u003e, \u003ca href=\"#Page_178\" class=\"pginternal\"\u003e178\u003c/a\u003e, \u003ca href=\"#Page_229\" class=\"pginternal\"\u003e229\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eBerkeley, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_64\" class=\"pginternal\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_102\" class=\"pginternal\"\u003e102\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eBolzano, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eBoole, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eBradley, \u003ca href=\"#Page_6\" class=\"pginternal\"\u003e6\u003c/a\u003e, \u003ca href=\"#Page_39\" class=\"pginternal\"\u003e39\u003c/a\u003e, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eBroad, \u003ca href=\"#Footnote_42\" class=\"pginternal\"\u003e172 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eBrochard, \u003ca href=\"#Footnote_38\" class=\"pginternal\"\u003e169 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eBurnet, \u003ca href=\"#Footnote_6\" class=\"pginternal\"\u003e19 n.\u003c/a\u003e, \u003ca href=\"#Footnote_22\" class=\"pginternal\"\u003e160 n.\u003c/a\u003e, \u003ca href=\"#Footnote_25\" class=\"pginternal\"\u003e161 n.\u003c/a\u003e, \u003ca href=\"#Footnote_39\" class=\"pginternal\"\u003e170 n.\u003c/a\u003e, \u003ca href=\"#Page_171\" class=\"pginternal\"\u003e171\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eCalderon, \u003ca href=\"#Page_95\" class=\"pginternal\"\u003e95\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCantor, \u003ca href=\"#Page_vi\" class=\"pginternal\"\u003evi\u003c/a\u003e, \u003ca href=\"#Page_vii\" class=\"pginternal\"\u003evii\u003c/a\u003e, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e, \u003ca href=\"#Page_190\" class=\"pginternal\"\u003e190\u003c/a\u003e, \u003ca href=\"#Page_194\" class=\"pginternal\"\u003e194\u003c/a\u003e, \u003ca href=\"#Page_199\" class=\"pginternal\"\u003e199\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCategories, \u003ca href=\"#Page_38\" class=\"pginternal\"\u003e38\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCausal laws, \u003ca href=\"#Page_109\" class=\"pginternal\"\u003e109\u003c/a\u003e, \u003ca href=\"#Page_212\" class=\"pginternal\"\u003e212\u003c/a\u003e ff.\u003cbr\u003e\r\n evidence for, \u003ca href=\"#Page_216\" class=\"pginternal\"\u003e216\u003c/a\u003e ff.\u003cbr\u003e\r\n in psychology, \u003ca href=\"#Page_219\" class=\"pginternal\"\u003e219\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCausation, \u003ca href=\"#Page_34\" class=\"pginternal\"\u003e34\u003c/a\u003e ff., \u003ca href=\"#Page_79\" class=\"pginternal\"\u003e79\u003c/a\u003e, \u003ca href=\"#Page_212\" class=\"pginternal\"\u003e212\u003c/a\u003e ff.\u003cbr\u003e\r\n law of, \u003ca href=\"#Page_221\" class=\"pginternal\"\u003e221\u003c/a\u003e.\u003cbr\u003e\r\n not \u003ci lang=\"la\"\u003ea priori\u003c/i\u003e, \u003ca href=\"#Page_223\" class=\"pginternal\"\u003e223\u003c/a\u003e, \u003ca href=\"#Page_232\" class=\"pginternal\"\u003e232\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCause, \u003ca href=\"#Page_220\" class=\"pginternal\"\u003e220\u003c/a\u003e, \u003ca href=\"#Page_223\" class=\"pginternal\"\u003e223\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCertainty, degrees of, \u003ca href=\"#Page_67\" class=\"pginternal\"\u003e67\u003c/a\u003e, \u003ca href=\"#Page_68\" class=\"pginternal\"\u003e68\u003c/a\u003e, \u003ca href=\"#Page_212\" class=\"pginternal\"\u003e212\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eChange,\u003cbr\u003e\r\n demands analysis, \u003ca href=\"#Page_151\" class=\"pginternal\"\u003e151\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCinematograph, \u003ca href=\"#Page_148\" class=\"pginternal\"\u003e148\u003c/a\u003e, \u003ca href=\"#Page_174\" class=\"pginternal\"\u003e174\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eClasses, \u003ca href=\"#Page_202\" class=\"pginternal\"\u003e202\u003c/a\u003e.\u003cbr\u003e\r\n non-existence of, \u003ca href=\"#Page_205\" class=\"pginternal\"\u003e205\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eClassical tradition, \u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e ff., \u003ca href=\"#Page_58\" class=\"pginternal\"\u003e58\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eComplexity, \u003ca href=\"#Page_145\" class=\"pginternal\"\u003e145\u003c/a\u003e, \u003ca href=\"#Page_157\" class=\"pginternal\"\u003e157\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eCompulsion, \u003ca href=\"#Page_229\" class=\"pginternal\"\u003e229\u003c/a\u003e, \u003ca href=\"#Page_233\" class=\"pginternal\"\u003e233\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eCongruence, \u003ca href=\"#Page_195\" class=\"pginternal\"\u003e195\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eConsecutiveness, \u003ca href=\"#Page_134\" class=\"pginternal\"\u003e134\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eConservation, \u003ca href=\"#Page_105\" class=\"pginternal\"\u003e105\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eConstituents of facts, \u003ca href=\"#Page_51\" class=\"pginternal\"\u003e51\u003c/a\u003e, \u003ca href=\"#Page_145\" class=\"pginternal\"\u003e145\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eConstruction \u003ci\u003ev.\u003c/i\u003e inference, iv.\u003c/p\u003e\r\n\u003cp\u003eContemporaries, initial, \u003ca href=\"#Page_119\" class=\"pginternal\"\u003e119\u003c/a\u003e, \u003ca href=\"#Footnote_17\" class=\"pginternal\"\u003e120 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eContinuity, \u003ca href=\"#Page_64\" class=\"pginternal\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_129\" class=\"pginternal\"\u003e129\u003c/a\u003e ff., \u003ca href=\"#Page_141\" class=\"pginternal\"\u003e141\u003c/a\u003e ff., \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff.\u003cbr\u003e\r\n of change, \u003ca href=\"#Page_106\" class=\"pginternal\"\u003e106\u003c/a\u003e, \u003ca href=\"#Page_108\" class=\"pginternal\"\u003e108\u003c/a\u003e, \u003ca href=\"#Page_130\" class=\"pginternal\"\u003e130\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eCorrelation of mental and physical, \u003ca href=\"#Page_233\" class=\"pginternal\"\u003e233\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCounting, \u003ca href=\"#Page_164\" class=\"pginternal\"\u003e164\u003c/a\u003e, \u003ca href=\"#Page_181\" class=\"pginternal\"\u003e181\u003c/a\u003e, \u003ca href=\"#Page_187\" class=\"pginternal\"\u003e187\u003c/a\u003e ff., \u003ca href=\"#Page_203\" class=\"pginternal\"\u003e203\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eCouturat, \u003ca href=\"#Footnote_13\" class=\"pginternal\"\u003e40 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eDante, \u003ca href=\"#Page_10\" class=\"pginternal\"\u003e10\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDarwin, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e, \u003ca href=\"#Page_23\" class=\"pginternal\"\u003e23\u003c/a\u003e, \u003ca href=\"#Page_30\" class=\"pginternal\"\u003e30\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eData, \u003ca href=\"#Page_65\" class=\"pginternal\"\u003e65\u003c/a\u003e ff., \u003ca href=\"#Page_211\" class=\"pginternal\"\u003e211\u003c/a\u003e.\u003cbr\u003e\r\n “hard” and “soft,” \u003ca href=\"#Page_70\" class=\"pginternal\"\u003e70\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eDates, \u003ca href=\"#Page_117\" class=\"pginternal\"\u003e117\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDefinition, \u003ca href=\"#Page_204\" class=\"pginternal\"\u003e204\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDescartes, \u003ca href=\"#Page_5\" class=\"pginternal\"\u003e5\u003c/a\u003e, \u003ca href=\"#Page_73\" class=\"pginternal\"\u003e73\u003c/a\u003e, \u003ca href=\"#Page_238\" class=\"pginternal\"\u003e238\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDescriptions, \u003ca href=\"#Page_201\" class=\"pginternal\"\u003e201\u003c/a\u003e, \u003ca href=\"#Page_214\" class=\"pginternal\"\u003e214\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDesire, \u003ca href=\"#Page_227\" class=\"pginternal\"\u003e227\u003c/a\u003e, \u003ca href=\"#Page_235\" class=\"pginternal\"\u003e235\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDeterminism, \u003ca href=\"#Page_233\" class=\"pginternal\"\u003e233\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDoubt, \u003ca href=\"#Page_237\" class=\"pginternal\"\u003e237\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDreams, \u003ca href=\"#Page_85\" class=\"pginternal\"\u003e85\u003c/a\u003e, \u003ca href=\"#Page_93\" class=\"pginternal\"\u003e93\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eDuration, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e, \u003ca href=\"#Page_149\" class=\"pginternal\"\u003e149\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eEarlier and later, \u003ca href=\"#Page_116\" class=\"pginternal\"\u003e116\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEffect, \u003ca href=\"#Page_220\" class=\"pginternal\"\u003e220\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEleatics, \u003ca href=\"#Page_19\" class=\"pginternal\"\u003e19\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEmpiricism, \u003ca href=\"#Page_37\" class=\"pginternal\"\u003e37\u003c/a\u003e, \u003ca href=\"#Page_222\" class=\"pginternal\"\u003e222\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEnclosure, \u003ca href=\"#Page_114\" class=\"pginternal\"\u003e114\u003c/a\u003e ff., \u003ca href=\"#Page_120\" class=\"pginternal\"\u003e120\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEnumeration, \u003ca href=\"#Page_202\" class=\"pginternal\"\u003e202\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEuclid, \u003ca href=\"#Page_160\" class=\"pginternal\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_164\" class=\"pginternal\"\u003e164\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"244\" id=\"Page_244\"\u003e \u003c/a\u003eEvellin, \u003ca href=\"#Page_169\" class=\"pginternal\"\u003e169\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eEvolutionism, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eExtension, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e, \u003ca href=\"#Page_149\" class=\"pginternal\"\u003e149\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eExternal world, knowledge of, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eFact, \u003ca href=\"#Page_51\" class=\"pginternal\"\u003e51\u003c/a\u003e.\u003cbr\u003e\r\n atomic, \u003ca href=\"#Page_52\" class=\"pginternal\"\u003e52\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eFinalism, \u003ca href=\"#Page_13\" class=\"pginternal\"\u003e13\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eForm, logical, \u003ca href=\"#Page_42\" class=\"pginternal\"\u003e42\u003c/a\u003e ff., \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e, \u003ca href=\"#Page_208\" class=\"pginternal\"\u003e208\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eFractions, \u003ca href=\"#Page_132\" class=\"pginternal\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_179\" class=\"pginternal\"\u003e179\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eFree will, \u003ca href=\"#Page_213\" class=\"pginternal\"\u003e213\u003c/a\u003e, \u003ca href=\"#Page_227\" class=\"pginternal\"\u003e227\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eFrege, \u003ca href=\"#Page_5\" class=\"pginternal\"\u003e5\u003c/a\u003e, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e, \u003ca href=\"#Page_199\" class=\"pginternal\"\u003e199\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eGalileo, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_59\" class=\"pginternal\"\u003e59\u003c/a\u003e, \u003ca href=\"#Page_192\" class=\"pginternal\"\u003e192\u003c/a\u003e, \u003ca href=\"#Page_194\" class=\"pginternal\"\u003e194\u003c/a\u003e, \u003ca href=\"#Page_239\" class=\"pginternal\"\u003e239\u003c/a\u003e, \u003ca href=\"#Page_240\" class=\"pginternal\"\u003e240\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eGaye, \u003ca href=\"#Footnote_35\" class=\"pginternal\"\u003e169 n.\u003c/a\u003e, \u003ca href=\"#Page_175\" class=\"pginternal\"\u003e175\u003c/a\u003e, \u003ca href=\"#Page_177\" class=\"pginternal\"\u003e177\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eGeometry, \u003ca href=\"#Page_5\" class=\"pginternal\"\u003e5\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eGiles, \u003ca href=\"#Footnote_53\" class=\"pginternal\"\u003e206 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eGreater and less, \u003ca href=\"#Page_195\" class=\"pginternal\"\u003e195\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eHarvard, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHegel, \u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e, \u003ca href=\"#Page_37\" class=\"pginternal\"\u003e37\u003c/a\u003e ff., \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_166\" class=\"pginternal\"\u003e166\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e“Here,” \u003ca href=\"#Page_73\" class=\"pginternal\"\u003e73\u003c/a\u003e, \u003ca href=\"#Page_92\" class=\"pginternal\"\u003e92\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHereditary properties, \u003ca href=\"#Page_195\" class=\"pginternal\"\u003e195\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHippasos, \u003ca href=\"#Page_163\" class=\"pginternal\"\u003e163\u003c/a\u003e, \u003ca href=\"#Page_237\" class=\"pginternal\"\u003e237\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHui Tzŭ, \u003ca href=\"#Page_206\" class=\"pginternal\"\u003e206\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHume, \u003ca href=\"#Page_217\" class=\"pginternal\"\u003e217\u003c/a\u003e, \u003ca href=\"#Page_221\" class=\"pginternal\"\u003e221\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eHypotheses in philosophy, \u003ca href=\"#Page_239\" class=\"pginternal\"\u003e239\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eIllusions, \u003ca href=\"#Page_85\" class=\"pginternal\"\u003e85\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eIncommensurables, \u003ca href=\"#Page_162\" class=\"pginternal\"\u003e162\u003c/a\u003e ff., \u003ca href=\"#Page_237\" class=\"pginternal\"\u003e237\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eIndependence, \u003ca href=\"#Page_73\" class=\"pginternal\"\u003e73\u003c/a\u003e, \u003ca href=\"#Page_74\" class=\"pginternal\"\u003e74\u003c/a\u003e.\u003cbr\u003e\r\n causal and logical, \u003ca href=\"#Page_74\" class=\"pginternal\"\u003e74\u003c/a\u003e, \u003ca href=\"#Page_75\" class=\"pginternal\"\u003e75\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eIndiscernibility, \u003ca href=\"#Page_141\" class=\"pginternal\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_148\" class=\"pginternal\"\u003e148\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eIndivisibles, \u003ca href=\"#Page_160\" class=\"pginternal\"\u003e160\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eInduction, \u003ca href=\"#Page_34\" class=\"pginternal\"\u003e34\u003c/a\u003e, \u003ca href=\"#Page_222\" class=\"pginternal\"\u003e222\u003c/a\u003e.\u003cbr\u003e\r\n mathematical, \u003ca href=\"#Page_195\" class=\"pginternal\"\u003e195\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eInductiveness, \u003ca href=\"#Page_190\" class=\"pginternal\"\u003e190\u003c/a\u003e, \u003ca href=\"#Page_195\" class=\"pginternal\"\u003e195\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eInference, \u003ca href=\"#Page_44\" class=\"pginternal\"\u003e44\u003c/a\u003e, \u003ca href=\"#Page_54\" class=\"pginternal\"\u003e54\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eInfinite, \u003ca href=\"#Page_vi\" class=\"pginternal\"\u003evi\u003c/a\u003e, \u003ca href=\"#Page_64\" class=\"pginternal\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_133\" class=\"pginternal\"\u003e133\u003c/a\u003e, \u003ca href=\"#Page_149\" class=\"pginternal\"\u003e149\u003c/a\u003e.\u003cbr\u003e\r\n historically considered, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff.\u003cbr\u003e\r\n “true,” \u003ca href=\"#Page_179\" class=\"pginternal\"\u003e179\u003c/a\u003e, \u003ca href=\"#Page_180\" class=\"pginternal\"\u003e180\u003c/a\u003e.\u003cbr\u003e\r\n positive theory of, \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eInfinitesimals, \u003ca href=\"#Page_135\" class=\"pginternal\"\u003e135\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eInstants, \u003ca href=\"#Page_116\" class=\"pginternal\"\u003e116\u003c/a\u003e ff., \u003ca href=\"#Page_129\" class=\"pginternal\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_151\" class=\"pginternal\"\u003e151\u003c/a\u003e, \u003ca href=\"#Page_216\" class=\"pginternal\"\u003e216\u003c/a\u003e.\u003cbr\u003e\r\n defined, \u003ca href=\"#Page_118\" class=\"pginternal\"\u003e118\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eInstinct \u003ci\u003ev.\u003c/i\u003e Reason, \u003ca href=\"#Page_20\" class=\"pginternal\"\u003e20\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eIntellect, \u003ca href=\"#Page_22\" class=\"pginternal\"\u003e22\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eIntelligence, how displayed by friends, \u003cins title=\"93\"\u003e\u003ca href=\"#Page_93\" class=\"pginternal\"\u003e93\u003c/a\u003e.\u003c/ins\u003e\u003cbr\u003e\r\n inadequacy of display, \u003ca href=\"#Page_96\" class=\"pginternal\"\u003e96\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eInterpretation, \u003ca href=\"#Page_144\" class=\"pginternal\"\u003e144\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eJames, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_10\" class=\"pginternal\"\u003e10\u003c/a\u003e, \u003ca href=\"#Page_13\" class=\"pginternal\"\u003e13\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eJourdain, \u003ca href=\"#Footnote_28\" class=\"pginternal\"\u003e165 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eJowett, \u003ca href=\"#Page_167\" class=\"pginternal\"\u003e167\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eJudgment, \u003ca href=\"#Page_58\" class=\"pginternal\"\u003e58\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eKant, \u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e, \u003ca href=\"#Page_112\" class=\"pginternal\"\u003e112\u003c/a\u003e, \u003ca href=\"#Page_116\" class=\"pginternal\"\u003e116\u003c/a\u003e, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff., \u003ca href=\"#Page_200\" class=\"pginternal\"\u003e200\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eKnowledge about, \u003ca href=\"#Page_144\" class=\"pginternal\"\u003e144\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eLanguage, bad, \u003ca href=\"#Page_82\" class=\"pginternal\"\u003e82\u003c/a\u003e, \u003ca href=\"#Page_135\" class=\"pginternal\"\u003e135\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eLaplace, \u003ca href=\"#Page_12\" class=\"pginternal\"\u003e12\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eLaws of nature, \u003ca href=\"#Page_218\" class=\"pginternal\"\u003e218\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eLeibniz, \u003ca href=\"#Page_13\" class=\"pginternal\"\u003e13\u003c/a\u003e, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e, \u003ca href=\"#Page_87\" class=\"pginternal\"\u003e87\u003c/a\u003e, \u003ca href=\"#Page_186\" class=\"pginternal\"\u003e186\u003c/a\u003e, \u003ca href=\"#Page_191\" class=\"pginternal\"\u003e191\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eLogic, \u003ca href=\"#Page_201\" class=\"pginternal\"\u003e201\u003c/a\u003e.\u003cbr\u003e\r\n analytic not constructive, \u003ca href=\"#Page_8\" class=\"pginternal\"\u003e8\u003c/a\u003e.\u003cbr\u003e\r\n Aristotelian, \u003ca href=\"#Page_5\" class=\"pginternal\"\u003e5\u003c/a\u003e.\u003cbr\u003e\r\n and fact, \u003ca href=\"#Page_53\" class=\"pginternal\"\u003e53\u003c/a\u003e.\u003cbr\u003e\r\n inductive, \u003ca href=\"#Page_34\" class=\"pginternal\"\u003e34\u003c/a\u003e, \u003ca href=\"#Page_222\" class=\"pginternal\"\u003e222\u003c/a\u003e.\u003cbr\u003e\r\n mathematical, \u003ca href=\"#Page_vi\" class=\"pginternal\"\u003evi\u003c/a\u003e, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e ff.\u003cbr\u003e\r\n mystical, \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e.\u003cbr\u003e\r\n and philosophy, \u003ca href=\"#Page_8\" class=\"pginternal\"\u003e8\u003c/a\u003e, \u003ca href=\"#Page_33\" class=\"pginternal\"\u003e33\u003c/a\u003e ff., \u003ca href=\"#Page_239\" class=\"pginternal\"\u003e239\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eLogical constants, \u003ca href=\"#Page_208\" class=\"pginternal\"\u003e208\u003c/a\u003e, \u003ca href=\"#Page_213\" class=\"pginternal\"\u003e213\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eMach, \u003ca href=\"#Page_123\" class=\"pginternal\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_224\" class=\"pginternal\"\u003e224\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMacran, \u003ca href=\"#Footnote_12\" class=\"pginternal\"\u003e39 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eMathematics, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e, \u003ca href=\"#Page_57\" class=\"pginternal\"\u003e57\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMatter, \u003ca href=\"#Page_75\" class=\"pginternal\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_101\" class=\"pginternal\"\u003e101\u003c/a\u003e ff.\u003cbr\u003e\r\n permanence of, \u003ca href=\"#Page_102\" class=\"pginternal\"\u003e102\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eMeasurement, \u003ca href=\"#Page_164\" class=\"pginternal\"\u003e164\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMemory, \u003ca href=\"#Page_230\" class=\"pginternal\"\u003e230\u003c/a\u003e, \u003ca href=\"#Page_234\" class=\"pginternal\"\u003e234\u003c/a\u003e, \u003ca href=\"#Page_236\" class=\"pginternal\"\u003e236\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMethod, deductive, \u003ca href=\"#Page_5\" class=\"pginternal\"\u003e5\u003c/a\u003e.\u003cbr\u003e\r\n logical-analytic, \u003ca href=\"#Page_v\" class=\"pginternal\"\u003ev\u003c/a\u003e, \u003ca href=\"#Page_65\" class=\"pginternal\"\u003e65\u003c/a\u003e, \u003ca href=\"#Page_211\" class=\"pginternal\"\u003e211\u003c/a\u003e, \u003ca href=\"#Page_236\" class=\"pginternal\"\u003e236\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eMilhaud, \u003ca href=\"#Footnote_32\" class=\"pginternal\"\u003e168 n.\u003c/a\u003e, \u003ca href=\"#Footnote_34\" class=\"pginternal\"\u003e169 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eMill, \u003ca href=\"#Page_34\" class=\"pginternal\"\u003e34\u003c/a\u003e, \u003ca href=\"#Page_200\" class=\"pginternal\"\u003e200\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMontaigne, \u003ca href=\"#Page_28\" class=\"pginternal\"\u003e28\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eMotion, \u003ca href=\"#Page_130\" class=\"pginternal\"\u003e130\u003c/a\u003e, \u003ca href=\"#Page_216\" class=\"pginternal\"\u003e216\u003c/a\u003e.\u003cbr\u003e\r\n continuous, \u003ca href=\"#Page_133\" class=\"pginternal\"\u003e133\u003c/a\u003e, \u003ca href=\"#Page_136\" class=\"pginternal\"\u003e136\u003c/a\u003e.\u003cbr\u003e\r\n mathematical theory of, \u003ca href=\"#Page_133\" class=\"pginternal\"\u003e133\u003c/a\u003e.\u003cbr\u003e\r\n perception of, \u003ca href=\"#Page_137\" class=\"pginternal\"\u003e137\u003c/a\u003e ff.\u003cbr\u003e\r\n Zeno\u0027s arguments on, \u003ca href=\"#Page_168\" class=\"pginternal\"\u003e168\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eMysticism, \u003ca href=\"#Page_19\" class=\"pginternal\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_95\" class=\"pginternal\"\u003e95\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eNewton, \u003ca href=\"#Page_30\" class=\"pginternal\"\u003e30\u003c/a\u003e, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eNietzsche, \u003ca href=\"#Page_10\" class=\"pginternal\"\u003e10\u003c/a\u003e, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eNoël, \u003ca href=\"#Page_169\" class=\"pginternal\"\u003e169\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eNumber, cardinal, \u003cins title=\"131\"\u003e\u003ca href=\"#Page_131\" class=\"pginternal\"\u003e131\u003c/a\u003e,\u003c/ins\u003e \u003ca href=\"#Page_186\" class=\"pginternal\"\u003e186\u003c/a\u003e ff.\u003cbr\u003e\r\n defined, \u003ca href=\"#Page_199\" class=\"pginternal\"\u003e199\u003c/a\u003e ff.\u003cbr\u003e\r\n finite, \u003ca href=\"#Page_160\" class=\"pginternal\"\u003e160\u003c/a\u003e, \u003ca href=\"#Page_190\" class=\"pginternal\"\u003e190\u003c/a\u003e ff.\u003cbr\u003e\r\n inductive, \u003ca href=\"#Page_197\" class=\"pginternal\"\u003e197\u003c/a\u003e.\u003cbr\u003e\r\n infinite, \u003ca href=\"#Page_178\" class=\"pginternal\"\u003e178\u003c/a\u003e, \u003ca href=\"#Page_180\" class=\"pginternal\"\u003e180\u003c/a\u003e, \u003ca href=\"#Page_188\" class=\"pginternal\"\u003e188\u003c/a\u003e ff., \u003ca href=\"#Page_197\" class=\"pginternal\"\u003e197\u003c/a\u003e.\u003cbr\u003e\r\n reflexive, \u003ca href=\"#Page_190\" class=\"pginternal\"\u003e190\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eOccam, \u003ca href=\"#Page_107\" class=\"pginternal\"\u003e107\u003c/a\u003e, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eOne and many, \u003ca href=\"#Page_167\" class=\"pginternal\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_170\" class=\"pginternal\"\u003e170\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eOrder, \u003ca href=\"#Page_131\" class=\"pginternal\"\u003e131\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eParmenides, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e ff., \u003ca href=\"#Page_178\" class=\"pginternal\"\u003e178\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePast and future, \u003ca href=\"#Page_224\" class=\"pginternal\"\u003e224\u003c/a\u003e, \u003ca href=\"#Page_234\" class=\"pginternal\"\u003e234\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003e\u003ca class=\"pagenum\" title=\"245\" id=\"Page_245\"\u003e \u003c/a\u003ePeano, \u003ca href=\"#Page_40\" class=\"pginternal\"\u003e40\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePerspectives, \u003ca href=\"#Page_88\" class=\"pginternal\"\u003e88\u003c/a\u003e ff., \u003ca href=\"#Page_111\" class=\"pginternal\"\u003e111\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePhiloponus, \u003ca href=\"#Footnote_41\" class=\"pginternal\"\u003e171 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003ePhilosophy and ethics, \u003ca href=\"#Page_26\" class=\"pginternal\"\u003e26\u003c/a\u003e ff.\u003cbr\u003e\r\n and mathematics, \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e ff.\u003cbr\u003e\r\n province of, \u003ca href=\"#Page_17\" class=\"pginternal\"\u003e17\u003c/a\u003e, \u003ca href=\"#Page_26\" class=\"pginternal\"\u003e26\u003c/a\u003e, \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e, \u003ca href=\"#Page_236\" class=\"pginternal\"\u003e236\u003c/a\u003e.\u003cbr\u003e\r\n scientific, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e, \u003ca href=\"#Page_16\" class=\"pginternal\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_18\" class=\"pginternal\"\u003e18\u003c/a\u003e, \u003ca href=\"#Page_29\" class=\"pginternal\"\u003e29\u003c/a\u003e, \u003ca href=\"#Page_236\" class=\"pginternal\"\u003e236\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003ePhysics, \u003ca href=\"#Page_101\" class=\"pginternal\"\u003e101\u003c/a\u003e ff., \u003ca href=\"#Page_147\" class=\"pginternal\"\u003e147\u003c/a\u003e, \u003ca href=\"#Page_239\" class=\"pginternal\"\u003e239\u003c/a\u003e, \u003ca href=\"#Page_242\" class=\"pginternal\"\u003e242\u003c/a\u003e.\u003cbr\u003e\r\n descriptive, \u003ca href=\"#Page_224\" class=\"pginternal\"\u003e224\u003c/a\u003e.\u003cbr\u003e\r\n verifiability of, \u003ca href=\"#Page_81\" class=\"pginternal\"\u003e81\u003c/a\u003e, \u003ca href=\"#Page_110\" class=\"pginternal\"\u003e110\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePlace, \u003ca href=\"#Page_86\" class=\"pginternal\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_90\" class=\"pginternal\"\u003e90\u003c/a\u003e.\u003cbr\u003e\r\n\u003cem\u003eat\u003c/em\u003e and \u003cem\u003efrom\u003c/em\u003e, \u003ca href=\"#Page_92\" class=\"pginternal\"\u003e92\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePlato, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_19\" class=\"pginternal\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_27\" class=\"pginternal\"\u003e27\u003c/a\u003e, \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e, \u003ca href=\"#Footnote_29\" class=\"pginternal\"\u003e165 n.\u003c/a\u003e, \u003ca href=\"#Page_166\" class=\"pginternal\"\u003e166\u003c/a\u003e, \u003ca href=\"#Page_167\" class=\"pginternal\"\u003e167\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePoincaré, \u003ca href=\"#Page_123\" class=\"pginternal\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_141\" class=\"pginternal\"\u003e141\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePoints, \u003ca href=\"#Page_113\" class=\"pginternal\"\u003e113\u003c/a\u003e ff., \u003ca href=\"#Page_129\" class=\"pginternal\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_158\" class=\"pginternal\"\u003e158\u003c/a\u003e.\u003cbr\u003e\r\n definition of, \u003ca href=\"#Page_vi\" class=\"pginternal\"\u003evi\u003c/a\u003e, \u003ca href=\"#Page_115\" class=\"pginternal\"\u003e115\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePragmatism, \u003ca href=\"#Page_11\" class=\"pginternal\"\u003e11\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePrantl, \u003ca href=\"#Page_174\" class=\"pginternal\"\u003e174\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePredictability, \u003ca href=\"#Page_229\" class=\"pginternal\"\u003e229\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003ePremisses, \u003ca href=\"#Page_211\" class=\"pginternal\"\u003e211\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eProbability, \u003ca href=\"#Page_36\" class=\"pginternal\"\u003e36\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePropositions, \u003ca href=\"#Page_52\" class=\"pginternal\"\u003e52\u003c/a\u003e.\u003cbr\u003e\r\n atomic, \u003ca href=\"#Page_52\" class=\"pginternal\"\u003e52\u003c/a\u003e.\u003cbr\u003e\r\n general, \u003ca href=\"#Page_55\" class=\"pginternal\"\u003e55\u003c/a\u003e.\u003cbr\u003e\r\n molecular, \u003ca href=\"#Page_54\" class=\"pginternal\"\u003e54\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003ePythagoras, \u003ca href=\"#Page_19\" class=\"pginternal\"\u003e19\u003c/a\u003e, \u003ca href=\"#Page_160\" class=\"pginternal\"\u003e160\u003c/a\u003e ff., \u003ca href=\"#Page_237\" class=\"pginternal\"\u003e237\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eRace-course, Zeno\u0027s argument of, \u003ca href=\"#Page_171\" class=\"pginternal\"\u003e171\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eRealism, new, \u003ca href=\"#Page_6\" class=\"pginternal\"\u003e6\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eReflexiveness, \u003ca href=\"#Page_190\" class=\"pginternal\"\u003e190\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eRelations, \u003ca href=\"#Page_45\" class=\"pginternal\"\u003e45\u003c/a\u003e.\u003cbr\u003e\r\n asymmetrical, \u003ca href=\"#Page_47\" class=\"pginternal\"\u003e47\u003c/a\u003e.\u003cbr\u003e\r\n Bradley\u0027s reasons against, \u003ca href=\"#Page_6\" class=\"pginternal\"\u003e6\u003c/a\u003e.\u003cbr\u003e\r\n external, \u003ca href=\"#Page_150\" class=\"pginternal\"\u003e150\u003c/a\u003e.\u003cbr\u003e\r\n intransitive, \u003ca href=\"#Page_48\" class=\"pginternal\"\u003e48\u003c/a\u003e.\u003cbr\u003e\r\n multiple, \u003ca href=\"#Page_50\" class=\"pginternal\"\u003e50\u003c/a\u003e.\u003cbr\u003e\r\n one-one, \u003ca href=\"#Page_203\" class=\"pginternal\"\u003e203\u003c/a\u003e.\u003cbr\u003e\r\n reality of, \u003ca href=\"#Page_49\" class=\"pginternal\"\u003e49\u003c/a\u003e.\u003cbr\u003e\r\n symmetrical, \u003ca href=\"#Page_47\" class=\"pginternal\"\u003e47\u003c/a\u003e, \u003ca href=\"#Page_124\" class=\"pginternal\"\u003e124\u003c/a\u003e.\u003cbr\u003e\r\n transitive, \u003ca href=\"#Page_48\" class=\"pginternal\"\u003e48\u003c/a\u003e, \u003ca href=\"#Page_124\" class=\"pginternal\"\u003e124\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eRelativity, \u003ca href=\"#Page_103\" class=\"pginternal\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_242\" class=\"pginternal\"\u003e242\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eRepetitions, \u003ca href=\"#Page_230\" class=\"pginternal\"\u003e230\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eRest, \u003ca href=\"#Page_136\" class=\"pginternal\"\u003e136\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eRitter and Preller, \u003ca href=\"#Footnote_26\" class=\"pginternal\"\u003e161 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eRobertson, D. S., \u003ca href=\"#Footnote_22\" class=\"pginternal\"\u003e160 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eRousseau, \u003ca href=\"#Page_20\" class=\"pginternal\"\u003e20\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eRoyce, \u003ca href=\"#Page_50\" class=\"pginternal\"\u003e50\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eSantayana, \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eScepticism, \u003ca href=\"#Page_66\" class=\"pginternal\"\u003e66\u003c/a\u003e, \u003ca href=\"#Page_67\" class=\"pginternal\"\u003e67\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSeeing double, \u003ca href=\"#Page_86\" class=\"pginternal\"\u003e86\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSelf, \u003ca href=\"#Page_73\" class=\"pginternal\"\u003e73\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSensation, \u003ca href=\"#Page_25\" class=\"pginternal\"\u003e25\u003c/a\u003e, \u003ca href=\"#Page_75\" class=\"pginternal\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_123\" class=\"pginternal\"\u003e123\u003c/a\u003e.\u003cbr\u003e\r\n and stimulus, \u003ca href=\"#Page_139\" class=\"pginternal\"\u003e139\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSense-data, \u003ca href=\"#Page_56\" class=\"pginternal\"\u003e56\u003c/a\u003e, \u003ca href=\"#Page_63\" class=\"pginternal\"\u003e63\u003c/a\u003e, \u003ca href=\"#Page_67\" class=\"pginternal\"\u003e67\u003c/a\u003e, \u003ca href=\"#Page_75\" class=\"pginternal\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_110\" class=\"pginternal\"\u003e110\u003c/a\u003e, \u003ca href=\"#Page_141\" class=\"pginternal\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_143\" class=\"pginternal\"\u003e143\u003c/a\u003e, \u003ca href=\"#Page_213\" class=\"pginternal\"\u003e213\u003c/a\u003e.\u003cbr\u003e\r\n and physics, \u003ca href=\"#Page_v\" class=\"pginternal\"\u003ev\u003c/a\u003e, \u003ca href=\"#Page_64\" class=\"pginternal\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_81\" class=\"pginternal\"\u003e81\u003c/a\u003e, \u003ca href=\"#Page_97\" class=\"pginternal\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_101\" class=\"pginternal\"\u003e101\u003c/a\u003e ff., \u003ca href=\"#Page_140\" class=\"pginternal\"\u003e140\u003c/a\u003e.\u003cbr\u003e\r\n infinitely numerous? \u003ca href=\"#Page_149\" class=\"pginternal\"\u003e149\u003c/a\u003e, \u003ca href=\"#Page_159\" class=\"pginternal\"\u003e159\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSense-perception, \u003ca href=\"#Page_53\" class=\"pginternal\"\u003e53\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSeries, \u003ca href=\"#Page_49\" class=\"pginternal\"\u003e49\u003c/a\u003e.\u003cbr\u003e\r\n compact, \u003ca href=\"#Page_132\" class=\"pginternal\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_142\" class=\"pginternal\"\u003e142\u003c/a\u003e, \u003ca href=\"#Page_178\" class=\"pginternal\"\u003e178\u003c/a\u003e.\u003cbr\u003e\r\n continuous, \u003ca href=\"#Page_131\" class=\"pginternal\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_132\" class=\"pginternal\"\u003e132\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSigwart, \u003ca href=\"#Page_187\" class=\"pginternal\"\u003e187\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSimplicius, \u003ca href=\"#Footnote_39\" class=\"pginternal\"\u003e170 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eSimultaneity, \u003ca href=\"#Page_116\" class=\"pginternal\"\u003e116\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSpace, \u003ca href=\"#Page_73\" class=\"pginternal\"\u003e73\u003c/a\u003e, \u003ca href=\"#Page_88\" class=\"pginternal\"\u003e88\u003c/a\u003e, \u003ca href=\"#Page_103\" class=\"pginternal\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_112\" class=\"pginternal\"\u003e112\u003c/a\u003e ff., \u003ca href=\"#Page_130\" class=\"pginternal\"\u003e130\u003c/a\u003e.\u003cbr\u003e\r\n absolute and relative, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e, \u003ca href=\"#Page_159\" class=\"pginternal\"\u003e159\u003c/a\u003e.\u003cbr\u003e\r\n antinomies of, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff.\u003cbr\u003e\r\n perception of, \u003ca href=\"#Page_68\" class=\"pginternal\"\u003e68\u003c/a\u003e.\u003cbr\u003e\r\n of perspectives, \u003ca href=\"#Page_88\" class=\"pginternal\"\u003e88\u003c/a\u003e ff.\u003cbr\u003e\r\n private, \u003ca href=\"#Page_89\" class=\"pginternal\"\u003e89\u003c/a\u003e, \u003ca href=\"#Page_90\" class=\"pginternal\"\u003e90\u003c/a\u003e.\u003cbr\u003e\r\n of touch and sight, \u003ca href=\"#Page_78\" class=\"pginternal\"\u003e78\u003c/a\u003e, \u003ca href=\"#Page_113\" class=\"pginternal\"\u003e113\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSpencer, \u003ca href=\"#Page_4\" class=\"pginternal\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_12\" class=\"pginternal\"\u003e12\u003c/a\u003e, \u003ca href=\"#Page_236\" class=\"pginternal\"\u003e236\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSpinoza, \u003ca href=\"#Page_46\" class=\"pginternal\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_166\" class=\"pginternal\"\u003e166\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eStadium, Zeno\u0027s argument of, \u003ca href=\"#Footnote_18\" class=\"pginternal\"\u003e134 n.\u003c/a\u003e, \u003ca href=\"#Page_175\" class=\"pginternal\"\u003e175\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003cp\u003eSubject-predicate, \u003ca href=\"#Page_45\" class=\"pginternal\"\u003e45\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eSynthesis, \u003ca href=\"#Page_157\" class=\"pginternal\"\u003e157\u003c/a\u003e, \u003ca href=\"#Page_185\" class=\"pginternal\"\u003e185\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eTannery, Paul, \u003ca href=\"#Footnote_35\" class=\"pginternal\"\u003e169 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eTeleology, \u003ca href=\"#Page_223\" class=\"pginternal\"\u003e223\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eTestimony, \u003ca href=\"#Page_67\" class=\"pginternal\"\u003e67\u003c/a\u003e, \u003ca href=\"#Page_72\" class=\"pginternal\"\u003e72\u003c/a\u003e, \u003ca href=\"#Page_82\" class=\"pginternal\"\u003e82\u003c/a\u003e, \u003ca href=\"#Page_87\" class=\"pginternal\"\u003e87\u003c/a\u003e, \u003ca href=\"#Page_96\" class=\"pginternal\"\u003e96\u003c/a\u003e, \u003ca href=\"#Page_212\" class=\"pginternal\"\u003e212\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eThales, \u003ca href=\"#Page_3\" class=\"pginternal\"\u003e3\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eThing-in-itself, \u003ca href=\"#Page_75\" class=\"pginternal\"\u003e75\u003c/a\u003e, \u003ca href=\"#Page_84\" class=\"pginternal\"\u003e84\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eThings, \u003ca href=\"#Page_89\" class=\"pginternal\"\u003e89\u003c/a\u003e ff., \u003ca href=\"#Page_104\" class=\"pginternal\"\u003e104\u003c/a\u003e ff., \u003ca href=\"#Page_213\" class=\"pginternal\"\u003e213\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eTime, \u003ca href=\"#Page_103\" class=\"pginternal\"\u003e103\u003c/a\u003e, \u003ca href=\"#Page_116\" class=\"pginternal\"\u003e116\u003c/a\u003e ff., \u003ca href=\"#Page_130\" class=\"pginternal\"\u003e130\u003c/a\u003e, \u003ca href=\"#Page_155\" class=\"pginternal\"\u003e155\u003c/a\u003e ff., \u003ca href=\"#Page_166\" class=\"pginternal\"\u003e166\u003c/a\u003e, \u003ca href=\"#Page_215\" class=\"pginternal\"\u003e215\u003c/a\u003e.\u003cbr\u003e\r\n absolute or relative, \u003ca href=\"#Page_146\" class=\"pginternal\"\u003e146\u003c/a\u003e.\u003cbr\u003e\r\n local, \u003ca href=\"#Page_103\" class=\"pginternal\"\u003e103\u003c/a\u003e.\u003cbr\u003e\r\n private, \u003ca href=\"#Page_121\" class=\"pginternal\"\u003e121\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eUniformities, \u003ca href=\"#Page_217\" class=\"pginternal\"\u003e217\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eUnity, organic, \u003ca href=\"#Page_9\" class=\"pginternal\"\u003e9\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eUniversal and particular, \u003ca href=\"#Footnote_12\" class=\"pginternal\"\u003e39 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eVolition, \u003ca href=\"#Page_223\" class=\"pginternal\"\u003e223\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eWhitehead, \u003ca href=\"#Page_vi\" class=\"pginternal\"\u003evi\u003c/a\u003e, \u003ca href=\"#Page_207\" class=\"pginternal\"\u003e207\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eWittgenstein, \u003ca href=\"#Page_vii\" class=\"pginternal\"\u003evii\u003c/a\u003e, \u003ca href=\"#Footnote_55\" class=\"pginternal\"\u003e208 n.\u003c/a\u003e\u003c/p\u003e\r\n\u003cp\u003eWorlds, actual and ideal, \u003ca href=\"#Page_111\" class=\"pginternal\"\u003e111\u003c/a\u003e.\u003cbr\u003e\r\n possible, \u003ca href=\"#Page_186\" class=\"pginternal\"\u003e186\u003c/a\u003e.\u003cbr\u003e\r\n private, \u003ca href=\"#Page_88\" class=\"pginternal\"\u003e88\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"index\"\u003e\r\n\u003cp\u003eZeller, \u003ca href=\"#Page_173\" class=\"pginternal\"\u003e173\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003eZeno, \u003ca href=\"#Page_129\" class=\"pginternal\"\u003e129\u003c/a\u003e, \u003ca href=\"#Page_134\" class=\"pginternal\"\u003e134\u003c/a\u003e, \u003ca href=\"#Page_136\" class=\"pginternal\"\u003e136\u003c/a\u003e, \u003ca href=\"#Page_165\" class=\"pginternal\"\u003e165\u003c/a\u003e ff.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp class=\"center\"\u003ePRINTED BY NEILL AND CO., LTD., EDINBURGH.\u003c/p\u003e\r\n\u003cdiv class=\"footnotes\"\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_1\" id=\"Footnote_1\"\u003e[1]\u003c/a\u003e\r\nDelivered as Lowell Lectures in Boston, in March and April 1914.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_2\" id=\"Footnote_2\"\u003e[2]\u003c/a\u003e\r\nLondon and New York, 1912 (“Home University Library”).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_3\" id=\"Footnote_3\"\u003e[3]\u003c/a\u003e\r\nThe first volume was published at Cambridge in 1910, the second in\r\n1912, and the third in 1913.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_4\" id=\"Footnote_4\"\u003e[4]\u003c/a\u003e\r\n\u003ccite\u003eAppearance and Reality\u003c/cite\u003e, pp. 32–33.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_5\" id=\"Footnote_5\"\u003e[5]\u003c/a\u003e\r\n\u003ccite\u003eCreative Evolution\u003c/cite\u003e, English translation, p. 41.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_6\" id=\"Footnote_6\"\u003e[6]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e Burnet, \u003ccite\u003eEarly Greek Philosophy\u003c/cite\u003e, pp. 85 ff.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_7\" id=\"Footnote_7\"\u003e[7]\u003c/a\u003e\r\n\u003ccite\u003eIntroduction to Metaphysics\u003c/cite\u003e, p. 1.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_8\" id=\"Footnote_8\"\u003e[8]\u003c/a\u003e\r\n\u003ccite\u003eLogic\u003c/cite\u003e, book iii., chapter iii., § 2.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_9\" id=\"Footnote_9\"\u003e[9]\u003c/a\u003e\r\nBook iii., chapter xxi., § 3.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_10\" id=\"Footnote_10\"\u003e[10]\u003c/a\u003e\r\nOr rather a propositional function.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_11\" id=\"Footnote_11\"\u003e[11]\u003c/a\u003e\r\nThe subject of causality and induction will be discussed again in\r\n\u003ca href=\"#Lecture_8\" class=\"pginternal\"\u003eLecture VIII\u003c/a\u003e.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_12\" id=\"Footnote_12\"\u003e[12]\u003c/a\u003e\r\nSee the translation by H. S. Macran, \u003ccite\u003eHegel\u0027s Doctrine of Formal\r\nLogic\u003c/cite\u003e, Oxford, 1912. Hegel\u0027s argument in this portion of his “Logic”\r\ndepends throughout upon confusing the “is” of predication, as in\r\n“Socrates is mortal,” with the “is” of identity, as in “Socrates is the\r\nphilosopher who drank the hemlock.” Owing to this confusion, he thinks\r\nthat “Socrates” and “mortal” must be identical. Seeing that they are\r\ndifferent, he does not infer, as others would, that there is a mistake somewhere,\r\nbut that they exhibit “identity in difference.” Again, Socrates is\r\nparticular, “mortal” is universal. Therefore, he says, since Socrates is\r\nmortal, it follows that the particular is the universal—taking the “is” to\r\nbe throughout expressive of identity. But to say “the particular is the\r\nuniversal” is self-contradictory. Again Hegel does not suspect a mistake\r\nbut proceeds to synthesise particular and universal in the individual, or\r\nconcrete universal. This is an example of how, for want of care at the\r\nstart, vast and imposing systems of philosophy are built upon stupid and\r\ntrivial confusions, which, but for the almost incredible fact that they are\r\nunintentional, one would be tempted to characterise as puns.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_13\" id=\"Footnote_13\"\u003e[13]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e Couturat, \u003ccite lang=\"fr\"\u003eLa Logique de Leibniz\u003c/cite\u003e, pp. 361, 386.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_14\" id=\"Footnote_14\"\u003e[14]\u003c/a\u003e\r\nIt was often recognised that there was \u003cem\u003esome\u003c/em\u003e difference between them,\r\nbut it was not recognised that the difference is fundamental, and of very\r\ngreat importance.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_15\" id=\"Footnote_15\"\u003e[15]\u003c/a\u003e\r\n\u003ccite\u003eEncyclopædia of the Philosophical Sciences\u003c/cite\u003e, vol. i. p. 97.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_16\" id=\"Footnote_16\"\u003e[16]\u003c/a\u003e\r\nThis perhaps requires modification in order to include such facts as\r\nbeliefs and wishes, since such facts apparently contain propositions as\r\ncomponents. Such facts, though not strictly atomic, must be supposed\r\nincluded if the statement in the text is to be true.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_17\" id=\"Footnote_17\"\u003e[17]\u003c/a\u003e\r\nThe assumptions made concerning time-relations in the above are as\r\nfollows:—\r\n\u003c/p\u003e\r\n\u003cp class=\"indent1\"\u003eI. In order to secure that instants form a series, we assume:\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003ea\u003c/i\u003e) No event wholly precedes itself. (An “event” is defined as\r\nwhatever is simultaneous with something or other.)\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003eb\u003c/i\u003e) If one event wholly precedes another, and the other wholly\r\nprecedes a third, then the first wholly precedes the third.\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003ec\u003c/i\u003e) If one event wholly precedes another, it is not simultaneous\r\nwith it.\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003ed\u003c/i\u003e) Of two events which are not simultaneous, one must wholly\r\nprecede the other.\u003c/p\u003e\r\n\u003cp class=\"indent1\"\u003eII. In order to secure that the initial contemporaries of a given event\r\nshould form an instant, we assume:\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003ee\u003c/i\u003e) An event wholly after some contemporary of a given event is\r\nwholly after some \u003cem\u003einitial\u003c/em\u003e contemporary of the given event.\u003c/p\u003e\r\n\u003cp class=\"indent1\"\u003eIII. In order to secure that the series of instants shall be compact,\r\nwe assume:\u003c/p\u003e\r\n\u003cp class=\"indent2\"\u003e(\u003ci\u003ef\u003c/i\u003e) If one event wholly precedes another, there is an event\r\nwholly after the one and simultaneous with something\r\nwholly before the other.\u003c/p\u003e\r\n\u003cp\u003e\r\nThis assumption entails the consequence that if one event covers the\r\nwhole of a stretch of time immediately preceding another event, then\r\nit must have at least one instant in common with the other event; \u003ci\u003ei.e.\u003c/i\u003e it\r\nis impossible for one event to cease just before another begins. I do\r\nnot know whether this should be regarded as inadmissible. For a\r\nmathematico-logical treatment of the above topics, \u003ci\u003ecf.\u003c/i\u003e N. Wilner, “A\r\nContribution to the Theory of Relative Position,” \u003ccite\u003eProc. Camb. Phil. Soc.\u003c/cite\u003e,\r\nxvii. 5, pp. 441–449.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_18\" id=\"Footnote_18\"\u003e[18]\u003c/a\u003e\r\nThe above paradox is essentially the same as Zeno\u0027s argument of the\r\nstadium which will be considered in our \u003ca href=\"#Lecture_6\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_19\" id=\"Footnote_19\"\u003e[19]\u003c/a\u003e\r\nSee \u003ca href=\"#Lecture_6\" class=\"pginternal\"\u003enext lecture\u003c/a\u003e.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_20\" id=\"Footnote_20\"\u003e[20]\u003c/a\u003e\r\n\u003ccite\u003eMonist\u003c/cite\u003e, July 1912, pp. 337–341.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_21\" id=\"Footnote_21\"\u003e[21]\u003c/a\u003e\r\n“Le continu mathématique,” \u003ccite lang=\"fr\"\u003eRevue de Métaphysique et de Morale\u003c/cite\u003e,\r\nvol. i. p. 29.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_22\" id=\"Footnote_22\"\u003e[22]\u003c/a\u003e\r\nIn what concerns the early Greek philosophers, my knowledge is\r\nlargely derived from Burnet\u0027s valuable work, \u003ccite\u003eEarly Greek Philosophy\u003c/cite\u003e\r\n(2nd ed., London, 1908). I have also been greatly assisted by Mr D. S.\r\nRobertson of Trinity College, who has supplied the deficiencies of my\r\nknowledge of Greek, and brought important references to my notice.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_23\" id=\"Footnote_23\"\u003e[23]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e Aristotle, \u003ccite\u003eMetaphysics\u003c/cite\u003e, M. 6, 1080\u003ci\u003eb\u003c/i\u003e, 18 \u003ci\u003esqq.\u003c/i\u003e, and 1083\u003ci\u003eb\u003c/i\u003e, 8 \u003ci\u003esqq.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_24\" id=\"Footnote_24\"\u003e[24]\u003c/a\u003e\r\nThere is some reason to think that the Pythagoreans distinguished\r\nbetween discrete and continuous quantity. G. J. Allman, in his \u003ccite\u003eGreek\r\nGeometry from Thales to Euclid\u003c/cite\u003e, says (p. 23): “The Pythagoreans made\r\na fourfold division of mathematical science, attributing one of its parts to\r\nthe how many, \u003cspan class=\"greek\" title=\"to poson\" lang=\"grc\"\u003eτὸ πόσον\u003c/span\u003e, and the other to the how much, \u003cspan class=\"greek\" title=\"to pêlikon\" lang=\"grc\"\u003eτὸ πηλίκον\u003c/span\u003e; and\r\nthey assigned to each of these parts a twofold division. For they said\r\nthat discrete quantity, or the \u003cem\u003ehow many\u003c/em\u003e, either subsists by itself or must\r\nbe considered with relation to some other; but that continued quantity, or\r\nthe \u003cem\u003ehow much\u003c/em\u003e, is either stable or in motion. Hence they affirmed that\r\narithmetic contemplates that discrete quantity which subsists by itself, but\r\nmusic that which is related to another; and that geometry considers\r\ncontinued quantity so far as it is immovable; but astronomy (\u003cspan class=\"greek\" title=\"tên sphairikên\" lang=\"grc\"\u003eτὴν σφαιρικήν\u003c/span\u003e)\r\ncontemplates continued quantity so far as it is of a self-motive nature.\r\n(Proclus, ed. Friedlein, p. 35. As to the distinction between \u003cspan class=\"greek\" title=\"to pêlikon\" lang=\"grc\"\u003eτὸ πηλίκον\u003c/span\u003e,\r\ncontinuous, and \u003cspan class=\"greek\" title=\"to poson\" lang=\"grc\"\u003eτὸ πόσον\u003c/span\u003e, discrete quantity, see Iambl., \u003ccite lang=\"la\"\u003ein Nicomachi Geraseni\r\nArithmeticam introductionem\u003c/cite\u003e, ed. Tennulius, p. 148.)” \u003ci\u003eCf.\u003c/i\u003e p. 48.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_25\" id=\"Footnote_25\"\u003e[25]\u003c/a\u003e\r\nReferred to by Burnet, \u003ci\u003eop. cit.\u003c/i\u003e, p. 120.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_26\" id=\"Footnote_26\"\u003e[26]\u003c/a\u003e\r\niv., 6. 213\u003ci\u003eb\u003c/i\u003e, 22; H. Ritter and L. Preller, \u003ccite lang=\"la\"\u003eHistoria Philosophiæ\r\nGræcæ\u003c/cite\u003e, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in\r\nfuture as “R. P.”).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_27\" id=\"Footnote_27\"\u003e[27]\u003c/a\u003e\r\nThe Pythagorean proof is roughly as follows. If possible, let the\r\nratio of the diagonal to the side of a square be \u003ci\u003em\u003c/i\u003e/\u003ci\u003en\u003c/i\u003e, where \u003ci\u003em\u003c/i\u003e and \u003ci\u003en\u003c/i\u003e are\r\nwhole numbers having no common factor. Then we must have \u003ci\u003em\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e = 2\u003ci\u003en\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e.\r\nNow the square of an odd number is odd, but \u003ci\u003em\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e, being equal to 2\u003ci\u003en\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e, is\r\neven. Hence \u003ci\u003em\u003c/i\u003e must be even. But the square of an even number divides\r\nby 4, therefore \u003ci\u003en\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e, which is half of \u003ci\u003em\u003c/i\u003e\u003csup\u003e2\u003c/sup\u003e, must be even. Therefore \u003ci\u003en\u003c/i\u003e must\r\nbe even. But, since \u003ci\u003em\u003c/i\u003e is even, and \u003ci\u003em\u003c/i\u003e and \u003ci\u003en\u003c/i\u003e have no common factor, \u003ci\u003en\u003c/i\u003e\r\nmust be odd. Thus \u003ci\u003en\u003c/i\u003e must be both odd and even, which is impossible;\r\nand therefore the diagonal and the side cannot have a rational ratio.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_28\" id=\"Footnote_28\"\u003e[28]\u003c/a\u003e\r\nIn regard to Zeno and the Pythagoreans, I have derived much valuable\r\ninformation and criticism from Mr P. E. B. Jourdain.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_29\" id=\"Footnote_29\"\u003e[29]\u003c/a\u003e\r\nSo Plato makes Zeno say in the \u003ccite\u003eParmenides\u003c/cite\u003e, apropos of his philosophy\r\nas a whole; and all internal and external evidence supports this view.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_30\" id=\"Footnote_30\"\u003e[30]\u003c/a\u003e\r\n“With Parmenides,” Hegel says, “philosophising proper began.”\r\n\u003ccite lang=\"de\"\u003eWerke\u003c/cite\u003e (edition of 1840), vol. xiii. p. 274.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_31\" id=\"Footnote_31\"\u003e[31]\u003c/a\u003e\r\n\u003ccite\u003eParmenides\u003c/cite\u003e, 128 \u003cspan class=\"small-caps all-upper\"\u003eA\u003c/span\u003e–\u003cspan class=\"small-caps all-upper\"\u003eD\u003c/span\u003e.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_32\" id=\"Footnote_32\"\u003e[32]\u003c/a\u003e\r\nThis interpretation is combated by Milhaud, \u003ccite lang=\"fr\"\u003eLes philosophes-géomètres\r\nde la Grèce\u003c/cite\u003e, p. 140 n., but his reasons do not seem to me convincing. All\r\nthe interpretations in what follows are open to question, but all have the\r\nsupport of reputable authorities.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_33\" id=\"Footnote_33\"\u003e[33]\u003c/a\u003e\r\n\u003ccite\u003ePhysics\u003c/cite\u003e, vi. 9. 2396 (R.P. 136–139).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_34\" id=\"Footnote_34\"\u003e[34]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e Gaston Milhaud, \u003ccite lang=\"fr\"\u003eLes philosophes-géomètres de la Grèce\u003c/cite\u003e, p. 140 n.;\r\nPaul Tannery, \u003ccite lang=\"fr\"\u003ePour l\u0027histoire de la science hellène\u003c/cite\u003e, p. 249; Burnet,\r\n\u003ci\u003eop. cit.\u003c/i\u003e, p. 362.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_35\" id=\"Footnote_35\"\u003e[35]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e R. K. Gaye, “On Aristotle, \u003ccite\u003ePhysics\u003c/cite\u003e, Z ix.” \u003ccite\u003eJournal of Philology\u003c/cite\u003e,\r\nvol. xxxi., esp. p. 111. Also Moritz Cantor, \u003ccite lang=\"de\"\u003eVorlesungen über Geschichte\r\nder Mathematik\u003c/cite\u003e, 1st ed., vol. i., 1880, p. 168, who, however, subsequently\r\nadopted Paul Tannery\u0027s opinion, \u003ccite lang=\"de\"\u003eVorlesungen\u003c/cite\u003e, 3rd ed. (vol. i. p. 200).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_36\" id=\"Footnote_36\"\u003e[36]\u003c/a\u003e\r\n“Le mouvement et les partisans des indivisibles,” \u003ccite lang=\"fr\"\u003eRevue de Métaphysique\r\net de Morale\u003c/cite\u003e, vol. i. pp. 382–395.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_37\" id=\"Footnote_37\"\u003e[37]\u003c/a\u003e\r\n“Le mouvement et les arguments de Zénon d\u0027Élée,” \u003ccite lang=\"fr\"\u003eRevue de Métaphysique\r\net de Morale\u003c/cite\u003e, vol. i. pp. 107–125.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_38\" id=\"Footnote_38\"\u003e[38]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e M. Brochard, “Les prétendus sophismes de Zénon d\u0027Élée,” \u003ccite lang=\"fr\"\u003eRevue\r\nde Métaphysique et de Morale\u003c/cite\u003e, vol. i. pp. 209–215.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_39\" id=\"Footnote_39\"\u003e[39]\u003c/a\u003e\r\nSimplicius, \u003ccite\u003ePhys.\u003c/cite\u003e, 140, 28 \u003cspan class=\"small-caps all-upper\"\u003eD\u003c/span\u003e (R.P. 133); Burnet, \u003ci\u003eop. cit.\u003c/i\u003e, pp. 364–365.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_40\" id=\"Footnote_40\"\u003e[40]\u003c/a\u003e\r\n\u003ci\u003eOp. cit.\u003c/i\u003e, p. 367.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_41\" id=\"Footnote_41\"\u003e[41]\u003c/a\u003e\r\nAristotle\u0027s words are: “The first is the one on the non-existence of\r\nmotion on the ground that what is moved must always attain the middle\r\npoint sooner than the end-point, on which we gave our opinion in the\r\nearlier part of our discourse.” \u003ccite\u003ePhys.\u003c/cite\u003e, vi. 9. 939\u003cspan class=\"small-caps all-upper\"\u003eB\u003c/span\u003e (R.P. 136). Aristotle\r\nseems to refer to \u003ccite\u003ePhys.\u003c/cite\u003e, vi. 2. 223\u003cspan class=\"small-caps all-upper\"\u003eAB\u003c/span\u003e [R.P. 136\u003cspan class=\"small-caps all-upper\"\u003eA\u003c/span\u003e]: “All space is continuous,\r\nfor time and space are divided into the same and equal divisions….\r\nWherefore also Zeno\u0027s argument is fallacious, that it is impossible to go\r\nthrough an infinite collection or to touch an infinite collection one by one\r\nin a finite time. For there are two senses in which the term ‘infinite’\r\nis applied both to length and to time, and in fact to all continuous things,\r\neither in regard to divisibility, or in regard to the ends. Now it is not\r\npossible to touch things infinite in regard to number in a finite time, but\r\nit is possible to touch things infinite in regard to divisibility: for time\r\nitself also is infinite in this sense. So that in fact we go through an infinite,\r\n[space] in an infinite [time] and not in a finite [time], and we touch infinite\r\nthings with infinite things, not with finite things.” Philoponus, a sixth-century\r\ncommentator (R.P. 136\u003cspan class=\"small-caps all-upper\"\u003eA\u003c/span\u003e, \u003ccite lang=\"la\"\u003eExc. Paris Philop. in Arist. Phys.\u003c/cite\u003e,\r\n803, 2. Vit.), gives the following illustration: “For if a thing were moved\r\nthe space of a cubit in one hour, since in every space there are an infinite\r\nnumber of points, the thing moved must needs touch all the points of\r\nthe space: it will then go through an infinite collection in a finite time,\r\nwhich is impossible.”\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_42\" id=\"Footnote_42\"\u003e[42]\u003c/a\u003e\r\n\u003ci\u003eCf.\u003c/i\u003e Mr C. D. Broad, “Note on Achilles and the Tortoise,” \u003ccite\u003eMind\u003c/cite\u003e, N.S.,\r\nvol. xxii. pp. 318–9.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_43\" id=\"Footnote_43\"\u003e[43]\u003c/a\u003e\r\n\u003ci\u003eOp. cit.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_44\" id=\"Footnote_44\"\u003e[44]\u003c/a\u003e\r\nAristotle\u0027s words are: “The second is the so-called Achilles. It\r\nconsists in this, that the slower will never be overtaken in its course by\r\nthe quickest, for the pursuer must always come first to the point from\r\nwhich the pursued has just departed, so that the slower must necessarily\r\nbe always still more or less in advance.” \u003ccite\u003ePhys.\u003c/cite\u003e, vi. 9. 239\u003cspan class=\"small-caps all-upper\"\u003eB\u003c/span\u003e (R.P. 137).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_45\" id=\"Footnote_45\"\u003e[45]\u003c/a\u003e\r\n\u003ccite\u003ePhys.\u003c/cite\u003e, vi. 9. 239\u003cspan class=\"small-caps all-upper\"\u003eB\u003c/span\u003e (R.P. 138).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_46\" id=\"Footnote_46\"\u003e[46]\u003c/a\u003e\r\n\u003ccite\u003ePhys.\u003c/cite\u003e, vi. 9. 239\u003cspan class=\"small-caps all-upper\"\u003eB\u003c/span\u003e (R.P. 139).\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_47\" id=\"Footnote_47\"\u003e[47]\u003c/a\u003e\r\n\u003ci\u003eLoc. cit.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_48\" id=\"Footnote_48\"\u003e[48]\u003c/a\u003e\r\n\u003ci\u003eLoc. cit.\u003c/i\u003e, p. 105.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_49\" id=\"Footnote_49\"\u003e[49]\u003c/a\u003e\r\n\u003ccite lang=\"de\"\u003ePhil. Werke\u003c/cite\u003e, Gerhardt\u0027s edition, vol. i. p. 338.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_50\" id=\"Footnote_50\"\u003e[50]\u003c/a\u003e\r\n\u003ccite\u003eMathematical Discourses concerning two new sciences relating to\r\nmechanics and local motion, in four dialogues.\u003c/cite\u003e By Galileo Galilei,\r\nChief Philosopher and Mathematician to the Grand Duke of Tuscany.\r\nDone into English from the Italian, by Tho. Weston, late Master, and\r\nnow published by John Weston, present Master, of the Academy at\r\nGreenwich. See pp. 46 ff.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_51\" id=\"Footnote_51\"\u003e[51]\u003c/a\u003e\r\nIn his \u003ccite lang=\"de\"\u003eGrundlagen einer allgemeinen Mannichfaltigkeitslehre\u003c/cite\u003e and in\r\narticles in \u003ccite lang=\"la\"\u003eActa Mathematica\u003c/cite\u003e, vol. ii.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_52\" id=\"Footnote_52\"\u003e[52]\u003c/a\u003e\r\nThe definition of number contained in this book, and elaborated in\r\nthe \u003ccite lang=\"de\"\u003eGrundgesetze der Arithmetik\u003c/cite\u003e (vol. i., 1893; vol. ii., 1903), was rediscovered\r\nby me in ignorance of Frege\u0027s work. I wish to state as\r\nemphatically as possible—what seems still often ignored—that his discovery\r\nantedated mine by eighteen years.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_53\" id=\"Footnote_53\"\u003e[53]\u003c/a\u003e\r\nGiles, \u003ccite\u003eThe Civilisation of China\u003c/cite\u003e (Home University Library), p. 147.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_54\" id=\"Footnote_54\"\u003e[54]\u003c/a\u003e\r\nCf. \u003ccite lang=\"la\"\u003ePrincipia Mathematica\u003c/cite\u003e, § 20, and Introduction, chapter iii.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_55\" id=\"Footnote_55\"\u003e[55]\u003c/a\u003e\r\nIn the above remarks I am making use of unpublished work by my\r\nfriend Ludwig Wittgenstein.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_56\" id=\"Footnote_56\"\u003e[56]\u003c/a\u003e\r\nThus we are not using “thing” here in the sense of a class of correlated\r\n“aspects,” as we did in \u003ca href=\"#Lecture_3\" class=\"pginternal\"\u003eLecture III\u003c/a\u003e. Each “aspect” will count\r\nseparately in stating causal laws.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cdiv class=\"footnote\"\u003e\r\n\u003cp\u003e\u003ca class=\"label pginternal\" href=\"#FNanchor_57\" id=\"Footnote_57\"\u003e[57]\u003c/a\u003e\r\nThe above remarks, for purposes of illustration, adopt one of several\r\npossible opinions on each of several disputed points.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003c/div\u003e\r\n\u003cdiv id=\"tnote-bottom\"\u003e\r\n\u003cp class=\"center\"\u003e\u003ca id=\"tn-bottom\"\u003e\u003cb\u003eTranscriber\u0027s Note:\u003c/b\u003e\u003c/a\u003e\u003c/p\u003e\r\n\u003cp class=\"no-indent\"\u003eThe following is a list of corrections made to the original.\r\nThe first passage is the original passage, the second the corrected one.\u003c/p\u003e\r\n\u003cul id=\"corrections\"\u003e\r\n\u003cli\u003e\u003ca href=\"#advertisement\" class=\"pginternal\"\u003eAdvertisement\u003c/a\u003e:\u003cbr\u003e\r\n\u003ci\u003eSecond \u003cspan class=\"correction\"\u003eImpression\u003c/span\u003e\u003c/i\u003e \u003cspan class=\"float-right\"\u003eCr. 8vo, 6s. net.\u003c/span\u003e\u003cbr\u003e\r\n\u003ci\u003eSecond \u003cspan class=\"correction\"\u003eImpression.\u003c/span\u003e\u003c/i\u003e \u003cspan class=\"float-right\"\u003eCr. 8vo, 6s. net.\u003c/span\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_8\" class=\"pginternal\"\u003ePage 8\u003c/a\u003e:\u003cbr\u003e\r\nimpossibility of alternatives which seemed \u003ci lang=\"la\"\u003e\u003cspan class=\"correction\"\u003eprima\u003c/span\u003e facie\u003c/i\u003e\u003cbr\u003e\r\nimpossibility of alternatives which seemed \u003ci lang=\"la\"\u003e\u003cspan class=\"correction\"\u003eprimâ\u003c/span\u003e facie\u003c/i\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_119\" class=\"pginternal\"\u003ePage 119\u003c/a\u003e:\u003cbr\u003e\r\nwith it. We will call these the “initial \u003cspan class=\"correction\"\u003econtemporaries\u003c/span\u003e\u003cbr\u003e\r\nwith it. We will call these the “initial \u003cspan class=\"correction\"\u003econtemporaries”\u003c/span\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_197\" class=\"pginternal\"\u003ePage 197\u003c/a\u003e:\u003cbr\u003e\r\nof infinite \u003cspan class=\"correction\"\u003enumber\u003c/span\u003e. Many of the most\u003cbr\u003e\r\nof infinite \u003cspan class=\"correction\"\u003enumbers\u003c/span\u003e. Many of the most\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_200\" class=\"pginternal\"\u003ePage 200\u003c/a\u003e:\u003cbr\u003e\r\n\u003cspan class=\"correction\"\u003epscyhical\u003c/span\u003e processes as the North Sea…. The botanist\u003cbr\u003e\r\n\u003cspan class=\"correction\"\u003epsychical\u003c/span\u003e processes as the North Sea…. The botanist\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_215\" class=\"pginternal\"\u003ePage 215\u003c/a\u003e:\u003cbr\u003e\r\nsomething which existed a quarter of \u003cspan class=\"correction\"\u003ea\u003c/span\u003e hour ago, the\u003cbr\u003e\r\nsomething which existed a quarter of \u003cspan class=\"correction\"\u003ean\u003c/span\u003e hour ago, the\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_244\" class=\"pginternal\"\u003ePage 244\u003c/a\u003e:\u003cbr\u003e\r\nIntelligence, how displayed by friends, \u003cspan class=\"correction\"\u003e93\u003c/span\u003e\u003cbr\u003e\r\nIntelligence, how displayed by friends, \u003cspan class=\"correction\"\u003e93.\u003c/span\u003e\r\n\u003c/li\u003e\r\n\u003cli\u003e\u003ca href=\"#Page_244\" class=\"pginternal\"\u003ePage 244\u003c/a\u003e:\u003cbr\u003e\r\nNumber, cardinal, \u003cspan class=\"correction\"\u003e131\u003c/span\u003e 186 ff.\u003cbr\u003e\r\nNumber, cardinal, \u003cspan class=\"correction\"\u003e131,\u003c/span\u003e 186 ff.\r\n\u003c/li\u003e\r\n\u003c/ul\u003e\r\n\u003c/div\u003e\r\n\u003cpre\u003e\u003c/pre\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}}