Introduction to Mathematical Philosophy
{"WorkMasterId":5231,"WpPageId":252990,"ParentWpPageId":189742,"Slug":"introduction-to-mathematical-philosophy","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/introduction-to-mathematical-philosophy/","RelativeUrl":"theos/humanities/philosophy/philosophers/bertrand-russell/introduction-to-mathematical-philosophy/","HasFullText":true,"RawHtmlLength":1108281,"CleanHtmlLength":1052171,"Kicker":"Philosophy Work","Title":"Introduction to Mathematical Philosophy","Deck":"Russell offers a more accessible account of number, order, infinity, classes, descriptions, and the logicist treatment of mathematics.","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":"1919 CE","Url":"","DateText":""}],"DateNote":"Displayed year is the publication year, after Russell composed the book during his Brixton prison sentence.","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:logic"},{"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 #41654 .","Url":"","Label":"","Kicker":"","Cards":[]},"CoreThesis":["Russell offers a more accessible account of number, order, infinity, classes, descriptions, and the logicist treatment of mathematics."],"Classification":{"AlternateTitles":"","KeyConcepts":"Introduction to Mathematical Philosophy; 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 public exposition of Russell\u0027s logical philosophy."],"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 #41654\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/41654\"\u003eOpen full version\u003c/a\u003e\n \u003c/article\u003e\n \u003c/div\u003e"},{"Kind":"TextSection","Title":"Core Thesis","Paragraphs":["Russell offers a more accessible account of number, order, infinity, classes, descriptions, and the logicist treatment of mathematics."]},{"Kind":"FieldSection","Title":"Classification","Fields":[{"Label":"Alternate Titles","Value":""},{"Label":"Key Concepts","Value":"Introduction to Mathematical Philosophy; 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 public exposition of Russell\u0027s logical philosophy."]},{"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/41654\"\u003eProject Gutenberg eBook #41654\u003c/a\u003e.\u003c/p\u003e\n \u003carticle class=\"dz-philo__full-text-body\"\u003e\r\n\u003cdiv class=\"figcenter width500\" style=\"width: 1524px;\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-cover.jpg\" width=\"1524\" height=\"2560\" alt=\"Title page of the book Introduction to Mathematical Philosophy.\"\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003cp class=\"center\"\u003eLibrary of Philosophy\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003ci\u003eEDITED BY J. H. MUIRHEAD, LL.D.\u003c/i\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003cb\u003eINTRODUCTION TO MATHEMATICAL\u003cbr\u003e\r\nPHILOSOPHY\u003c/b\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003cp class=\"center\"\u003e\u003ci\u003eBy the same Author.\u003c/i\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\nPRINCIPLES OF SOCIAL RECONSTRUCTION.\r\n\u003ci\u003e3rd Impression.\u003c/i\u003e Demy 8vo. 7s. 6d.\r\nnet.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\"Mr Russell has written a big and living book.\"\u0026mdash;\u003ci\u003eThe\r\nNation\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\nROADS TO FREEDOM: SOCIALISM,\r\nANARCHISM, AND SYNDICALISM. Demy 8vo.\r\n7s. 6d. net.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\nAn attempt to extract the essence of these three doctrines,\r\nfirst historically, then as guidance for the coming reconstruction.\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003ci\u003eLondon: George Allen \u0026amp; Unwin, Ltd.\u003c/i\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_iii\"\u003e[Pg iii]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch1\u003eINTRODUCTION TO\r\nMATHEMATICAL PHILOSOPHY\u003c/h1\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003cb\u003eBY\u003c/b\u003e\u003c/p\u003e\r\n\r\n\u003cdiv style=\u0027text-align:center; font-size:1.2em;\u0027\u003eBERTRAND RUSSELL\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003cb\u003eLONDON: GEORGE ALLEN \u0026amp; UNWIN, LTD.\u003c/b\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003cb\u003eNEW YORK: THE MACMILLAN CO.\u003c/b\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_iv\"\u003e[Pg iv]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003ci\u003eFirst published May\u003c/i\u003e 1919\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\u003ci\u003eSecond Edition April\u003c/i\u003e 1920\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e[All rights reserved]\r\n\u003cspan class=\"pagenum\" id=\"Page_v\"\u003e[Pg v]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027PREFACE\u0027\u003e\u003ca id=\"PREFACE\"\u003ePREFACE\u003c/a\u003e\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHIS book is intended essentially as an \"Introduction,\" and\r\ndoes not aim at giving an exhaustive discussion of the problems\r\nwith which it deals. It seemed desirable to set forth certain\r\nresults, hitherto only available to those who have mastered\r\nlogical symbolism, in a form offering the minimum of difficulty\r\nto the beginner. The utmost endeavour has been made to\r\navoid dogmatism on such questions as are still open to serious\r\ndoubt, and this endeavour has to some extent dominated the\r\nchoice of topics considered. The beginnings of mathematical\r\nlogic are less definitely known than its later portions, but are of\r\nat least equal philosophical interest. Much of what is set forth\r\nin the following chapters is not properly to be called \"philosophy,\"\r\nthough the matters concerned were included in philosophy so\r\nlong as no satisfactory science of them existed. The nature of\r\ninfinity and continuity, for example, belonged in former days\r\nto philosophy, but belongs now to mathematics. Mathematical\r\n\u003ci\u003ephilosophy\u003c/i\u003e, in the strict sense, cannot, perhaps, be held to include\r\nsuch definite scientific results as have been obtained in this\r\nregion; the philosophy of mathematics will naturally be expected\r\nto deal with questions on the frontier of knowledge, as\r\nto which comparative certainty is not yet attained. But\r\nspeculation on such questions is hardly likely to be fruitful\r\nunless the more scientific parts of the principles of mathematics\r\nare known. A book dealing with those parts may, therefore,\r\nclaim to be an \u003ci\u003eintroduction\u003c/i\u003e to mathematical philosophy, though\r\nit can hardly claim, except where it steps outside its province,\r\n\u003cspan class=\"pagenum\" id=\"Page_vi\"\u003e[Pg vi]\u003c/span\u003e\r\nto be actually dealing with a part of philosophy. It does deal,\r\nhowever, with a body of knowledge which, to those who accept\r\nit, appears to invalidate much traditional philosophy, and even\r\na good deal of what is current in the present day. In this way,\r\nas well as by its bearing on still unsolved problems, mathematical\r\nlogic is relevant to philosophy. For this reason, as well as on\r\naccount of the intrinsic importance of the subject, some purpose\r\nmay be served by a succinct account of the main results of\r\nmathematical logic in a form requiring neither a knowledge of\r\nmathematics nor an aptitude for mathematical symbolism.\r\nHere, however, as elsewhere, the method is more important than\r\nthe results, from the point of view of further research; and the\r\nmethod cannot well be explained within the framework of such\r\na book as the following. It is to be hoped that some readers\r\nmay be sufficiently interested to advance to a study of the\r\nmethod by which mathematical logic can be made helpful in\r\ninvestigating the traditional problems of philosophy. But that\r\nis a topic with which the following pages have not attempted\r\nto deal.\r\n\u003c/p\u003e\r\n\r\n\u003cp style=\"text-align:right\"\u003eBERTRAND RUSSELL.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_vii\"\u003e[Pg vii]\u003c/span\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027EDITORS NOTE\u0027\u003e\u003ca id=\"EDITORS_NOTE\"\u003eEDITOR\u0027S NOTE\u003c/a\u003e\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHOSE who, relying on the distinction between Mathematical\r\nPhilosophy and the Philosophy of Mathematics, think that this\r\nbook is out of place in the present Library, may be referred to\r\nwhat the author himself says on this head in the Preface. It is\r\nnot necessary to agree with what he there suggests as to the\r\nreadjustment of the field of philosophy by the transference from\r\nit to mathematics of such problems as those of class, continuity,\r\ninfinity, in order to perceive the bearing of the definitions and\r\ndiscussions that follow on the work of \"traditional philosophy.\"\r\nIf philosophers cannot consent to relegate the criticism of these\r\ncategories to any of the special sciences, it is essential, at any\r\nrate, that they should know the precise meaning that the science\r\nof mathematics, in which these concepts play so large a part,\r\nassigns to them. If, on the other hand, there be mathematicians\r\nto whom these definitions and discussions seem to be an elaboration\r\nand complication of the simple, it may be well to remind\r\nthem from the side of philosophy that here, as elsewhere, apparent\r\nsimplicity may conceal a complexity which it is the business of\r\nsomebody, whether philosopher or mathematician, or, like the\r\nauthor of this volume, both in one, to unravel.\r\n\u003cspan class=\"pagenum\" id=\"Page_viii\"\u003e[Pg viii]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2\u003eCONTENTS\u003c/h2\u003e\r\n\u003cp class=\"nind\"\u003e\r\nCHAP.\u003cbr\u003e\r\n\u003ca href=\"#PREFACE\"\u003ePREFACE\u003c/a\u003e\u003cbr\u003e\r\n\r\n\u003ca href=\"#EDITORS_NOTE\"\u003eEDITOR\u0027S NOTE\u003c/a\u003e\u003cbr\u003e\r\n\r\n1. \u003ca href=\"#chap01\"\u003eTHE SERIES OF NATURAL NUMBERS\u003c/a\u003e\u003cbr\u003e\r\n\r\n2. \u003ca href=\"#chap02\"\u003eDEFINITION OF NUMBER\u003c/a\u003e\u003cbr\u003e\r\n\r\n3. \u003ca href=\"#chap03\"\u003eFINITUDE AND MATHEMATICAL INDUCTION\u003c/a\u003e\u003cbr\u003e\r\n\r\n4. \u003ca href=\"#chap04\"\u003eTHE DEFINITION OF ORDER\u003c/a\u003e\u003cbr\u003e\r\n\r\n5. \u003ca href=\"#chap05\"\u003eKINDS OF RELATIONS\u003c/a\u003e\u003cbr\u003e\r\n\r\n6. \u003ca href=\"#chap06\"\u003eSIMILARITY OF RELATIONS\u003c/a\u003e\u003cbr\u003e\r\n\r\n7. \u003ca href=\"#chap07\"\u003eRATIONAL, REAL, AND COMPLEX NUMBERS\u003c/a\u003e\u003cbr\u003e\r\n\r\n8. \u003ca href=\"#chap08\"\u003eINFINITE CARDINAL NUMBERS\u003c/a\u003e\u003cbr\u003e\r\n\r\n9. \u003ca href=\"#chap09\"\u003eINFINITE SERIES AND ORDINALS\u003c/a\u003e\u003cbr\u003e\r\n\r\n10. \u003ca href=\"#chap10\"\u003eLIMITS AND CONTINUITY\u003c/a\u003e\u003cbr\u003e\r\n\r\n11. \u003ca href=\"#chap11\"\u003eLIMITS AND CONTINUITY OF FUNCTIONS\u003c/a\u003e\u003cbr\u003e\r\n\r\n12. \u003ca href=\"#chap12\"\u003eSELECTIONS AND THE MULTIPLICATIVE AXIOM\u003c/a\u003e\u003cbr\u003e\r\n\r\n13. \u003ca href=\"#chap13\"\u003eTHE AXIOM OF INFINITY AND LOGICAL TYPES\u003c/a\u003e\u003cbr\u003e\r\n\r\n14. \u003ca href=\"#chap14\"\u003eINCOMPATIBILITY AND THE THEORY OF DEDUCTION\u003c/a\u003e\u003cbr\u003e\r\n\r\n15. \u003ca href=\"#chap15\"\u003ePROPOSITIONAL FUNCTIONS\u003c/a\u003e\u003cbr\u003e\r\n\r\n16. \u003ca href=\"#chap16\"\u003eDESCRIPTIONS\u003c/a\u003e\u003cbr\u003e\r\n\r\n17. \u003ca href=\"#chap17\"\u003eCLASSES\u003c/a\u003e\u003cbr\u003e\r\n\r\n18. \u003ca href=\"#chap18\"\u003eMATHEMATICS AND LOGIC\u003c/a\u003e\u003cbr\u003e\r\n\r\n\u003ca href=\"#INDEX\"\u003eINDEX\u003c/a\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_ix\"\u003e[Pg ix]\u003c/span\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2\u003eINTRODUCTION TO MATHEMATICAL PHILOSOPHY\u003c/h2\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027I: THE SERIES OF NATURAL NUMBERS\u0027\u003e\u003ca id=\"chap01\"\u003e\u003c/a\u003eCHAPTER I\r\n\u003cbr\u003e\u003cbr\u003e\r\nTHE SERIES OF NATURAL NUMBERS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nMATHEMATICS is a study which, when we start from its most\r\nfamiliar portions, may be pursued in either of two opposite\r\ndirections. The more familiar direction is constructive, towards\r\ngradually increasing complexity: from integers to fractions,\r\nreal numbers, complex numbers; from addition and multiplication\r\nto differentiation and integration, and on to higher\r\nmathematics. The other direction, which is less familiar,\r\nproceeds, by analysing, to greater and greater abstractness\r\nand logical simplicity; instead of asking what can be defined\r\nand deduced from what is assumed to begin with, we ask instead\r\nwhat more general ideas and principles can be found, in terms\r\nof which what was our starting-point can be defined or deduced.\r\nIt is the fact of pursuing this opposite direction that characterises\r\nmathematical philosophy as opposed to ordinary mathematics.\r\nBut it should be understood that the distinction is one, not in\r\nthe subject matter, but in the state of mind of the investigator.\r\nEarly Greek geometers, passing from the empirical rules of\r\nEgyptian land-surveying to the general propositions by which\r\nthose rules were found to be justifiable, and thence to Euclid\u0027s\r\naxioms and postulates, were engaged in mathematical philosophy,\r\naccording to the above definition; but when once the\r\naxioms and postulates had been reached, their deductive employment,\r\nas we find it in Euclid, belonged to mathematics in the\r\n\u003cspan class=\"pagenum\" id=\"Page_1\"\u003e[Pg 1]\u003c/span\u003e\r\nordinary sense. The distinction between mathematics and\r\nmathematical philosophy is one which depends upon the interest\r\ninspiring the research, and upon the stage which the research\r\nhas reached; not upon the propositions with which the research\r\nis concerned.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may state the same distinction in another way. The\r\nmost obvious and easy things in mathematics are not those that\r\ncome logically at the beginning; they are things that, from\r\nthe point of view of logical deduction, come somewhere in the\r\nmiddle. Just as the easiest bodies to see are those that are\r\nneither very near nor very far, neither very small nor very\r\ngreat, so the easiest conceptions to grasp are those that are\r\nneither very complex nor very simple (using \"simple\" in a\r\n\u003ci\u003elogical\u003c/i\u003e sense). And as we need two sorts of instruments, the\r\ntelescope and the microscope, for the enlargement of our visual\r\npowers, so we need two sorts of instruments for the enlargement\r\nof our logical powers, one to take us forward to the higher\r\nmathematics, the other to take us backward to the logical\r\nfoundations of the things that we are inclined to take for granted\r\nin mathematics. We shall find that by analysing our ordinary\r\nmathematical notions we acquire fresh insight, new powers,\r\nand the means of reaching whole new mathematical subjects\r\nby adopting fresh lines of advance after our backward journey.\r\nIt is the purpose of this book to explain mathematical philosophy\r\nsimply and untechnically, without enlarging upon those\r\nportions which are so doubtful or difficult that an elementary\r\ntreatment is scarcely possible. A full treatment will be found\r\nin \u003ci\u003ePrincipia Mathematica\u003c/i\u003e;\u003ca id=\"FNanchor_1_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_1_1\" class=\"fnanchor\"\u003e[1]\u003c/a\u003e\r\nthe treatment in the present volume is intended merely as an\r\nintroduction.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_1_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_1_1\"\u003e\u003cspan class=\"label\"\u003e[1]\u003c/span\u003e\u003c/a\u003eCambridge University Press, vol. I., 1910; vol. II., 1911; vol. III., 1913.\r\nBy Whitehead and Russell.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nTo the average educated person of the present day, the\r\nobvious starting-point of mathematics would be the series of\r\nwhole numbers,\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 18.301ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-1.png\" alt=\"\" data-tex=\"\r\n1,\\ 2,\\ 3,\\ 4,\\ \\dots\\ \\text{etc.}\r\n\"\u003e\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_2\"\u003e[Pg 2]\u003c/span\u003e\r\nProbably only a person with some mathematical knowledge\r\nwould think of beginning with 0 instead of with 1, but we will\r\npresume this degree of knowledge; we will take as our starting-point\r\nthe series:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 26.624ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-2.png\" alt=\"\" data-tex=\"\r\n0,\\ 1,\\ 2,\\ 3,\\ \\dots\\ n,\\ n + 1,\\dots\r\n\"\u003e\u003c/span\u003e\r\nand it is this series that we shall mean when we speak of the\r\n\"series of natural numbers.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is only at a high stage of civilisation that we could take\r\nthis series as our starting-point. It must have required many\r\nages to discover that a brace of pheasants and a couple of days\r\nwere both instances of the number 2: the degree of abstraction\r\ninvolved is far from easy. And the discovery that 1 is a number\r\nmust have been difficult. As for 0, it is a very recent addition;\r\nthe Greeks and Romans had no such digit. If we had been\r\nembarking upon mathematical philosophy in earlier days, we\r\nshould have had to start with something less abstract than the\r\nseries of natural numbers, which we should reach as a stage on\r\nour backward journey. When the logical foundations of mathematics\r\nhave grown more familiar, we shall be able to start further\r\nback, at what is now a late stage in our analysis. But for the\r\nmoment the natural numbers seem to represent what is easiest\r\nand most familiar in mathematics.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut though familiar, they are not understood. Very few\r\npeople are prepared with a definition of what is meant by\r\n\"number,\" or \"0,\" or \"1.\" It is not very difficult to see that,\r\nstarting from 0, any other of the natural numbers can be reached\r\nby repeated additions of 1, but we shall have to define what\r\nwe mean by \"adding 1,\" and what we mean by \"repeated.\"\r\nThese questions are by no means easy. It was believed until\r\nrecently that some, at least, of these first notions of arithmetic\r\nmust be accepted as too simple and primitive to be defined.\r\nSince all terms that are defined are defined by means of other\r\nterms, it is clear that human knowledge must always be content\r\nto accept some terms as intelligible without definition, in order\r\n\u003cspan class=\"pagenum\" id=\"Page_3\"\u003e[Pg 3]\u003c/span\u003e\r\nto have a starting-point for its definitions. It is not clear that\r\nthere must be terms which are \u003ci\u003eincapable\u003c/i\u003e of definition: it is\r\npossible that, however far back we go in defining, we always\r\n\u003ci\u003emight\u003c/i\u003e go further still. On the other hand, it is also possible\r\nthat, when analysis has been pushed far enough, we can reach\r\nterms that really are simple, and therefore logically incapable\r\nof the sort of definition that consists in analysing. This is a\r\nquestion which it is not necessary for us to decide; for our\r\npurposes it is sufficient to observe that, since human powers\r\nare finite, the definitions known to us must always begin somewhere,\r\nwith terms undefined for the moment, though perhaps\r\nnot permanently.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAll traditional pure mathematics, including analytical geometry,\r\nmay be regarded as consisting wholly of propositions\r\nabout the natural numbers. That is to say, the terms which\r\noccur can be defined by means of the natural numbers, and\r\nthe propositions can be deduced from the properties of the\r\nnatural numbers\u0026mdash;with the addition, in each case, of the ideas\r\nand propositions of pure logic.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThat all traditional pure mathematics can be derived from\r\nthe natural numbers is a fairly recent discovery, though it had\r\nlong been suspected. Pythagoras, who believed that not only\r\nmathematics, but everything else could be deduced from\r\nnumbers, was the discoverer of the most serious obstacle in\r\nthe way of what is called the \"arithmetising\" of mathematics.\r\nIt was Pythagoras who discovered the existence of incommensurables,\r\nand, in particular, the incommensurability of the\r\nside of a square and the diagonal. If the length of the side is\r\n1 inch, the number of inches in the diagonal is the square root\r\nof 2, which appeared not to be a number at all. The problem\r\nthus raised was solved only in our own day, and was only solved\r\n\u003ci\u003ecompletely\u003c/i\u003e by the help of the reduction of arithmetic to logic,\r\nwhich will be explained in following chapters. For the present,\r\nwe shall take for granted the arithmetisation of mathematics,\r\nthough this was a feat of the very greatest importance.\r\n\u003cspan class=\"pagenum\" id=\"Page_4\"\u003e[Pg 4]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nHaving reduced all traditional pure mathematics to the\r\ntheory of the natural numbers, the next step in logical analysis\r\nwas to reduce this theory itself to the smallest set of premisses\r\nand undefined terms from which it could be derived. This work\r\nwas accomplished by Peano. He showed that the entire theory\r\nof the natural numbers could be derived from three primitive\r\nideas and five primitive propositions in addition to those of\r\npure logic. These three ideas and five propositions thus became,\r\nas it were, hostages for the whole of traditional pure mathematics.\r\nIf they could be defined and proved in terms of others,\r\nso could all pure mathematics. Their logical \"weight,\" if one\r\nmay use such an expression, is equal to that of the whole series\r\nof sciences that have been deduced from the theory of the natural\r\nnumbers; the truth of this whole series is assured if the truth\r\nof the five primitive propositions is guaranteed, provided, of\r\ncourse, that there is nothing erroneous in the purely logical\r\napparatus which is also involved. The work of analysing mathematics\r\nis extraordinarily facilitated by this work of Peano\u0027s.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe three primitive ideas in Peano\u0027s arithmetic are:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 20.663ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-3.png\" alt=\"\" data-tex=\"\r\n\\text{0, number, successor.}\r\n\"\u003e\u003c/span\u003e\r\nBy \"successor\" he means the next number in the natural\r\norder. That is to say, the successor of 0 is 1, the successor of\r\n1 is 2, and so on. By \"number\" he means, in this connection,\r\nthe class of the natural numbers.\u003ca id=\"FNanchor_2_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_2_1\" class=\"fnanchor\"\u003e[2]\u003c/a\u003e\r\nHe is not assuming that\r\nwe know all the members of this class, but only that we know\r\nwhat we mean when we say that this or that is a number, just\r\nas we know what we mean when we say \"Jones is a man,\"\r\nthough we do not know all men individually.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_2_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_2_1\"\u003e\u003cspan class=\"label\"\u003e[2]\u003c/span\u003e\u003c/a\u003eWe shall use \"number\" in this sense in the present chapter. Afterwards\r\nthe word will be used in a more general sense.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe five primitive propositions which Peano assumes are:\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(1) 0 is a number.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(2) The successor of any number is a number.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(3) No two numbers have the same successor.\r\n\u003cspan class=\"pagenum\" id=\"Page_5\"\u003e[Pg 5]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(4) 0 is not the successor of any number.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(5) Any property which belongs to 0, and also to the successor\r\nof every number which has the property, belongs to all\r\nnumbers.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nThe last of these is the principle of mathematical induction.\r\nWe shall have much to say concerning mathematical induction\r\nin the sequel; for the present, we are concerned with it only\r\nas it occurs in Peano\u0027s analysis of arithmetic.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us consider briefly the kind of way in which the theory\r\nof the natural numbers results from these three ideas and five\r\npropositions. To begin with, we define 1 as \"the successor of 0,\"\r\n2 as \"the successor of 1,\" and so on. We can obviously go\r\non as long as we like with these definitions, since, in virtue of (2),\r\nevery number that we reach will have a successor, and, in\r\nvirtue of (3), this cannot be any of the numbers already defined,\r\nbecause, if it were, two different numbers would have the same\r\nsuccessor; and in virtue of (4) none of the numbers we reach\r\nin the series of successors can be 0. Thus the series of successors\r\ngives us an endless series of continually new numbers. In virtue\r\nof (5) all numbers come in this series, which begins with 0 and\r\ntravels on through successive successors: for (\u003ci\u003ea\u003c/i\u003e) 0 belongs to\r\nthis series, and (\u003ci\u003eb\u003c/i\u003e) if a number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e belongs to it, so does its successor,\r\nwhence, by mathematical induction, every number belongs to\r\nthe series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSuppose we wish to define the sum of two numbers. Taking\r\nany number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e,\u003c/span\u003e we define \u003cimg style=\"vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-48.png\" alt=\"\" data-tex=\"m + 0\"\u003e as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e,\u003c/span\u003e and\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 11.767ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-49.png\" alt=\"\" data-tex=\"m + (n + 1)\"\u003e as the\r\nsuccessor of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-50.png\" alt=\"\" data-tex=\"m + n\"\u003e.\u003c/span\u003e In virtue of (5) this gives a definition of\r\nthe sum of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e whatever number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be. Similarly\r\nwe can define the product of any two numbers. The reader can\r\neasily convince himself that any ordinary elementary proposition\r\nof arithmetic can be proved by means of our five premisses,\r\nand if he has any difficulty he can find the proof in Peano.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is time now to turn to the considerations which make it\r\nnecessary to advance beyond the standpoint of Peano, who\r\n\u003cspan class=\"pagenum\" id=\"Page_6\"\u003e[Pg 6]\u003c/span\u003e\r\nrepresents the last perfection of the \"arithmetisation\" of\r\nmathematics, to that of Frege, who first succeeded in \"logicising\"\r\nmathematics, \u003ci\u003ei.e.\u003c/i\u003e in reducing to logic the arithmetical notions\r\nwhich his predecessors had shown to be sufficient for mathematics.\r\nWe shall not, in this chapter, actually give Frege\u0027s definition of\r\nnumber and of particular numbers, but we shall give some of the\r\nreasons why Peano\u0027s treatment is less final than it appears to be.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the first place, Peano\u0027s three primitive ideas\u0026mdash;namely, \"0,\"\r\n\"number,\" and \"successor\"\u0026mdash;are capable of an infinite number\r\nof different interpretations, all of which will satisfy the five\r\nprimitive propositions. We will give some examples.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) Let \"0\" be taken to mean 100, and let \"number\" be\r\ntaken to mean the numbers from 100 onward in the series of\r\nnatural numbers. Then all our primitive propositions are\r\nsatisfied, even the fourth, for, though 100 is the successor of 99,\r\n99 is not a \"number\" in the sense which we are now giving\r\nto the word \"number.\" It is obvious that any number may be\r\nsubstituted for 100 in this example.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) Let \"0\" have its usual meaning, but let \"number\"\r\nmean what we usually call \"even numbers,\" and let the\r\n\"successor\" of a number be what results from adding two to\r\nit. Then \"1\" will stand for the number two, \"2\" will stand\r\nfor the number four, and so on; the series of \"numbers\" now\r\nwill be\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.466ex; width: 25.175ex; height: 2.061ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-4.png\" alt=\"\" data-tex=\"\r\n\\text{0, two, four, six, eight} \\dots.\r\n\"\u003e\u003c/span\u003e\r\nAll Peano\u0027s five premisses are satisfied still.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) Let \"0\" mean the number one, let \"number\" mean\r\nthe set\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -1.602ex; width: 21.656ex; height: 4.638ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-5.png\" alt=\"\" data-tex=\"\r\n1,\\ \\dfrac{1}{2},\\ \\dfrac{1}{4},\\ \\dfrac{1}{8},\\ \\dfrac{1}{16},\\ \\dots\r\n\"\u003e\u003c/span\u003e\r\nand let \"successor\" mean \"half.\" Then all Peano\u0027s five\r\naxioms will be true of this set.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that such examples might be multiplied indefinitely.\r\nIn fact, given any series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 26.427ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-6.png\" alt=\"\" data-tex=\"\r\nx_{0},\\ x_{1},\\ x_{2},\\ x_{3},\\ \\dots\\ x_{n},\\ \\dots\r\n\"\u003e\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_7\"\u003e[Pg 7]\u003c/span\u003e\r\nwhich is endless, contains no repetitions, has a beginning, and\r\nhas no terms that cannot be reached from the beginning in a\r\nfinite number of steps, we have a set of terms verifying Peano\u0027s\r\naxioms. This is easily seen, though the formal proof is somewhat\r\nlong. Let \"0\" mean \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-51.png\" alt=\"\" data-tex=\"x_{0}\"\u003e,\u003c/span\u003e let \"number\" mean the whole\r\nset of terms, and let the \"successor\" of \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-52.png\" alt=\"\" data-tex=\"x_{n}\"\u003e mean \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-53.png\" alt=\"\" data-tex=\"x_{n+1}\"\u003e.\u003c/span\u003e Then\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) \"0 is a number,\" \u003ci\u003ei.e.\u003c/i\u003e \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-51.png\" alt=\"\" data-tex=\"x_{0}\"\u003e is a member of the set.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) \"The successor of any number is a number,\" \u003ci\u003ei.e.\u003c/i\u003e taking\r\nany term \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-52.png\" alt=\"\" data-tex=\"x_{n}\"\u003e in the set, \u003cimg style=\"vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-53.png\" alt=\"\" data-tex=\"x_{n+1}\"\u003e is also in the set.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) \"No two numbers have the same successor,\" \u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.357ex; width: 2.887ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-54.png\" alt=\"\" data-tex=\"x_{m}\"\u003e\r\nand \u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-52.png\" alt=\"\" data-tex=\"x_{n}\"\u003e are two different members of the set, \u003cimg style=\"vertical-align: -0.471ex; width: 4.931ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-55.png\" alt=\"\" data-tex=\"x_{m+1}\"\u003e and\r\n\u003cimg style=\"vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-53.png\" alt=\"\" data-tex=\"x_{n+1}\"\u003e are\r\ndifferent; this results from the fact that (by hypothesis) there\r\nare no repetitions in the set.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(4) \"0 is not the successor of any number,\" \u003ci\u003ei.e.\u003c/i\u003e no term in\r\nthe set comes before \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-51.png\" alt=\"\" data-tex=\"x_{0}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(5) This becomes: Any property which belongs to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-51.png\" alt=\"\" data-tex=\"x_{0}\"\u003e,\u003c/span\u003e and\r\nbelongs to \u003cimg style=\"vertical-align: -0.471ex; width: 4.486ex; height: 1.471ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-53.png\" alt=\"\" data-tex=\"x_{n+1}\"\u003e provided it belongs to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-52.png\" alt=\"\" data-tex=\"x_{n}\"\u003e,\u003c/span\u003e belongs\r\nto all the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003es.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis follows from the corresponding property for numbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series of the form\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 22.574ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-7.png\" alt=\"\" data-tex=\"\r\nx_{0},\\ x_{1},\\ x_{2},\\ \\dots\\ x_{n},\\ \\dots\r\n\"\u003e\u003c/span\u003e\r\nin which there is a first term, a successor to each term (so that\r\nthere is no last term), no repetitions, and every term can be\r\nreached from the start in a finite number of steps, is called a\r\n\u003ci\u003eprogression\u003c/i\u003e. Progressions are of great importance in the principles\r\nof mathematics. As we have just seen, every progression\r\nverifies Peano\u0027s five axioms. It can be proved, conversely,\r\nthat every series which verifies Peano\u0027s five axioms is a progression.\r\nHence these five axioms may be used to define the\r\nclass of progressions: \"progressions\" are \"those series which\r\nverify these five axioms.\" Any progression may be taken as\r\nthe basis of pure mathematics: we may give the name \"0\"\r\nto its first term, the name \"number\" to the whole set of its\r\nterms, and the name \"successor\" to the next in the progression.\r\nThe progression need not be composed of numbers: it may be\r\n\u003cspan class=\"pagenum\" id=\"Page_8\"\u003e[Pg 8]\u003c/span\u003e\r\ncomposed of points in space, or moments of time, or any other\r\nterms of which there is an infinite supply. Each different\r\nprogression will give rise to a different interpretation of all the\r\npropositions of traditional pure mathematics; all these possible\r\ninterpretations will be equally true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn Peano\u0027s system there is nothing to enable us to distinguish\r\nbetween these different interpretations of his primitive ideas.\r\nIt is assumed that we know what is meant by \"0,\" and that\r\nwe shall not suppose that this symbol means 100 or Cleopatra\u0027s\r\nNeedle or any of the other things that it might mean.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis point, that \"0\" and \"number\" and \"successor\"\r\ncannot be defined by means of Peano\u0027s five axioms, but must\r\nbe independently understood, is important. We want our\r\nnumbers not merely to verify mathematical formulæ, but to\r\napply in the right way to common objects. We want to have\r\nten fingers and two eyes and one nose. A system in which \"1\"\r\nmeant 100, and \"2\" meant 101, and so on, might be all right\r\nfor pure mathematics, but would not suit daily life. We want\r\n\"0\" and \"number\" and \"successor\" to have meanings which\r\nwill give us the right allowance of fingers and eyes and noses.\r\nWe have already some knowledge (though not sufficiently\r\narticulate or analytic) of what we mean by \"1\" and \"2\" and\r\nso on, and our use of numbers in arithmetic must conform to\r\nthis knowledge. We cannot secure that this shall be the case\r\nby Peano\u0027s method; all that we can do, if we adopt his method,\r\nis to say \"we know what we mean by \u00270\u0027 and \u0027number\u0027 and\r\n\u0027successor,\u0027 though we cannot explain what we mean in terms\r\nof other simpler concepts.\" It is quite legitimate to say this\r\nwhen we must, and at \u003ci\u003esome\u003c/i\u003e point we all must; but it is the\r\nobject of mathematical philosophy to put off saying it as long\r\nas possible. By the logical theory of arithmetic we are able to\r\nput it off for a very long time.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt might be suggested that, instead of setting up \"0\" and\r\n\"number\" and \"successor\" as terms of which we know the\r\nmeaning although we cannot define them, we might let them\r\n\u003cspan class=\"pagenum\" id=\"Page_9\"\u003e[Pg 9]\u003c/span\u003e\r\nstand for \u003ci\u003eany\u003c/i\u003e three terms that verify Peano\u0027s five axioms. They\r\nwill then no longer be terms which have a meaning that is definite\r\nthough undefined: they will be \"variables,\" terms concerning\r\nwhich we make certain hypotheses, namely, those stated in the\r\nfive axioms, but which are otherwise undetermined. If we adopt\r\nthis plan, our theorems will not be proved concerning an ascertained\r\nset of terms called \"the natural numbers,\" but concerning\r\nall sets of terms having certain properties. Such a procedure\r\nis not fallacious; indeed for certain purposes it represents a\r\nvaluable generalisation. But from two points of view it fails\r\nto give an adequate basis for arithmetic. In the first place, it\r\ndoes not enable us to know whether there are any sets of terms\r\nverifying Peano\u0027s axioms; it does not even give the faintest\r\nsuggestion of any way of discovering whether there are such sets.\r\nIn the second place, as already observed, we want our numbers\r\nto be such as can be used for counting common objects, and this\r\nrequires that our numbers should have a \u003ci\u003edefinite\u003c/i\u003e meaning, not\r\nmerely that they should have certain formal properties. This\r\ndefinite meaning is defined by the logical theory of arithmetic.\r\n\u003cspan class=\"pagenum\" id=\"Page_10\"\u003e[Pg 10]\u003c/span\u003e\r\n\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027II: DEFINITION OF NUMBER\u0027\u003e\u003ca id=\"chap02\"\u003e\u003c/a\u003eCHAPTER II\r\n\u003cbr\u003e\u003cbr\u003e\r\nDEFINITION OF NUMBER\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHE question \"What is a number?\" is one which has been\r\noften asked, but has only been correctly answered in our own\r\ntime. The answer was given by Frege in 1884, in his \u003ci\u003eGrundlagen\r\nder Arithmetik\u003c/i\u003e.\u003ca id=\"FNanchor_3_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_3_1\" class=\"fnanchor\"\u003e[3]\u003c/a\u003e\r\nAlthough this book is quite short, not difficult,\r\nand of the very highest importance, it attracted almost no\r\nattention, and the definition of number which it contains remained\r\npractically unknown until it was rediscovered by the\r\npresent author in 1901.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_3_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_3_1\"\u003e\u003cspan class=\"label\"\u003e[3]\u003c/span\u003e\u003c/a\u003eThe same answer is given more fully and with more development in\r\nhis \u003ci\u003eGrundgesetze der Arithmetik\u003c/i\u003e, vol. I., 1893.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIn seeking a definition of number, the first thing to be clear\r\nabout is what we may call the grammar of our inquiry. Many\r\nphilosophers, when attempting to define number, are really\r\nsetting to work to define plurality, which is quite a different\r\nthing. \u003ci\u003eNumber\u003c/i\u003e is what is characteristic of numbers, as \u003ci\u003eman\u003c/i\u003e\r\nis what is characteristic of men. A plurality is not an instance\r\nof number, but of some particular number. A trio of men,\r\nfor example, is an instance of the number 3, and the number 3\r\nis an instance of number; but the trio is not an instance of\r\nnumber. This point may seem elementary and scarcely worth\r\nmentioning; yet it has proved too subtle for the philosophers,\r\nwith few exceptions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA particular number is not identical with any collection of\r\nterms having that number: the number 3 is not identical with\r\n\u003cspan class=\"pagenum\" id=\"Page_11\"\u003e[Pg 11]\u003c/span\u003e\r\nthe trio consisting of Brown, Jones, and Robinson. The number 3\r\nis something which all trios have in common, and which distinguishes\r\nthem from other collections. A number is something\r\nthat characterises certain collections, namely, those that have\r\nthat number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nInstead of speaking of a \"collection,\" we shall as a rule speak\r\nof a \"class,\" or sometimes a \"set.\" Other words used in\r\nmathematics for the same thing are \"aggregate\" and \"manifold.\"\r\nWe shall have much to say later on about classes. For\r\nthe present, we will say as little as possible. But there are\r\nsome remarks that must be made immediately.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA class or collection may be defined in two ways that at first\r\nsight seem quite distinct. We may enumerate its members, as\r\nwhen we say, \"The collection I mean is Brown, Jones, and\r\nRobinson.\" Or we may mention a defining property, as when\r\nwe speak of \"mankind\" or \"the inhabitants of London.\" The\r\ndefinition which enumerates is called a definition by \"extension,\"\r\nand the one which mentions a defining property is called\r\na definition by \"intension.\" Of these two kinds of definition,\r\nthe one by intension is logically more fundamental. This is\r\nshown by two considerations: (1) that the extensional definition\r\ncan always be reduced to an intensional one; (2) that the\r\nintensional one often cannot even theoretically be reduced to\r\nthe extensional one. Each of these points needs a word of\r\nexplanation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) Brown, Jones, and Robinson all of them possess a certain\r\nproperty which is possessed by nothing else in the whole universe,\r\nnamely, the property of being either Brown or Jones or Robinson.\r\nThis property can be used to give a definition by intension of\r\nthe class consisting of Brown and Jones and Robinson. Consider\r\nsuch a formula as \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is Brown or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is Jones or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is Robinson.\"\r\nThis formula will be true for just three \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003es, namely, Brown and\r\nJones and Robinson. In this respect it resembles a cubic equation\r\nwith its three roots. It may be taken as assigning a property\r\ncommon to the members of the class consisting of these three\r\n\u003cspan class=\"pagenum\" id=\"Page_12\"\u003e[Pg 12]\u003c/span\u003e\r\nmen, and peculiar to them. A similar treatment can obviously\r\nbe applied to any other class given in extension.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) It is obvious that in practice we can often know a great\r\ndeal about a class without being able to enumerate its members.\r\nNo one man could actually enumerate all men, or even all the\r\ninhabitants of London, yet a great deal is known about each of\r\nthese classes. This is enough to show that definition by extension\r\nis not \u003ci\u003enecessary\u003c/i\u003e to knowledge about a class. But when we come\r\nto consider infinite classes, we find that enumeration is not even\r\ntheoretically possible for beings who only live for a finite time.\r\nWe cannot enumerate all the natural numbers: they are 0, 1, 2,\r\n3, \u003ci\u003eand so on\u003c/i\u003e. At some point we must content ourselves with\r\n\"and so on.\" We cannot enumerate all fractions or all irrational\r\nnumbers, or all of any other infinite collection. Thus our knowledge\r\nin regard to all such collections can only be derived from a\r\ndefinition by intension.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese remarks are relevant, when we are seeking the definition\r\nof number, in three different ways. In the first place, numbers\r\nthemselves form an infinite collection, and cannot therefore\r\nbe defined by enumeration. In the second place, the collections\r\nhaving a given number of terms themselves presumably form an\r\ninfinite collection: it is to be presumed, for example, that there\r\nare an infinite collection of trios in the world, for if this were\r\nnot the case the total number of things in the world would be\r\nfinite, which, though possible, seems unlikely. In the third\r\nplace, we wish to define \"number\" in such a way that infinite\r\nnumbers may be possible; thus we must be able to speak of\r\nthe number of terms in an infinite collection, and such a collection\r\nmust be defined by intension, \u003ci\u003ei.e.\u003c/i\u003e by a property common to all\r\nits members and peculiar to them.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFor many purposes, a class and a defining characteristic of\r\nit are practically interchangeable. The vital difference between\r\nthe two consists in the fact that there is only one class having a\r\ngiven set of members, whereas there are always many different\r\ncharacteristics by which a given class may be defined. Men\r\n\u003cspan class=\"pagenum\" id=\"Page_13\"\u003e[Pg 13]\u003c/span\u003e\r\nmay be defined as featherless bipeds, or as rational animals,\r\nor (more correctly) by the traits by which Swift delineates the\r\nYahoos. It is this fact that a defining characteristic is never\r\nunique which makes classes useful; otherwise we could be\r\ncontent with the properties common and peculiar to their\r\nmembers.\u003ca id=\"FNanchor_4_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_4_1\" class=\"fnanchor\"\u003e[4]\u003c/a\u003e\r\nAny one of these properties can be used in place\r\nof the class whenever uniqueness is not important.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_4_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_4_1\"\u003e\u003cspan class=\"label\"\u003e[4]\u003c/span\u003e\u003c/a\u003eAs will be explained later, classes may be regarded as logical fictions,\r\nmanufactured out of defining characteristics. But for the present it will\r\nsimplify our exposition to treat classes as if they were real.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nReturning now to the definition of number, it is clear that\r\nnumber is a way of bringing together certain collections, namely,\r\nthose that have a given number of terms. We can suppose\r\nall couples in one bundle, all trios in another, and so on. In\r\nthis way we obtain various bundles of collections, each bundle\r\nconsisting of all the collections that have a certain number of\r\nterms. Each bundle is a class whose members are collections,\r\n\u003ci\u003ei.e.\u003c/i\u003e classes; thus each is a class of classes. The bundle consisting\r\nof all couples, for example, is a class of classes: each\r\ncouple is a class with two members, and the whole bundle of\r\ncouples is a class with an infinite number of members, each of\r\nwhich is a class of two members.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nHow shall we decide whether two collections are to belong\r\nto the same bundle? The answer that suggests itself is: \"Find\r\nout how many members each has, and put them in the same\r\nbundle if they have the same number of members.\" But this\r\npresupposes that we have defined numbers, and that we know\r\nhow to discover how many terms a collection has. We are so\r\nused to the operation of counting that such a presupposition\r\nmight easily pass unnoticed. In fact, however, counting,\r\nthough familiar, is logically a very complex operation; moreover\r\nit is only available, as a means of discovering how many\r\nterms a collection has, when the collection is finite. Our definition\r\nof number must not assume in advance that all numbers\r\nare finite; and we cannot in any case, without a vicious circle,\r\n\u003cspan class=\"pagenum\" id=\"Page_14\"\u003e[Pg 14]\u003c/span\u003e\r\nuse counting to define numbers, because numbers are used in\r\ncounting. We need, therefore, some other method of deciding\r\nwhen two collections have the same number of terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn actual fact, it is simpler logically to find out whether two\r\ncollections have the same number of terms than it is to define\r\nwhat that number is. An illustration will make this clear.\r\nIf there were no polygamy or polyandry anywhere in the world,\r\nit is clear that the number of husbands living at any moment\r\nwould be exactly the same as the number of wives. We do\r\nnot need a census to assure us of this, nor do we need to know\r\nwhat is the actual number of husbands and of wives. We know\r\nthe number must be the same in both collections, because each\r\nhusband has one wife and each wife has one husband. The\r\nrelation of husband and wife is what is called \"one-one.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA relation is said to be \"one-one\" when, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation\r\nin question to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e no other term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003e has the same relation to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\r\nand \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e does not have the same relation to any term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\u0027\u003c/span\u003e other\r\nthan \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e When only the first of these two conditions is fulfilled,\r\nthe relation is called \"one-many\"; when only the second is\r\nfulfilled, it is called \"many-one.\" It should be observed that\r\nthe number 1 is not used in these definitions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn Christian countries, the relation of husband to wife is\r\none-one; in Mahometan countries it is one-many; in Tibet\r\nit is many-one. The relation of father to son is one-many;\r\nthat of son to father is many-one, but that of eldest son to father\r\nis one-one. If \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any number, the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e is\r\none-one; so is the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-59.png\" alt=\"\" data-tex=\"2n\"\u003e or to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 2.489ex; height: 1.554ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-60.png\" alt=\"\" data-tex=\"3n\"\u003e.\u003c/span\u003e When we are\r\nconsidering only positive numbers, the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 2.345ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-61.png\" alt=\"\" data-tex=\"n^{2}\"\u003e is\r\none-one; but when negative numbers are admitted, it becomes\r\ntwo-one, since \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e and \u003cimg style=\"vertical-align: -0.186ex; width: 3.118ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-62.png\" alt=\"\" data-tex=\"-n\"\u003e have the same square. These instances\r\nshould suffice to make clear the notions of one-one, one-many,\r\nand many-one relations, which play a great part in the principles\r\nof mathematics, not only in relation to the definition of\r\nnumbers, but in many other connections.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTwo classes are said to be \"similar\" when there is a one-one\r\n\u003cspan class=\"pagenum\" id=\"Page_15\"\u003e[Pg 15]\u003c/span\u003e\r\nrelation which correlates the terms of the one class each with\r\none term of the other class, in the same manner in which the\r\nrelation of marriage correlates husbands with wives. A few\r\npreliminary definitions will help us to state this definition more\r\nprecisely. The class of those terms that have a given relation\r\nto something or other is called the \u003ci\u003edomain\u003c/i\u003e of that relation:\r\nthus fathers are the domain of the relation of father to child,\r\nhusbands are the domain of the relation of husband to wife,\r\nwives are the domain of the relation of wife to husband, and\r\nhusbands and wives together are the domain of the relation of\r\nmarriage. The relation of wife to husband is called the \u003ci\u003econverse\u003c/i\u003e\r\nof the relation of husband to wife. Similarly \u003ci\u003eless\u003c/i\u003e is the converse\r\nof \u003ci\u003egreater\u003c/i\u003e, \u003ci\u003elater\u003c/i\u003e is the converse of \u003ci\u003eearlier\u003c/i\u003e, and so on. Generally,\r\nthe converse of a given relation is that relation which holds\r\nbetween \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e whenever the given relation holds between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e The \u003ci\u003econverse domain\u003c/i\u003e of a relation is the domain of\r\nits converse: thus the class of wives is the converse domain\r\nof the relation of husband to wife. We may now state our\r\ndefinition of similarity as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eOne class is said to be \"similar\" to another when there is a\r\none-one relation of which the one class is the domain, while the\r\nother is the converse domain.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to prove (1) that every class is similar to itself, (2) that\r\nif a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is similar to a class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e is\r\nsimilar to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e (3) that\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is similar to \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-65.png\" alt=\"\" data-tex=\"\\gamma\"\u003e,\u003c/span\u003e\r\nthen \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is similar to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-65.png\" alt=\"\" data-tex=\"\\gamma\"\u003e.\u003c/span\u003e A\r\nrelation is said to be \u003ci\u003ereflexive\u003c/i\u003e when it possesses the first of these\r\nproperties, \u003ci\u003esymmetrical\u003c/i\u003e when it possesses the second, and \u003ci\u003etransitive\u003c/i\u003e\r\nwhen it possesses the third. It is obvious that a relation\r\nwhich is symmetrical and transitive must be reflexive throughout\r\nits domain. Relations which possess these properties are an\r\nimportant kind, and it is worth while to note that similarity is\r\none of this kind of relations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is obvious to common sense that two finite classes have\r\nthe same number of terms if they are similar, but not otherwise.\r\nThe act of counting consists in establishing a one-one correlation\r\n\u003cspan class=\"pagenum\" id=\"Page_16\"\u003e[Pg 16]\u003c/span\u003e\r\nbetween the set of objects counted and the natural numbers\r\n(excluding 0) that are used up in the process. Accordingly\r\ncommon sense concludes that there are as many objects in the\r\nset to be counted as there are numbers up to the last number\r\nused in the counting. And we also know that, so long as we\r\nconfine ourselves to finite numbers, there are just \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e numbers\r\nfrom 1 up to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e Hence it follows that the last number used in\r\ncounting a collection is the number of terms in the collection,\r\nprovided the collection is finite. But this result, besides being\r\nonly applicable to finite collections, depends upon and assumes\r\nthe fact that two classes which are similar have the same number\r\nof terms; for what we do when we count (say) 10 objects is to\r\nshow that the set of these objects is similar to the set of numbers\r\n1 to 10. The notion of similarity is logically presupposed in\r\nthe operation of counting, and is logically simpler though less\r\nfamiliar. In counting, it is necessary to take the objects counted\r\nin a certain order, as first, second, third, etc., but order is not\r\nof the essence of number: it is an irrelevant addition, an unnecessary\r\ncomplication from the logical point of view. The\r\nnotion of similarity does not demand an order: for example,\r\nwe saw that the number of husbands is the same as the number\r\nof wives, without having to establish an order of precedence\r\namong them. The notion of similarity also does not require\r\nthat the classes which are similar should be finite. Take, for\r\nexample, the natural numbers (excluding 0) on the one hand,\r\nand the fractions which have 1 for their numerator on the other\r\nhand: it is obvious that we can correlate 2 with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -1.552ex; width: 2.127ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-66.png\" alt=\"\" data-tex=\"\\dfrac{1}{2}\"\u003e,\u003c/span\u003e 3\r\nwith \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -1.602ex; width: 2.127ex; height: 4.638ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-67.png\" alt=\"\" data-tex=\"\\dfrac{1}{3}\"\u003e,\u003c/span\u003e and\r\nso on, thus proving that the two classes are similar.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may thus use the notion of \"similarity\" to decide when\r\ntwo collections are to belong to the same bundle, in the sense\r\nin which we were asking this question earlier in this chapter.\r\nWe want to make one bundle containing the class that has no\r\nmembers: this will be for the number 0. Then we want a bundle\r\nof all the classes that have one member: this will be for the\r\nnumber 1. Then, for the number 2, we want a bundle consisting\r\n\u003cspan class=\"pagenum\" id=\"Page_17\"\u003e[Pg 17]\u003c/span\u003e\r\nof all couples; then one of all trios; and so on. Given any collection,\r\nwe can define the bundle it is to belong to as being the class\r\nof all those collections that are \"similar\" to it. It is very easy\r\nto see that if (for example) a collection has three members, the\r\nclass of all those collections that are similar to it will be the\r\nclass of trios. And whatever number of terms a collection may\r\nhave, those collections that are \"similar\" to it will have the same\r\nnumber of terms. We may take this as a \u003ci\u003edefinition\u003c/i\u003e of \"having\r\nthe same number of terms.\" It is obvious that it gives results\r\nconformable to usage so long as we confine ourselves to finite\r\ncollections.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSo far we have not suggested anything in the slightest degree\r\nparadoxical. But when we come to the actual definition of\r\nnumbers we cannot avoid what must at first sight seem a paradox,\r\nthough this impression will soon wear off. We naturally think\r\nthat the class of couples (for example) is something different\r\nfrom the number 2. But there is no doubt about the class of\r\ncouples: it is indubitable and not difficult to define, whereas\r\nthe number 2, in any other sense, is a metaphysical entity about\r\nwhich we can never feel sure that it exists or that we have tracked\r\nit down. It is therefore more prudent to content ourselves with\r\nthe class of couples, which we are sure of, than to hunt for a\r\nproblematical number 2 which must always remain elusive.\r\nAccordingly we set up the following definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eThe number of a class is the class of all those classes that are\r\nsimilar to it.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the number of a couple will be the class of all couples.\r\nIn fact, the class of all couples will \u003ci\u003ebe\u003c/i\u003e the number 2, according\r\nto our definition. At the expense of a little oddity, this definition\r\nsecures definiteness and indubitableness; and it is not difficult\r\nto prove that numbers so defined have all the properties that we\r\nexpect numbers to have.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may now go on to define numbers in general as any one of\r\nthe bundles into which similarity collects classes. A number\r\nwill be a set of classes such as that any two are similar to each\r\n\u003cspan class=\"pagenum\" id=\"Page_18\"\u003e[Pg 18]\u003c/span\u003e\r\nother, and none outside the set are similar to any inside the set.\r\nIn other words, a number (in general) is any collection which is\r\nthe number of one of its members; or, more simply still:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eA number is anything which is the number of some class.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSuch a definition has a verbal appearance of being circular,\r\nbut in fact it is not. We define \"the number of a given class\"\r\nwithout using the notion of number in general; therefore we may\r\ndefine number in general in terms of \"the number of a given\r\nclass\" without committing any logical error.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nDefinitions of this sort are in fact very common. The class\r\nof fathers, for example, would have to be defined by first defining\r\nwhat it is to be the father of somebody; then the class of fathers\r\nwill be all those who are somebody\u0027s father. Similarly if we want\r\nto define square numbers (say), we must first define what we\r\nmean by saying that one number is the square of another, and\r\nthen define square numbers as those that are the squares of\r\nother numbers. This kind of procedure is very common, and\r\nit is important to realise that it is legitimate and even often\r\nnecessary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have now given a definition of numbers which will serve\r\nfor finite collections. It remains to be seen how it will serve\r\nfor infinite collections. But first we must decide what we mean\r\nby \"finite\" and \"infinite,\" which cannot be done within the\r\nlimits of the present chapter.\r\n\u003cspan class=\"pagenum\" id=\"Page_19\"\u003e[Pg 19]\u003c/span\u003e\r\n\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027III: FINITUDE AND MATHEMATICAL INDUCTION\u0027\u003e\u003ca id=\"chap03\"\u003e\u003c/a\u003eCHAPTER III\r\n\u003cbr\u003e\u003cbr\u003e\r\nFINITUDE AND MATHEMATICAL INDUCTION\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHE series of natural numbers, as we saw in Chapter I., can all\r\nbe defined if we know what we mean by the three terms \"0,\"\r\n\"number,\" and \"successor.\" But we may go a step farther:\r\nwe can define all the natural numbers if we know what we mean\r\nby \"0\" and \"successor.\" It will help us to understand the\r\ndifference between finite and infinite to see how this can be done,\r\nand why the method by which it is done cannot be extended\r\nbeyond the finite. We will not yet consider how \"0\" and \"successor\"\r\nare to be defined: we will for the moment assume that\r\nwe know what these terms mean, and show how thence all other\r\nnatural numbers can be obtained.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to see that we can reach any assigned number, say\r\n30,000. We first define \"1\" as \"the successor of 0,\" then we\r\ndefine \"2\" as \"the successor of 1,\" and so on. In the case of\r\nan assigned number, such as 30,000, the proof that we can reach\r\nit by proceeding step by step in this fashion may be made, if we\r\nhave the patience, by actual experiment: we can go on until\r\nwe actually arrive at 30,000. But although the method of\r\nexperiment is available for each particular natural number, it\r\nis not available for proving the general proposition that \u003ci\u003eall\u003c/i\u003e such\r\nnumbers can be reached in this way, \u003ci\u003ei.e.\u003c/i\u003e by proceeding from 0\r\nstep by step from each number to its successor. Is there any\r\nother way by which this can be proved?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us consider the question the other way round. What are\r\nthe numbers that can be reached, given the terms \"0\" and\r\n\u003cspan class=\"pagenum\" id=\"Page_20\"\u003e[Pg 20]\u003c/span\u003e\r\n\"successor\"? Is there any way by which we can define the\r\nwhole class of such numbers? We reach 1, as the successor of 0;\r\n2, as the successor of 1; 3, as the successor of 2; and so on. It\r\nis this \"and so on\" that we wish to replace by something less\r\nvague and indefinite. We might be tempted to say that \"and\r\nso on\" means that the process of proceeding to the successor\r\nmay be repeated \u003ci\u003eany finite number\u003c/i\u003e of times; but the problem\r\nupon which we are engaged is the problem of defining \"finite\r\nnumber,\" and therefore we must not use this notion in our definition.\r\nOur definition must not assume that we know what a\r\nfinite number is.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe key to our problem lies in \u003ci\u003emathematical induction\u003c/i\u003e. It will\r\nbe remembered that, in Chapter I., this was the fifth of the five\r\nprimitive propositions which we laid down about the natural\r\nnumbers. It stated that any property which belongs to 0, and\r\nto the successor of any number which has the property, belongs\r\nto all the natural numbers. This was then presented as a principle,\r\nbut we shall now adopt it as a definition. It is not difficult\r\nto see that the terms obeying it are the same as the numbers\r\nthat can be reached from 0 by successive steps from next to\r\nnext, but as the point is important we will set forth the matter\r\nin some detail.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe shall do well to begin with some definitions, which will be\r\nuseful in other connections also.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA property is said to be \"hereditary\" in the natural-number\r\nseries if, whenever it belongs to a number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e it also belongs to\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e\r\nthe successor of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e Similarly a class is said to be \"hereditary\"\r\nif, whenever \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is a member of the class, so is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e It is\r\neasy to \u003ci\u003esee\u003c/i\u003e, though we are not yet supposed to know, that to say\r\na property is hereditary is equivalent to saying that it belongs\r\nto all the natural numbers not less than some one of them, \u003ci\u003ee.g.\u003c/i\u003e\r\nit must belong to all that are not less than 100, or all that are\r\nless than 1000, or it may be that it belongs to all that are not\r\nless than 0, \u003ci\u003ei.e.\u003c/i\u003e to all without exception.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA property is said to be \"inductive\" when it is a hereditary\r\n\u003cspan class=\"pagenum\" id=\"Page_21\"\u003e[Pg 21]\u003c/span\u003e\r\nproperty which belongs to 0. Similarly a class is \"inductive\"\r\nwhen it is a hereditary class of which 0 is a member.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven a hereditary class of which 0 is a member, it follows\r\nthat 1 is a member of it, because a hereditary class contains the\r\nsuccessors of its members, and 1 is the successor of 0. Similarly,\r\ngiven a hereditary class of which 1 is a member, it follows that\r\n2 is a member of it; and so on. Thus we can prove by a step-by-step\r\nprocedure that any assigned natural number, say 30,000,\r\nis a member of every inductive class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe will define the \"posterity\" of a given natural number\r\nwith respect to the relation \"immediate predecessor\" (which\r\nis the converse of \"successor\") as all those terms that belong\r\nto every hereditary class to which the given number belongs. It\r\nis again easy to \u003ci\u003esee\u003c/i\u003e that the posterity of a natural number consists\r\nof itself and all greater natural numbers; but this also we\r\ndo not yet officially know.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBy the above definitions, the posterity of 0 will consist of those\r\nterms which belong to every inductive class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is now not difficult to make it obvious that the posterity of 0\r\nis the same set as those terms that can be reached from 0 by\r\nsuccessive steps from next to next. For, in the first place, 0 belongs\r\nto both these sets (in the sense in which we have defined\r\nour terms); in the second place, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e belongs to both sets,\r\nso does \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e\r\nIt is to be observed that we are dealing here with the\r\nkind of matter that does not admit of precise proof, namely, the\r\ncomparison of a relatively vague idea with a relatively precise\r\none. The notion of \"those terms that can be reached from 0\r\nby successive steps from next to next\" is vague, though it \u003ci\u003eseems\u003c/i\u003e\r\nas if it conveyed a definite meaning; on the other hand, \"the\r\nposterity of 0\" is precise and explicit just where the other idea\r\nis hazy. It may be taken as giving what we \u003ci\u003emeant\u003c/i\u003e to mean\r\nwhen we spoke of the terms that can be reached from 0 by\r\nsuccessive steps.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe now lay down the following definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eThe \"natural numbers\" are the posterity of 0 with respect to the\r\n\u003cspan class=\"pagenum\" id=\"Page_22\"\u003e[Pg 22]\u003c/span\u003e\r\nrelation \"immediate predecessor\"\u003c/i\u003e (which is the converse of\r\n\"successor\").\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have thus arrived at a definition of one of Peano\u0027s three\r\nprimitive ideas in terms of the other two. As a result of this\r\ndefinition, two of his primitive propositions\u0026mdash;namely, the one\r\nasserting that 0 is a number and the one asserting mathematical\r\ninduction\u0026mdash;become unnecessary, since they result from the definition.\r\nThe one asserting that the successor of a natural number\r\nis a natural number is only needed in the weakened form \"every\r\nnatural number has a successor.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can, of course, easily define \"0\" and \"successor\" by means\r\nof the definition of number in general which we arrived at in\r\nChapter II. The number 0 is the number of terms in a class\r\nwhich has no members, \u003ci\u003ei.e.\u003c/i\u003e in the class which is called the \"null-class.\"\r\nBy the general definition of number, the number of terms\r\nin the null-class is the set of all classes similar to the null-class,\r\n\u003ci\u003ei.e.\u003c/i\u003e (as is easily proved) the set consisting of the null-class all\r\nalone, \u003ci\u003ei.e.\u003c/i\u003e the class whose only member is the null-class. (This\r\nis not identical with the null-class: it has one member, namely,\r\nthe null-class, whereas the null-class itself has no members. A\r\nclass which has one member is never identical with that one\r\nmember, as we shall explain when we come to the theory of\r\nclasses.) Thus we have the following purely logical definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003e0 is the class whose only member is the null-class.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt remains to define \"successor.\" Given any number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e let\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e be a class which has \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e members, and let \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e be a term which\r\nis not a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e Then the class consisting of\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nadded on will have \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e members. Thus we have the following\r\ndefinition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eThe successor of the number of terms in the class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is the number\r\nof terms in the class consisting of a together with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\r\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is any\r\nterm not belonging to the class.\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nCertain niceties are required to make this definition perfect,\r\nbut they need not concern us.\u003ca id=\"FNanchor_5_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_5_1\" class=\"fnanchor\"\u003e[5]\u003c/a\u003e\r\nIt will be remembered that we\r\n\u003cspan class=\"pagenum\" id=\"Page_23\"\u003e[Pg 23]\u003c/span\u003e\r\nhave already given (in Chapter II.) a logical definition of the\r\nnumber of terms in a class, namely, we defined it as the set of all\r\nclasses that are similar to the given class.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_5_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_5_1\"\u003e\u003cspan class=\"label\"\u003e[5]\u003c/span\u003e\u003c/a\u003eSee \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. II. * 110.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe have thus reduced Peano\u0027s three primitive ideas to ideas\r\nof logic: we have given definitions of them which make them\r\ndefinite, no longer capable of an infinity of different meanings,\r\nas they were when they were only determinate to the extent of\r\nobeying Peano\u0027s five axioms. We have removed them from the\r\nfundamental apparatus of terms that must be merely apprehended,\r\nand have thus increased the deductive articulation of\r\nmathematics.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAs regards the five primitive propositions, we have already\r\nsucceeded in making two of them demonstrable by our definition\r\nof \"natural number.\" How stands it with the remaining three?\r\nIt is very easy to prove that 0 is not the successor of any number,\r\nand that the successor of any number is a number. But there\r\nis a difficulty about the remaining primitive proposition, namely,\r\n\"no two numbers have the same successor.\" The difficulty\r\ndoes not arise unless the total number of individuals in the\r\nuniverse is finite; for given two numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e neither of\r\nwhich is the total number of individuals in the universe, it is\r\neasy to prove that we cannot have \u003cimg style=\"vertical-align: -0.186ex; width: 14.155ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-68.png\" alt=\"\" data-tex=\"m + 1 = n + 1\"\u003e unless we have\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 6.361ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-69.png\" alt=\"\" data-tex=\"m = n\"\u003e.\u003c/span\u003e But let us suppose that the total number of individuals\r\nin the universe were (say) 10; then there would be no class of\r\n11 individuals, and the number 11 would be the null-class. So\r\nwould the number 12. Thus we should have 11 = 12; therefore\r\nthe successor of 10 would be the same as the successor of 11,\r\nalthough 10 would not be the same as 11. Thus we should have\r\ntwo different numbers with the same successor. This failure of\r\nthe third axiom cannot arise, however, if the number of individuals\r\nin the world is not finite. We shall return to this topic\r\nat a later stage.\u003ca id=\"FNanchor_6_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_6_1\" class=\"fnanchor\"\u003e[6]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_6_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_6_1\"\u003e\u003cspan class=\"label\"\u003e[6]\u003c/span\u003e\u003c/a\u003eSee Chapter XIII.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nAssuming that the number of individuals in the universe is\r\nnot finite, we have now succeeded not only in defining Peano\u0027s\r\n\u003cspan class=\"pagenum\" id=\"Page_24\"\u003e[Pg 24]\u003c/span\u003e\r\nthree primitive ideas, but in seeing how to prove his five primitive\r\npropositions, by means of primitive ideas and propositions belonging\r\nto logic. It follows that all pure mathematics, in so far\r\nas it is deducible from the theory of the natural numbers, is only\r\na prolongation of logic. The extension of this result to those\r\nmodern branches of mathematics which are not deducible from\r\nthe theory of the natural numbers offers no difficulty of principle,\r\nas we have shown elsewhere.\u003ca id=\"FNanchor_7_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_7_1\" class=\"fnanchor\"\u003e[7]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_7_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_7_1\"\u003e\u003cspan class=\"label\"\u003e[7]\u003c/span\u003e\u003c/a\u003eFor geometry, in so far as it is not purely analytical, see \u003ci\u003ePrinciples of\r\nMathematics\u003c/i\u003e, part VI.; for rational dynamics, \u003ci\u003eibid.\u003c/i\u003e, part VII.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe process of mathematical induction, by means of which\r\nwe defined the natural numbers, is capable of generalisation.\r\nWe defined the natural numbers as the \"posterity\" of 0 with\r\nrespect to the relation of a number to its immediate successor.\r\nIf we call this relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e,\u003c/span\u003e any number \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e will have this relation\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-71.png\" alt=\"\" data-tex=\"m + 1\"\u003e.\u003c/span\u003e A property is \"hereditary with respect to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e,\u003c/span\u003e\" or\r\nsimply \"\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e-hereditary,\" if, whenever the property belongs to a\r\nnumber \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e,\u003c/span\u003e it also belongs to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.883ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-71.png\" alt=\"\" data-tex=\"m + 1\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e to the number to which\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e.\u003c/span\u003e And a number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e will be said to belong to\r\nthe \"posterity\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e with respect to the relation \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e has\r\nevery \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e-hereditary property belonging to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e.\u003c/span\u003e These definitions\r\ncan all be applied to any other relation just as well as to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-70.png\" alt=\"\" data-tex=\"\\mathrm N\"\u003e.\u003c/span\u003e Thus\r\nif \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is any relation whatever, we can lay down the following\r\ndefinitions:\u003ca id=\"FNanchor_8_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_8_1\" class=\"fnanchor\"\u003e[8]\u003c/a\u003e\u0026mdash;\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_8_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_8_1\"\u003e\u003cspan class=\"label\"\u003e[8]\u003c/span\u003e\u003c/a\u003eThese definitions, and the generalised theory of induction, are due to\r\nFrege, and were published so long ago as 1879 in his \u003ci\u003eBegriffsschrift\u003c/i\u003e. In\r\nspite of the great value of this work, I was, I believe, the first person who\r\never read it\u0026mdash;more than twenty years after its publication.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nA property is called \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-hereditary\" when, if it belongs to\r\na term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\r\nthen it belongs to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA class is \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-hereditary when its defining property\r\nis \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-hereditary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is said to be an \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" of the\r\nterm \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e if \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e has\r\nevery \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-hereditary property that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has,\r\nprovided \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a term\r\nwhich has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to something or to which something\r\nhas the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e (This is only to exclude trivial cases.)\r\n\u003cspan class=\"pagenum\" id=\"Page_25\"\u003e[Pg 25]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-posterity\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is all the terms of\r\nwhich \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have framed the above definitions so that if a term is the\r\nancestor of anything it is its own ancestor and belongs to its own\r\nposterity. This is merely for convenience.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be observed that if we take for \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e the relation \"parent,\"\r\n\"ancestor\" and \"posterity\" will have the usual meanings,\r\nexcept that a person will be included among his own ancestors\r\nand posterity. It is, of course, obvious at once that \"ancestor\"\r\nmust be capable of definition in terms of \"parent,\" but until\r\nFrege developed his generalised theory of induction, no one could\r\nhave defined \"ancestor\" precisely in terms of \"parent.\" A\r\nbrief consideration of this point will serve to show the importance\r\nof the theory. A person confronted for the first time with the\r\nproblem of defining \"ancestor\" in terms of \"parent\" would\r\nnaturally say that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e is an ancestor of \u003cimg style=\"vertical-align: 0; width: 1.382ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-74.png\" alt=\"\" data-tex=\"\\mathrm Z\"\u003e if,\r\nbetween \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.382ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-74.png\" alt=\"\" data-tex=\"\\mathrm Z\"\u003e,\u003c/span\u003e\r\nthere are a certain number of people, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.048ex; width: 1.633ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-76.png\" alt=\"\" data-tex=\"\\mathrm C\"\u003e,\u003c/span\u003e …, of whom\r\n\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e is a child of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e each is a parent of the next, until the last, who\r\nis a parent of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.382ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-74.png\" alt=\"\" data-tex=\"\\mathrm Z\"\u003e.\u003c/span\u003e But this definition is not adequate unless we\r\nadd that the number of intermediate terms is to be finite. Take,\r\nfor example, such a series as the following:\u0026mdash;\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.816ex; width: 36.934ex; height: 2.773ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-8.png\" alt=\"\" data-tex=\"\r\n1,\\ -\\tfrac{1}{2},\\ -\\tfrac{1}{4},\\ -\\tfrac{1}{8},\\ \\dots\\\r\n\\tfrac{1}{8},\\ \\tfrac{1}{4},\\ \\tfrac{1}{2},\\ 1.\r\n\"\u003e\u003c/span\u003e\r\nHere we have first a series of negative fractions with no end,\r\nand then a series of positive fractions with no beginning. Shall\r\nwe say that, in this series, \u003cimg style=\"vertical-align: -0.816ex; width: 3.556ex; height: 2.773ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-77.png\" alt=\"\" data-tex=\"-\\frac{1}{8}\"\u003e is an ancestor of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.816ex; width: 1.795ex; height: 2.773ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-78.png\" alt=\"\" data-tex=\"\\frac{1}{8}\"\u003e?\u003c/span\u003e It will be\r\nso according to the beginner\u0027s definition suggested above, but\r\nit will not be so according to any definition which will give the\r\nkind of idea that we wish to define. For this purpose, it is\r\nessential that the number of intermediaries should be finite.\r\nBut, as we saw, \"finite\" is to be defined by means of mathematical\r\ninduction, and it is simpler to define the ancestral relation\r\ngenerally at once than to define it first only for the case of the\r\nrelation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e and then extend it to other cases. Here,\r\nas constantly elsewhere, generality from the first, though it may\r\n\u003cspan class=\"pagenum\" id=\"Page_26\"\u003e[Pg 26]\u003c/span\u003e\r\nrequire more thought at the start, will be found in the long run\r\nto economise thought and increase logical power.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe use of mathematical induction in demonstrations was,\r\nin the past, something of a mystery. There seemed no reasonable\r\ndoubt that it was a valid method of proof, but no one quite\r\nknew why it was valid. Some believed it to be really a case\r\nof induction, in the sense in which that word is used in logic.\r\nPoincaré\u003ca id=\"FNanchor_9_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_9_1\" class=\"fnanchor\"\u003e[9]\u003c/a\u003e\r\nconsidered it to be a principle of the utmost importance,\r\nby means of which an infinite number of syllogisms could be\r\ncondensed into one argument. We now know that all such views\r\nare mistaken, and that mathematical induction is a definition,\r\nnot a principle. There are some numbers to which it can be\r\napplied, and there are others (as we shall see in Chapter VIII.)\r\nto which it cannot be applied. We \u003ci\u003edefine\u003c/i\u003e the \"natural numbers\"\r\nas those to which proofs by mathematical induction can be\r\napplied, \u003ci\u003ei.e.\u003c/i\u003e as those that possess all inductive properties. It\r\nfollows that such proofs can be applied to the natural numbers,\r\nnot in virtue of any mysterious intuition or axiom or principle,\r\nbut as a purely verbal proposition. If \"quadrupeds\" are\r\ndefined as animals having four legs, it will follow that animals\r\nthat have four legs are quadrupeds; and the case of numbers\r\nthat obey mathematical induction is exactly similar.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_9_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_9_1\"\u003e\u003cspan class=\"label\"\u003e[9]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eScience and Method\u003c/i\u003e, chap. IV.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe shall use the phrase \"inductive numbers\" to mean the\r\nsame set as we have hitherto spoken of as the \"natural numbers.\"\r\nThe phrase \"inductive numbers\" is preferable as affording a\r\nreminder that the definition of this set of numbers is obtained\r\nfrom mathematical induction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMathematical induction affords, more than anything else,\r\nthe essential characteristic by which the finite is distinguished\r\nfrom the infinite. The principle of mathematical induction\r\nmight be stated popularly in some such form as \"what can be\r\ninferred from next to next can be inferred from first to last.\"\r\nThis is true when the number of intermediate steps between\r\nfirst and last is finite, not otherwise. Anyone who has ever\r\n\u003cspan class=\"pagenum\" id=\"Page_27\"\u003e[Pg 27]\u003c/span\u003e\r\nwatched a goods train beginning to move will have noticed how\r\nthe impulse is communicated with a jerk from each truck to\r\nthe next, until at last even the hindmost truck is in motion.\r\nWhen the train is very long, it is a very long time before the last\r\ntruck moves. If the train were infinitely long, there would be\r\nan infinite succession of jerks, and the time would never come\r\nwhen the whole train would be in motion. Nevertheless, if\r\nthere were a series of trucks no longer than the series of inductive\r\nnumbers (which, as we shall see, is an instance of the smallest\r\nof infinites), every truck would begin to move sooner or later\r\nif the engine persevered, though there would always be other\r\ntrucks further back which had not yet begun to move. This\r\nimage will help to elucidate the argument from next to next,\r\nand its connection with finitude. When we come to infinite\r\nnumbers, where arguments from mathematical induction will\r\nbe no longer valid, the properties of such numbers will help to\r\nmake clear, by contrast, the almost unconscious use that is made\r\nof mathematical induction where finite numbers are concerned.\r\n\u003cspan class=\"pagenum\" id=\"Page_28\"\u003e[Pg 28]\u003c/span\u003e\r\n\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027IV: THE DEFINITION OF ORDER\u0027\u003e\u003ca id=\"chap04\"\u003e\u003c/a\u003eCHAPTER IV\r\n\u003cbr\u003e\u003cbr\u003e\r\nTHE DEFINITION OF ORDER\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWE have now carried our analysis of the series of natural numbers\r\nto the point where we have obtained logical definitions of the\r\nmembers of this series, of the whole class of its members, and\r\nof the relation of a number to its immediate successor. We\r\nmust now consider the \u003ci\u003eserial\u003c/i\u003e character of the natural numbers\r\nin the order 0, 1, 2, 3,…. We ordinarily think of the numbers\r\nas in this \u003ci\u003eorder\u003c/i\u003e, and it is an essential part of the work\r\nof analysing our data to seek a definition of \"order\" or \"series\"\r\nin logical terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe notion of order is one which has enormous importance\r\nin mathematics. Not only the integers, but also rational fractions\r\nand all real numbers have an order of magnitude, and\r\nthis is essential to most of their mathematical properties. The\r\norder of points on a line is essential to geometry; so is the\r\nslightly more complicated order of lines through a point in a\r\nplane, or of planes through a line. Dimensions, in geometry,\r\nare a development of order. The conception of a \u003ci\u003elimit\u003c/i\u003e, which\r\nunderlies all higher mathematics, is a serial conception. There\r\nare parts of mathematics which do not depend upon the notion\r\nof order, but they are very few in comparison with the parts\r\nin which this notion is involved.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn seeking a definition of order, the first thing to realise is\r\nthat no set of terms has just \u003ci\u003eone\u003c/i\u003e order to the exclusion of others.\r\nA set of terms has all the orders of which it is capable. Sometimes\r\none order is so much more familiar and natural to our\r\n\u003cspan class=\"pagenum\" id=\"Page_29\"\u003e[Pg 29]\u003c/span\u003e\r\nthoughts that we are inclined to regard it as \u003ci\u003ethe\u003c/i\u003e order of that\r\nset of terms; but this is a mistake. The natural numbers\u0026mdash;or\r\nthe \"inductive\" numbers, as we shall also call them\u0026mdash;occur\r\nto us most readily in order of magnitude; but they are capable\r\nof an infinite number of other arrangements. We might, for\r\nexample, consider first all the odd numbers and then all the\r\neven numbers; or first 1, then all the even numbers, then all\r\nthe odd multiples of 3, then all the multiples of 5 but not of\r\n2 or 3, then all the multiples of 7 but not of 2 or 3 or 5, and so\r\non through the whole series of primes. When we say that we\r\n\"arrange\" the numbers in these various orders, that is an\r\ninaccurate expression: what we really do is to turn our attention\r\nto certain relations between the natural numbers, which themselves\r\ngenerate such-and-such an arrangement. We can no\r\nmore \"arrange\" the natural numbers than we can the starry\r\nheavens; but just as we may notice among the fixed stars\r\neither their order of brightness or their distribution in the sky,\r\nso there are various relations among numbers which may be\r\nobserved, and which give rise to various different orders among\r\nnumbers, all equally legitimate. And what is true of numbers\r\nis equally true of points on a line or of the moments of time:\r\none order is more familiar, but others are equally valid. We\r\nmight, for example, take first, on a line, all the points that have\r\nintegral co-ordinates, then all those that have non-integral\r\nrational co-ordinates, then all those that have algebraic non-rational\r\nco-ordinates, and so on, through any set of complications\r\nwe please. The resulting order will be one which the\r\npoints of the line certainly have, whether we choose to notice\r\nit or not; the only thing that is arbitrary about the various\r\norders of a set of terms is our attention, for the terms themselves\r\nhave always all the orders of which they are capable.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne important result of this consideration is that we must\r\nnot look for the definition of order in the nature of the set of\r\nterms to be ordered, since one set of terms has many orders.\r\nThe order lies, not in the \u003ci\u003eclass\u003c/i\u003e of terms, but in a relation among\r\n\u003cspan class=\"pagenum\" id=\"Page_30\"\u003e[Pg 30]\u003c/span\u003e\r\nthe members of the class, in respect of which some appear as\r\nearlier and some as later. The fact that a class may have many\r\norders is due to the fact that there can be many relations holding\r\namong the members of one single class. What properties must\r\na relation have in order to give rise to an order?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe essential characteristics of a relation which is to give rise\r\nto order may be discovered by considering that in respect of\r\nsuch a relation we must be able to say, of any two terms in\r\nthe class which is to be ordered, that one \"precedes\" and the\r\nother \"follows.\" Now, in order that we may be able to use\r\nthese words in the way in which we should naturally understand\r\nthem, we require that the ordering relation should have three\r\nproperties:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e precedes \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e must not also precede \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e This is an\r\nobvious characteristic of the kind of relations that lead to series.\r\nIf \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is not also less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\r\nIf \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is earlier in\r\ntime than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is not also earlier than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is\r\nto the left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is not to the left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e On the other hand, relations which\r\ndo not give rise to series often do not have this property. If\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a brother or sister of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is a brother or\r\nsister of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is\r\nof the same height as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is of the same height as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is of a\r\ndifferent height from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is of a different height from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e In\r\nall these cases, when the relation holds between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e it also\r\nholds between \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e But with serial relations such a thing\r\ncannot happen. A relation having this first property is called\r\n\u003ci\u003easymmetrical\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e precedes \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e precedes \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e must precede \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e This\r\nmay be illustrated by the same instances as before: \u003ci\u003eless\u003c/i\u003e, \u003ci\u003eearlier\u003c/i\u003e,\r\n\u003ci\u003eleft of\u003c/i\u003e. But as instances of relations which do \u003ci\u003enot\u003c/i\u003e have this\r\nproperty only two of our previous three instances will serve.\r\nIf \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is brother or sister of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e may not be brother\r\nor sister of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e since \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e may be the same person. The same\r\napplies to difference of height, but not to sameness of height,\r\nwhich has our second property but not our first. The relation\r\n\"father,\" on the other hand, has our first property but not\r\n\u003cspan class=\"pagenum\" id=\"Page_31\"\u003e[Pg 31]\u003c/span\u003e\r\nour second. A relation having our second property is called\r\n\u003ci\u003etransitive\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) Given any two terms of the class which is to be ordered,\r\nthere must be one which precedes and the other which follows.\r\nFor example, of any two integers, or fractions, or real numbers,\r\none is smaller and the other greater; but of any two complex\r\nnumbers this is not true. Of any two moments in time, one\r\nmust be earlier than the other; but of events, which may be\r\nsimultaneous, this cannot be said. Of two points on a line,\r\none must be to the left of the other. A relation having this\r\nthird property is called \u003ci\u003econnected\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen a relation possesses these three properties, it is of the\r\nsort to give rise to an order among the terms between which it\r\nholds; and wherever an order exists, some relation having these\r\nthree properties can be found generating it.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBefore illustrating this thesis, we will introduce a few\r\ndefinitions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) A relation is said to be an aliorelative,\u003ca id=\"FNanchor_10_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_10_1\" class=\"fnanchor\"\u003e[10]\u003c/a\u003e\r\nor to \u003ci\u003ebe contained\r\nin\u003c/i\u003e or \u003ci\u003eimply diversity\u003c/i\u003e, if no term has this relation to itself.\r\nThus, for example, \"greater,\" \"different in size,\" \"brother,\"\r\n\"husband,\" \"father\" are aliorelatives; but \"equal,\" \"born\r\nof the same parents,\" \"dear friend\" are not.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_10_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_10_1\"\u003e\u003cspan class=\"label\"\u003e[10]\u003c/span\u003e\u003c/a\u003eThis term is due to C. S. Peirce.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\n(2) The \u003ci\u003esquare\u003c/i\u003e of a relation is that relation which holds between\r\ntwo terms \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e when there is an intermediate term \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e such\r\nthat the given relation holds between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and between\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e Thus \"paternal grandfather\" is the square of \"father,\"\r\n\"greater by 2\" is the square of \"greater by 1,\" and so on.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) The \u003ci\u003edomain\u003c/i\u003e of a relation consists of all those terms that\r\nhave the relation to something or other, and the \u003ci\u003econverse domain\u003c/i\u003e\r\nconsists of all those terms to which something or other has the\r\nrelation. These words have been already defined, but are\r\nrecalled here for the sake of the following definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(4) The \u003ci\u003efield\u003c/i\u003e of a relation consists of its domain and converse\r\ndomain together.\r\n\u003cspan class=\"pagenum\" id=\"Page_32\"\u003e[Pg 32]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(5) One relation is said to \u003ci\u003econtain\u003c/i\u003e or \u003ci\u003ebe implied by\u003c/i\u003e another if\r\nit holds whenever the other holds.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be seen that an \u003ci\u003easymmetrical\u003c/i\u003e relation is the same thing\r\nas a relation whose square is an aliorelative. It often happens\r\nthat a relation is an aliorelative without being asymmetrical,\r\nthough an asymmetrical relation is always an aliorelative. For\r\nexample, \"spouse\" is an aliorelative, but is symmetrical,\r\nsince if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the spouse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the spouse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e But among\r\n\u003ci\u003etransitive\u003c/i\u003e relations, all aliorelatives are asymmetrical as well\r\nas \u003ci\u003evice versa\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFrom the definitions it will be seen that a \u003ci\u003etransitive\u003c/i\u003e relation\r\nis one which is implied by its square, or, as we also say, \"contains\"\r\nits square. Thus \"ancestor\" is transitive, because\r\nan ancestor\u0027s ancestor is an ancestor; but \"father\" is not\r\ntransitive, because a father\u0027s father is not a father. A transitive\r\naliorelative is one which contains its square and is contained\r\nin diversity; or, what comes to the same thing, one whose\r\nsquare implies both it and diversity\u0026mdash;because, when a relation\r\nis transitive, asymmetry is equivalent to being an aliorelative.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA relation is \u003ci\u003econnected\u003c/i\u003e when, given any two different terms\r\nof its field, the relation holds between the first and the second\r\nor between the second and the first (not excluding the possibility\r\nthat both may happen, though both cannot happen if the relation\r\nis asymmetrical).\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be seen that the relation \"ancestor,\" for example,\r\nis an aliorelative and transitive, but not connected; it is because\r\nit is not connected that it does not suffice to arrange the human\r\nrace in a series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe relation \"less than or equal to,\" among numbers, is\r\ntransitive and connected, but not asymmetrical or an aliorelative.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe relation \"greater or less\" among numbers is an aliorelative\r\nand is connected, but is not transitive, for if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is greater\r\nor less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is greater or less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e it may happen\r\nthat \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e are the same number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the three properties of being (1) an aliorelative, (2) transitive,\r\n\u003cspan class=\"pagenum\" id=\"Page_33\"\u003e[Pg 33]\u003c/span\u003e\r\nand (3) connected, are mutually independent, since\r\na relation may have any two without having the third.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe now lay down the following definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA relation is \u003ci\u003eserial\u003c/i\u003e when it is an aliorelative, transitive, and\r\nconnected; or, what is equivalent, when it is asymmetrical,\r\ntransitive, and connected.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \u003ci\u003eseries\u003c/i\u003e is the same thing as a serial relation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt might have been thought that a series should be the \u003ci\u003efield\u003c/i\u003e\r\nof a serial relation, not the serial relation itself. But this would\r\nbe an error. For example,\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 55.564ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-9.png\" alt=\"\" data-tex=\"\r\n1,\\ 2,\\ 3;\\quad\r\n1,\\ 3,\\ 2;\\quad\r\n2,\\ 3,\\ 1;\\quad\r\n2,\\ 1,\\ 3;\\quad\r\n3,\\ 1,\\ 2;\\quad\r\n3,\\ 2,\\ 1\r\n\"\u003e\u003c/span\u003e\r\nare six different series which all have the same field. If the\r\nfield \u003ci\u003ewere\u003c/i\u003e the series, there could only be one series with a given\r\nfield. What distinguishes the above six series is simply the\r\ndifferent ordering relations in the six cases. Given the ordering\r\nrelation, the field and the order are both determinate. Thus\r\nthe ordering relation may be taken to \u003ci\u003ebe\u003c/i\u003e the series, but the field\r\ncannot be so taken.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven any serial relation, say \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e we shall say that, in respect\r\nof this relation, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e \"precedes\" \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation\r\n\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\r\nwhich we shall write \"\u003cimg style=\"vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-81.png\" alt=\"\" data-tex=\"x\\mathrm Py\"\u003e\" for short. The three characteristics\r\nwhich \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e must have in order to be serial are:\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(1) We must never have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 4.129ex; height: 1.57ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-82.png\" alt=\"\" data-tex=\"x\\mathrm Px\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e no term must precede\r\nitself.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(2) \u003cimg style=\"vertical-align: 0; width: 2.528ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-83.png\" alt=\"\" data-tex=\"\\mathrm P^{2}\"\u003e must imply \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e precedes \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e precedes \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e must\r\nprecede \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(3) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are two different terms in the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e we shall\r\nhave \u003cimg style=\"vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-81.png\" alt=\"\" data-tex=\"x\\mathrm Py\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 3.943ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-84.png\" alt=\"\" data-tex=\"y\\mathrm Px\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e one of the two must precede the\r\nother.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nThe reader can easily convince himself that, where these three\r\nproperties are found in an ordering relation, the characteristics\r\nwe expect of series will also be found, and \u003ci\u003evice versa\u003c/i\u003e. We are\r\ntherefore justified in taking the above as a definition of order\r\n\u003cspan class=\"pagenum\" id=\"Page_34\"\u003e[Pg 34]\u003c/span\u003e\r\nor series. And it will be observed that the definition is effected\r\nin purely logical terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAlthough a transitive asymmetrical connected relation always\r\nexists wherever there is a series, it is not always the relation\r\nwhich would most naturally be regarded as generating the series.\r\nThe natural-number series may serve as an illustration. The\r\nrelation we assumed in considering the natural numbers was\r\nthe relation of immediate succession, \u003ci\u003ei.e.\u003c/i\u003e the relation between\r\nconsecutive integers. This relation is asymmetrical, but not\r\ntransitive or connected. We can, however, derive from it,\r\nby the method of mathematical induction, the \"ancestral\"\r\nrelation which we considered in the preceding chapter. This\r\nrelation will be the same as \"less than or equal to\" among\r\ninductive integers. For purposes of generating the series of\r\nnatural numbers, we want the relation \"less than,\" excluding\r\n\"equal to.\" This is the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e when \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is an ancestor\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e but not identical with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e or (what comes to the same thing)\r\nwhen the successor of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is an ancestor of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e in the sense in which\r\na number is its own ancestor. That is to say, we shall lay down\r\nthe following definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAn inductive number \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is said to be \u003ci\u003eless than\u003c/i\u003e another number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e\r\nwhen \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e possesses every hereditary property possessed by the\r\nsuccessor of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to see, and not difficult to prove, that the relation\r\n\"less than,\" so defined, is asymmetrical, transitive, and connected,\r\nand has the inductive numbers for its field. Thus by\r\nmeans of this relation the inductive numbers acquire an order\r\nin the sense in which we defined the term \"order,\" and this order\r\nis the so-called \"natural\" order, or order of magnitude.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe generation of series by means of relations more or less\r\nresembling that of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e is very common. The series of the\r\nKings of England, for example, is generated by relations of each\r\nto his successor. This is probably the easiest way, where it is\r\napplicable, of conceiving the generation of a series. In this\r\nmethod we pass on from each term to the next, as long as there\r\n\u003cspan class=\"pagenum\" id=\"Page_35\"\u003e[Pg 35]\u003c/span\u003e\r\nis a next, or back to the one before, as long as there is one before.\r\nThis method always requires the generalised form of mathematical\r\ninduction in order to enable us to define \"earlier\" and\r\n\"later\" in a series so generated. On the analogy of \"proper\r\nfractions,\" let us give the name \"proper posterity of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e with respect\r\nto \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\" to the class of those terms that belong to the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-posterity\r\nof some term to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e in the sense which\r\nwe gave before to \"posterity,\" which includes a term in its own\r\nposterity. Reverting to the fundamental definitions, we find that\r\nthe \"proper posterity\" may be defined as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"proper posterity\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e with respect to \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e consists of\r\nall terms that possess every \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-hereditary property possessed by\r\nevery term to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is to be observed that this definition has to be so framed\r\nas to be applicable not only when there is only one term to which\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e but also in cases (as \u003ci\u003ee.g.\u003c/i\u003e that of father and\r\nchild) where there may be many terms to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e\r\nWe define further:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a \"proper ancestor\" of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e with respect to\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e if \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e belongs\r\nto the proper posterity of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e with respect to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe shall speak for short of \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-posterity\" and \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestors\"\r\nwhen these terms seem more convenient.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nReverting now to the generation of series by the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nbetween consecutive terms, we see that, if this method is to be\r\npossible, the relation \"proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" must be an aliorelative,\r\ntransitive, and connected. Under what circumstances will\r\nthis occur? It will always be transitive: no matter what sort\r\nof relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e may be, \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" and \"proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\"\r\nare always both transitive. But it is only under certain circumstances\r\nthat it will be an aliorelative or connected. Consider,\r\nfor example, the relation to one\u0027s left-hand neighbour at a round\r\ndinner-table at which there are twelve people. If we call this\r\nrelation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e the proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-posterity of a person consists of all who\r\ncan be reached by going round the table from right to left. This\r\nincludes everybody at the table, including the person himself, since\r\n\u003cspan class=\"pagenum\" id=\"Page_36\"\u003e[Pg 36]\u003c/span\u003e\r\ntwelve steps bring us back to our starting-point. Thus in such\r\na case, though the relation \"proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" is connected,\r\nand though \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e itself is an aliorelative, we do not get a series\r\nbecause \"proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" is not an aliorelative. It is for\r\nthis reason that we cannot say that one person comes before\r\nanother with respect to the relation \"right of\" or to its ancestral\r\nderivative.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe above was an instance in which the ancestral relation was\r\nconnected but not contained in diversity. An instance where\r\nit is contained in diversity but not connected is derived from the\r\nordinary sense of the word \"ancestor.\" If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a proper ancestor\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e cannot be the same person; but it is not true that\r\nof any two persons one must be an ancestor of the other.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe question of the circumstances under which series can be\r\ngenerated by ancestral relations derived from relations of consecutiveness\r\nis often important. Some of the most important\r\ncases are the following: Let \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e be a many-one relation, and let\r\nus confine our attention to the posterity of some term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e When\r\nso confined, the relation \"proper \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" must be connected;\r\ntherefore all that remains to ensure its being serial is that it shall\r\nbe contained in diversity. This is a generalisation of the instance\r\nof the dinner-table. Another generalisation consists in taking\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to be a one-one relation, and including the ancestry of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e as\r\nwell as the posterity. Here again, the one condition required\r\nto secure the generation of a series is that the relation \"proper\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-ancestor\" shall be contained in diversity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe generation of order by means of relations of consecutiveness,\r\nthough important in its own sphere, is less general than the\r\nmethod which uses a transitive relation to define the order. It\r\noften happens in a series that there are an infinite number of intermediate\r\nterms between any two that may be selected, however\r\nnear together these may be. Take, for instance, fractions in order\r\nof magnitude. Between any two fractions there are others\u0026mdash;for\r\nexample, the arithmetic mean of the two. Consequently there is\r\nno such thing as a pair of consecutive fractions. If we depended\r\n\u003cspan class=\"pagenum\" id=\"Page_37\"\u003e[Pg 37]\u003c/span\u003e\r\nupon consecutiveness for defining order, we should not be able\r\nto define the order of magnitude among fractions. But in fact\r\nthe relations of greater and less among fractions do not demand\r\ngeneration from relations of consecutiveness, and the relations\r\nof greater and less among fractions have the three characteristics\r\nwhich we need for defining serial relations. In all such cases\r\nthe order must be defined by means of a \u003ci\u003etransitive\u003c/i\u003e relation, since\r\nonly such a relation is able to leap over an infinite number of\r\nintermediate terms. The method of consecutiveness, like that\r\nof counting for discovering the number of a collection, is appropriate\r\nto the finite; it may even be extended to certain infinite\r\nseries, namely, those in which, though the total number of terms is\r\ninfinite, the number of terms between any two is always finite;\r\nbut it must not be regarded as general. Not only so, but care\r\nmust be taken to eradicate from the imagination all habits of\r\nthought resulting from supposing it general. If this is not done,\r\nseries in which there are no consecutive terms will remain difficult\r\nand puzzling. And such series are of vital importance for the\r\nunderstanding of continuity, space, time, and motion.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere are many ways in which series may be generated, but\r\nall depend upon the finding or construction of an asymmetrical\r\ntransitive connected relation. Some of these ways have considerable\r\nimportance. We may take as illustrative the generation\r\nof series by means of a three-term relation which we may\r\ncall \"between.\" This method is very useful in geometry, and\r\nmay serve as an introduction to relations having more than two\r\nterms; it is best introduced in connection with elementary\r\ngeometry.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven any three points on a straight line in ordinary space,\r\nthere must be one of them which is \u003ci\u003ebetween\u003c/i\u003e the other two. This\r\nwill not be the case with the points on a circle or any other closed\r\ncurve, because, given any three points on a circle, we can travel\r\nfrom any one to any other without passing through the third.\r\nIn fact, the notion \"between\" is characteristic of open series\u0026mdash;or\r\nseries in the strict sense\u0026mdash;as opposed to what may be called\r\n\u003cspan class=\"pagenum\" id=\"Page_38\"\u003e[Pg 38]\u003c/span\u003e\r\n\"cyclic\" series, where, as with people at the dinner-table, a\r\nsufficient journey brings us back to our starting-point. This\r\nnotion of \"between\" may be chosen as the fundamental notion\r\nof ordinary geometry; but for the present we will only consider\r\nits application to a single straight line and to the ordering of the\r\npoints on a straight line.\u003ca id=\"FNanchor_11_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_11_1\" class=\"fnanchor\"\u003e[11]\u003c/a\u003e\r\nTaking any two points \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e the line \u003cimg style=\"vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-87.png\" alt=\"\" data-tex=\"(ab)\"\u003e\r\nconsists of three parts (besides \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e themselves):\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_11_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_11_1\"\u003e\u003cspan class=\"label\"\u003e[11]\u003c/span\u003e\u003c/a\u003eCf. \u003ci\u003eRivista di Matematica\u003c/i\u003e, IV. pp. 55 ff.; \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e, p. 394\r\n(\u0026sect; 375).\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n(1) Points between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(2) Points \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e such that \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(3) Points \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is between \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the line \u003cimg style=\"vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-87.png\" alt=\"\" data-tex=\"(ab)\"\u003e can be defined in terms of the relation\r\n\"between.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn order that this relation \"between\" may arrange the points\r\nof the line in an order from left to right, we need certain assumptions,\r\nnamely, the following:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(1) If anything is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e are not identical.\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(2) Anything between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is also between \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(3) Anything between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is not identical with \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (nor,\r\nconsequently, with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e in virtue of (2)).\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(4) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e anything between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is also\r\nbetween \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(5) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is\r\nbetween \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(6) If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e then either \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are\r\nidentical, or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(7) If \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and also between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e then either\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are identical, or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e or \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese seven properties are obviously verified in the case of points\r\non a straight line in ordinary space. Any three-term relation\r\nwhich verifies them gives rise to series, as may be seen from the\r\nfollowing definitions. For the sake of definiteness, let us assume\r\n\u003cspan class=\"pagenum\" id=\"Page_39\"\u003e[Pg 39]\u003c/span\u003e\r\nthat \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is to the left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e Then the points of the line \u003cimg style=\"vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-87.png\" alt=\"\" data-tex=\"(ab)\"\u003e are (1) those\r\nbetween which and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e lies\u0026mdash;these we will call to the left\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e;\u003c/span\u003e (2) \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e itself; (3) those between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e (4) \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e itself; (5) those\r\nbetween which and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e lies \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e\u0026mdash;these we will call to the right\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e We may now define generally that of two points \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e on\r\nthe line \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.928ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-87.png\" alt=\"\" data-tex=\"(ab)\"\u003e,\u003c/span\u003e we shall say that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \"to the left of\" \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e in any\r\nof the following cases:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(1) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are both to the left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(2) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is to the left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e or between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e or to the right of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(3) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e or is \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e or is to the\r\nright of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(4) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are both between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(5) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e or to the right of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(6) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is to the right of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n(7) When \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are both to the right of \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be found that, from the seven properties which we have\r\nassigned to the relation \"between,\" it can be deduced that the\r\nrelation \"to the left of,\" as above defined, is a \u003ci\u003eserial\u003c/i\u003e relation as\r\nwe defined that term. It is important to notice that nothing\r\nin the definitions or the argument depends upon our meaning\r\nby \"between\" the actual relation of that name which occurs in\r\nempirical space: any three-term relation having the above seven\r\npurely formal properties will serve the purpose of the argument\r\nequally well.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nCyclic order, such as that of the points on a circle, cannot be\r\ngenerated by means of three-term relations of \"between.\" We\r\nneed a relation of four terms, which may be called \"separation\r\nof couples.\" The point may be illustrated by considering a\r\njourney round the world. One may go from England to New\r\nZealand by way of Suez or by way of San Francisco; we cannot\r\n\u003cspan class=\"pagenum\" id=\"Page_40\"\u003e[Pg 40]\u003c/span\u003e\r\nsay definitely that either of these two places is \"between\"\r\nEngland and New Zealand. But if a man chooses that route\r\nto go round the world, whichever way round he goes, his times in\r\nEngland and New Zealand are separated from each other by his\r\ntimes in Suez and San Francisco, and conversely. Generalising,\r\nif we take any four points on a circle, we can separate them into\r\ntwo couples, say \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e such that, in order to get\r\nfrom \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e one must pass through either \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e and in order to\r\nget from \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e one must pass through either \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e.\u003c/span\u003e Under these\r\ncircumstances we say that the couple \u003cimg style=\"vertical-align: -0.566ex; width: 4.934ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-88.png\" alt=\"\" data-tex=\"(a, b)\"\u003e are \"separated\" by\r\nthe couple \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-89.png\" alt=\"\" data-tex=\"(x, y)\"\u003e.\u003c/span\u003e Out of this relation a cyclic order can be generated,\r\nin a way resembling that in which we generated an open\r\norder from \"between,\" but somewhat more complicated.\u003ca id=\"FNanchor_12_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_12_1\" class=\"fnanchor\"\u003e[12]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_12_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_12_1\"\u003e\u003cspan class=\"label\"\u003e[12]\u003c/span\u003e\u003c/a\u003eCf. \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e, p. 205 (\u0026sect; 194), and references there given.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe purpose of the latter half of this chapter has been to suggest\r\nthe subject which one may call \"generation of serial relations.\"\r\nWhen such relations have been defined, the generation of them\r\nfrom other relations possessing only some of the properties\r\nrequired for series becomes very important, especially in the\r\nphilosophy of geometry and physics. But we cannot, within\r\nthe limits of the present volume, do more than make the reader\r\naware that such a subject exists.\r\n\u003cspan class=\"pagenum\" id=\"Page_41\"\u003e[Pg 41]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027V: KINDS OF RELATIONS\u0027\u003e\u003ca id=\"chap05\"\u003e\u003c/a\u003eCHAPTER V\r\n\u003cbr\u003e\u003cbr\u003e\r\nKINDS OF RELATIONS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nA great part of the philosophy of mathematics is concerned with\r\n\u003ci\u003erelations\u003c/i\u003e, and many different kinds of relations have different\r\nkinds of uses. It often happens that a property which belongs\r\nto \u003ci\u003eall\u003c/i\u003e relations is only important as regards relations of certain\r\nsorts; in these cases the reader will not see the bearing of the\r\nproposition asserting such a property unless he has in mind the\r\nsorts of relations for which it is useful. For reasons of this\r\ndescription, as well as from the intrinsic interest of the subject,\r\nit is well to have in our minds a rough list of the more\r\nmathematically serviceable varieties of relations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe dealt in the preceding chapter with a supremely important\r\nclass, namely, \u003ci\u003eserial\u003c/i\u003e relations. Each of the three properties which\r\nwe combined in defining series\u0026mdash;namely, \u003ci\u003easymmetry\u003c/i\u003e, \u003ci\u003etransitiveness\u003c/i\u003e,\r\nand \u003ci\u003econnexity\u003c/i\u003e\u0026mdash;has its own importance. We will begin by saying\r\nsomething on each of these three.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eAsymmetry\u003c/i\u003e, \u003ci\u003ei.e.\u003c/i\u003e the property of being incompatible with the\r\nconverse, is a characteristic of the very greatest interest and\r\nimportance. In order to develop its functions, we will consider\r\nvarious examples. The relation \u003ci\u003ehusband\u003c/i\u003e is asymmetrical, and\r\nso is the relation \u003ci\u003ewife\u003c/i\u003e; \u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is husband of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e cannot be husband\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e and similarly in the case of \u003ci\u003ewife\u003c/i\u003e. On the other hand, the\r\nrelation \"spouse\" is symmetrical: if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is spouse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e is\r\nspouse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e Suppose now we are given the relation \u003ci\u003espouse\u003c/i\u003e, and\r\nwe wish to derive the relation \u003ci\u003ehusband\u003c/i\u003e. \u003ci\u003eHusband\u003c/i\u003e is the same as\r\n\u003ci\u003emale spouse\u003c/i\u003e or \u003ci\u003espouse of a female\u003c/i\u003e; thus the relation \u003ci\u003ehusband\u003c/i\u003e can\r\n\u003cspan class=\"pagenum\" id=\"Page_42\"\u003e[Pg 42]\u003c/span\u003e\r\nbe derived from \u003ci\u003espouse\u003c/i\u003e either by limiting the domain to males\r\nor by limiting the converse to females. We see from this instance\r\nthat, when a symmetrical relation is given, it is sometimes possible,\r\nwithout the help of any further relation, to separate it into two\r\nasymmetrical relations. But the cases where this is possible are\r\nrare and exceptional: they are cases where there are two mutually\r\nexclusive classes, say \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e such that whenever the relation\r\nholds between two terms, one of the terms is a member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and\r\nthe other is a member of \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0026mdash;as, in the case of \u003ci\u003espouse\u003c/i\u003e, one term\r\nof the relation belongs to the class of males and one to the class\r\nof females. In such a case, the relation with its domain confined\r\nto \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e will be asymmetrical, and so will the relation with its domain\r\nconfined to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e But such cases are not of the sort that occur\r\nwhen we are dealing with series of more than two terms; for in\r\na series, all terms, except the first and last (if these exist), belong\r\nboth to the domain and to the converse domain of the generating\r\nrelation, so that a relation like \u003ci\u003ehusband\u003c/i\u003e, where the domain and\r\nconverse domain do not overlap, is excluded.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe question how to \u003ci\u003econstruct\u003c/i\u003e relations having some useful\r\nproperty by means of operations upon relations which only have\r\nrudiments of the property is one of considerable importance.\r\nTransitiveness and connexity are easily constructed in many cases\r\nwhere the originally given relation does not possess them: for\r\nexample, if \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is any relation whatever, the ancestral relation\r\nderived from \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e by generalised induction is transitive; and if \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is\r\na many-one relation, the ancestral relation will be connected\r\nif confined to the posterity of a given term. But asymmetry is\r\na much more difficult property to secure by construction. The\r\nmethod by which we derived \u003ci\u003ehusband\u003c/i\u003e from \u003ci\u003espouse\u003c/i\u003e is, as we have\r\nseen, not available in the most important cases, such as \u003ci\u003egreater\u003c/i\u003e,\r\n\u003ci\u003ebefore\u003c/i\u003e, \u003ci\u003eto the right of\u003c/i\u003e, where domain and converse domain overlap.\r\nIn all these cases, we can of course obtain a symmetrical relation\r\nby adding together the given relation and its converse, but we\r\ncannot pass back from this symmetrical relation to the original\r\nasymmetrical relation except by the help of some asymmetrical\r\n\u003cspan class=\"pagenum\" id=\"Page_43\"\u003e[Pg 43]\u003c/span\u003e\r\nrelation. Take, for example, the relation \u003ci\u003egreater\u003c/i\u003e: the relation\r\n\u003ci\u003egreater or less\u003c/i\u003e\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e \u003ci\u003eunequal\u003c/i\u003e\u0026mdash;is symmetrical, but there is nothing\r\nin this relation to show that it is the sum of two asymmetrical\r\nrelations. Take such a relation as \"differing in shape.\" This\r\nis not the sum of an asymmetrical relation and its converse, since\r\nshapes do not form a single series; but there is nothing to show\r\nthat it differs from \"differing in magnitude\" if we did not already\r\nknow that magnitudes have relations of greater and less. This\r\nillustrates the fundamental character of asymmetry as a property\r\nof relations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFrom the point of view of the classification of relations, being\r\nasymmetrical is a much more important characteristic than\r\nimplying diversity. Asymmetrical relations imply diversity,\r\nbut the converse is not the case. \"Unequal,\" for example,\r\nimplies diversity, but is symmetrical. Broadly speaking, we\r\nmay say that, if we wished as far as possible to dispense with\r\nrelational propositions and replace them by such as ascribed\r\npredicates to subjects, we could succeed in this so long as we\r\nconfined ourselves to \u003ci\u003esymmetrical\u003c/i\u003e relations: those that do not\r\nimply diversity, if they are transitive, may be regarded as asserting\r\na common predicate, while those that do imply diversity\r\nmay be regarded as asserting incompatible predicates. For\r\nexample, consider the relation of \u003ci\u003esimilarity between classes\u003c/i\u003e,\r\nby means of which we defined numbers. This relation is symmetrical\r\nand transitive and does not imply diversity. It would\r\nbe possible, though less simple than the procedure we adopted,\r\nto regard the number of a collection as a predicate of the collection:\r\nthen two similar classes will be two that have the same\r\nnumerical predicate, while two that are not similar will be two\r\nthat have different numerical predicates. Such a method of\r\nreplacing relations by predicates is formally possible (though\r\noften very inconvenient) so long as the relations concerned are\r\nsymmetrical; but it is formally impossible when the relations\r\nare asymmetrical, because both sameness and difference of predicates\r\nare symmetrical. Asymmetrical relations are, we may\r\n\u003cspan class=\"pagenum\" id=\"Page_44\"\u003e[Pg 44]\u003c/span\u003e\r\nsay, the most characteristically relational of relations, and the\r\nmost important to the philosopher who wishes to study the\r\nultimate logical nature of relations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother class of relations that is of the greatest use is the\r\nclass of one-many relations, \u003ci\u003ei.e.\u003c/i\u003e relations which at most one\r\nterm can have to a given term. Such are father, mother,\r\nhusband (except in Tibet), square of, sine of, and so on. But\r\nparent, square root, and so on, are not one-many. It is possible,\r\nformally, to replace all relations by one-many relations by means\r\nof a device. Take (say) the relation \u003ci\u003eless\u003c/i\u003e among the inductive\r\nnumbers. Given any number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e greater than 1, there will not\r\nbe only one number having the relation \u003ci\u003eless\u003c/i\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e but we can\r\nform the whole class of numbers that are less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e This\r\nis one class, and its relation to \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not shared by any other class.\r\nWe may call the class of numbers that are less than \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e the \"proper\r\nancestry\" of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e in the sense in which we spoke of ancestry and\r\nposterity in connection with mathematical induction. Then\r\n\"proper ancestry\" is a one-many relation (\u003ci\u003eone-many\u003c/i\u003e will always\r\nbe used so as to include \u003ci\u003eone-one\u003c/i\u003e), since each number determines\r\na single class of numbers as constituting its proper ancestry.\r\nThus the relation \u003ci\u003eless than\u003c/i\u003e can be replaced by \u003ci\u003ebeing a member of\r\nthe proper ancestry of\u003c/i\u003e. In this way a one-many relation in which\r\nthe one is a class, together with membership of this class, can\r\nalways formally replace a relation which is not one-many. Peano,\r\nwho for some reason always instinctively conceives of a relation\r\nas one-many, deals in this way with those that are naturally\r\nnot so. Reduction to one-many relations by this method,\r\nhowever, though possible as a matter of form, does not represent\r\na technical simplification, and there is every reason to think\r\nthat it does not represent a philosophical analysis, if only because\r\nclasses must be regarded as \"logical fictions.\" We shall therefore\r\ncontinue to regard one-many relations as a special kind of\r\nrelations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne-many relations are involved in all phrases of the form\r\n\"the so-and-so of such-and-such.\" \"The King of England,\"\r\n\u003cspan class=\"pagenum\" id=\"Page_45\"\u003e[Pg 45]\u003c/span\u003e\r\n\"the wife of Socrates,\" \"the father of John Stuart Mill,\" and\r\nso on, all describe some person by means of a one-many relation\r\nto a given term. A person cannot have more than one father,\r\ntherefore \"the father of John Stuart Mill\" described some one\r\nperson, even if we did not know whom. There is much to\r\nsay on the subject of descriptions, but for the present it is\r\nrelations that we are concerned with, and descriptions are only\r\nrelevant as exemplifying the uses of one-many relations. It\r\nshould be observed that all mathematical functions result from\r\none-many relations: the logarithm of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e the cosine of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e etc.,\r\nare, like the father of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e terms described by means of a one-many\r\nrelation (logarithm, cosine, etc.) to a given term (\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e)\u003c/span\u003e. The\r\nnotion of \u003ci\u003efunction\u003c/i\u003e need not be confined to numbers, or to the\r\nuses to which mathematicians have accustomed us; it can be\r\nextended to all cases of one-many relations, and \"the father of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\"\r\nis just as legitimately a function of which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the argument as\r\nis \"the logarithm of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\" Functions in this sense are \u003ci\u003edescriptive\u003c/i\u003e\r\nfunctions. As we shall see later, there are functions of a still\r\nmore general and more fundamental sort, namely, \u003ci\u003epropositional\u003c/i\u003e\r\nfunctions; but for the present we shall confine our attention\r\nto descriptive functions, \u003ci\u003ei.e.\u003c/i\u003e \"the term having the\r\nrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\" or, for short, \"the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\" where \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is any one-many\r\nrelation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be observed that if \"the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is to describe a definite\r\nterm, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e must be a term to which something has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e\r\nand there must not be more than one term having the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e since \"the,\" correctly used, must imply uniqueness.\r\nThus we may speak of \"the father of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is any human being\r\nexcept Adam and Eve; but we cannot speak of \"the father\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a table or a chair or anything else that does not\r\nhave a father. We shall say that the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e \"exists\" when\r\nthere is just one term, and no more, having the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\r\nThus if \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is a one-many relation, the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e exists whenever\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e belongs to the converse domain of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e and not otherwise.\r\nRegarding \"the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" as a function in the mathematical\r\n\u003cspan class=\"pagenum\" id=\"Page_46\"\u003e[Pg 46]\u003c/span\u003e\r\nsense, we say that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the \"argument\" of the function, and if\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the term which has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e\r\nif \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\r\nthen \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the \"value\" of the function for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e If\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is a one-many relation, the range of possible arguments to\r\nthe function is the converse domain of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e and the range of values\r\nis the domain. Thus the range of possible arguments to the\r\nfunction \"the father of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is all who have fathers, \u003ci\u003ei.e.\u003c/i\u003e the converse\r\ndomain of the relation \u003ci\u003efather\u003c/i\u003e, while the range of possible\r\nvalues for the function is all fathers, \u003ci\u003ei.e.\u003c/i\u003e the domain of the relation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMany of the most important notions in the logic of relations\r\nare descriptive functions, for example: \u003ci\u003econverse\u003c/i\u003e, \u003ci\u003edomain\u003c/i\u003e, \u003ci\u003econverse\r\ndomain\u003c/i\u003e, \u003ci\u003efield\u003c/i\u003e. Other examples will occur as we proceed.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAmong one-many relations, \u003ci\u003eone-one\u003c/i\u003e relations are a specially\r\nimportant class. We have already had occasion to speak of\r\none-one relations in connection with the definition of number,\r\nbut it is necessary to be familiar with them, and not merely\r\nto know their formal definition. Their formal definition may\r\nbe derived from that of one-many relations: they may be\r\ndefined as one-many relations which are also the converses of\r\none-many relations, \u003ci\u003ei.e.\u003c/i\u003e as relations which are both one-many\r\nand many-one. One-many relations may be defined as relations\r\nsuch that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation in question to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e there is no other\r\nterm \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003e which also has the relation to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e Or, again, they may\r\nbe defined as follows: Given two terms \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003e, the terms to\r\nwhich \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the given relation and those to which \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003e has it have\r\nno member in common. Or, again, they may be defined as\r\nrelations such that the relative product of one of them and\r\nits converse implies identity, where the \"relative product\"\r\nof two relations \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e and \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is that relation which holds between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e when there is an intermediate term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has\r\nthe relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e has the relation \u003cimg style=\"vertical-align: -0.025ex; width: 0.891ex; height: 1.038ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-91.png\" alt=\"\" data-tex=\"\\mathrm s\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e Thus, for\r\nexample, if \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is the relation of father to son, the relative product\r\nof \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e and its converse will be the relation which holds between\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and a man \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e when there is a person \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the father\r\nof \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the son of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e It is obvious that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e must be\r\n\u003cspan class=\"pagenum\" id=\"Page_47\"\u003e[Pg 47]\u003c/span\u003e\r\nthe same person. If, on the other hand, we take the relation\r\nof parent and child, which is not one-many, we can no longer\r\nargue that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a parent of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is a child of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e must\r\nbe the same person, because one may be the father of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and the\r\nother the mother. This illustrates that it is characteristic of\r\none-many relations when the relative product of a relation and\r\nits converse implies identity. In the case of one-one relations\r\nthis happens, and also the relative product of the converse and\r\nthe relation implies identity. Given a relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e it is convenient,\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e to think of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e as being reached from \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nby an \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-step\" or an \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-vector.\" In the same case \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e will\r\nbe reached from \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e by a \"backward \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-step.\" Thus we may\r\nstate the characteristic of one-many relations with which we\r\nhave been dealing by saying that an \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-step followed by a backward\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e-step must bring us back to our starting-point. With\r\nother relations, this is by no means the case; for example, if\r\n\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is the relation of child to parent, the relative product of \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e and\r\nits converse is the relation \"self or brother or sister,\" and if \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is\r\nthe relation of grandchild to grandparent, the relative product\r\nof \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e and its converse is \"self or brother or sister or first cousin.\"\r\nIt will be observed that the relative product of two relations\r\nis not in general commutative, \u003ci\u003ei.e.\u003c/i\u003e the relative product of \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nand \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is not in general the same relation as the relative product\r\nof \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e \u003ci\u003eE.g.\u003c/i\u003e the relative product of parent and brother is\r\nuncle, but the relative product of brother and parent is parent.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne-one relations give a correlation of two classes, term for\r\nterm, so that each term in either class has its correlate in the\r\nother. Such correlations are simplest to grasp when the two\r\nclasses have no members in common, like the class of husbands\r\nand the class of wives; for in that case we know at once whether\r\na term is to be considered as one \u003ci\u003efrom\u003c/i\u003e which the correlating\r\nrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e goes, or as one \u003ci\u003eto\u003c/i\u003e which it goes. It is convenient\r\nto use the word \u003ci\u003ereferent\u003c/i\u003e for the term \u003ci\u003efrom\u003c/i\u003e which the relation\r\ngoes, and the term \u003ci\u003erelatum\u003c/i\u003e for the term \u003ci\u003eto\u003c/i\u003e which it goes. Thus\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are husband and wife, then, with respect to the relation\r\n\u003cspan class=\"pagenum\" id=\"Page_48\"\u003e[Pg 48]\u003c/span\u003e\r\n\"husband,\" \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is referent and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e relatum, but with respect to the\r\nrelation \"wife,\" \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is referent and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e relatum. We say that a\r\nrelation and its converse have opposite \"senses\"; thus the\r\n\"sense\" of a relation that goes from \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is the opposite of\r\nthat of the corresponding relation from \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e The fact that a\r\nrelation has a \"sense\" is fundamental, and is part of the reason\r\nwhy order can be generated by suitable relations. It will be\r\nobserved that the class of all possible referents to a given relation\r\nis its domain, and the class of all possible relata is its converse\r\ndomain.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut it very often happens that the domain and converse\r\ndomain of a one-one relation overlap. Take, for example,\r\nthe first ten integers (excluding 0), and add 1 to each; thus\r\ninstead of the first ten integers we now have the integers\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 28.348ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-10.png\" alt=\"\" data-tex=\"\r\n2,\\ 3,\\ 4,\\ 5,\\ 6,\\ 7,\\ 8,\\ 9,\\ 10,\\ 11.\r\n\"\u003e\u003c/span\u003e\r\nThese are the same as those we had before, except that 1 has\r\nbeen cut off at the beginning and 11 has been joined on at the\r\nend. There are still ten integers: they are correlated with\r\nthe previous ten by the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e which is a one-one\r\nrelation. Or, again, instead of adding 1 to each of our original\r\nten integers, we could have doubled each of them, thus obtaining\r\nthe integers\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 32.873ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-11.png\" alt=\"\" data-tex=\"\r\n2,\\ 4,\\ 6,\\ 8,\\ 10,\\ 12,\\ 14,\\ 16,\\ 18,\\ 20.\r\n\"\u003e\u003c/span\u003e\r\nHere we still have five of our previous set of integers, namely,\r\n2, 4, 6, 8, 10. The correlating relation in this case is the relation\r\nof a number to its double, which is again a one-one relation.\r\nOr we might have replaced each number by its square, thus\r\nobtaining the set\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 35.136ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-12.png\" alt=\"\" data-tex=\"\r\n1,\\ 4,\\ 9,\\ 16,\\ 25,\\ 36,\\ 49,\\ 64,\\ 81,\\ 100.\r\n\"\u003e\u003c/span\u003e\r\nOn this occasion only three of our original set are left, namely,\r\n1, 4, 9. Such processes of correlation may be varied endlessly.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe most interesting case of the above kind is the case where\r\nour one-one relation has a converse domain which is part, but\r\n\u003cspan class=\"pagenum\" id=\"Page_49\"\u003e[Pg 49]\u003c/span\u003e\r\nnot the whole, of the domain. If, instead of confining the domain\r\nto the first ten integers, we had considered the whole of the\r\ninductive numbers, the above instances would have illustrated\r\nthis case. We may place the numbers concerned in two rows,\r\nputting the correlate directly under the number whose correlate\r\nit is. Thus when the correlator is the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e we\r\nhave the two rows:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 28.346ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-13.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\u00261,\\ 2,\\ 3,\\ 4,\\ 5,\\ \\dots\\ n,\\ \\dots \\\\\r\n\u00262,\\ 3,\\ 4,\\ 5,\\ 6,\\ \\dots\\ n + 1,\\ \\dots.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nWhen the correlator is the relation of a number to its double,\r\nwe have the two rows:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 26.712ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-14.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\u00261,\\ 2,\\ 3,\\ 4,\\,\\,\\,\\, 5,\\ \\dots\\ n,\\ \\dots \\\\\r\n\u00262,\\ 4,\\ 6,\\ 8,\\ 10,\\ \\dots\\ 2n,\\ \\dots.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nWhen the correlator is the relation of a number to its square,\r\nthe rows are:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.188ex; width: 27.7ex; height: 5.507ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-15.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\u00261,\\ 2,\\ 3,\\ \\,4,\\ \\,\\,\\,\\,5,\\ \\dots\\ n,\\ \\dots \\\\\r\n\u00261,\\ 4,\\ 9,\\ 16,\\ 25,\\ \\dots\\ n^{2},\\ \\dots.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nIn all these cases, all inductive numbers occur in the top row,\r\nand only some in the bottom row.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nCases of this sort, where the converse domain is a \"proper\r\npart\" of the domain (\u003ci\u003ei.e.\u003c/i\u003e a part not the whole), will occupy us\r\nagain when we come to deal with infinity. For the present, we\r\nwish only to note that they exist and demand consideration.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother class of correlations which are often important is\r\nthe class called \"permutations,\" where the domain and converse\r\ndomain are identical. Consider, for example, the six possible\r\narrangements of three letters:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -7.919ex; width: 6.919ex; height: 16.968ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-16.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\na,\\ b,\\ c; \\\\\r\na,\\ c,\\ b; \\\\\r\nb,\\ c,\\ a; \\\\\r\nb,\\ a,\\ c; \\\\\r\nc,\\ a,\\ b; \\\\\r\nc,\\ b,\\ a.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_50\"\u003e[Pg 50]\u003c/span\u003e\r\nEach of these can be obtained from any one of the others by\r\nmeans of a correlation. Take, for example, the first and last,\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 6.919ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-92.png\" alt=\"\" data-tex=\"(a, b, c)\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.919ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-93.png\" alt=\"\" data-tex=\"(c, b, a)\"\u003e.\u003c/span\u003e Here \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is correlated with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e with itself,\r\nand \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e It is obvious that the combination of two permutations\r\nis again a permutation, \u003ci\u003ei.e.\u003c/i\u003e the permutations of a given\r\nclass form what is called a \"group.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese various kinds of correlations have importance in various\r\nconnections, some for one purpose, some for another. The\r\ngeneral notion of one-one correlations has boundless importance\r\nin the philosophy of mathematics, as we have partly seen already,\r\nbut shall see much more fully as we proceed. One of its uses\r\nwill occupy us in our next chapter.\r\n\u003cspan class=\"pagenum\" id=\"Page_51\"\u003e[Pg 51]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027VI: SIMILARITY OF RELATIONS\u0027\u003e\u003ca id=\"chap06\"\u003e\u003c/a\u003eCHAPTER VI\r\n\u003cbr\u003e\u003cbr\u003e\r\nSIMILARITY OF RELATIONS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWE saw in Chapter II. that two classes have the same number\r\nof terms when they are \"similar,\" \u003ci\u003ei.e.\u003c/i\u003e when there is a one-one\r\nrelation whose domain is the one class and whose converse\r\ndomain is the other. In such a case we say that there is a\r\n\"one-one correlation\" between the two classes.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the present chapter we have to define a relation between\r\nrelations, which will play the same part for them that similarity\r\nof classes plays for classes. We will call this relation \"similarity\r\nof relations,\" or \"likeness\" when it seems desirable to use a\r\ndifferent word from that which we use for classes. How is\r\nlikeness to be defined?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe shall employ still the notion of correlation: we shall\r\nassume that the domain of the one relation can be correlated\r\nwith the domain of the other, and the converse domain with the\r\nconverse domain; but that is not enough for the sort of resemblance\r\nwhich we desire to have between our two relations.\r\nWhat we desire is that, whenever either relation holds between\r\ntwo terms, the other relation shall hold between the correlates\r\nof these two terms. The easiest example of the sort of thing\r\nwe desire is a map. When one place is north of another, the\r\nplace on the map corresponding to the one is above the place\r\non the map corresponding to the other; when one place is west\r\nof another, the place on the map corresponding to the one is\r\nto the left of the place on the map corresponding to the other;\r\nand so on. The structure of the map corresponds with that of\r\n\u003cspan class=\"pagenum\" id=\"Page_52\"\u003e[Pg 52]\u003c/span\u003e\r\nthe country of which it is a map. The space-relations in the\r\nmap have \"likeness\" to the space-relations in the country\r\nmapped. It is this kind of connection between relations that\r\nwe wish to define.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may, in the first place, profitably introduce a certain\r\nrestriction. We will confine ourselves, in defining likeness, to\r\nsuch relations as have \"fields,\" \u003ci\u003ei.e.\u003c/i\u003e to such as permit of the\r\nformation of a single class out of the domain and the converse\r\ndomain. This is not always the case. Take, for example,\r\nthe relation \"domain,\" \u003ci\u003ei.e.\u003c/i\u003e the relation which the domain of a\r\nrelation has to the relation. This relation has all classes for its\r\ndomain, since every class is the domain of some relation; and\r\nit has all relations for its converse domain, since every relation\r\nhas a domain. But classes and relations cannot be added together\r\nto form a new single class, because they are of different\r\nlogical \"types.\" We do not need to enter upon the difficult\r\ndoctrine of types, but it is well to know when we are abstaining\r\nfrom entering upon it. We may say, without entering upon\r\nthe grounds for the assertion, that a relation only has a \"field\"\r\nwhen it is what we call \"homogeneous,\" \u003ci\u003ei.e.\u003c/i\u003e when its domain\r\nand converse domain are of the same logical type; and as a\r\nrough-and-ready indication of what we mean by a \"type,\"\r\nwe may say that individuals, classes of individuals, relations\r\nbetween individuals, relations between classes, relations of\r\nclasses to individuals, and so on, are different types. Now the\r\nnotion of likeness is not very useful as applied to relations that\r\nare not homogeneous; we shall, therefore, in defining likeness,\r\nsimplify our problem by speaking of the \"field\" of one of the\r\nrelations concerned. This somewhat limits the generality of\r\nour definition, but the limitation is not of any practical importance.\r\nAnd having been stated, it need no longer be remembered.\r\nWe may define two relations \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e as \"similar,\" or as\r\nhaving \"likeness,\" when there is a one-one relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e whose\r\ndomain is the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and whose converse domain is the field\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e,\u003c/span\u003e and which is such that, if one term has the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_53\"\u003e[Pg 53]\u003c/span\u003e\r\nto another, the correlate of the one has the relation \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e to the\r\ncorrelate of the other, and \u003ci\u003evice versa\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-figure01.jpg\" class=\u0027floatleft\u0027 alt=\u0027fig1\u0027\u003e\r\nA figure will make this\r\nclearer. Let \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e be two\r\nterms having the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\nThen there are to be two terms\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e,\u003c/span\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e has the relation \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e.\u003c/span\u003e If this happens with\r\nevery pair of terms such as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nand \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e and if the converse happens with every pair of terms such\r\nas \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e,\u003c/span\u003e it is clear that for every instance in which the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\r\nholds there is a corresponding instance in which the relation \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e\r\nholds, and \u003ci\u003evice versa\u003c/i\u003e; and this is what we desire to secure by\r\nour definition. We can eliminate some redundancies in the\r\nabove sketch of a definition, by observing that, when the above\r\nconditions are realised, the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is the same as the relative\r\nproduct of \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e and the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e,\u003c/span\u003e\r\n\u003ci\u003ei.e.\u003c/i\u003e the \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e-step from\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e may be replaced by the succession of the \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e-step from\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e,\u003c/span\u003e the \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-step from \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e,\u003c/span\u003e and the backward \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e-step from\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.62ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-96.png\" alt=\"\" data-tex=\"w\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e Thus we may set up the following definitions:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is said to be a \"correlator\" or an \"ordinal\r\ncorrelator\" of two relations \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e if \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is one-one, has the\r\nfield of \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e for its converse domain, and is such that \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is the\r\nrelative product of \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e and the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTwo relations \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e are said to be \"similar,\" or to have\r\n\"likeness,\" when there is at least one correlator of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese definitions will be found to yield what we above decided\r\nto be necessary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be found that, when two relations are similar, they\r\nshare all properties which do not depend upon the actual terms\r\nin their fields. For instance, if one implies diversity, so does\r\nthe other; if one is transitive, so is the other; if one is connected,\r\nso is the other. Hence if one is serial, so is the other.\r\nAgain, if one is one-many or one-one, the other is one-many\r\n\u003cspan class=\"pagenum\" id=\"Page_54\"\u003e[Pg 54]\u003c/span\u003e\r\nor one-one; and so on, through all the general properties of\r\nrelations. Even statements involving the actual terms of the\r\nfield of a relation, though they may not be true as they stand\r\nwhen applied to a similar relation, will always be capable of\r\ntranslation into statements that are analogous. We are led\r\nby such considerations to a problem which has, in mathematical\r\nphilosophy, an importance by no means adequately recognised\r\nhitherto. Our problem may be stated as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven some statement in a language of which we know the\r\ngrammar and the syntax, but not the vocabulary, what are the\r\npossible meanings of such a statement, and what are the meanings\r\nof the unknown words that would make it true?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe reason that this question is important is that it represents,\r\nmuch more nearly than might be supposed, the state of our\r\nknowledge of nature. We know that certain scientific propositions\u0026mdash;which,\r\nin the most advanced sciences, are expressed\r\nin mathematical symbols\u0026mdash;are more or less true of the world,\r\nbut we are very much at sea as to the interpretation to be put\r\nupon the terms which occur in these propositions. We know\r\nmuch more (to use, for a moment, an old-fashioned pair of\r\nterms) about the \u003ci\u003eform\u003c/i\u003e of nature than about the \u003ci\u003ematter\u003c/i\u003e.\r\nAccordingly, what we really know when we enunciate a law\r\nof nature is only that there is probably \u003ci\u003esome\u003c/i\u003e interpretation of\r\nour terms which will make the law approximately true. Thus\r\ngreat importance attaches to the question: What are the\r\npossible meanings of a law expressed in terms of which we do\r\nnot know the substantive meaning, but only the grammar and\r\nsyntax? And this question is the one suggested above.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFor the present we will ignore the general question, which\r\nwill occupy us again at a later stage; the subject of likeness\r\nitself must first be further investigated.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOwing to the fact that, when two relations are similar, their\r\nproperties are the same except when they depend upon the\r\nfields being composed of just the terms of which they are composed,\r\nit is desirable to have a nomenclature which collects\r\n\u003cspan class=\"pagenum\" id=\"Page_55\"\u003e[Pg 55]\u003c/span\u003e\r\ntogether all the relations that are similar to a given relation.\r\nJust as we called the set of those classes that are similar to a\r\ngiven class the \"number\" of that class, so we may call the set\r\nof all those relations that are similar to a given relation the\r\n\"number\" of that relation. But in order to avoid confusion with\r\nthe numbers appropriate to classes, we will speak, in this case, of\r\na \"relation-number.\" Thus we have the following definitions:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"relation-number\" of a given relation is the class of all\r\nthose relations that are similar to the given relation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"Relation-numbers\" are the set of all those classes of relations\r\nthat are relation-numbers of various relations; or, what comes to\r\nthe same thing, a relation number is a class of relations consisting\r\nof all those relations that are similar to one member of the class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen it is necessary to speak of the numbers of classes in\r\na way which makes it impossible to confuse them with relation-numbers,\r\nwe shall call them \"cardinal numbers.\" Thus cardinal\r\nnumbers are the numbers appropriate to classes. These include\r\nthe ordinary integers of daily life, and also certain infinite\r\nnumbers, of which we shall speak later. When we speak of\r\n\"numbers\" without qualification, we are to be understood as\r\nmeaning \u003ci\u003ecardinal\u003c/i\u003e numbers. The definition of a cardinal number,\r\nit will be remembered, is as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"cardinal number\" of a given class is the set of all\r\nthose classes that are similar to the given class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe most obvious application of relation-numbers is to \u003ci\u003eseries\u003c/i\u003e.\r\nTwo series may be regarded as equally long when they have\r\nthe same relation-number. Two \u003ci\u003efinite\u003c/i\u003e series will have the\r\nsame relation-number when their fields have the same cardinal\r\nnumber of terms, and only then\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e a series of (say) 15 terms\r\nwill have the same relation-number as any other series of fifteen\r\nterms, but will not have the same relation-number as a series\r\nof 14 or 16 terms, nor, of course, the same relation-number\r\nas a relation which is not serial. Thus, in the quite special case\r\nof finite series, there is parallelism between cardinal and relation-numbers.\r\nThe relation-numbers applicable to series may be\r\n\u003cspan class=\"pagenum\" id=\"Page_56\"\u003e[Pg 56]\u003c/span\u003e\r\ncalled \"serial numbers\" (what are commonly called \"ordinal\r\nnumbers\" are a sub-class of these); thus a finite serial number\r\nis determinate when we know the cardinal number of terms\r\nin the field of a series having the serial number in question.\r\nIf \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is a finite cardinal number, the relation-number of a series\r\nwhich has \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms is called the \"ordinal\" number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e (There\r\nare also infinite ordinal numbers, but of them we shall speak\r\nin a later chapter.) When the cardinal number of terms in\r\nthe field of a series is infinite, the relation-number of the series\r\nis not determined merely by the cardinal number, indeed an\r\ninfinite number of relation-numbers exist for one infinite cardinal\r\nnumber, as we shall see when we come to consider infinite series.\r\nWhen a series is infinite, what we may call its \"length,\" \u003ci\u003ei.e.\u003c/i\u003e\r\nits relation-number, may vary without change in the cardinal\r\nnumber; but when a series is finite, this cannot happen.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can define addition and multiplication for relation-numbers\r\nas well as for cardinal numbers, and a whole arithmetic\r\nof relation-numbers can be developed. The manner in which\r\nthis is to be done is easily seen by considering the case of series.\r\nSuppose, for example, that we wish to define the sum of two\r\nnon-overlapping series in such a way that the relation-number\r\nof the sum shall be capable of being defined as the sum of the\r\nrelation-numbers of the two series. In the first place, it is clear\r\nthat there is an \u003ci\u003eorder\u003c/i\u003e involved as between the two series: one\r\nof them must be placed before the other. Thus if \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e\r\nare the generating relations of the two series, in the series which\r\nis their sum with \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e put before \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e,\u003c/span\u003e every member of the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\r\nwill precede every member of the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e Thus the serial\r\nrelation which is to be defined as the sum of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e is not\r\n\"\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e or \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e\" simply, but \"\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e or \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e or the relation of any member\r\nof the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to any member of the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e\" Assuming\r\nthat \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e do not overlap, this relation is serial,\r\nbut \"\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e or \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e\"\r\nis not serial, being not connected, since it does not hold between\r\na member of the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and a member of the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e Thus\r\nthe sum of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e,\u003c/span\u003e as above defined, is what we need in order\r\n\u003cspan class=\"pagenum\" id=\"Page_57\"\u003e[Pg 57]\u003c/span\u003e\r\nto define the sum of two relation-numbers. Similar modifications\r\nare needed for products and powers. The resulting arithmetic\r\ndoes not obey the commutative law: the sum or product\r\nwhich they are taken. But it obeys the associative law, one\r\nform of the distributive law, and two of the formal laws for\r\npowers, not only as applied to serial numbers, but as applied to\r\nrelation-numbers generally. Relation-arithmetic, in fact, though\r\nrecent, is a thoroughly respectable branch of mathematics.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt must not be supposed, merely because series afford the\r\nmost obvious application of the idea of likeness, that there are\r\nno other applications that are important. We have already\r\nmentioned maps, and we might extend our thoughts from this\r\nillustration to geometry generally. If the system of relations\r\nby which a geometry is applied to a certain set of terms can be\r\nbrought fully into relations of likeness with a system applying\r\nto another set of terms, then the geometry of the two sets is\r\nindistinguishable from the mathematical point of view, \u003ci\u003ei.e.\u003c/i\u003e all\r\nthe propositions are the same, except for the fact that they are\r\napplied in one case to one set of terms and in the other to another.\r\nWe may illustrate this by the relations of the sort that may be\r\ncalled \"between,\" which we considered in Chapter IV. We\r\nthere saw that, provided a three-term relation has certain formal\r\nlogical properties, it will give rise to series, and may be called\r\na \"between-relation.\" Given any two points, we can use the\r\nbetween-relation to define the straight line determined by those\r\ntwo points; it consists of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e together with all points \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\r\nsuch that the between-relation holds between the three points\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e in some order or other. It has been shown by O. Veblen\r\nthat we may regard our whole space as the field of a three-term\r\nbetween-relation, and define our geometry by the properties we\r\nassign to our between-relation.\u003ca id=\"FNanchor_13_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_13_1\" class=\"fnanchor\"\u003e[13]\u003c/a\u003e\r\nNow likeness is just as easily\r\n\u003cspan class=\"pagenum\" id=\"Page_58\"\u003e[Pg 58]\u003c/span\u003e\r\ndefinable between three-term relations as between two-term\r\nrelations. If \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e are two between-relations, so that\r\n\"\u003cimg style=\"vertical-align: -0.566ex; width: 7.823ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-97.png\" alt=\"\" data-tex=\"x\\mathrm B(y, z)\"\u003e\" means \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is between \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e with respect to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e,\u003c/span\u003e\"\r\nwe shall call \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e a correlator of \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e if\r\nit has the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e\r\nfor its converse domain, and is such that the relation \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e holds\r\nbetween three terms when \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e holds between their \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e-correlates,\r\nand only then. And we shall say that \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e is like \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e when there\r\nis at least one correlator of \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e. The reader can easily\r\nconvince himself that, if \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e is like \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e in this sense, there can be\r\nno difference between the geometry generated by\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e and that\r\ngenerated by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\u0027\u003c/span\u003e.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_13_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_13_1\"\u003e\u003cspan class=\"label\"\u003e[13]\u003c/span\u003e\u003c/a\u003eThis does not apply to elliptic space, but only to spaces in which\r\nthe straight line is an open series. \u003ci\u003eModern Mathematics\u003c/i\u003e, edited by\r\nJ. W. A. Young, pp. 3-51 (monograph by O. Veblen on \"The Foundations of\r\nGeometry\").\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIt follows from this that the mathematician need not concern\r\nhimself with the particular being or intrinsic nature of his points,\r\nlines, and planes, even when he is speculating as an \u003ci\u003eapplied\u003c/i\u003e\r\nmathematician. We may say that there is empirical evidence\r\nof the approximate truth of such parts of geometry as are not\r\nmatters of definition. But there is no empirical evidence as to\r\nwhat a \"point\" is to be. It has to be something that as nearly\r\nas possible satisfies our axioms, but it does not have to be \"very\r\nsmall\" or \"without parts.\" Whether or not it is those things\r\nis a matter of indifference, so long as it satisfies the axioms. If\r\nwe can, out of empirical material, construct a logical structure,\r\nno matter how complicated, which will satisfy our geometrical\r\naxioms, that structure may legitimately be called a \"point.\"\r\nWe must not say that there is nothing else that could legitimately\r\nbe called a \"point\"; we must only say: \"This object we have\r\nconstructed is sufficient for the geometer; it may be one of\r\nmany objects, any of which would be sufficient, but that is no\r\nconcern of ours, since this object is enough to vindicate the\r\nempirical truth of geometry, in so far as geometry is not a\r\nmatter of definition.\" This is only an illustration of the general\r\nprinciple that what matters in mathematics, and to a very great\r\nextent in physical science, is not the intrinsic nature of our\r\nterms, but the logical nature of their interrelations.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may say, of two similar relations, that they have the same\r\n\u003cspan class=\"pagenum\" id=\"Page_59\"\u003e[Pg 59]\u003c/span\u003e\r\n\"structure.\" For mathematical purposes (though not for those\r\nof pure philosophy) the only thing of importance about a relation\r\nis the cases in which it holds, not its intrinsic nature. Just as a\r\nclass may be defined by various different but co-extensive concepts\u0026mdash;\u003ci\u003ee.g.\u003c/i\u003e\r\n\"man\" and \"featherless biped,\"\u0026mdash;so two relations which\r\nare conceptually different may hold in the same set of instances.\r\nAn \"instance\" in which a relation holds is to be conceived as a\r\ncouple of terms, with an order, so that one of the terms comes\r\nfirst and the other second; the couple is to be, of course,\r\nsuch that its first term has the relation in question to its second.\r\nTake (say) the relation \"father\": we can define what we may\r\ncall the \"extension\" of this relation as the class of all ordered\r\ncouples \u003cimg style=\"vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-89.png\" alt=\"\" data-tex=\"(x, y)\"\u003e which are such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the father of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e From\r\nthe mathematical point of view, the only thing of importance\r\nabout the relation \"father\" is that it defines this set of ordered\r\ncouples. Speaking generally, we say:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"extension\" of a relation is the class of those ordered\r\ncouples \u003cimg style=\"vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-89.png\" alt=\"\" data-tex=\"(x, y)\"\u003e which are such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation in question\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can now go a step further in the process of abstraction,\r\nand consider what we mean by \"structure.\" Given any relation,\r\nwe can, if it is a sufficiently simple one, construct a map of it.\r\nFor the sake of definiteness, let us take a relation of which the\r\nextension is the following couples: \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.167ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-98.png\" alt=\"\" data-tex=\"ab\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.176ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-99.png\" alt=\"\" data-tex=\"ac\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 2.373ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-100.png\" alt=\"\" data-tex=\"ad\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.95ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-101.png\" alt=\"\" data-tex=\"bc\"\u003e,\u003c/span\u003e\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.034ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-102.png\" alt=\"\" data-tex=\"ce\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.156ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-103.png\" alt=\"\" data-tex=\"dc\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.231ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-104.png\" alt=\"\" data-tex=\"de\"\u003e,\u003c/span\u003e where\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.176ex; height: 1.593ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-105.png\" alt=\"\" data-tex=\"d\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.054ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-106.png\" alt=\"\" data-tex=\"e\"\u003e are five terms, no matter what.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-figure02.jpg\" class=\u0027floatleft\u0027 alt=\u0027fig2\u0027\u003e\r\nWe may make a\r\n\"map\" of this relation by taking five points\r\non a plane and connecting them by arrows,\r\nas in the accompanying figure. What is\r\nrevealed by the map is what we call the\r\n\"structure\" of the relation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that the \"structure\" of the\r\nrelation does not depend upon the particular\r\nterms that make up the field of the relation.\r\nThe field may be changed without changing the structure, and\r\nthe structure may be changed without changing the field\u0026mdash;for\r\n\u003cspan class=\"pagenum\" id=\"Page_60\"\u003e[Pg 60]\u003c/span\u003e\r\nexample, if we were to add the couple \u003cimg style=\"vertical-align: -0.025ex; width: 2.251ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-107.png\" alt=\"\" data-tex=\"ae\"\u003e in the above illustration\r\nwe should alter the structure but not the field. Two relations\r\nhave the same \"structure,\" we shall say, when the same map\r\nwill do for both\u0026mdash;or, what comes to the same thing, when either\r\ncan be a map for the other (since every relation can be its own\r\nmap). And that, as a moment\u0027s reflection shows, is the very\r\nsame thing as what we have called \"likeness.\" That is to say,\r\ntwo relations have the same structure when they have likeness,\r\n\u003ci\u003ei.e.\u003c/i\u003e when they have the same relation-number. Thus what we\r\ndefined as the \"relation-number\" is the very same thing as is\r\nobscurely intended by the word \"structure\"\u0026mdash;a word which,\r\nimportant as it is, is never (so far as we know) defined in precise\r\nterms by those who use it.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere has been a great deal of speculation in traditional\r\nphilosophy which might have been avoided if the importance of\r\nstructure, and the difficulty of getting behind it, had been realised.\r\nFor example, it is often said that space and time are subjective,\r\nbut they have objective counterparts; or that phenomena are\r\nsubjective, but are caused by things in themselves, which must\r\nhave differences \u003ci\u003einter se\u003c/i\u003e corresponding with the differences in\r\nthe phenomena to which they give rise. Where such hypotheses\r\nare made, it is generally supposed that we can know very little\r\nabout the objective counterparts. In actual fact, however, if\r\nthe hypotheses as stated were correct, the objective counterparts\r\nwould form a world having the same structure as the phenomenal\r\nworld, and allowing us to infer from phenomena the truth of all\r\npropositions that can be stated in abstract terms and are known\r\nto be true of phenomena. If the phenomenal world has three\r\ndimensions, so must the world behind phenomena; if the phenomenal\r\nworld is Euclidean, so must the other be; and so on.\r\nIn short, every proposition having a communicable significance\r\nmust be true of both worlds or of neither: the only difference\r\nmust lie in just that essence of individuality which always eludes\r\nwords and baffles description, but which, for that very reason,\r\nis irrelevant to science. Now the only purpose that philosophers\r\n\u003cspan class=\"pagenum\" id=\"Page_61\"\u003e[Pg 61]\u003c/span\u003e\r\nhave in view in condemning phenomena is in order to persuade\r\nthemselves and others that the real world is very different from\r\nthe world of appearance. We can all sympathise with their wish\r\nto prove such a very desirable proposition, but we cannot congratulate\r\nthem on their success. It is true that many of them\r\ndo not assert objective counterparts to phenomena, and these\r\nescape from the above argument. Those who do assert counterparts\r\nare, as a rule, very reticent on the subject, probably because\r\nthey feel instinctively that, if pursued, it will bring about too\r\nmuch of a \u003ci\u003erapprochement\u003c/i\u003e between the real and the phenomenal\r\nworld. If they were to pursue the topic, they could hardly avoid\r\nthe conclusions which we have been suggesting. In such ways,\r\nas well as in many others, the notion of structure or relation-number\r\nis important.\r\n\u003cspan class=\"pagenum\" id=\"Page_62\"\u003e[Pg 62]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027VII: RATIONAL, REAL, AND COMPLEX NUMBERS\u0027\u003e\u003ca id=\"chap07\"\u003e\u003c/a\u003eCHAPTER VII\r\n\u003cbr\u003e\u003cbr\u003e\r\nRATIONAL, REAL, AND COMPLEX NUMBERS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWE have now seen how to define cardinal numbers, and also\r\nrelation-numbers, of which what are commonly called ordinal\r\nnumbers are a particular species. It will be found that each\r\nof these kinds of number may be infinite just as well as finite.\r\nBut neither is capable, as it stands, of the more familiar extensions\r\nof the idea of number, namely, the extensions to negative,\r\nfractional, irrational, and complex numbers. In the present\r\nchapter we shall briefly supply logical definitions of these various\r\nextensions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne of the mistakes that have delayed the discovery of correct\r\ndefinitions in this region is the common idea that each extension\r\nof number included the previous sorts as special cases. It was\r\nthought that, in dealing with positive and negative integers, the\r\npositive integers might be identified with the original signless\r\nintegers. Again it was thought that a fraction whose denominator\r\nis 1 may be identified with the natural number which is its\r\nnumerator. And the irrational numbers, such as the square\r\nroot of 2, were supposed to find their place among rational fractions,\r\nas being greater than some of them and less than the others,\r\nso that rational and irrational numbers could be taken together\r\nas one class, called \"real numbers.\" And when the idea of\r\nnumber was further extended so as to include \"complex\"\r\nnumbers, \u003ci\u003ei.e.\u003c/i\u003e numbers involving the square root of -1, it was\r\nthought that real numbers could be regarded as those among\r\ncomplex numbers in which the imaginary part (\u003ci\u003ei.e.\u003c/i\u003e the part\r\n\u003cspan class=\"pagenum\" id=\"Page_63\"\u003e[Pg 63]\u003c/span\u003e\r\nwhich was a multiple of the square root of -1) was zero. All\r\nthese suppositions were erroneous, and must be discarded, as we\r\nshall find, if correct definitions are to be given.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us begin with \u003ci\u003epositive and negative integers\u003c/i\u003e. It is obvious\r\non a moment\u0027s consideration that +1 and -1 must both be\r\nrelations, and in fact must be each other\u0027s converses. The\r\nobvious and sufficient definition is that +1 is the relation of\r\n\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e and -1 is the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e\r\nGenerally, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is\r\nany inductive number, \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-108.png\" alt=\"\" data-tex=\"+m\"\u003e will be the relation of \u003cimg style=\"vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-109.png\" alt=\"\" data-tex=\"n + m\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e\r\n(for any \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e)\u003c/span\u003e, and \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-110.png\" alt=\"\" data-tex=\"-m\"\u003e will be the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 6.11ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-109.png\" alt=\"\" data-tex=\"n + m\"\u003e.\u003c/span\u003e According\r\nto this definition, \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-108.png\" alt=\"\" data-tex=\"+m\"\u003e is a relation which is one-one so\r\nlong as \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is a cardinal number (finite or infinite) and \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is an\r\ninductive cardinal number. But \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-108.png\" alt=\"\" data-tex=\"+m\"\u003e is under no circumstances\r\ncapable of being identified with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e,\u003c/span\u003e which is not a relation, but\r\na class of classes. Indeed, \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-108.png\" alt=\"\" data-tex=\"+m\"\u003e is every bit as distinct from \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e\r\nas \u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-110.png\" alt=\"\" data-tex=\"-m\"\u003e is.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eFractions\u003c/i\u003e are more interesting than positive or negative integers.\r\nWe need fractions for many purposes, but perhaps most obviously\r\nfor purposes of measurement. My friend and collaborator Dr\r\nA. N. Whitehead has developed a theory of fractions specially\r\nadapted for their application to measurement, which is set forth\r\nin \u003ci\u003ePrincipia Mathematica\u003c/i\u003e.\u003ca id=\"FNanchor_14_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_14_1\" class=\"fnanchor\"\u003e[14]\u003c/a\u003e\r\nBut if all that is needed is to define\r\nobjects having the required purely mathematical properties, this\r\npurpose can be achieved by a simpler method, which we shall\r\nhere adopt. We shall define the fraction \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e as being that\r\nrelation which holds between two inductive numbers \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e when\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 8.764ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-112.png\" alt=\"\" data-tex=\"xn = ym\"\u003e.\u003c/span\u003e This definition enables us to prove that \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e is a one-one\r\nrelation, provided neither \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is zero. And of course \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-113.png\" alt=\"\" data-tex=\"n/m\"\u003e is\r\nthe converse relation to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_14_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_14_1\"\u003e\u003cspan class=\"label\"\u003e[14]\u003c/span\u003e\u003c/a\u003eVol. III. * 300 ff., especially 303.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nFrom the above definition it is clear that the fraction \u003cimg style=\"vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-114.png\" alt=\"\" data-tex=\"m/1\"\u003e is\r\nthat relation between two integers \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e which consists in the\r\nfact that \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 7.406ex; height: 1.783ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-115.png\" alt=\"\" data-tex=\"x = my\"\u003e.\u003c/span\u003e This relation, like the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 3.747ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-108.png\" alt=\"\" data-tex=\"+m\"\u003e,\u003c/span\u003e is by no\r\nmeans capable of being identified with the inductive cardinal\r\nnumber \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e,\u003c/span\u003e because a relation and a class of classes are objects\r\n\u003cspan class=\"pagenum\" id=\"Page_64\"\u003e[Pg 64]\u003c/span\u003e\r\nof utterly different kinds.\u003ca id=\"FNanchor_15_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_15_1\" class=\"fnanchor\"\u003e[15]\u003c/a\u003e\r\nIt will be seen that \u003cimg style=\"vertical-align: -0.566ex; width: 3.62ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-116.png\" alt=\"\" data-tex=\"0/n\"\u003e is always the\r\nsame relation, whatever inductive number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be; it is, in short,\r\nthe relation of 0 to any other inductive cardinal. We may call\r\nthis the zero of rational numbers; it is not, of course, identical\r\nwith the cardinal number 0. Conversely, the relation \u003cimg style=\"vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-117.png\" alt=\"\" data-tex=\"m/0\"\u003e is\r\nalways the same, whatever inductive number \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e may be. There\r\nis not any inductive cardinal to correspond to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 4.249ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-117.png\" alt=\"\" data-tex=\"m/0\"\u003e.\u003c/span\u003e We may call\r\nit \"the infinity of rationals.\" It is an instance of the sort of\r\ninfinite that is traditional in mathematics, and that is represented\r\nby \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.262ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-118.png\" alt=\"\" data-tex=\"\\infty\"\u003e.\u003c/span\u003e\" This is a totally different sort from the true Cantorian\r\ninfinite, which we shall consider in our next chapter. The infinity\r\nof rationals does not demand, for its definition or use, any\r\ninfinite classes or infinite integers. It is not, in actual fact, a\r\nvery important notion, and we could dispense with it altogether\r\nif there were any object in doing so. The Cantorian infinite, on\r\nthe other hand, is of the greatest and most fundamental importance;\r\nthe understanding of it opens the way to whole new realms\r\nof mathematics and philosophy.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_15_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_15_1\"\u003e\u003cspan class=\"label\"\u003e[15]\u003c/span\u003e\u003c/a\u003eOf course in practice we shall continue to speak of a fraction as (say)\r\ngreater or less than 1, meaning greater or less than the ratio \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.394ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-119.png\" alt=\"\" data-tex=\"1/1\"\u003e.\u003c/span\u003e So\r\nlong as it is understood that the ratio \u003cimg style=\"vertical-align: -0.566ex; width: 3.394ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-119.png\" alt=\"\" data-tex=\"1/1\"\u003e and the cardinal number 1 are\r\ndifferent, it is not necessary to be always pedantic in emphasising the\r\ndifference.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIt will be observed that zero and infinity, alone among ratios,\r\nare not one-one. Zero is one-many, and infinity is many-one.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is not any difficulty in defining \u003ci\u003egreater\u003c/i\u003e and \u003ci\u003eless\u003c/i\u003e among\r\nratios (or fractions). Given two ratios \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e,\u003c/span\u003e we shall say\r\nthat \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e is \u003ci\u003eless\u003c/i\u003e than \u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e if \u003cimg style=\"vertical-align: -0.439ex; width: 3.027ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-121.png\" alt=\"\" data-tex=\"mq\"\u003e is less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.495ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-122.png\" alt=\"\" data-tex=\"pn\"\u003e.\u003c/span\u003e There is no\r\ndifficulty in proving that the relation \"less than,\" so defined, is\r\nserial, so that the ratios form a series in order of magnitude. In\r\nthis series, zero is the smallest term and infinity is the largest.\r\nIf we omit zero and infinity from our series, there is no longer\r\nany smallest or largest ratio; it is obvious that if \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e is any ratio\r\nother than zero and infinity, \u003cimg style=\"vertical-align: -0.566ex; width: 5.606ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-123.png\" alt=\"\" data-tex=\"m/2n\"\u003e is smaller and \u003cimg style=\"vertical-align: -0.566ex; width: 5.606ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-124.png\" alt=\"\" data-tex=\"2m/n\"\u003e is larger,\r\nthough neither is zero or infinity, so that \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e is neither the smallest\r\n\u003cspan class=\"pagenum\" id=\"Page_65\"\u003e[Pg 65]\u003c/span\u003e\r\nnor the largest ratio, and therefore (when zero and infinity are\r\nomitted) there is no smallest or largest, since \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e was chosen\r\narbitrarily. In like manner we can prove that however nearly\r\nequal two fractions may be, there are always other fractions\r\nbetween them. For, let \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e and \u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e be two fractions, of which\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003eis the greater. Then it is easy to see (or to prove) that\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 15.706ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-125.png\" alt=\"\" data-tex=\"(m + p)/(n + q)\"\u003e will be greater than \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e and less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e.\u003c/span\u003e Thus\r\nthe series of ratios is one in which no two terms are consecutive,\r\nbut there are always other terms between any two. Since there\r\nare other terms between these others, and so on \u003ci\u003ead infinitum\u003c/i\u003e, it\r\nis obvious that there are an infinite number of ratios between\r\nany two, however nearly equal these two may be.\u003ca id=\"FNanchor_16_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_16_1\" class=\"fnanchor\"\u003e[16]\u003c/a\u003e\r\nA series having the property that there are always other terms between\r\nany two, so that no two are consecutive, is called \"compact.\"\r\nThus the ratios in order of magnitude form a \"compact\" series.\r\nSuch series have many important properties, and it is important\r\nto observe that ratios afford an instance of a compact series\r\ngenerated purely logically, without any appeal to space or time\r\nor any other empirical datum.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_16_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_16_1\"\u003e\u003cspan class=\"label\"\u003e[16]\u003c/span\u003e\u003c/a\u003eStrictly speaking, this statement, as well as those following to the end\r\nof the paragraph, involves what is called the \"axiom of infinity,\" which\r\nwill be discussed in a later chapter.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nPositive and negative ratios can be defined in a way analogous\r\nto that in which we defined positive and negative integers.\r\nHaving first defined the sum of two ratios \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e and \u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e as\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 13.578ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-126.png\" alt=\"\" data-tex=\"(mq + pn)/nq\"\u003e,\u003c/span\u003e we define \u003cimg style=\"vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-127.png\" alt=\"\" data-tex=\"+p/q\"\u003e as the relation of \u003cimg style=\"vertical-align: -0.566ex; width: 10.551ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-128.png\" alt=\"\" data-tex=\"m/n + p/q\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e,\u003c/span\u003e\r\nwhere \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e is any ratio; and \u003cimg style=\"vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-129.png\" alt=\"\" data-tex=\"-p/q\"\u003e is of course the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 5.07ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-127.png\" alt=\"\" data-tex=\"+p/q\"\u003e.\u003c/span\u003e\r\nThis is not the only possible way of defining positive and\r\nnegative ratios, but it is a way which, for our purpose, has the\r\nmerit of being an obvious adaptation of the way we adopted in\r\nthe case of integers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe come now to a more interesting extension of the idea of\r\nnumber, \u003ci\u003ei.e.\u003c/i\u003e the extension to what are called \"real\" numbers,\r\nwhich are the kind that embrace irrationals. In Chapter I. we\r\nhad occasion to mention \"incommensurables\" and their discovery\r\n\u003cspan class=\"pagenum\" id=\"Page_66\"\u003e[Pg 66]\u003c/span\u003e\r\nby Pythagoras. It was through them, \u003ci\u003ei.e.\u003c/i\u003e through\r\ngeometry, that irrational numbers were first thought of. A\r\nsquare of which the side is one inch long will have a diagonal of\r\nwhich the length is the square root of 2 inches. But, as the\r\nancients discovered, there is no fraction of which the square is 2.\r\nThis proposition is proved in the tenth book of Euclid, which is\r\none of those books that schoolboys supposed to be fortunately lost\r\nin the days when Euclid was still used as a text-book. The proof\r\nis extraordinarily simple. If possible, let \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e be the square root\r\nof 2, so that \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 10.599ex; height: 2.452ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-130.png\" alt=\"\" data-tex=\"m^{2}/n^{2} = 2\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.468ex; height: 2.072ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-131.png\" alt=\"\" data-tex=\"m^{2} = 2n^{2}\"\u003e.\u003c/span\u003e\r\nThus \u003cimg style=\"vertical-align: -0.025ex; width: 2.974ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-132.png\" alt=\"\" data-tex=\"m^{2}\"\u003e is an even number,\r\nand therefore \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e must be an even number, because the square of\r\nan odd number is odd. Now if \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e is even, \u003cimg style=\"vertical-align: -0.025ex; width: 2.974ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-132.png\" alt=\"\" data-tex=\"m^{2}\"\u003e must divide by 4,\r\nfor if \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 7.273ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-133.png\" alt=\"\" data-tex=\"m = 2p\"\u003e,\u003c/span\u003e then \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 9.248ex; height: 2.326ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-134.png\" alt=\"\" data-tex=\"m^{2} = 4p^{2}\"\u003e.\u003c/span\u003e Thus we shall have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 9.75ex; height: 2.326ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-135.png\" alt=\"\" data-tex=\"4p^{2} = 2n^{2}\"\u003e,\u003c/span\u003e where\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is half of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e.\u003c/span\u003e Hence \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 8.619ex; height: 2.326ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-137.png\" alt=\"\" data-tex=\"2p^{2} = n^{2}\"\u003e,\u003c/span\u003e and therefore \u003cimg style=\"vertical-align: -0.566ex; width: 3.627ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-138.png\" alt=\"\" data-tex=\"n/p\"\u003e will also be the\r\nsquare root of 2. But then we can repeat the argument: if\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 6.547ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-139.png\" alt=\"\" data-tex=\"n = 2q\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e will also be the square root of 2, and so on, through\r\nan unending series of numbers that are each half of its predecessor.\r\nBut this is impossible; if we divide a number by 2, and then\r\nhalve the half, and so on, we must reach an odd number after a\r\nfinite number of steps. Or we may put the argument even more\r\nsimply by assuming that the \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e we start with is in its lowest\r\nterms; in that case, \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e cannot both be even; yet we have\r\nseen that, if \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 10.599ex; height: 2.452ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-130.png\" alt=\"\" data-tex=\"m^{2}/n^{2} = 2\"\u003e,\u003c/span\u003e they must be. Thus there cannot be any\r\nfraction \u003cimg style=\"vertical-align: -0.566ex; width: 4.475ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-111.png\" alt=\"\" data-tex=\"m/n\"\u003e whose square is 2.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus no fraction will express exactly the length of the diagonal\r\nof a square whose side is one inch long. This seems like a\r\nchallenge thrown out by nature to arithmetic. However the\r\narithmetician may boast (as Pythagoras did) about the power\r\nof numbers, nature seems able to baffle him by exhibiting lengths\r\nwhich no numbers can estimate in terms of the unit. But the\r\nproblem did not remain in this geometrical form. As soon as\r\nalgebra was invented, the same problem arose as regards the\r\nsolution of equations, though here it took on a wider form,\r\nsince it also involved complex numbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that fractions can be found which approach nearer\r\n\u003cspan class=\"pagenum\" id=\"Page_67\"\u003e[Pg 67]\u003c/span\u003e\r\nand nearer to having their square equal to 2. We can form an\r\nascending series of fractions all of which have their squares\r\nless than 2, but differing from 2 in their later members by\r\nless than any assigned amount. That is to say, suppose I assign\r\nsome small amount in advance, say one-billionth, it will be\r\nfound that all the terms of our series after a certain one, say the\r\ntenth, have squares that differ from 2 by less than this amount.\r\nAnd if I had assigned a still smaller amount, it might have been\r\nnecessary to go further along the series, but we should have\r\nreached sooner or later a term in the series, say the twentieth,\r\nafter which all terms would have had squares differing from 2\r\nby less than this still smaller amount. If we set to work to\r\nextract the square root of 2 by the usual arithmetical rule, we\r\nshall obtain an unending decimal which, taken to so-and-so\r\nmany places, exactly fulfils the above conditions. We can\r\nequally well form a descending series of fractions whose squares\r\nare all greater than 2, but greater by continually smaller amounts\r\nas we come to later terms of the series, and differing, sooner or\r\nlater, by less than any assigned amount. In this way we seem\r\nto be drawing a cordon round the square root of 2, and it may\r\nseem difficult to believe that it can permanently escape us.\r\nNevertheless, it is not by this method that we shall actually\r\nreach the square root of 2.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf we divide \u003ci\u003eall\u003c/i\u003e ratios into two classes, according as their\r\nsquares are less than 2 or not, we find that, among those whose\r\nsquares are \u003ci\u003enot\u003c/i\u003e less than 2, all have their squares greater than 2.\r\nThere is no maximum to the ratios whose square is less than 2,\r\nand no minimum to those whose square is greater than 2. There\r\nis no lower limit short of zero to the difference between the\r\nnumbers whose square is a little less than 2 and the numbers\r\nwhose square is a little greater than 2. We can, in short, divide\r\n\u003ci\u003eall\u003c/i\u003e ratios into two classes such that all the terms in one class\r\nare less than all in the other, there is no maximum to the one\r\nclass, and there is no minimum to the other. Between these\r\ntwo classes, where \u003cimg style=\"vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-140.png\" alt=\"\" data-tex=\"\\sqrt{2}\"\u003e ought to be, there is nothing. Thus our\r\n\u003cspan class=\"pagenum\" id=\"Page_68\"\u003e[Pg 68]\u003c/span\u003e\r\ncordon, though we have drawn it as tight as possible, has been\r\ndrawn in the wrong place, and has not caught \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-140.png\" alt=\"\" data-tex=\"\\sqrt{2}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe above method of dividing all the terms of a series into\r\ntwo classes, of which the one wholly precedes the other, was\r\nbrought into prominence by Dedekind,\u003ca id=\"FNanchor_17_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_17_1\" class=\"fnanchor\"\u003e[17]\u003c/a\u003e\r\nand is therefore called\r\na \"Dedekind cut.\" With respect to what happens at the point\r\nof section, there are four possibilities: (1) there may be a\r\nmaximum to the lower section and a minimum to the upper\r\nsection, (2) there may be a maximum to the one and no minimum\r\nto the other, (3) there may be no maximum to the one, but a\r\nminimum to the other, (4) there may be neither a maximum to\r\nthe one nor a minimum to the other. Of these four cases, the\r\nfirst is illustrated by any series in which there are consecutive\r\nterms: in the series of integers, for instance, a lower section\r\nmust end with some number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e and the upper section must\r\nthen begin with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e The second case will be illustrated\r\nin the series of ratios if we take as our lower section all ratios\r\nup to and including 1, and in our upper section all ratios greater\r\nthan 1. The third case is illustrated if we take for our lower\r\nsection all ratios less than 1, and for our upper section all ratios\r\nfrom 1 upward (including 1 itself). The fourth case, as we have\r\nseen, is illustrated if we put in our lower section all ratios whose\r\nsquare is less than 2, and in our upper section all ratios whose\r\nsquare is greater than 2.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_17_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_17_1\"\u003e\u003cspan class=\"label\"\u003e[17]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eStetigkeit und irrationale Zahlen\u003c/i\u003e, 2nd edition, Brunswick, 1892.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe may neglect the first of our four cases, since it only arises\r\nin series where there are consecutive terms. In the second of\r\nour four cases, we say that the maximum of the lower section\r\nis the \u003ci\u003elower limit\u003c/i\u003e of the upper section, or of any set of terms\r\nchosen out of the upper section in such a way that no term of\r\nthe upper section is before all of them. In the third of our\r\nfour cases, we say that the minimum of the upper section is the\r\nupper limit of the lower section, or of any set of terms chosen\r\nout of the lower section in such a way that no term of the lower\r\nsection is after all of them. In the fourth case, we say that\r\n\u003cspan class=\"pagenum\" id=\"Page_69\"\u003e[Pg 69]\u003c/span\u003e\r\nthere is a \"gap\": neither the upper section nor the lower has\r\na limit or a last term. In this case, we may also say that we\r\nhave an \"irrational section,\" since sections of the series of ratios\r\nhave \"gaps\" when they correspond to irrationals.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhat delayed the true theory of irrationals was a mistaken\r\nbelief that there must be \"limits\" of series of ratios. The\r\nnotion of \"limit\" is of the utmost importance, and before\r\nproceeding further it will be well to define it.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is said to be an \"upper limit\" of a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with\r\nrespect to a relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e if (1) \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has no maximum\r\nin \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e (2) every\r\nmember of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e which belongs to the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e precedes \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e (3) every\r\nmember of the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e which precedes \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e precedes some member\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e (By \"precedes\" we mean \"has the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to.\")\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis presupposes the following definition of a \"maximum\":\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is said to be a \"maximum\" of a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect\r\nto a relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and\r\nof the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and does\r\nnot have the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to any other member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese definitions do not demand that the terms to which\r\nthey are applied should be quantitative. For example, given\r\na series of moments of time arranged by earlier and later, their\r\n\"maximum\" (if any) will be the last of the moments; but if\r\nthey are arranged by later and earlier, their \"maximum\" (if\r\nany) will be the first of the moments.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"minimum\" of a class with respect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is its maximum\r\nwith respect to the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e;\u003c/span\u003e and the \"lower limit\" with\r\nrespect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is the upper limit with respect to the\r\nconverse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe notions of limit and maximum do not essentially demand\r\nthat the relation in respect to which they are defined should\r\nbe serial, but they have few important applications except to\r\ncases when the relation is serial or quasi-serial. A notion which\r\nis often important is the notion \"upper limit or maximum,\"\r\nto which we may give the name \"upper boundary.\" Thus the\r\n\"upper boundary\" of a set of terms chosen out of a series is\r\ntheir last member if they have one, but, if not, it is the first\r\nterm after all of them, if there is such a term. If there is neither\r\n\u003cspan class=\"pagenum\" id=\"Page_70\"\u003e[Pg 70]\u003c/span\u003e\r\na maximum nor a limit, there is no upper boundary. The\r\n\"lower boundary\" is the lower limit or minimum.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nReverting to the four kinds of Dedekind section, we see that\r\nin the case of the first three kinds each section has a boundary\r\n(upper or lower as the case may be), while in the fourth kind\r\nneither has a boundary. It is also clear that, whenever the\r\nlower section has an upper boundary, the upper section has\r\na lower boundary. In the second and third cases, the two\r\nboundaries are identical; in the first, they are consecutive\r\nterms of the series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series is called \"Dedekindian\" when every section has a\r\nboundary, upper or lower as the case may be.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have seen that the series of ratios in order of magnitude\r\nis not Dedekindian.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFrom the habit of being influenced by spatial imagination,\r\npeople have supposed that series \u003ci\u003emust\u003c/i\u003e have limits in cases where\r\nit seems odd if they do not. Thus, perceiving that there was\r\nno \u003ci\u003erational\u003c/i\u003e limit to the ratios whose square is less than 2, they\r\nallowed themselves to \"postulate\" an \u003ci\u003eirrational\u003c/i\u003e limit, which\r\nwas to fill the Dedekind gap. Dedekind, in the above-mentioned\r\nwork, set up the axiom that the gap must always be filled, \u003ci\u003ei.e.\u003c/i\u003e\r\nthat every section must have a boundary. It is for this reason\r\nthat series where his axiom is verified are called \"Dedekindian.\"\r\nBut there are an infinite number of series for which it is not\r\nverified.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe method of \"postulating\" what we want has many advantages;\r\nthey are the same as the advantages of theft over honest\r\ntoil. Let us leave them to others and proceed with our honest toil.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that an irrational Dedekind cut in some way \"represents\"\r\nan irrational. In order to make use of this, which to\r\nbegin with is no more than a vague feeling, we must find some\r\nway of eliciting from it a precise definition; and in order to do\r\nthis, we must disabuse our minds of the notion that an irrational\r\nmust be the limit of a set of ratios. Just as ratios whose denominator\r\nis 1 are not identical with integers, so those rational\r\n\u003cspan class=\"pagenum\" id=\"Page_71\"\u003e[Pg 71]\u003c/span\u003e\r\nnumbers which can be greater or less than irrationals, or can\r\nhave irrationals as their limits, must not be identified with ratios.\r\nWe have to define a new kind of numbers called \"real numbers,\"\r\nof which some will be rational and some irrational. Those that\r\nare rational \"correspond\" to ratios, in the same kind of way\r\nin which the ratio \u003cimg style=\"vertical-align: -0.566ex; width: 3.62ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-141.png\" alt=\"\" data-tex=\"n/1\"\u003e corresponds to the integer \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e;\u003c/span\u003e but they are\r\nnot the same as ratios. In order to decide what they are to be,\r\nlet us observe that an irrational is represented by an irrational\r\ncut, and a cut is represented by its lower section. Let us confine\r\nourselves to cuts in which the lower section has no maximum;\r\nin this case we will call the lower section a \"segment.\" Then\r\nthose segments that correspond to ratios are those that consist\r\nof all ratios less than the ratio they correspond to, which is\r\ntheir boundary; while those that represent irrationals are those\r\nthat have no boundary. Segments, both those that have\r\nboundaries and those that do not, are such that, of any two\r\npertaining to one series, one must be part of the other; hence\r\nthey can all be arranged in a series by the relation of whole and\r\npart. A series in which there are Dedekind gaps, \u003ci\u003ei.e.\u003c/i\u003e in which\r\nthere are segments that have no boundary, will give rise to more\r\nsegments than it has terms, since each term will define a segment\r\nhaving that term for boundary, and then the segments without\r\nboundaries will be extra.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe are now in a position to define a real number and an\r\nirrational number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"real number\" is a segment of the series of ratios in order\r\nof magnitude.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAn \"irrational number\" is a segment of the series of ratios\r\nwhich has no boundary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"rational real number\" is a segment of the series of ratios\r\nwhich has a boundary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus a rational real number consists of all ratios less than a\r\ncertain ratio, and it is the rational real number corresponding\r\nto that ratio. The real number 1, for instance, is the class of\r\nproper fractions.\r\n\u003cspan class=\"pagenum\" id=\"Page_72\"\u003e[Pg 72]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the cases in which we naturally supposed that an irrational\r\nmust be the limit of a set of ratios, the truth is that it is the limit\r\nof the corresponding set of rational real numbers in the series\r\nof segments ordered by whole and part. For example, \u003cimg style=\"vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-140.png\" alt=\"\" data-tex=\"\\sqrt{2}\"\u003e is\r\nthe upper limit of all those segments of the series of ratios that\r\ncorrespond to ratios whose square is less than 2. More simply\r\nstill, \u003cimg style=\"vertical-align: -0.225ex; width: 3.061ex; height: 2.398ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-140.png\" alt=\"\" data-tex=\"\\sqrt{2}\"\u003e is the segment \u003ci\u003econsisting\u003c/i\u003e of all\r\nthose ratios whose square is less than 2.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to prove that the series of segments of any series\r\nis Dedekindian. For, given any set of segments, their boundary\r\nwill be their logical sum, \u003ci\u003ei.e.\u003c/i\u003e the class of all those terms that\r\nbelong to at least one segment of the set.\u003ca id=\"FNanchor_18_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_18_1\" class=\"fnanchor\"\u003e[18]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_18_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_18_1\"\u003e\u003cspan class=\"label\"\u003e[18]\u003c/span\u003e\u003c/a\u003eFor a fuller treatment of the subject of segments and Dedekindian\r\nrelations, see \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. II. * 210-214. For a fuller\r\ntreatment of real numbers, see \u003ci\u003eibid.\u003c/i\u003e, vol. III. * 310 ff., and\r\n\u003ci\u003ePrinciples of Mathematics\u003c/i\u003e, chaps. XXXIII. and XXXIV.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe above definition of real numbers is an example of \"construction\"\r\nas against \"postulation,\" of which we had another\r\nexample in the definition of cardinal numbers. The great\r\nadvantage of this method is that it requires no new assumptions,\r\nbut enables us to proceed deductively from the original apparatus\r\nof logic.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is no difficulty in defining addition and multiplication\r\nfor real numbers as above defined. Given two real numbers\r\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e,\u003c/span\u003e each being a class of ratios, take any member of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and\r\nany member of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e and add them together according to the rule\r\nfor the addition of ratios. Form the class of all such sums\r\nobtainable by varying the selected members of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e.\u003c/span\u003e This\r\ngives a new class of ratios, and it is easy to prove that this new\r\nclass is a segment of the series of ratios. We define it as the\r\nsum of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e.\u003c/span\u003e We may state the definition more shortly as\r\nfollows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \u003ci\u003earithmetical sum of two real numbers\u003c/i\u003e is the class of the\r\narithmetical sums of a member of the one and a member of the\r\nother chosen in all possible ways.\r\n\u003cspan class=\"pagenum\" id=\"Page_73\"\u003e[Pg 73]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can define the arithmetical product of two real numbers\r\nin exactly the same way, by multiplying a member of the one by\r\na member of the other in all possible ways. The class of ratios\r\nthus generated is defined as the product of the two real numbers.\r\n(In all such definitions, the series of ratios is to be defined as\r\nexcluding 0 and infinity.)\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is no difficulty in extending our definitions to positive\r\nand negative real numbers and their addition and multiplication.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt remains to give the definition of complex numbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nComplex numbers, though capable of a geometrical interpretation,\r\nare not demanded by geometry in the same imperative way\r\nin which irrationals are demanded. A \"complex\" number means\r\na number involving the square root of a negative number, whether\r\nintegral, fractional, or real. Since the square of a negative\r\nnumber is positive, a number whose square is to be negative has\r\nto be a new sort of number. Using the letter \u003cimg style=\"vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-144.png\" alt=\"\" data-tex=\"i\"\u003e for the square\r\nroot of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 2.891ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-145.png\" alt=\"\" data-tex=\"-1\"\u003e,\u003c/span\u003e any number involving the square root of a negative\r\nnumber can be expressed in the form \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 5.949ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-146.png\" alt=\"\" data-tex=\"x + yi\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are\r\nreal. The part \u003cimg style=\"vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-147.png\" alt=\"\" data-tex=\"yi\"\u003e is called the \"imaginary\" part of this number,\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e being the \"real\" part. (The reason for the phrase \"real\r\nnumbers\" is that they are contrasted with such as are \"imaginary.\")\r\nComplex numbers have been for a long time habitually\r\nused by mathematicians, in spite of the absence of any precise\r\ndefinition. It has been simply assumed that they would obey\r\nthe usual arithmetical rules, and on this assumption their employment\r\nhas been found profitable. They are required less for\r\ngeometry than for algebra and analysis. We desire, for example,\r\nto be able to say that every quadratic equation has two roots,\r\nand every cubic equation has three, and so on. But if we are\r\nconfined to real numbers, such an equation as \u003cimg style=\"vertical-align: -0.186ex; width: 10.327ex; height: 2.072ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-148.png\" alt=\"\" data-tex=\"x^{2} + 1 = 0\"\u003e has no\r\nroots, and such an equation as \u003cimg style=\"vertical-align: -0.186ex; width: 10.327ex; height: 2.071ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-149.png\" alt=\"\" data-tex=\"x^{3} – 1 = 0\"\u003e has only one. Every\r\ngeneralisation of number has first presented itself as needed for\r\nsome simple problem: negative numbers were needed in order\r\nthat subtraction might be always possible, since otherwise \u003cimg style=\"vertical-align: -0.186ex; width: 4.933ex; height: 1.756ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-150.png\" alt=\"\" data-tex=\"a – b\"\u003e\r\nwould be meaningless if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e were less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e;\u003c/span\u003e fractions were needed\r\n\u003cspan class=\"pagenum\" id=\"Page_74\"\u003e[Pg 74]\u003c/span\u003e\r\nin order that division might be always possible; and complex\r\nnumbers are needed in order that extraction of roots and solution\r\nof equations may be always possible. But extensions of\r\nnumber are not \u003ci\u003ecreated\u003c/i\u003e by the mere need for them: they are\r\ncreated by the definition, and it is to the definition of complex\r\nnumbers that we must now turn our attention.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA complex number may be regarded and defined as simply an\r\nordered couple of real numbers. Here, as elsewhere, many\r\ndefinitions are possible. All that is necessary is that the definitions\r\nadopted shall lead to certain properties. In the case of\r\ncomplex numbers, if they are defined as ordered couples of real\r\nnumbers, we secure at once some of the properties required,\r\nnamely, that two real numbers are required to determine a complex\r\nnumber, and that among these we can distinguish a first\r\nand a second, and that two complex numbers are only identical\r\nwhen the first real number involved in the one is equal to the\r\nfirst involved in the other, and the second to the second. What\r\nis needed further can be secured by defining the rules of addition\r\nand multiplication. We are to have\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.17ex; width: 47.805ex; height: 5.47ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-17.png\" alt=\"\" data-tex=\"\r\n\\begin{alignat*}{2}\r\n\u0026(x + yi) + (x\u0027 + y\u0027i) \u0026\u0026= (x + x\u0027) + (y + y\u0027)i, \\\\\r\n\u0026(x + yi) (x\u0027 + y\u0027i) \u0026\u0026= (xx\u0027 – yy\u0027) + (xy\u0027 + x\u0027y)i.\r\n\\end{alignat*}\r\n\"\u003e\u003c/span\u003e\r\nThus we shall define that, given two ordered couples of real\r\nnumbers, \u003cimg style=\"vertical-align: -0.566ex; width: 5.169ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-89.png\" alt=\"\" data-tex=\"(x, y)\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.424ex; height: 2.283ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-151.png\" alt=\"\" data-tex=\"(x\u0027, y\u0027)\"\u003e,\u003c/span\u003e their sum is to be the couple\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 14.359ex; height: 2.283ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-152.png\" alt=\"\" data-tex=\"(x + x\u0027, y + y\u0027)\"\u003e,\u003c/span\u003e\r\nand their product is to be the couple \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 20.419ex; height: 2.283ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-153.png\" alt=\"\" data-tex=\"(xx\u0027 – yy\u0027, xy\u0027 + x\u0027y)\"\u003e.\u003c/span\u003e\r\nBy these definitions we shall secure that our ordered couples\r\nshall have the properties we desire. For example, take the\r\nproduct of the two couples \u003cimg style=\"vertical-align: -0.566ex; width: 5.006ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-154.png\" alt=\"\" data-tex=\"(0, y)\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 5.634ex; height: 2.283ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-155.png\" alt=\"\" data-tex=\"(0, y\u0027)\"\u003e.\u003c/span\u003e This will, by the\r\nabove rule, be the couple \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 8.503ex; height: 2.283ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-156.png\" alt=\"\" data-tex=\"(-yy\u0027, 0)\"\u003e.\u003c/span\u003e Thus the square of the\r\ncouple \u003cimg style=\"vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-157.png\" alt=\"\" data-tex=\"(0, 1)\"\u003e will be the couple \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-158.png\" alt=\"\" data-tex=\"(-1, 0)\"\u003e.\u003c/span\u003e Now those couples in\r\nwhich the second term is 0 are those which, according to the usual\r\nnomenclature, have their imaginary part zero; in the notation\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 5.949ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-146.png\" alt=\"\" data-tex=\"x + yi\"\u003e,\u003c/span\u003e they are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.972ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-159.png\" alt=\"\" data-tex=\"x + 0i\"\u003e,\u003c/span\u003e which it is natural to write simply \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e Just\r\nas it is natural (but erroneous) to identify ratios whose denominator\r\nis unity with integers, so it is natural (but erroneous)\r\n\u003cspan class=\"pagenum\" id=\"Page_75\"\u003e[Pg 75]\u003c/span\u003e\r\nto identify complex numbers whose imaginary part is zero with\r\nreal numbers. Although this is an error in theory, it is a convenience\r\nin practice; \"\u003cimg style=\"vertical-align: -0.186ex; width: 5.972ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-159.png\" alt=\"\" data-tex=\"x + 0i\"\u003e\" may be replaced simply by \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\"\r\nand \"\u003cimg style=\"vertical-align: -0.464ex; width: 5.786ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-160.png\" alt=\"\" data-tex=\"0 + yi\"\u003e\" by \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-147.png\" alt=\"\" data-tex=\"yi\"\u003e,\u003c/span\u003e\" provided we remember that the \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is\r\nnot really a real number, but a special case of a complex number.\r\nAnd when \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is 1, \"\u003cimg style=\"vertical-align: -0.464ex; width: 1.889ex; height: 1.959ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-147.png\" alt=\"\" data-tex=\"yi\"\u003e\" may of course be replaced by \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-144.png\" alt=\"\" data-tex=\"i\"\u003e.\u003c/span\u003e\" Thus\r\nthe couple \u003cimg style=\"vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-157.png\" alt=\"\" data-tex=\"(0, 1)\"\u003e is represented by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-144.png\" alt=\"\" data-tex=\"i\"\u003e,\u003c/span\u003e and the couple \u003cimg style=\"vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-158.png\" alt=\"\" data-tex=\"(-1, 0)\"\u003e is\r\nrepresented by -1. Now our rules of multiplication make the\r\nsquare of \u003cimg style=\"vertical-align: -0.566ex; width: 5.029ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-157.png\" alt=\"\" data-tex=\"(0, 1)\"\u003e equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.789ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-158.png\" alt=\"\" data-tex=\"(-1, 0)\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e the\r\nsquare of \u003cimg style=\"vertical-align: -0.025ex; width: 0.781ex; height: 1.52ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-144.png\" alt=\"\" data-tex=\"i\"\u003e is -1. This\r\nis what we desired to secure. Thus our definitions serve all\r\nnecessary purposes.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to give a geometrical interpretation of complex\r\nnumbers in the geometry of the plane. This subject was agreeably\r\nexpounded by W. K. Clifford in his \u003ci\u003eCommon Sense of the\r\nExact Sciences\u003c/i\u003e, a book of great merit, but written before the\r\nimportance of purely logical definitions had been realised.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nComplex numbers of a higher order, though much less useful\r\nand important than those what we have been defining, have\r\ncertain uses that are not without importance in geometry, as\r\nmay be seen, for example, in Dr Whitehead\u0027s \u003ci\u003eUniversal Algebra\u003c/i\u003e.\r\nThe definition of complex numbers of order \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is obtained by an\r\nobvious extension of the definition we have given. We define a\r\ncomplex number of order \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e as a one-many relation whose domain\r\nconsists of certain real numbers and whose converse domain\r\nconsists of the integers from 1 to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e\u003ca id=\"FNanchor_19_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_19_1\" class=\"fnanchor\"\u003e[19]\u003c/a\u003e\r\nThis is what would ordinarily\r\nbe indicated by the notation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 18.1ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-161.png\" alt=\"\" data-tex=\"(x_{1}, x_{2}, x_{3}, \\dots, x_{n})\"\u003e,\u003c/span\u003e where the\r\nsuffixes denote correlation with the integers used as suffixes, and\r\nthe correlation is one-many, not necessarily one-one, because \u003cimg style=\"vertical-align: -0.357ex; width: 2.203ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-162.png\" alt=\"\" data-tex=\"x_{r}\"\u003e\r\nand \u003cimg style=\"vertical-align: -0.355ex; width: 2.232ex; height: 1.355ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-163.png\" alt=\"\" data-tex=\"x_{s}\"\u003e may be equal when \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e are not equal. The above\r\ndefinition, with a suitable rule of multiplication, will serve all\r\npurposes for which complex numbers of higher orders are needed.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_19_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_19_1\"\u003e\u003cspan class=\"label\"\u003e[19]\u003c/span\u003e\u003c/a\u003eCf. \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e, \u0026sect; 360, p. 379.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe have now completed our review of those extensions of\r\nnumber which do not involve infinity. The application of number\r\nto infinite collections must be our next topic.\r\n\u003cspan class=\"pagenum\" id=\"Page_76\"\u003e[Pg 76]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027VIII: INFINITE CARDINAL NUMBERS\u0027\u003e\u003ca id=\"chap08\"\u003e\u003c/a\u003eCHAPTER VIII\r\n\u003cbr\u003e\u003cbr\u003e\r\nINFINITE CARDINAL NUMBERS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHE definition of cardinal numbers which we gave in Chapter II.\r\nwas applied in Chapter III. to finite numbers, \u003ci\u003ei.e.\u003c/i\u003e to the ordinary\r\nnatural numbers. To these we gave the name \"inductive\r\nnumbers,\" because we found that they are to be defined as\r\nnumbers which obey mathematical induction starting from 0.\r\nBut we have not yet considered collections which do not have an\r\ninductive number of terms, nor have we inquired whether such\r\ncollections can be said to have a number at all. This is an\r\nancient problem, which has been solved in our own day, chiefly\r\nby Georg Cantor. In the present chapter we shall attempt to\r\nexplain the theory of transfinite or infinite cardinal numbers as\r\nit results from a combination of his discoveries with those of\r\nFrege on the logical theory of numbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt cannot be said to be \u003ci\u003ecertain\u003c/i\u003e that there are in fact any infinite\r\ncollections in the world. The assumption that there are is what\r\nwe call the \"axiom of infinity.\" Although various ways suggest\r\nthemselves by which we might hope to prove this axiom, there\r\nis reason to fear that they are all fallacious, and that there is no\r\nconclusive logical reason for believing it to be true. At the same\r\ntime, there is certainly no logical reason \u003ci\u003eagainst\u003c/i\u003e infinite collections,\r\nand we are therefore justified, in logic, in investigating the hypothesis\r\nthat there are such collections. The practical form of this\r\nhypothesis, for our present purposes, is the assumption that, if\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any inductive number, \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e Various\r\nsubtleties arise in identifying this form of our assumption with\r\n\u003cspan class=\"pagenum\" id=\"Page_77\"\u003e[Pg 77]\u003c/span\u003e\r\nthe form that asserts the existence of infinite collections; but\r\nwe will leave these out of account until, in a later chapter, we\r\ncome to consider the axiom of infinity on its own account. For\r\nthe present we shall merely assume that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive\r\nnumber, \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e This is involved in Peano\u0027s\r\nassumption that no two inductive numbers have the same successor;\r\nfor, if \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-166.png\" alt=\"\" data-tex=\"n = n + 1\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-167.png\" alt=\"\" data-tex=\"n – 1\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e have the same successor,\r\nnamely \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e Thus we are assuming nothing that was not involved\r\nin Peano\u0027s primitive propositions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us now consider the collection of the inductive numbers\r\nthemselves. This is a perfectly well-defined class. In the first\r\nplace, a cardinal number is a set of classes which are all similar\r\nto each other and are not similar to anything except each other.\r\nWe then define as the \"inductive numbers\" those among\r\ncardinals which belong to the posterity of 0 with respect to the\r\nrelation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e those which possess every property\r\npossessed by 0 and by the successors of possessors, meaning by\r\nthe \"successor\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e the number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e Thus the class of\r\n\"inductive numbers\" is perfectly definite. By our general\r\ndefinition of cardinal numbers, the number of terms in the class\r\nof inductive numbers is to be defined as \"all those classes that\r\nare similar to the class of inductive numbers\"\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e this set of\r\nclasses \u003ci\u003eis\u003c/i\u003e the number of the inductive numbers according to our\r\ndefinitions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nNow it is easy to see that this number is not one of the inductive\r\nnumbers. If \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any inductive number, the number of numbers\r\nfrom 0 to \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e (both included) is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e;\u003c/span\u003e therefore the total number\r\nof inductive numbers is greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e no matter which of the\r\ninductive numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be. If we arrange the inductive\r\nnumbers in a series in order of magnitude, this series has no last\r\nterm; but if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive number, every series whose field\r\nhas \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms has a last term, as it is easy to prove. Such differences\r\nmight be multiplied \u003ci\u003ead lib\u003c/i\u003e. Thus the number of inductive\r\nnumbers is a new number, different from all of them, not possessing\r\nall inductive properties. It may happen that 0 has a certain\r\n\u003cspan class=\"pagenum\" id=\"Page_78\"\u003e[Pg 78]\u003c/span\u003e\r\nproperty, and that if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e has it so has \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e and yet that this new\r\nnumber does not have it. The difficulties that so long delayed\r\nthe theory of infinite numbers were largely due to the fact that\r\nsome, at least, of the inductive properties were wrongly judged\r\nto be such as \u003ci\u003emust\u003c/i\u003e belong to all numbers; indeed it was thought\r\nthat they could not be denied without contradiction. The first\r\nstep in understanding infinite numbers consists in realising the\r\nmistakenness of this view.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe most noteworthy and astonishing difference between an\r\ninductive number and this new number is that this new number\r\nis unchanged by adding 1 or subtracting 1 or doubling or halving\r\nor any of a number of other operations which we think of as\r\nnecessarily making a number larger or smaller. The fact of being\r\nnot altered by the addition of 1 is used by Cantor for the definition\r\nof what he calls \"transfinite\" cardinal numbers; but for\r\nvarious reasons, some of which will appear as we proceed, it is\r\nbetter to define an infinite cardinal number as one which does\r\nnot possess all inductive properties, \u003ci\u003ei.e.\u003c/i\u003e simply as one which is\r\nnot an inductive number. Nevertheless, the property of being\r\nunchanged by the addition of 1 is a very important one, and we\r\nmust dwell on it for a time.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTo say that a class has a number which is not altered by the\r\naddition of 1 is the same thing as to say that, if we take a term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nwhich does not belong to the class, we can find a one-one relation\r\nwhose domain is the class and whose converse domain is obtained\r\nby adding \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to the class. For in that case, the class is similar\r\nto the sum of itself and the term \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e to a class having one extra\r\nterm; so that it has the same number as a class with one extra\r\nterm, so that if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is this number, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-166.png\" alt=\"\" data-tex=\"n = n + 1\"\u003e.\u003c/span\u003e In this case, we shall\r\nalso have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-168.png\" alt=\"\" data-tex=\"n = n – 1\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e there will be one-one relations whose\r\ndomains consist of the whole class and whose converse domains\r\nconsist of just one term short of the whole class. It can be shown\r\nthat the cases in which this happens are the same as the apparently\r\nmore general cases in which \u003ci\u003esome\u003c/i\u003e part (short of the whole) can be\r\nput into one-one relation with the whole. When this can be done,\r\n\u003cspan class=\"pagenum\" id=\"Page_79\"\u003e[Pg 79]\u003c/span\u003e\r\nthe correlator by which it is done may be said to \"reflect\" the\r\nwhole class into a part of itself; for this reason, such classes will\r\nbe called \"reflexive.\" Thus:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"reflexive\" class is one which is similar to a proper part\r\nof itself. (A \"proper part\" is a part short of the whole.)\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"reflexive\" cardinal number is the cardinal number of a\r\nreflexive class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have now to consider this property of reflexiveness.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne of the most striking instances of a \"reflexion\" is Royce\u0027s\r\nillustration of the map: he imagines it decided to make a map\r\nof England upon a part of the surface of England. A map, if\r\nit is accurate, has a perfect one-one correspondence with its\r\noriginal; thus our map, which is part, is in one-one relation with\r\nthe whole, and must contain the same number of points as the\r\nwhole, which must therefore be a reflexive number. Royce is\r\ninterested in the fact that the map, if it is correct, must contain\r\na map of the map, which must in turn contain a map of the map\r\nof the map, and so on \u003ci\u003ead infinitum\u003c/i\u003e. This point is interesting,\r\nbut need not occupy us at this moment. In fact, we shall do\r\nwell to pass from picturesque illustrations to such as are more\r\ncompletely definite, and for this purpose we cannot do better\r\nthan consider the number-series itself.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e confined to inductive numbers, is\r\none-one, has the whole of the inductive numbers for its domain,\r\nand all except 0 for its converse domain. Thus the whole class\r\nof inductive numbers is similar to what the same class becomes\r\nwhen we omit 0. Consequently it is a \"reflexive\" class according\r\nto the definition, and the number of its terms is a \"reflexive\"\r\nnumber. Again, the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-59.png\" alt=\"\" data-tex=\"2n\"\u003e,\u003c/span\u003e confined to inductive\r\nnumbers, is one-one, has the whole of the inductive numbers for\r\nits domain, and the even inductive numbers alone for its converse\r\ndomain. Hence the total number of inductive numbers is the\r\nsame as the number of even inductive numbers. This property\r\nwas used by Leibniz (and many others) as a proof that infinite\r\nnumbers are impossible; it was thought self-contradictory that\r\n\u003cspan class=\"pagenum\" id=\"Page_80\"\u003e[Pg 80]\u003c/span\u003e\r\n\"the part should be equal to the whole.\" But this is one of those\r\nphrases that depend for their plausibility upon an unperceived\r\nvagueness: the word \"equal\" has many meanings, but if it is\r\ntaken to mean what we have called \"similar,\" there is no contradiction,\r\nsince an infinite collection can perfectly well have parts\r\nsimilar to itself. Those who regard this as impossible have,\r\nunconsciously as a rule, attributed to numbers in general properties\r\nwhich can only be proved by mathematical induction,\r\nand which only their familiarity makes us regard, mistakenly,\r\nas true beyond the region of the finite.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhenever we can \"reflect\" a class into a part of itself, the\r\nsame relation will necessarily reflect that part into a smaller\r\npart, and so on \u003ci\u003ead infinitum\u003c/i\u003e. For example, we can reflect,\r\nas we have just seen, all the inductive numbers into the even\r\nnumbers; we can, by the same relation (that of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.489ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-59.png\" alt=\"\" data-tex=\"2n\"\u003e)\u003c/span\u003e reflect\r\nthe even numbers into the multiples of 4, these into the multiples\r\nof 8, and so on. This is an abstract analogue to Royce\u0027s problem\r\nof the map. The even numbers are a \"map\" of all the inductive\r\nnumbers; the multiples of 4 are a map of the map; the multiples\r\nof 8 are a map of the map of the map; and so on. If we had\r\napplied the same process to the relation of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e our \"map\"\r\nwould have consisted of all the inductive numbers except 0;\r\nthe map of the map would have consisted of all from 2 onward,\r\nthe map of the map of the map of all from 3 onward; and so on.\r\nThe chief use of such illustrations is in order to become familiar\r\nwith the idea of reflexive classes, so that apparently paradoxical\r\narithmetical propositions can be readily translated into the\r\nlanguage of reflexions and classes, in which the air of paradox\r\nis much less.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be useful to give a definition of the number which is\r\nthat of the inductive cardinals. For this purpose we will\r\nfirst define the kind of series exemplified by the inductive cardinals\r\nin order of magnitude. The kind of series which is called a\r\n\"progression\" has already been considered in Chapter I. It is a\r\nseries which can be generated by a relation of consecutiveness:\r\n\u003cspan class=\"pagenum\" id=\"Page_81\"\u003e[Pg 81]\u003c/span\u003e\r\nevery member of the series is to have a successor, but there is\r\nto be just one which has no predecessor, and every member of\r\nthe series is to be in the posterity of this term with respect to\r\nthe relation \"immediate predecessor.\" These characteristics\r\nmay be summed up in the following definition:\u003ca id=\"FNanchor_20_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_20_1\" class=\"fnanchor\"\u003e[20]\u003c/a\u003e\u0026mdash;\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_20_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_20_1\"\u003e\u003cspan class=\"label\"\u003e[20]\u003c/span\u003e\u003c/a\u003eCf. \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. II. * 123.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nA \"progession\" is a one-one relation such that there is just\r\none term belonging to the domain but not to the converse domain,\r\nand the domain is identical with the posterity of this one term.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to see that a progression, so defined, satisfies Peano\u0027s\r\nfive axioms. The term belonging to the domain but not to the\r\nconverse domain will be what he calls \"0\"; the term to which\r\na term has the one-one relation will be the \"successor\" of the\r\nterm; and the domain of the one-one relation will be what\r\nhe calls \"number.\" Taking his five axioms in turn, we have\r\nthe following translations:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) \"0 is a number\" becomes: \"The member of the domain\r\nwhich is not a member of the converse domain is a member of\r\nthe domain.\" This is equivalent to the existence of such a\r\nmember, which is given in our definition. We will call this\r\nmember \"the first term.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) \"The successor of any number is a number\" becomes:\r\n\"The term to which a given member of the domain has the relation\r\nin question is again a member of the domain.\" This is\r\nproved as follows: By the definition, every member of the\r\ndomain is a member of the posterity of the first term; hence\r\nthe successor of a member of the domain must be a member of\r\nthe posterity of the first term (because the posterity of a term\r\nalways contains its own successors, by the general definition of\r\nposterity), and therefore a member of the domain, because by\r\nthe definition the posterity of the first term is the same as the\r\ndomain.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) \"No two numbers have the same successor.\" This is\r\nonly to say that the relation is one-many, which it is by definition\r\n(being one-one).\r\n\u003cspan class=\"pagenum\" id=\"Page_82\"\u003e[Pg 82]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(4) \"0 is not the successor of any number\" becomes: \"The\r\nfirst term is not a member of the converse domain,\" which is\r\nagain an immediate result of the definition.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(5) This is mathematical induction, and becomes: \"Every\r\nmember of the domain belongs to the posterity of the first term,\"\r\nwhich was part of our definition.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus progressions as we have defined them have the five\r\nformal properties from which Peano deduces arithmetic. It is\r\neasy to show that two progessions are \"similar\" in the sense\r\ndefined for similarity of relations in Chapter VI. We can, of\r\ncourse, derive a relation which is serial from the one-one relation\r\nby which we define a progression: the method used is that\r\nexplained in Chapter IV., and the relation is that of a term to\r\na member of its proper posterity with respect to the original\r\none-one relation.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTwo transitive asymmetrical relations which generate progressions\r\nare similar, for the same reasons for which the corresponding\r\none-one relations are similar. The class of all such\r\ntransitive generators of progressions is a \"serial number\" in\r\nthe sense of Chapter VI.; it is in fact the smallest of infinite\r\nserial numbers, the number to which Cantor has given the name \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-169.png\" alt=\"\" data-tex=\"\\omega\"\u003e,\u003c/span\u003e\r\nby which he has made it famous.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut we are concerned, for the moment, with \u003ci\u003ecardinal\u003c/i\u003e numbers.\r\nSince two progressions are similar relations, it follows that their\r\ndomains (or their fields, which are the same as their domains)\r\nare similar classes. The domains of progressions form a cardinal\r\nnumber, since every class which is similar to the domain of a\r\nprogression is easily shown to be itself the domain of a progression.\r\nThis cardinal number is the smallest of the infinite cardinal\r\nnumbers; it is the one to which Cantor has appropriated the\r\nHebrew Aleph with the suffix 0, to distinguish it from larger\r\ninfinite cardinals, which have other suffixes. Thus the name of\r\nthe smallest of infinite cardinals is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTo say that a class has \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms is the same thing as to say\r\nthat it is a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e and this is the same thing as to say\r\n\u003cspan class=\"pagenum\" id=\"Page_83\"\u003e[Pg 83]\u003c/span\u003e\r\nthat the members of the class can be arranged in a progression.\r\nIt is obvious that any progression remains a progression if we\r\nomit a finite number of terms from it, or every other term, or\r\nall except every tenth term or every hundredth term. These\r\nmethods of thinning out a progression do not make it cease to\r\nbe a progression, and therefore do not diminish the number of\r\nits terms, which remains \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e In fact, any selection from a progression\r\nis a progression if it has no last term, however sparsely\r\nit may be distributed. Take (say) inductive numbers of the form \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.505ex; height: 1.553ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-171.png\" alt=\"\" data-tex=\"n^{n}\"\u003e,\u003c/span\u003e\r\nor \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 3.317ex; height: 1.927ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-172.png\" alt=\"\" data-tex=\"n^{n^{n}}\"\u003e.\u003c/span\u003e Such numbers grow very rare in the higher parts\r\nof the number series, and yet there are just as many of them as\r\nthere are inductive numbers altogether, namely, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nConversely, we can add terms to the inductive numbers without\r\nincreasing their number. Take, for example, ratios. One\r\nmight be inclined to think that there must be many more ratios\r\nthan integers, since ratios whose denominator is 1 correspond\r\nto the integers, and seem to be only an infinitesimal proportion\r\nof ratios. But in actual fact the number of ratios (or fractions)\r\nis exactly the same as the number of inductive numbers, namely, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e\r\nThis is easily seen by arranging ratios in a series on the\r\nfollowing plan: If the sum of numerator and denominator in\r\none is less than in the other, put the one before the other; if\r\nthe sum is equal in the two, put first the one with the smaller\r\nnumerator. This gives us the series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 44.638ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-18.png\" alt=\"\" data-tex=\"\r\n1,\\ 1/2,\\ 2,\\ 1/3,\\ 3,\\ 1/4,\\ 2/3,\\ 3/2,\\ 4,\\ 1/5,\\ \\dots.\r\n\"\u003e\u003c/span\u003e\r\nThis series is a progression, and all ratios occur in it sooner or\r\nlater. Hence we can arrange all ratios in a progression, and\r\ntheir number is therefore \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is not the case, however, that \u003ci\u003eall\u003c/i\u003e infinite collections have\r\n\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms. The number of real numbers, for example, is greater\r\nthan \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e;\u003c/span\u003e it is, in fact, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-173.png\" alt=\"\" data-tex=\"2^{\\aleph_{0}}\"\u003e,\u003c/span\u003e and it is not\r\nhard to prove that \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e is\r\ngreater than \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e even when \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is infinite. The easiest way of\r\nproving this is to prove, first, that if a class has \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e members, it\r\ncontains \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e sub-classes\u0026mdash;in other words, that there are \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e ways\r\n\u003cspan class=\"pagenum\" id=\"Page_84\"\u003e[Pg 84]\u003c/span\u003e\r\nof selecting some of its members (including the extreme cases\r\nwhere we select all or none); and secondly, that the number of\r\nsub-classes contained in a class is always greater than the number\r\nof members of the class. Of these two propositions, the first\r\nis familiar in the case of finite numbers, and is not hard to extend\r\nto infinite numbers. The proof of the second is so simple and\r\nso instructive that we shall give it:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the first place, it is clear that the number of sub-classes\r\nof a given class (say \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e)\u003c/span\u003e is at least as great as the number of\r\nmembers, since each member constitutes a sub-class, and we thus\r\nhave a correlation of all the members with some of the sub-classes.\r\nHence it follows that, if the number of sub-classes is\r\nnot \u003ci\u003eequal\u003c/i\u003e to the number of members, it must be \u003ci\u003egreater\u003c/i\u003e. Now\r\nit is easy to prove that the number is not equal, by showing that,\r\ngiven any one-one relation whose domain is the members and\r\nwhose converse domain is contained among the set of sub-classes,\r\nthere must be at least one sub-class not belonging to\r\nthe converse domain. The proof is as follows:\u003ca id=\"FNanchor_21_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_21_1\" class=\"fnanchor\"\u003e[21]\u003c/a\u003e\r\nWhen a one-one\r\ncorrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is established between all the members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\r\nand some of the sub-classes, it may happen that a given member \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nis correlated with a sub-class of which it is a member; or,\r\nagain, it may happen that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is correlated with a sub-class of\r\nwhich it is not a member. Let us form the whole class, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e say,\r\nof those members \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e which are correlated with sub-classes of which\r\nthey are not members. This is a sub-class of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e and it is not\r\ncorrelated with any member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e For, taking first the members\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e each of them is (by the definition of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e)\u003c/span\u003e correlated with\r\nsome sub-class of which it is not a member, and is therefore not\r\ncorrelated with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e Taking next the terms which are not members\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e each of them (by the definition of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e)\u003c/span\u003e is correlated with\r\nsome sub-class of which it is a member, and therefore again\r\nis not correlated with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e Thus no member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is correlated\r\nwith \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e Since \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e was \u003ci\u003eany\u003c/i\u003e one-one correlation of all members\r\n\u003cspan class=\"pagenum\" id=\"Page_85\"\u003e[Pg 85]\u003c/span\u003e\r\nwith some sub-classes, it follows that there is no correlation\r\nof all members with \u003ci\u003eall\u003c/i\u003e sub-classes. It does not matter to the\r\nproof if \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e has no members: all that happens in that case is that\r\nthe sub-class which is shown to be omitted is the null-class.\r\nHence in any case the number of sub-classes is not equal to the\r\nnumber of members, and therefore, by what was said earlier,\r\nit is greater. Combining this with the proposition that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is\r\nthe number of members, \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e is the number of sub-classes, we have\r\nthe theorem that \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e is always greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e even when \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is\r\ninfinite.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_21_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_21_1\"\u003e\u003cspan class=\"label\"\u003e[21]\u003c/span\u003e\u003c/a\u003eThis proof is taken from Cantor, with some simplifications: see\r\n\u003ci\u003eJahresbericht der deutschen Mathematiker-Vereinigung\u003c/i\u003e, I. (1892), p. 77.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIt follows from this proposition that there is no maximum\r\nto the infinite cardinal numbers. However great an infinite\r\nnumber \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be, \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e will be still greater. The arithmetic of\r\ninfinite numbers is somewhat surprising until one becomes\r\naccustomed to it. We have, for example,\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.672ex; width: 45.257ex; height: 8.476ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-19.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\\aleph_{0} + 1 \u0026= \\aleph_{0}, \\\\\r\n\\aleph_{0} + n \u0026= \\aleph_{0}, \\text{where $n$ is any inductive number,} \\\\\r\n\\aleph_{0}^{2} \u0026= \\aleph_{0}.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\n(This follows from the case of the ratios, for, since a ratio is\r\ndetermined by a pair of inductive numbers, it is easy to see that\r\nthe number of ratios is the square of the number of inductive\r\nnumbers, \u003ci\u003ei.e.\u003c/i\u003e it is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.687ex; width: 2.37ex; height: 2.573ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-175.png\" alt=\"\" data-tex=\"\\aleph_{0}^{2}\"\u003e;\u003c/span\u003e but we saw that it\r\nis also \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e)\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.908ex; width: 59.455ex; height: 8.947ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-20.png\" alt=\"\" data-tex=\"\r\n\\begin{alignat*}{2}\r\n\u0026\u0026\\llap{$\\aleph_{0}^{n}$} \u0026= \\aleph_{0}, \\text{where $n$ is any inductive number.} \\\\\r\n\u0026\\text{(This follows from } \\aleph_{0}^{2} \u0026\u0026= \\aleph_{0} \\text{ by induction; for if $\\aleph_{0}^{n} = \\aleph_{0}$,} \\\\\r\n\u0026\\text{then} \u0026\\llap{$\\aleph_{0}^{n+1}} \u0026= \\aleph_{0}^{2} = \\aleph_{0}.)\r\n\\end{alignat*}\r\n\"\u003e\u003c/span\u003e\r\nBut\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 9.011ex; height: 2.419ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-21.png\" alt=\"\" data-tex=\"\r\n2^{\\aleph_{0}} \u003e \\aleph_{0}.\r\n\"\u003e\u003c/span\u003e\r\nIn fact, as we shall see later, \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-173.png\" alt=\"\" data-tex=\"2^{\\aleph_{0}}\"\u003e is a very important number,\r\nnamely, the number of terms in a series which has \"continuity\"\r\nin the sense in which this word is used by Cantor. Assuming\r\nspace and time to be continuous in this sense (as we commonly\r\ndo in analytical geometry and kinematics), this will be the\r\nnumber of points in space or of instants in time; it will also be\r\nthe number of points in any finite portion of space, whether\r\n\u003cspan class=\"pagenum\" id=\"Page_86\"\u003e[Pg 86]\u003c/span\u003e\r\nline, area, or volume. After \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-173.png\" alt=\"\" data-tex=\"2^{\\aleph_{0}}\"\u003e is the most important and\r\ninteresting of infinite cardinal numbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAlthough addition and multiplication are always possible\r\nwith infinite cardinals, subtraction and division no longer give\r\ndefinite results, and cannot therefore be employed as they are\r\nemployed in elementary arithmetic. Take subtraction to begin\r\nwith: so long as the number subtracted is finite, all goes well;\r\nif the other number is reflexive, it remains unchanged. Thus\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 11.88ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-176.png\" alt=\"\" data-tex=\"\\aleph_{0} – n = \\aleph_{0}\"\u003e,\u003c/span\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is finite; so far, subtraction gives a perfectly\r\ndefinite result. But it is otherwise when we subtract \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e from\r\nitself; we may then get any result, from 0 up to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e This is\r\neasily seen by examples. From the inductive, numbers, take\r\naway the following collections of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) All the inductive numbers\u0026mdash;remainder, zero.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) All the inductive numbers from \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e onwards\u0026mdash;remainder,\r\nthe numbers from 0 to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-167.png\" alt=\"\" data-tex=\"n – 1\"\u003e,\u003c/span\u003e numbering \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms in all.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) All the odd numbers\u0026mdash;remainder, all the even numbers,\r\nnumbering \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAll these are different ways of subtracting \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e and\r\nall give different results.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAs regards division, very similar results follow from the fact\r\nthat \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e is unchanged when multiplied by 2 or 3 or any finite\r\nnumber \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e or by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e It follows that \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e divided\r\nby \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e may have\r\nany value from 1 up to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFrom the ambiguity of subtraction and division it results\r\nthat negative numbers and ratios cannot be extended to infinite\r\nnumbers. Addition, multiplication, and exponentiation proceed\r\nquite satisfactorily, but the inverse operations\u0026mdash;subtraction,\r\ndivision, and extraction of roots\u0026mdash;are ambiguous, and the notions\r\nthat depend upon them fail when infinite numbers are concerned.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe characteristic by which we defined finitude was mathematical\r\ninduction, \u003ci\u003ei.e.\u003c/i\u003e we defined a number as finite when it\r\nobeys mathematical induction starting from 0, and a class as\r\nfinite when its number is finite. This definition yields the sort\r\nof result that a definition ought to yield, namely, that the finite\r\n\u003cspan class=\"pagenum\" id=\"Page_87\"\u003e[Pg 87]\u003c/span\u003e\r\nnumbers are those that occur in the ordinary number-series\r\n0, 1, 2, 3, … But in the present chapter, the infinite numbers\r\nwe have discussed have not merely been non-inductive:\r\nthey have also been \u003ci\u003ereflexive\u003c/i\u003e. Cantor used reflexiveness as the\r\n\u003ci\u003edefinition\u003c/i\u003e of the infinite, and believes that it is equivalent to\r\nnon-inductiveness; that is to say, he believes that every class\r\nand every cardinal is either inductive or reflexive. This may be\r\ntrue, and may very possibly be capable of proof; but the proofs\r\nhitherto offered by Cantor and others (including the present\r\nauthor in former days) are fallacious, for reasons which will be\r\nexplained when we come to consider the \"multiplicative axiom.\"\r\nAt present, it is not known whether there are classes and cardinals\r\nwhich are neither reflexive nor inductive. If \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e were such a\r\ncardinal, we should not have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-166.png\" alt=\"\" data-tex=\"n = n + 1\"\u003e,\u003c/span\u003e but \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e would not be one\r\nof the \"natural numbers,\" and would be lacking in some of the\r\ninductive properties. All \u003ci\u003eknown\u003c/i\u003e infinite classes and cardinals\r\nare reflexive; but for the present it is well to preserve an open\r\nmind as to whether there are instances, hitherto unknown, of\r\nclasses and cardinals which are neither reflexive nor inductive.\r\nMeanwhile, we adopt the following definitions:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \u003ci\u003efinite\u003c/i\u003e class or cardinal is one which is \u003ci\u003einductive\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAn \u003ci\u003einfinite\u003c/i\u003e class or cardinal is one which is \u003ci\u003enot inductive\u003c/i\u003e.\r\nAll \u003ci\u003ereflexive\u003c/i\u003e classes and cardinals are infinite; but it is not known\r\nat present whether all infinite classes and cardinals are reflexive.\r\nWe shall return to this subject in Chapter XII.\r\n\u003cspan class=\"pagenum\" id=\"Page_88\"\u003e[Pg 88]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027IX: INFINITE SERIES AND ORDINALS\u0027\u003e\u003ca id=\"chap09\"\u003e\u003c/a\u003eCHAPTER IX\r\n\u003cbr\u003e\u003cbr\u003e\r\nINFINITE SERIES AND ORDINALS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nAN \"infinite series\" may be defined as a series of which the field\r\nis an infinite class. We have already had occasion to consider\r\none kind of infinite series, namely, progressions. In this chapter\r\nwe shall consider the subject more generally.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe most noteworthy characteristic of an infinite series is\r\nthat its serial number can be altered by merely re-arranging\r\nits terms. In this respect there is a certain oppositeness between\r\ncardinal and serial numbers. It is possible to keep the cardinal\r\nnumber of a reflexive class unchanged in spite of adding terms\r\nto it; on the other hand, it is possible to change the serial\r\nnumber of a series without adding or taking away any terms,\r\nby mere re-arrangement. At the same time, in the case of any\r\ninfinite series it is also possible, as with cardinals, to add terms\r\nwithout altering the serial number: everything depends upon\r\nthe way in which they are added.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn order to make matters clear, it will be best to begin with\r\nexamples. Let us first consider various different kinds of series\r\nwhich can be made out of the inductive numbers arranged on\r\nvarious plans. We start with the series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 21.747ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-22.png\" alt=\"\" data-tex=\"\r\n1,\\ 2,\\ 3,\\ 4,\\ \\dots\\ n,\\ \\dots,\r\n\"\u003e\u003c/span\u003e\r\nwhich, as we have already seen, represents the smallest of infinite\r\nserial numbers, the sort that Cantor calls \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-169.png\" alt=\"\" data-tex=\"\\omega\"\u003e.\u003c/span\u003e Let us\r\nproceed to thin out this series by repeatedly performing the\r\n\u003cspan class=\"pagenum\" id=\"Page_89\"\u003e[Pg 89]\u003c/span\u003e\r\noperation of removing to the end the first even number that\r\noccurs. We thus obtain in succession the various series:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.507ex; width: 32.746ex; height: 8.145ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-23.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\u00261,\\ 3,\\ 4,\\ 5,\\ \\dots\\ n,\\ \\dots\\ 2, \\\\\r\n\u00261,\\ 3,\\ 5,\\ 6,\\ \\dots\\ n + 1,\\ \\dots\\ 2,\\ 4, \\\\\r\n\u00261,\\ 3,\\ 5,\\ 7,\\ \\dots\\ n + 2,\\ \\dots\\ 2,\\ 4,\\ 6,\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nand so on. If we imagine this process carried on as long as\r\npossible, we finally reach the series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 49.589ex; height: 1.971ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-24.png\" alt=\"\" data-tex=\"\r\n1,\\ 3,\\ 5,\\ 7,\\ \\dots\\ 2n + 1,\\ \\dots\\ 2,\\ 4,\\ 6,\\ 8,\\ \\dots\\ 2n,\\ \\dots,\r\n\"\u003e\u003c/span\u003e\r\nin which we have first all the odd numbers and then all the even\r\nnumbers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe serial numbers of these various series are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 25.063ex; height: 1.946ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-177.png\" alt=\"\" data-tex=\"\\omega + 1, \\omega + 2,\r\n\\omega + 3, \\dots\\ 2\\omega\"\u003e.\u003c/span\u003e Each of these numbers is \"greater\" than any\r\nof its predecessors, in the following sense:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOne serial number is said to be \"greater\" than another if\r\nany series having the first number contains a part having the\r\nsecond number, but no series having the second number contains\r\na part having the first number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf we compare the two series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 27.34ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-25.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n\u00261,\\ 2,\\ 3,\\ 4,\\ \\dots\\ n,\\ \\dots, \\\\\r\n\u00261,\\ 3,\\ 4,\\ 5,\\ \\dots\\ n + 1,\\ \\dots\\ 2,\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nwe see that the first is similar to the part of the second which\r\nomits the last term, namely, the number 2, but the second is\r\nnot similar to any part of the first. (This is obvious, but is\r\neasily demonstrated.) Thus the second series has a greater\r\nserial number than the first, according to the definition\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e\r\n\u003cimg style=\"vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-178.png\" alt=\"\" data-tex=\"\\omega + 1\"\u003e is greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-169.png\" alt=\"\" data-tex=\"\\omega\"\u003e.\u003c/span\u003e But if we add a term at the beginning\r\nof a progression instead of the end, we still have a progression.\r\nThus \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.729ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-179.png\" alt=\"\" data-tex=\"1 + \\omega = \\omega\"\u003e.\u003c/span\u003e Thus \u003cimg style=\"vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-180.png\" alt=\"\" data-tex=\"1 + \\omega\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.304ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-178.png\" alt=\"\" data-tex=\"\\omega + 1\"\u003e.\u003c/span\u003e This is\r\ncharacteristic of relation-arithmetic generally: if \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e are\r\ntwo relation-numbers, the general rule is that \u003cimg style=\"vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-181.png\" alt=\"\" data-tex=\"\\mu + \\nu\"\u003e is not equal\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-182.png\" alt=\"\" data-tex=\"\\nu + \\mu\"\u003e.\u003c/span\u003e The case of finite ordinals, in which there is equality,\r\nis quite exceptional.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe series we finally reached just now consisted of first all the\r\nodd numbers and then all the even numbers, and its serial\r\n\u003cspan class=\"pagenum\" id=\"Page_90\"\u003e[Pg 90]\u003c/span\u003e\r\nnumber is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-183.png\" alt=\"\" data-tex=\"2\\omega\"\u003e.\u003c/span\u003e This number is greater than \u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-169.png\" alt=\"\" data-tex=\"\\omega\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.53ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-184.png\" alt=\"\" data-tex=\"\\omega + n\"\u003e,\u003c/span\u003e where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is finite. It is to be observed that, in accordance with the\r\ngeneral definition of order, each of these arrangements of integers\r\nis to be regarded as resulting from some definite relation. \u003ci\u003eE.g.\u003c/i\u003e\r\nthe one which merely removes 2 to the end will be defined by\r\nthe following relation: \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are finite integers, and either\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is 2 and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not 2, or neither is 2 and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\" The\r\none which puts first all the odd numbers and then all the even\r\nones will be defined by: \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are finite integers, and either\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is odd and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is even or \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is less than \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e and both are odd or both\r\nare even.\" We shall not trouble, as a rule, to give these formulæ\r\nin future; but the fact that they \u003ci\u003ecould\u003c/i\u003e be given is essential.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe number which we have called \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-183.png\" alt=\"\" data-tex=\"2\\omega\"\u003e,\u003c/span\u003e namely, the number of\r\na series consisting of two progressions, is sometimes called \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-185.png\" alt=\"\" data-tex=\"\\omega · 2\"\u003e.\u003c/span\u003e\r\nMultiplication, like addition, depends upon the order of the\r\nfactors: a progression of couples gives a series such as\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 38.411ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-26.png\" alt=\"\" data-tex=\"\r\nx_{1},\\ y_{1},\\ x_{2},\\ y_{2},\\ x_{3},\\ y_{3},\\ \\dots\\ x_{n},\\ y_{n},\\ \\dots,\r\n\"\u003e\u003c/span\u003e\r\nwhich is itself a progression; but a couple of progressions gives\r\na series which is twice as long as a progression. It is therefore\r\nnecessary to distinguish between \u003cimg style=\"vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-183.png\" alt=\"\" data-tex=\"2\\omega\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-185.png\" alt=\"\" data-tex=\"\\omega · 2\"\u003e.\u003c/span\u003e Usage is variable;\r\nwe shall use \u003cimg style=\"vertical-align: -0.025ex; width: 2.538ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-183.png\" alt=\"\" data-tex=\"2\\omega\"\u003e for a couple of progressions and \u003cimg style=\"vertical-align: -0.025ex; width: 4.173ex; height: 1.532ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-185.png\" alt=\"\" data-tex=\"\\omega · 2\"\u003e for a progression\r\nof couples, and this decision of course governs our\r\ngeneral interpretation of \"\u003cimg style=\"vertical-align: -0.439ex; width: 4.363ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-186.png\" alt=\"\" data-tex=\"\\alpha · \\beta\"\u003e\" when \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e are relation-numbers:\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 4.363ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-186.png\" alt=\"\" data-tex=\"\\alpha · \\beta\"\u003e\" will have to stand for a suitably constructed\r\nsum of a relations each having \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can proceed indefinitely with the process of thinning\r\nout the inductive numbers. For example, we can place first\r\nthe odd numbers, then their doubles, then the doubles of these,\r\nand so on. We thus obtain the series\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -2.036ex; width: 55.474ex; height: 5.204ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-27.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n1,\\ 3,\\ 5,\\ 7,\\ \\dots;\\quad\r\n2,\\ 6,\\ 10,\\ 14,\\ \\dots;\\quad\r\n4,\\ 12,\\ 20,\\ 28,\\ \\dots; \\\\\r\n8,\\ 24,\\ 40,\\ 56,\\ \\dots,\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nof which the number is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.395ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-187.png\" alt=\"\" data-tex=\"\\omega^{2}\"\u003e,\u003c/span\u003e since it is a progression of progressions.\r\nAny one of the progressions in this new series can of course be\r\n\u003cspan class=\"pagenum\" id=\"Page_91\"\u003e[Pg 91]\u003c/span\u003e\r\nthinned out as we thinned out our original progression. We can\r\nproceed to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.395ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-188.png\" alt=\"\" data-tex=\"\\omega^{3}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.395ex; height: 1.929ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-189.png\" alt=\"\" data-tex=\"\\omega^{4}\"\u003e,\u003c/span\u003e … \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.59ex; height: 1.555ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-190.png\" alt=\"\" data-tex=\"\\omega^{\\omega}\"\u003e,\u003c/span\u003e and so on; however far we have gone,\r\nwe can always go further.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe series of all the ordinals that can be obtained in this way,\r\n\u003ci\u003ei.e.\u003c/i\u003e all that can be obtained by thinning out a progression, is\r\nitself longer than any series that can be obtained by re-arranging\r\nthe terms of a progression. (This is not difficult to prove.)\r\nThe cardinal number of the class of such ordinals can be shown\r\nto be greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e;\u003c/span\u003e it is the number which Cantor calls \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-191.png\" alt=\"\" data-tex=\"\\aleph_{1}\"\u003e.\u003c/span\u003e\r\nThe ordinal number of the series of all ordinals that can\r\nbe made out of an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e taken in order of magnitude, is called \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-192.png\" alt=\"\" data-tex=\"\\omega_{1}\"\u003e.\u003c/span\u003e\r\nThus a series whose ordinal number is \u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-192.png\" alt=\"\" data-tex=\"\\omega_{1}\"\u003e has a field whose\r\ncardinal number is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-191.png\" alt=\"\" data-tex=\"\\aleph_{1}\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can proceed from \u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-192.png\" alt=\"\" data-tex=\"\\omega_{1}\"\u003e and \u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-191.png\" alt=\"\" data-tex=\"\\aleph_{1}\"\u003e to \u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-193.png\" alt=\"\" data-tex=\"\\omega_{2}\"\u003e and \u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-194.png\" alt=\"\" data-tex=\"\\aleph_{2}\"\u003e by a process\r\nexactly analogous to that by which we advanced from \u003cimg style=\"vertical-align: -0.025ex; width: 1.407ex; height: 1.027ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-169.png\" alt=\"\" data-tex=\"\\omega\"\u003e and \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e\r\nto \u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-192.png\" alt=\"\" data-tex=\"\\omega_{1}\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-191.png\" alt=\"\" data-tex=\"\\aleph_{1}\"\u003e.\u003c/span\u003e And there is nothing to prevent us from advancing\r\nindefinitely in this way to new cardinals and new ordinals. It\r\nis not known whether \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-173.png\" alt=\"\" data-tex=\"2^{\\aleph_{0}}\"\u003e is equal to any of the cardinals in the\r\nseries of Alephs. It is not even known whether it is comparable\r\nwith them in magnitude; for aught we know, it may be neither\r\nequal to nor greater nor less than any one of the Alephs. This\r\nquestion is connected with the multiplicative axiom, of which\r\nwe shall treat later.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAll the series we have been considering so far in this chapter\r\nhave been what is called \"well-ordered.\" A well-ordered\r\nseries is one which has a beginning, and has consecutive terms,\r\nand has a term \u003ci\u003enext\u003c/i\u003e after any selection of its terms, provided\r\nthere are any terms after the selection. This excludes, on the\r\none hand, compact series, in which there are terms between\r\nany two, and on the other hand series which have no beginning,\r\nor in which there are subordinate parts having no beginning.\r\nThe series of negative integers in order of magnitude, having\r\nno beginning, but ending with -1, is not well-ordered; but\r\ntaken in the reverse order, beginning with -1, it is well-ordered,\r\nbeing in fact a progression. The definition is:\r\n\u003cspan class=\"pagenum\" id=\"Page_92\"\u003e[Pg 92]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"well-ordered\" series is one in which every sub-class\r\n(except, of course, the null-class) has a first term.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAn \"ordinal\" number means the relation-number of a well-ordered\r\nseries. It is thus a species of serial number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAmong well-ordered series, a generalised form of mathematical\r\ninduction applies. A property may be said to be \"transfinitely\r\nhereditary\" if, when it belongs to a certain selection of the\r\nterms in a series, it belongs to their immediate successor provided\r\nthey have one. In a well-ordered series, a transfinitely\r\nhereditary property belonging to the first term of the series\r\nbelongs to the whole series. This makes it possible to prove\r\nmany propositions concerning well-ordered series which are not\r\ntrue of all series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to arrange the inductive numbers in series which\r\nare not well-ordered, and even to arrange them in compact\r\nseries. For example, we can adopt the following plan: consider\r\nthe decimals from .1 (inclusive) to 1 (exclusive), arranged in order\r\nof magnitude. These form a compact series; between any\r\ntwo there are always an infinite number of others. Now omit\r\nthe dot at the beginning of each, and we have a compact series\r\nconsisting of all finite integers except such as divide by 10. If\r\nwe wish to include those that divide by 10, there is no difficulty;\r\ninstead of starting with .1, we will include all decimals less than 1,\r\nbut when we remove the dot, we will transfer to the right any\r\n0\u0027s that occur at the beginning of our decimal. Omitting these,\r\nand returning to the ones that have no 0\u0027s at the beginning,\r\nwe can state the rule for the arrangement of our integers as\r\nfollows: Of two integers that do not begin with the same digit,\r\nthe one that begins with the smaller digit comes first. Of two\r\nthat do begin with the same digit, but differ at the second digit,\r\nthe one with the smaller second digit comes first, but first of all\r\nthe one with no second digit; and so on. Generally, if two\r\nintegers agree as regards the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e digits, but not as regards\r\nthe \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 8.701ex; height: 2.497ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-195.png\" alt=\"\" data-tex=\"(n + 1)^\\mathord{th}\"\u003e,\u003c/span\u003e that one comes first which has either no\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 8.701ex; height: 2.497ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-195.png\" alt=\"\" data-tex=\"(n + 1)^\\mathord{th}\"\u003e\r\ndigit or a smaller one than the other. This rule of arrangement,\r\n\u003cspan class=\"pagenum\" id=\"Page_93\"\u003e[Pg 93]\u003c/span\u003e\r\nas the reader can easily convince himself, gives rise to a compact\r\nseries containing all the integers not divisible by 10; and,\r\nas we saw, there is no difficulty about including those\r\nthat are divisible by 10. It follows from this example that\r\nit is possible to construct compact series having \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms.\r\nIn fact, we have already seen that there are \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e ratios, and\r\nratios in order of magnitude form a compact series; thus\r\nwe have here another example. We shall resume this topic\r\nin the next chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOf the usual formal laws of addition, multiplication, and exponentiation,\r\nall are obeyed by transfinite cardinals, but only\r\nsome are obeyed by transfinite ordinals, and those that are obeyed\r\nby them are obeyed by all relation-numbers. By the \"usual\r\nformal laws\" we mean the following:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nI. The commutative law:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 34.801ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-28.png\" alt=\"\" data-tex=\"\r\n\\alpha + \\beta = \\beta + \\alpha \\quad\\text{and}\\quad\r\n\\alpha × \\beta = \\beta × \\alpha.\r\n\"\u003e\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nII. The associative law:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 55.807ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-29.png\" alt=\"\" data-tex=\"\r\n(\\alpha + \\beta) + \\gamma = \\alpha + (\\beta + \\gamma) \\quad\\text{and}\\quad\r\n(\\alpha × \\beta) × \\gamma = \\alpha × (\\beta × \\gamma).\r\n\"\u003e\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIII. The distributive law:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-30.png\" alt=\"\" data-tex=\"\r\n\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma.\r\n\"\u003e\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen the commutative law does not hold, the above form\r\nof the distributive law must be distinguished from\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-31.png\" alt=\"\" data-tex=\"\r\n(\\beta + \\gamma)\\alpha = \\beta\\alpha + \\gamma\\alpha.\r\n\"\u003e\u003c/span\u003e\r\nAs we shall see immediately, one form may be true and the\r\nother false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIV. The laws of exponentiation:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 48.341ex; height: 2.628ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-32.png\" alt=\"\" data-tex=\"\r\n\\alpha^{\\beta} · \\alpha^{\\gamma} = \\alpha^{\\beta + \\gamma},\\quad\r\n\\alpha^{\\gamma} · \\beta^{\\gamma} = (\\alpha\\beta)^{\\gamma},\\quad\r\n(\\alpha^{\\beta})^{\\gamma} = \\alpha^{\\beta\\gamma}.\r\n\"\u003e\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAll these laws hold for cardinals, whether finite or infinite,\r\nand for \u003ci\u003efinite\u003c/i\u003e ordinals. But when we come to infinite ordinals,\r\nor indeed to relation-numbers in general, some hold and some\r\ndo not. The commutative law does not hold; the associative\r\nlaw does hold; the distributive law (adopting the convention\r\n\u003cspan class=\"pagenum\" id=\"Page_94\"\u003e[Pg 94]\u003c/span\u003e\r\nwe have adopted above as regards the order of the factors in a\r\nproduct) holds in the form\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-33.png\" alt=\"\" data-tex=\"\r\n(\\beta + \\gamma)\\alpha = \\beta\\alpha + \\gamma\\alpha,\r\n\"\u003e\u003c/span\u003e\r\nbut not in the form\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 20.3ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-34.png\" alt=\"\" data-tex=\"\r\n\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma;\r\n\"\u003e\u003c/span\u003e\r\nthe exponential laws\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 35.315ex; height: 2.628ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-35.png\" alt=\"\" data-tex=\"\r\n\\alpha^{\\beta} · \\alpha^{\\gamma} = \\alpha^{\\beta + \\gamma},\r\n\\quad\\text{and}\\quad\r\n(\\alpha^{\\beta})^{\\gamma} = \\alpha^{\\beta\\gamma}\r\n\"\u003e\u003c/span\u003e\r\nstill hold, but not the law\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 15.667ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-36.png\" alt=\"\" data-tex=\"\r\n\\alpha^{\\gamma} · \\beta^{\\gamma} = (\\alpha\\beta)^{\\gamma},\r\n\"\u003e\u003c/span\u003e\r\nwhich is obviously connected with the commutative law for\r\nmultiplication.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe definitions of multiplication and exponentiation that\r\nare assumed in the above propositions are somewhat complicated.\r\nThe reader who wishes to know what they are and how the\r\nabove laws are proved must consult the second volume of\r\n\u003ci\u003ePrincipia Mathematica\u003c/i\u003e, * 172-176.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOrdinal transfinite arithmetic was developed by Cantor at\r\nan earlier stage than cardinal transfinite arithmetic, because it\r\nhas various technical mathematical uses which led him to it.\r\nBut from the point of view of the philosophy of mathematics\r\nit is less important and less fundamental than the theory of\r\ntransfinite cardinals. Cardinals are essentially simpler than\r\nordinals, and it is a curious historical accident that they first\r\nappeared as an abstraction from the latter, and only gradually\r\ncame to be studied on their own account. This does not apply\r\nto Frege\u0027s work, in which cardinals, finite and transfinite, were\r\ntreated in complete independence of ordinals; but it was\r\nCantor\u0027s work that made the world aware of the subject, while\r\nFrege\u0027s remained almost unknown, probably in the main on\r\naccount of the difficulty of his symbolism. And mathematicians,\r\nlike other people, have more difficulty in understanding and\r\nusing notions which are comparatively \"simple\" in the logical\r\nsense than in manipulating more complex notions which are\r\n\u003cspan class=\"pagenum\" id=\"Page_95\"\u003e[Pg 95]\u003c/span\u003e\r\nmore akin to their ordinary practice. For these reasons, it was\r\nonly gradually that the true importance of cardinals in mathematical\r\nphilosophy was recognised. The importance of ordinals,\r\nthough by no means small, is distinctly less than that of cardinals,\r\nand is very largely merged in that of the more general conception\r\nof relation-numbers.\r\n\u003cspan class=\"pagenum\" id=\"Page_96\"\u003e[Pg 96]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027X: LIMITS AND CONTINUITY\u0027\u003e\u003ca id=\"chap10\"\u003e\u003c/a\u003eCHAPTER X\r\n\u003cbr\u003e\u003cbr\u003e\r\nLIMITS AND CONTINUITY\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHE conception of a \"limit\" is one of which the importance in\r\nmathematics has been found continually greater than had been\r\nthought. The whole of the differential and integral calculus,\r\nindeed practically everything in higher mathematics, depends\r\nupon limits. Formerly, it was supposed that infinitesimals were\r\ninvolved in the foundations of these subjects, but Weierstrass\r\nshowed that this is an error: wherever infinitesimals were thought\r\nto occur, what really occurs is a set of finite quantities having\r\nzero for their lower limit. It used to be thought that \"limit\"\r\nwas an essentially quantitative notion, namely, the notion of a\r\nquantity to which others approached nearer and nearer, so that\r\namong those others there would be some differing by less than any\r\nassigned quantity. But in fact the notion of \"limit\" is a purely\r\nordinal notion, not involving quantity at all (except by accident\r\nwhen the series concerned happens to be quantitative). A given\r\npoint on a line may be the limit of a set of points on the line,\r\nwithout its being necessary to bring in co-ordinates or measurement\r\nor anything quantitative. The cardinal number \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e is the\r\nlimit (in the order of magnitude) of the cardinal numbers 1, 2,\r\n3, … \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e …, although the numerical difference between \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e\r\nand a finite cardinal is constant and infinite: from a quantitative\r\npoint of view, finite numbers get no nearer to \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e as they grow\r\nlarger. What makes \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e the limit of the finite numbers is the\r\nfact that, in the series, it comes immediately after them, which\r\nis an \u003ci\u003eordinal\u003c/i\u003e fact, not a quantitative fact.\r\n\u003cspan class=\"pagenum\" id=\"Page_97\"\u003e[Pg 97]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere are various forms of the notion of \"limit,\" of increasing\r\ncomplexity. The simplest and most fundamental form,\r\nfrom which the rest are derived, has been already defined, but\r\nwe will here repeat the definitions which lead to it, in a general\r\nform in which they do not demand that the relation concerned\r\nshall be serial. The definitions are as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"minima\" of a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect to a relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are\r\nthose members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e (if any) to which no member\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"maxima\" with respect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are the minima with respect\r\nto the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"sequents\" of a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect to a relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are\r\nthe minima of the \"successors\" of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e and the \"successors\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\r\nare those members of the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to which every member of\r\nthe common part of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and the field of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e has the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"precedents\" with respect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are the sequents with\r\nrespect to the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"upper limits\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are the sequents\r\nprovided \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has no maximum; but if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has a maximum, it has no\r\nupper limits.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"lower limits\" with respect to \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e are the upper limits with\r\nrespect to the converse of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhenever \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e has connexity, a class can have at most one\r\nmaximum, one minimum, one sequent, etc. Thus, in the cases\r\nwe are concerned with in practice, we can speak of \"\u003ci\u003ethe\u003c/i\u003e limit\"\r\n(if any).\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is a serial relation, we can greatly simplify the above\r\ndefinition of a limit. We can, in that case, define first the\r\n\"boundary\" of a class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e its limits or maximum, and then\r\nproceed to distinguish the case where the boundary is the limit\r\nfrom the case where it is a maximum. For this purpose it is\r\nbest to use the notion of \"segment.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe will speak of the \"segment of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e defined by a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\" as\r\nall those terms that have the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to some one or more of\r\nthe members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e This will be a segment in the sense defined\r\n\u003cspan class=\"pagenum\" id=\"Page_98\"\u003e[Pg 98]\u003c/span\u003e\r\nin Chapter VII.; indeed, every segment in the sense there defined\r\nis the segment defined by \u003ci\u003esome\u003c/i\u003e class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is serial, the\r\nsegment defined by \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e consists of all the terms that precede\r\nsome term or other of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has a maximum, the segment will\r\nbe all the predecessors of the maximum. But if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has no\r\nmaximum, every member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e precedes some other member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e\r\nand the whole of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is therefore included in the segment defined\r\nby \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e Take, for example, the class consisting of the fractions\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.816ex; width: 18.303ex; height: 2.789ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-37.png\" alt=\"\" data-tex=\"\r\n\\tfrac{1}{2},\\ \\tfrac{3}{4},\\ \\tfrac{7}{8},\\ \\tfrac{15}{16},\\ \\dots,\r\n\"\u003e\u003c/span\u003e\r\n\u003ci\u003ei.e.\u003c/i\u003e of all fractions of the form \u003cimg style=\"vertical-align: -1.552ex; width: 7.171ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-196.png\" alt=\"\" data-tex=\"1 – \\dfrac{1}{2^{n}}\"\u003e for different finite values\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e This series of fractions has no maximum, and it is clear\r\nthat the segment which it defines (in the whole series of fractions\r\nin order of magnitude) is the class of all proper fractions. Or,\r\nagain, consider the prime numbers, considered as a selection from\r\nthe cardinals (finite and infinite) in order of magnitude. In this\r\ncase the segment defined consists of all finite integers.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAssuming that \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is serial, the \"boundary\" of a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e will be\r\nthe term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e (if it exists) whose predecessors are the segment\r\ndefined by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"maximum\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a boundary which is a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAn \"upper limit\" of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a boundary which is not a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf a class has no boundary, it has neither maximum nor limit.\r\nThis is the case of an \"irrational\" Dedekind cut, or of what is\r\ncalled a \"gap.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the \"upper limit\" of a set of terms \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect to a\r\nseries \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is that term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e (if it exists) which comes after all the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es,\r\nbut is such that every earlier term comes before some of the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may define all the \"upper limiting-points\" of a set of\r\nterms \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e as all those that are the upper limits of sets of terms\r\nchosen out of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e We shall, of course, have to distinguish upper\r\nlimiting-points from lower limiting-points. If we consider, for\r\nexample, the series of ordinal numbers:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 62.044ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-38.png\" alt=\"\" data-tex=\"\r\n1,\\ 2,\\ 3,\\ \\dots\\\r\n\\omega,\\ \\omega + 1,\\ \\dots\\\r\n2\\omega,\\ 2\\omega + 1,\\\r\n3\\omega,\\ \\dots\\ \\omega^{2},\\ \\dots\\ \\omega^{3},\\ \\dots,\r\n\"\u003e\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_99\"\u003e[Pg 99]\u003c/span\u003e\r\nthe upper limiting-points of the field of this series are those that\r\nhave no immediate predecessors, \u003ci\u003ei.e.\u003c/i\u003e\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 49.009ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-39.png\" alt=\"\" data-tex=\"\r\n1,\\ \\omega,\\ 2\\omega,\\ 3\\omega,\\ \\dots\\\r\n\\omega^{2},\\ \\omega^{2} + \\omega,\\ \\dots\\ 2\\omega^{2},\\ \\dots\\\r\n\\omega^{3}\\ \\dots\r\n\"\u003e\u003c/span\u003e\r\nThe upper limiting-points of the field of this new series will be\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 30.855ex; height: 2.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-40.png\" alt=\"\" data-tex=\"\r\n1,\\ \\omega^{2},\\ 2\\omega^{2},\\ \\dots\\\r\n\\omega^{3},\\ \\omega^{3} + \\omega^{2}\\ \\dots\r\n\"\u003e\u003c/span\u003e\r\nOn the other hand, the series of ordinals\u0026mdash;and indeed every well-ordered\r\nseries\u0026mdash;has no lower limiting-points, because there are\r\nno terms except the last that have no immediate successors. But\r\nif we consider such a series as the series of ratios, every member\r\nof this series is both an upper and a lower limiting-point for\r\nsuitably chosen sets. If we consider the series of real numbers,\r\nand select out of it the rational real numbers, this set (the\r\nrationals) will have all the real numbers as upper and lower\r\nlimiting-points. The limiting-points of a set are called its \"first\r\nderivative,\" and the limiting-points of the first derivative are\r\ncalled the second derivative, and so on.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWith regard to limits, we may distinguish various grades of\r\nwhat may be called \"continuity\" in a series. The word \"continuity\"\r\nhad been used for a long time, but had remained without\r\nany precise definition until the time of Dedekind and Cantor.\r\nEach of these two men gave a precise significance to the term,\r\nbut Cantor\u0027s definition is narrower than Dedekind\u0027s: a series\r\nwhich has Cantorian continuity must have Dedekindian continuity,\r\nbut the converse does not hold.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe first definition that would naturally occur to a man seeking\r\na precise meaning for the continuity of series would be to define\r\nit as consisting in what we have called \"compactness,\" \u003ci\u003ei.e.\u003c/i\u003e in the\r\nfact that between any two terms of the series there are others.\r\nBut this would be an inadequate definition, because of the\r\nexistence of \"gaps\" in series such as the series of ratios. We\r\nsaw in Chapter VII. that there are innumerable ways in which\r\nthe series of ratios can be divided into two parts, of which one\r\nwholly precedes the other, and of which the first has no last term,\r\n\u003cspan class=\"pagenum\" id=\"Page_100\"\u003e[Pg 100]\u003c/span\u003e\r\nwhile the second has no first term. Such a state of affairs seems\r\ncontrary to the vague feeling we have as to what should characterise\r\n\"continuity,\" and, what is more, it shows that the series of\r\nratios is not the sort of series that is needed for many mathematical\r\npurposes. Take geometry, for example: we wish to be able to\r\nsay that when two straight lines cross each other they have a\r\npoint in common, but if the series of points on a line were similar\r\nto the series of ratios, the two lines might cross in a \"gap\" and\r\nhave no point in common. This is a crude example, but many\r\nothers might be given to show that compactness is inadequate as\r\na mathematical definition of continuity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt was the needs of geometry, as much as anything, that led\r\nto the definition of \"Dedekindian\" continuity. It will be remembered\r\nthat we defined a series as Dedekindian when every\r\nsub-class of the field has a boundary. (It is sufficient to assume\r\nthat there is always an \u003ci\u003eupper\u003c/i\u003e boundary, or that there is always\r\na \u003ci\u003elower\u003c/i\u003e boundary. If one of these is assumed, the other can be\r\ndeduced.) That is to say, a series is Dedekindian when there\r\nare no gaps. The absence of gaps may arise either through\r\nterms having successors, or through the existence of limits in the\r\nabsence of maxima. Thus a finite series or a well-ordered series\r\nis Dedekindian, and so is the series of real numbers. The former\r\nsort of Dedekindian series is excluded by assuming that our\r\nseries is compact; in that case our series must have a property\r\nwhich may, for many purposes, be fittingly called continuity.\r\nThus we are led to the definition:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series has \"Dedekindian continuity\" when it is Dedekindian\r\nand compact.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut this definition is still too wide for many purposes. Suppose,\r\nfor example, that we desire to be able to assign such properties\r\nto geometrical space as shall make it certain that every point\r\ncan be specified by means of co-ordinates which are real numbers:\r\nthis is not insured by Dedekindian continuity alone. We want\r\nto be sure that every point which cannot be specified by \u003ci\u003erational\u003c/i\u003e\r\nco-ordinates can be specified as the limit of a \u003ci\u003eprogression\u003c/i\u003e of points\r\n\u003cspan class=\"pagenum\" id=\"Page_101\"\u003e[Pg 101]\u003c/span\u003e\r\nwhose co-ordinates are rational, and this is a further property\r\nwhich our definition does not enable us to deduce.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe are thus led to a closer investigation of series with respect\r\nto limits. This investigation was made by Cantor and formed\r\nthe basis of his definition of continuity, although, in its simplest\r\nform, this definition somewhat conceals the considerations which\r\nhave given rise to it. We shall, therefore, first travel through\r\nsome of Cantor\u0027s conceptions in this subject before giving his\r\ndefinition of continuity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nCantor defines a series as \"perfect\" when all its points are\r\nlimiting-points and all its limiting-points belong to it. But this\r\ndefinition does not express quite accurately what he means.\r\nThere is no correction required so far as concerns the property\r\nthat all its points are to be limiting-points; this is a property\r\nbelonging to compact series, and to no others if all points are to\r\nbe upper limiting- or all lower limiting-points. But if it is only\r\nassumed that they are limiting-points one way, without specifying\r\nwhich, there will be other series that will have the property\r\nin question\u0026mdash;for example, the series of decimals in which a decimal\r\nending in a recurring 9 is distinguished from the corresponding\r\nterminating decimal and placed immediately before it. Such a\r\nseries is very nearly compact, but has exceptional terms which\r\nare consecutive, and of which the first has no immediate predecessor,\r\nwhile the second has no immediate successor. Apart from\r\nsuch series, the series in which every point is a limiting-point\r\nare compact series; and this holds without qualification if it is\r\nspecified that every point is to be an upper limiting-point (or\r\nthat every point is to be a lower limiting-point).\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAlthough Cantor does not explicitly consider the matter, we\r\nmust distinguish different kinds of limiting-points according to\r\nthe nature of the smallest sub-series by which they can be defined.\r\nCantor assumes that they are to be defined by progressions, or\r\nby regressions (which are the converses of progressions). When\r\nevery member of our series is the limit of a progression or regression,\r\nCantor calls our series \"condensed in itself\" (\u003ci\u003einsichdicht\u003c/i\u003e).\r\n\u003cspan class=\"pagenum\" id=\"Page_102\"\u003e[Pg 102]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe come now to the second property by which perfection was\r\nto be defined, namely, the property which Cantor calls that of\r\nbeing \"closed\" (\u003ci\u003eabgeschlossen\u003c/i\u003e). This, as we saw, was first defined\r\nas consisting in the fact that all the limiting-points of a series\r\nbelong to it. But this only has any effective significance if our\r\nseries is \u003ci\u003egiven\u003c/i\u003e as contained in some other larger series (as is the\r\ncase, \u003ci\u003ee.g.\u003c/i\u003e, with a selection of real numbers), and limiting-points\r\nare taken in relation to the larger series. Otherwise, if a series\r\nis considered simply on its own account, it cannot fail to contain\r\nits limiting-points. What Cantor \u003ci\u003emeans\u003c/i\u003e is not exactly what\r\nhe says; indeed, on other occasions he says something rather\r\ndifferent, which \u003ci\u003eis\u003c/i\u003e what he means. What he really means is that\r\nevery subordinate series which is of the sort that might be expected\r\nto have a limit does have a limit within the given series;\r\n\u003ci\u003ei.e.\u003c/i\u003e every subordinate series which has no maximum has a limit,\r\n\u003ci\u003ei.e.\u003c/i\u003e every subordinate series has a boundary. But Cantor does\r\nnot state this for \u003ci\u003eevery\u003c/i\u003e subordinate series, but only for progressions\r\nand regressions. (It is not clear how far he recognises that\r\nthis is a limitation.) Thus, finally, we find that the definition we\r\nwant is the following:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series is said to be \"closed\" (\u003ci\u003eabgeschlossen\u003c/i\u003e) when every progression\r\nor regression contained in the series has a limit in the\r\nseries.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe then have the further definition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series is \"perfect\" when it is \u003ci\u003econdensed in itself\u003c/i\u003e and \u003ci\u003eclosed\u003c/i\u003e,\r\n\u003ci\u003ei.e.\u003c/i\u003e when every term is the limit of a progression or regression,\r\nand every progression or regression contained in the series has a\r\nlimit in the series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn seeking a definition of continuity, what Cantor has in mind\r\nis the search for a definition which shall apply to the series of\r\nreal numbers and to any series similar to that, but to no others.\r\nFor this purpose we have to add a further property. Among\r\nthe real numbers some are rational, some are irrational; although\r\nthe number of irrationals is greater than the number of rationals,\r\nyet there are rationals between any two real numbers, however\r\n\u003cspan class=\"pagenum\" id=\"Page_103\"\u003e[Pg 103]\u003c/span\u003e\r\nlittle the two may differ. The number of rationals, as we saw,\r\nis \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e This gives a further property which suffices to characterise\r\ncontinuity completely, namely, the property of containing a class\r\nof \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e members in such a way that some of this class occur\r\nbetween any two terms of our series, however near together.\r\nThis property, added to perfection, suffices to define a class of\r\nseries which are all similar and are in fact a serial number. This\r\nclass Cantor defines as that of continuous series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may slightly simplify his definition. To begin with,\r\nwe say:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"median class\" of a series is a sub-class of the field such\r\nthat members of it are to be found between any two terms of\r\nthe series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the rationals are a median class in the series of real\r\nnumbers. It is obvious that there cannot be median classes\r\nexcept in compact series.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe then find that Cantor\u0027s definition is equivalent to the\r\nfollowing:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA series is \"continuous\" when (1) it is Dedekindian, (2) it\r\ncontains a median class having \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTo avoid confusion, we shall speak of this kind as \"Cantorian\r\ncontinuity.\" It will be seen that it implies Dedekindian continuity,\r\nbut the converse is not the case. All series having\r\nCantorian continuity are similar, but not all series having\r\nDedekindian continuity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe notions of \u003ci\u003elimit\u003c/i\u003e and \u003ci\u003econtinuity\u003c/i\u003e which we have been defining\r\nmust not be confounded with the notions of the limit of a function\r\nfor approaches to a given argument, or the continuity of a function\r\nin the neighbourhood of a given argument. These are different\r\nnotions, very important, but derivative from the above and more\r\ncomplicated. The continuity of motion (if motion is continuous)\r\nis an instance of the continuity of a function; on the other hand,\r\nthe continuity of space and time (if they are continuous) is an\r\ninstance of the continuity of series, or (to speak more cautiously)\r\nof a kind of continuity which can, by sufficient mathematical\r\n\u003cspan class=\"pagenum\" id=\"Page_104\"\u003e[Pg 104]\u003c/span\u003e\r\nmanipulation, be reduced to the continuity of series. In view\r\nof the fundamental importance of motion in applied mathematics,\r\nas well as for other reasons, it will be well to deal\r\nbriefly with the notions of limits and continuity as applied\r\nto functions; but this subject will be best reserved for a\r\nseparate chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe definitions of continuity which we have been considering,\r\nnamely, those of Dedekind and Cantor, do not correspond very\r\nclosely to the vague idea which is associated with the word in\r\nthe mind of the man in the street or the philosopher. They\r\nconceive continuity rather as absence of separateness, the sort\r\nof general obliteration of distinctions which characterises a thick\r\nfog. A fog gives an impression of vastness without definite\r\nmultiplicity or division. It is this sort of thing that a metaphysician\r\nmeans by \"continuity,\" declaring it, very truly,\r\nto be characteristic of his mental life and of that of children\r\nand animals.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe general idea vaguely indicated by the word \"continuity\"\r\nwhen so employed, or by the word \"flux,\" is one which is certainly\r\nquite different from that which we have been defining. Take,\r\nfor example, the series of real numbers. Each is what it is,\r\nquite definitely and uncompromisingly; it does not pass over\r\nby imperceptible degrees into another; it is a hard, separate\r\nunit, and its distance from every other unit is finite, though\r\nit can be made less than any given finite amount assigned in\r\nadvance. The question of the relation between the kind of\r\ncontinuity existing among the real numbers and the kind exhibited,\r\n\u003ci\u003ee.g.\u003c/i\u003e by what we see at a given time, is a difficult and\r\nintricate one. It is not to be maintained that the two kinds\r\nare simply identical, but it may, I think, be very well maintained\r\nthat the mathematical conception which we have been\r\nconsidering in this chapter gives the abstract logical scheme to\r\nwhich it must be possible to bring empirical material by suitable\r\nmanipulation, if that material is to be called \"continuous\"\r\nin any precisely definable sense. It would be quite impossible\r\n\u003cspan class=\"pagenum\" id=\"Page_105\"\u003e[Pg 105]\u003c/span\u003e\r\nto justify this thesis within the limits of the present volume.\r\nThe reader who is interested may read an attempt to justify\r\nit as regards \u003ci\u003etime\u003c/i\u003e in particular by the present author in the\r\n\u003ci\u003eMonist\u003c/i\u003e for 1914-5, as well as in parts of \u003ci\u003eOur Knowledge of the\r\nExternal World\u003c/i\u003e. With these indications, we must leave this\r\nproblem, interesting as it is, in order to return to topics more\r\nclosely connected with mathematics.\r\n\u003cspan class=\"pagenum\" id=\"Page_106\"\u003e[Pg 106]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XI: LIMITS AND CONTINUITY OF FUNCTIONS\u0027\u003e\u003ca id=\"chap11\"\u003e\u003c/a\u003eCHAPTER XI\r\n\u003cbr\u003e\u003cbr\u003e\r\nLIMITS AND CONTINUITY OF FUNCTIONS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nIN this chapter we shall be concerned with the definition of the\r\nlimit of a function (if any) as the argument approaches a given\r\nvalue, and also with the definition of what is meant by a \"continuous\r\nfunction.\" Both of these ideas are somewhat technical,\r\nand would hardly demand treatment in a mere introduction\r\nto mathematical philosophy but for the fact that, especially\r\nthrough the so-called infinitesimal calculus, wrong views upon\r\nour present topics have become so firmly embedded in the minds\r\nof professional philosophers that a prolonged and considerable\r\neffort is required for their uprooting. It has been thought\r\never since the time of Leibniz that the differential and integral\r\ncalculus required infinitesimal quantities. Mathematicians\r\n(especially Weierstrass) proved that this is an error; but errors\r\nincorporated, \u003ci\u003ee.g.\u003c/i\u003e in what Hegel has to say about mathematics,\r\ndie hard, and philosophers have tended to ignore the work of\r\nsuch men as Weierstrass.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLimits and continuity of functions, in works on ordinary\r\nmathematics, are defined in terms involving number. This is\r\nnot essential, as Dr Whitehead has shown.\u003ca id=\"FNanchor_22_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_22_1\" class=\"fnanchor\"\u003e[22]\u003c/a\u003e\r\nWe will, however,\r\nbegin with the definitions in the text-books, and proceed afterwards\r\nto show how these definitions can be generalised so as to\r\napply to series in general, and not only to such as are numerical\r\nor numerically measurable.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_22_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_22_1\"\u003e\u003cspan class=\"label\"\u003e[22]\u003c/span\u003e\u003c/a\u003eSee \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. II. * 230-234.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nLet us consider any ordinary mathematical function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e,\u003c/span\u003e where\r\n\u003cspan class=\"pagenum\" id=\"Page_107\"\u003e[Pg 107]\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e are both real numbers, and \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e is one-valued\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e when\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is given, there is only one value that \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e can have. We call \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nthe \"argument,\" and \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e the \"value for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\" When\r\na function is what we call \"continuous,\" the rough idea for which\r\nwe are seeking a precise definition is that small differences in \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nshall correspond to small differences in \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e,\u003c/span\u003e and if we make the\r\ndifferences in \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e small enough, we can make the differences in \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e\r\nfall below any assigned amount. We do not want, if a function\r\nis to be continuous, that there shall be sudden jumps, so that,\r\nfor some value of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e any change, however small, will make a\r\nchange in \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e which exceeds some assigned finite amount. The\r\nordinary simple functions of mathematics have this property:\r\nit belongs, for example, to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.282ex; height: 1.912ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-198.png\" alt=\"\" data-tex=\"x^{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 2.282ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-199.png\" alt=\"\" data-tex=\"x^{3}\"\u003e,\u003c/span\u003e … \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.466ex; width: 4.563ex; height: 2.036ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-200.png\" alt=\"\" data-tex=\"\\log x\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 4.449ex; height: 1.538ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-201.png\" alt=\"\" data-tex=\"\\sin x\"\u003e,\u003c/span\u003e and so on.\r\nBut it is not at all difficult to define discontinuous functions.\r\nTake, as a non-mathematical example, \"the place of birth of\r\nthe youngest person living at time \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e.\u003c/span\u003e\" This is a function of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e;\u003c/span\u003e\r\nits value is constant from the time of one person\u0027s birth to the\r\ntime of the next birth, and then the value changes \u003ci\u003esuddenly\u003c/i\u003e\r\nfrom one birthplace to the other. An analogous mathematical\r\nexample would be \"the integer next below \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\" where \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a real\r\nnumber. This function remains constant from one integer to\r\nthe next, and then gives a sudden jump. The actual fact is\r\nthat, though continuous functions are more familiar, they are\r\nthe exceptions: there are infinitely more discontinuous functions\r\nthan continuous ones.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMany functions are discontinuous for one or several values of\r\nthe variable, but continuous for all other values. Take as an\r\nexample \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-203.png\" alt=\"\" data-tex=\"\\sin 1/x\"\u003e.\u003c/span\u003e The function \u003cimg style=\"vertical-align: -0.025ex; width: 4.216ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-204.png\" alt=\"\" data-tex=\"\\sin \\theta\"\u003e passes through all values\r\nfrom -1 to 1 every time that \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.618ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-205.png\" alt=\"\" data-tex=\"\\theta\"\u003e passes from \u003cimg style=\"vertical-align: -0.566ex; width: 5.312ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-206.png\" alt=\"\" data-tex=\"-\\pi/2\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-207.png\" alt=\"\" data-tex=\"\\pi/2\"\u003e,\u003c/span\u003e or from\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-207.png\" alt=\"\" data-tex=\"\\pi/2\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 4.683ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-208.png\" alt=\"\" data-tex=\"3\\pi/2\"\u003e,\u003c/span\u003e or generally from \u003cimg style=\"vertical-align: -0.566ex; width: 11.698ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-209.png\" alt=\"\" data-tex=\"(2n – 1)\\pi/2\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 11.698ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-210.png\" alt=\"\" data-tex=\"(2n + 1)\\pi/2\"\u003e,\u003c/span\u003e where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any integer. Now if we consider \u003cimg style=\"vertical-align: -0.566ex; width: 3.557ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-211.png\" alt=\"\" data-tex=\"1/x\"\u003e when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is very small,\r\nwe see that as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e diminishes \u003cimg style=\"vertical-align: -0.566ex; width: 3.557ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-211.png\" alt=\"\" data-tex=\"1/x\"\u003e grows faster and faster, so that\r\nit passes more and more quickly through the cycle of values from\r\none multiple of \u003cimg style=\"vertical-align: -0.566ex; width: 3.552ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-207.png\" alt=\"\" data-tex=\"\\pi/2\"\u003e to another as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e becomes smaller and smaller.\r\nConsequently \u003cimg style=\"vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-203.png\" alt=\"\" data-tex=\"\\sin 1/x\"\u003e passes more and more quickly from -1\r\n\u003cspan class=\"pagenum\" id=\"Page_108\"\u003e[Pg 108]\u003c/span\u003e\r\nto 1 and back again, as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e grows smaller. In fact, if we take\r\nany interval containing 0, say the interval from \u003cimg style=\"vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-212.png\" alt=\"\" data-tex=\"-\\epsilon\"\u003e to \u003cimg style=\"vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-213.png\" alt=\"\" data-tex=\"+\\epsilon\"\u003e where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e is some very small number, \u003cimg style=\"vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-203.png\" alt=\"\" data-tex=\"\\sin 1/x\"\u003e will go through an infinite\r\nnumber of oscillations in this interval, and we cannot diminish\r\nthe oscillations by making the interval smaller. Thus round\r\nabout the argument 0 the function is discontinuous. It is easy\r\nto manufacture functions which are discontinuous in several\r\nplaces, or in \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e places, or everywhere. Examples will be found\r\nin any book on the theory of functions of a real variable.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nProceeding now to seek a precise definition of what is meant\r\nby saying that a function is continuous for a given argument,\r\nwhen argument and value are both real numbers, let us first\r\ndefine a \"neighbourhood\" of a number \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e as all the numbers\r\nfrom \u003cimg style=\"vertical-align: -0.186ex; width: 4.978ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-215.png\" alt=\"\" data-tex=\"x – \\epsilon\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 4.978ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-216.png\" alt=\"\" data-tex=\"x + \\epsilon\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e is some number which, in important\r\ncases, will be very small. It is clear that continuity at a given\r\npoint has to do with what happens in \u003ci\u003eany\u003c/i\u003e neighbourhood of that\r\npoint, however small.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhat we desire is this: If \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is the argument for which we wish\r\nour function to be continuous, let us first define a neighbourhood\r\n(\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e say) containing the value \u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e which the function has for the\r\nargument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e;\u003c/span\u003e we desire that, if we take a sufficiently small\r\nneighbourhood containing \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e all values for arguments throughout\r\nthis neighbourhood shall be contained in the neighbourhood \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e\r\nno matter how small we may have made \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e That is to say, if\r\nwe decree that our function is not to differ from \u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e by more than\r\nsome very tiny amount, we can always find a stretch of real\r\nnumbers, having \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e in the middle of it, such that throughout\r\nthis stretch \u003cimg style=\"vertical-align: -0.464ex; width: 2.538ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-197.png\" alt=\"\" data-tex=\"fx\"\u003e will not differ from \u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e by more than the prescribed\r\ntiny amount. And this is to remain true whatever\r\ntiny amount we may select. Hence we are led to the following\r\ndefinition:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe function \u003cimg style=\"vertical-align: -0.566ex; width: 4.299ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-218.png\" alt=\"\" data-tex=\"f(x)\"\u003e is said to be \"continuous\" for the argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\r\nif, for every positive number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e,\u003c/span\u003e different from 0, but as\r\nsmall as we please, there exists a positive number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-219.png\" alt=\"\" data-tex=\"\\delta\"\u003e,\u003c/span\u003e different\r\nfrom 0, such that, for all values of \u003cimg style=\"vertical-align: -0.023ex; width: 1.005ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-219.png\" alt=\"\" data-tex=\"\\delta\"\u003e which are numerically\r\n\u003cspan class=\"pagenum\" id=\"Page_109\"\u003e[Pg 109]\u003c/span\u003e\r\nless\u003ca id=\"FNanchor_23_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_23_1\" class=\"fnanchor\"\u003e[23]\u003c/a\u003e\r\nthan \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e,\u003c/span\u003e the difference \u003cimg style=\"vertical-align: -0.566ex; width: 15.819ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-220.png\" alt=\"\" data-tex=\"f(a) + \\delta) – f(a)\"\u003e is numerically less\r\nthan \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_23_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_23_1\"\u003e\u003cspan class=\"label\"\u003e[23]\u003c/span\u003e\u003c/a\u003eA number is said to be \"numerically less\" than \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e when it lies between\r\n\u003cimg style=\"vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-212.png\" alt=\"\" data-tex=\"-\\epsilon\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-213.png\" alt=\"\" data-tex=\"+\\epsilon\"\u003e.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIn this definition, \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e first defines a neighbourhood of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 4.201ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-222.png\" alt=\"\" data-tex=\"f(a)\"\u003e,\u003c/span\u003e\r\nnamely, the neighbourhood from \u003cimg style=\"vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-223.png\" alt=\"\" data-tex=\"f(a) – \\sigma\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 7.886ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-224.png\" alt=\"\" data-tex=\"f(a) + \\epsilon\"\u003e.\u003c/span\u003e The definition\r\nthen proceeds to say that we can (by means of \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e define a\r\nneighbourhood, namely, that from \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-226.png\" alt=\"\" data-tex=\"a + \\epsilon\"\u003e,\u003c/span\u003e such that, for\r\nall arguments within this neighbourhood, the value of the function\r\nlies within the neighbourhood horn \u003cimg style=\"vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-223.png\" alt=\"\" data-tex=\"f(a) – \\sigma\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 8.259ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-227.png\" alt=\"\" data-tex=\"f(a) + \\sigma\"\u003e.\u003c/span\u003e If this\r\ncan be done, however \u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e may be chosen, the function is \"continuous\"\r\nfor the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSo far we have not defined the \"limit\" of a function for a\r\ngiven argument. If we had done so, we could have defined the\r\ncontinuity of a function differently: a function is continuous\r\nat a point where its value is the same as the limit of its value for\r\napproaches either from above or from below. But it is only\r\nthe exceptionally \"tame\" function that has a definite limit as\r\nthe argument approaches a given point. The general rule is\r\nthat a function oscillates, and that, given any neighbourhood\r\nof a given argument, however small, a whole stretch of values\r\nwill occur for arguments within this neighbourhood. As this\r\nis the general rule, let us consider it first.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us consider what may happen as the argument approaches\r\nsome value \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below. That is to say, we wish to consider\r\nwhat happens for arguments contained in the interval from\r\n\u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e is some number which, in important cases,\r\nwill be very small.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe values of the function for arguments from \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e excluded)\r\nwill be a set of real numbers which will define a certain\r\nsection of the set of real numbers, namely, the section consisting\r\nof those numbers that are not greater than \u003ci\u003eall\u003c/i\u003e the values for\r\narguments from \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e Given any number in this section,\r\nthere are values at least as great as this number for arguments\r\nbetween \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e for arguments that fall very little short\r\n\u003cspan class=\"pagenum\" id=\"Page_110\"\u003e[Pg 110]\u003c/span\u003e\r\nof \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (if \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e is very small). Let us take all possible \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e\u0027\u003c/span\u003es and all\r\npossible corresponding sections. The common part of all these\r\nsections we will call the \"ultimate section\" as the argument\r\napproaches \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e To say that a number \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e belongs to the ultimate\r\nsection is to say that, however small we may make \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e,\u003c/span\u003e there are\r\narguments between \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e for which the value of the function\r\nis not less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may apply exactly the same process to upper sections,\r\n\u003ci\u003ei.e.\u003c/i\u003e to sections that go from some point up to the top, instead of\r\nfrom the bottom up to some point. Here we take those numbers\r\nthat are not \u003ci\u003eless\u003c/i\u003e than all the values for arguments from \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e\r\nto \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e;\u003c/span\u003e this defines an upper section which will vary as \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e varies.\r\nTaking the common part of all such sections for all possible \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e\u0027\u003c/span\u003es,\r\nwe obtain the \"ultimate upper section.\" To say that a number \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e\r\nbelongs to the ultimate upper section is to say that, however\r\nsmall we make \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e,\u003c/span\u003e there are arguments between \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e for\r\nwhich the value of the function is not \u003ci\u003egreater\u003c/i\u003e than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf a term \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e belongs both to the ultimate section and to the\r\nultimate upper section, we shall say that it belongs to the\r\n\"ultimate oscillation.\" We may illustrate the matter by considering\r\nonce more the function \u003cimg style=\"vertical-align: -0.566ex; width: 6.712ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-203.png\" alt=\"\" data-tex=\"\\sin 1/x\"\u003e as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-228.png\" alt=\"\" data-tex=\"x \"\u003e approaches the\r\nvalue 0. We shall assume, in order to fit in with the above\r\ndefinitions, that this value is approached from below.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us begin with the \"ultimate section.\" Between \u003cimg style=\"vertical-align: -0.186ex; width: 2.679ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-212.png\" alt=\"\" data-tex=\"-\\epsilon\"\u003e\r\nand 0, whatever \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e may be, the function will assume the value 1\r\nfor certain arguments, but will never assume any greater value.\r\nHence the ultimate section consists of all real numbers, positive\r\nand negative, up to and including 1; \u003ci\u003ei.e.\u003c/i\u003e it consists of all negative\r\nnumbers together with 0, together with the positive numbers\r\nup to and including 1.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSimilarly the \"ultimate upper section\" consists of all positive\r\nnumbers together with 0, together with the negative numbers\r\ndown to and including -1.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus the \"ultimate oscillation\" consists of all real numbers\r\nfrom -1 to 1, both included.\r\n\u003cspan class=\"pagenum\" id=\"Page_111\"\u003e[Pg 111]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may say generally that the \"ultimate oscillation\" of\r\na function as the argument approaches \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below consists\r\nof all those numbers \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e which are such that, however near we\r\ncome to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e we shall still find values as great as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and values as\r\nsmall as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe ultimate oscillation may contain no terms, or one term,\r\nor many terms. In the first two cases the function has a definite\r\nlimit for approaches from below. If the ultimate oscillation\r\nhas one term, this is fairly obvious. It is equally true if it has\r\nnone; for it is not difficult to prove that, if the ultimate oscillation\r\nis null, the boundary of the ultimate section is the same as\r\nthat of the ultimate upper section, and may be defined as the\r\nlimit of the function for approaches from below. But if the\r\nultimate oscillation has many terms, there is no definite limit to\r\nthe function for approaches from below. In this case we can\r\ntake the lower and upper boundaries of the ultimate oscillation\r\n(\u003ci\u003ei.e.\u003c/i\u003e the lower boundary of the ultimate upper section and the\r\nupper boundary of the ultimate section) as the lower and upper\r\nlimits of its \"ultimate\" values for approaches from below.\r\nSimilarly we obtain lower and upper limits of the \"ultimate\"\r\nvalues for approaches from above. Thus we have, in the general\r\ncase, \u003ci\u003efour\u003c/i\u003e limits to a function for approaches to a given argument.\r\n\u003ci\u003eThe\u003c/i\u003e limit for a given argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e only exists when all these four\r\nare equal, and is then their common value. If it is also the\r\n\u003ci\u003evalue\u003c/i\u003e for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e the function is continuous for this\r\nargument. This may be taken as defining continuity: it is\r\nequivalent to our former definition.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can define the limit of a function for a given argument\r\n(if it exists) without passing through the ultimate oscillation\r\nand the four limits of the general case. The definition proceeds,\r\nin that case, just as the earlier definition of continuity proceeded.\r\nLet us define the limit for approaches from below. If there is to\r\nbe a definite limit for approaches to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below, it is necessary\r\nand sufficient that, given any small number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e,\u003c/span\u003e two values for\r\narguments sufficiently near to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (but both less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e)\u003c/span\u003e will differ\r\n\u003cspan class=\"pagenum\" id=\"Page_112\"\u003e[Pg 112]\u003c/span\u003e\r\nby less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e;\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 0.919ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-214.png\" alt=\"\" data-tex=\"\\epsilon\"\u003e is sufficiently small, and our arguments\r\nboth lie between \u003cimg style=\"vertical-align: -0.186ex; width: 4.881ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-225.png\" alt=\"\" data-tex=\"a – \\epsilon\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e excluded), then the difference\r\nbetween the values for these arguments will be less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e.\u003c/span\u003e\r\nThis is to hold for any \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.292ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-221.png\" alt=\"\" data-tex=\"\\sigma\"\u003e,\u003c/span\u003e however small; in that case the\r\nfunction has a limit for approaches from below. Similarly\r\nwe define the case when there is a limit for approaches from\r\nabove. These two limits, even when both exist, need not be\r\nidentical; and if they are identical, they still need not be identical\r\nwith the \u003ci\u003evalue\u003c/i\u003e for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e It is only in this last case\r\nthat we call the function \u003ci\u003econtinuous\u003c/i\u003e for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA function is called \"continuous\" (without qualification)\r\nwhen it is continuous for every argument.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother slightly different method of reaching the definition\r\nof continuity is the following:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us say that a function \"ultimately converges into a\r\nclass \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\" if there is some real number such that, for this argument\r\nand all arguments greater than this, the value of the function\r\nis a member of the class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 5.509ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-229.png\" alt=\"\" data-tex=\"alpha\"\u003e.\u003c/span\u003e Similarly we shall say that a function\r\n\"converges into \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e as the argument approaches \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e from below\"\r\nif there is some argument \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e less than \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e such that throughout\r\nthe interval from \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e (included) to \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e (excluded) the function has\r\nvalues which are members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 5.509ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-229.png\" alt=\"\" data-tex=\"alpha\"\u003e.\u003c/span\u003e We may now say that a\r\nfunction is continuous for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e for which it has the\r\nvalue \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e,\u003c/span\u003e if it satisfies four conditions, namely:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) Given any real number less than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e,\u003c/span\u003e the function converges\r\ninto the successors of this number as the argument\r\napproaches \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) Given any real number greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.441ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-217.png\" alt=\"\" data-tex=\"fa\"\u003e,\u003c/span\u003e the function converges\r\ninto the predecessors of this number as the argument\r\napproaches \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) and (4) Similar conditions for approaches to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from above.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe advantages of this form of definition is that it analyses\r\nthe conditions of continuity into four, derived from considering\r\narguments and values respectively greater or less than the\r\nargument and value for which continuity is to be defined.\r\n\u003cspan class=\"pagenum\" id=\"Page_113\"\u003e[Pg 113]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may now generalise our definitions so as to apply to series\r\nwhich are not numerical or known to be numerically measurable.\r\nThe case of motion is a convenient one to bear in mind. There\r\nis a story by H. G. Wells which will illustrate, from the case of\r\nmotion, the difference between the limit of a function for a given\r\nargument and its value for the same argument. The hero of\r\nthe story, who possessed, without his knowledge, the power of\r\nrealising his wishes, was being attacked by a policeman, but on\r\nejaculating \"Go to\u0026mdash;\u0026mdash;\" he found that the policeman disappeared.\r\nIf \u003cimg style=\"vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-230.png\" alt=\"\" data-tex=\"f(t)\"\u003e was the policeman\u0027s position at time \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-231.png\" alt=\"\" data-tex=\"t_{0}\"\u003e the moment\r\nof the ejaculation, the limit of the policeman\u0027s positions as \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e approached\r\nto \u003cimg style=\"vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-231.png\" alt=\"\" data-tex=\"t_{0}\"\u003e from below would be in contact with the hero,\r\nwhereas the value for the argument \u003cimg style=\"vertical-align: -0.375ex; width: 1.804ex; height: 1.791ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-231.png\" alt=\"\" data-tex=\"t_{0}\"\u003e was \u0026mdash;. But such occurrences\r\nare supposed to be rare in the real world, and it is assumed,\r\nthough without adequate evidence, that all motions are continuous,\r\n\u003ci\u003ei.e.\u003c/i\u003e that, given any body, if \u003cimg style=\"vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-230.png\" alt=\"\" data-tex=\"f(t)\"\u003e is its position at time \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.566ex; width: 3.821ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-230.png\" alt=\"\" data-tex=\"f(t)\"\u003e is\r\na continuous function of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e.\u003c/span\u003e It is the meaning of \"continuity\"\r\ninvolved in such statements which we now wish to define as\r\nsimply as possible.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe definitions given for the case of functions where argument\r\nand value are real numbers can readily be adapted for more\r\ngeneral use.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e and \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e be two relations, which it is well to imagine\r\nserial, though it is not necessary to our definitions that they\r\nshould be so. Let \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e be a one-many relation whose domain\r\nis contained in the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e while its converse domain is contained\r\nin the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e Then \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is (in a generalised sense) a\r\nfunction, whose arguments belong to the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e,\u003c/span\u003e while its\r\nvalues belong to the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e Suppose, for example, that we\r\nare dealing with a particle moving on a line: let \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e be the time-series,\r\n\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e the series of points on our line from left to right, \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e the\r\nrelation of the position of our particle on the line at time \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e to\r\nthe time \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e so that \"the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" is its position at time \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e.\u003c/span\u003e This\r\nillustration may be borne in mind throughout our definitions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe shall say that the function \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is continuous for the argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_114\"\u003e[Pg 114]\u003c/span\u003e\r\nif, given any interval \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e on the \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e-series containing the value\r\nof the function for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e there is an interval on the\r\n\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-series containing \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e not as an end-point and such that, throughout\r\nthis interval, the function has values which are members\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e (We mean by an \"interval\" all the terms between any\r\ntwo; \u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are two members of the field of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has\r\nthe relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e we shall mean by the \"\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e-interval \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\"\r\nall terms \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.052ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-79.png\" alt=\"\" data-tex=\"z\"\u003e has the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\r\nto \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\u0026mdash;together, when so stated, with \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e or \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e themselves.)\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can easily define the \"ultimate section\" and the \"ultimate\r\noscillation.\" To define the \"ultimate section\" for\r\napproaches to the argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below, take any argument \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\r\nwhich precedes \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (\u003ci\u003ei.e.\u003c/i\u003e has the relation \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e)\u003c/span\u003e, take the values\r\nof the function for all arguments up to and including \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e and\r\nform the section of \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e defined by these values, \u003ci\u003ei.e.\u003c/i\u003e those members\r\nof the \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e-series which are earlier than or identical with some of\r\nthese values. Form all such sections for all \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\u0027\u003c/span\u003es that precede \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e\r\nand take their common part; this will be the ultimate section.\r\nThe ultimate upper section and the ultimate oscillation are then\r\ndefined exactly as in the previous case.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe adaptation of the definition of convergence and the\r\nresulting alternative definition of continuity offers no difficulty\r\nof any kind.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe say that a function \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is \"ultimately \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-convergent into \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\"\r\nif there is a member \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e of the converse domain of \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e and the\r\nfield of \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e such that the value of the function for the argument \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\r\nand for any argument to which \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e has the relation \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a member\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e We say that \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e \"\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-converges into \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e as the argument\r\napproaches a given argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" if there is a term \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e having\r\nthe relation \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and belonging to the converse domain of \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nand such that the value of the function for any argument in the\r\n\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-interval from \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e (inclusive) to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e (exclusive) belongs to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nOf the four conditions that a function must fulfil in order\r\nto be continuous for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e the first is, putting \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e for\r\nthe value for the argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e:\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_115\"\u003e[Pg 115]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven any term having the relation \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e-converges\r\ninto the successors of \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e (with respect to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e)\u003c/span\u003e as the argument\r\napproaches \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e from below.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe second condition is obtained by replacing \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e by its\r\nconverse; the third and fourth are obtained from the first and\r\nsecond by replacing \u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e by its converse.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is thus nothing, in the notions of the limit of a function\r\nor the continuity of a function, that essentially involves number.\r\nBoth can be defined generally, and many propositions about\r\nthem can be proved for any two series (one being the argument-series\r\nand the other the value-series). It will be seen that the\r\ndefinitions do not involve infinitesimals. They involve infinite\r\nclasses of intervals, growing smaller without any limit short of\r\nzero, but they do not involve any intervals that are not finite.\r\nThis is analogous to the fact that if a line an inch long be halved,\r\nthen halved again, and so on indefinitely, we never reach infinitesimals\r\nin this way: after \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e bisections, the length of our bit is\r\n\u003cimg style=\"vertical-align: -1.552ex; width: 3.274ex; height: 4.588ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-232.png\" alt=\"\" data-tex=\"\\dfrac{1}{2^{n}}\"\u003e of an inch; and this is finite whatever finite number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may\r\nbe. The process of successive bisection does not lead to\r\ndivisions whose ordinal number is infinite, since it is essentially\r\na one-by-one process. Thus infinitesimals are not to be reached\r\nin this way. Confusions on such topics have had much to do\r\nwith the difficulties which have been found in the discussion of\r\ninfinity and continuity.\r\n\u003cspan class=\"pagenum\" id=\"Page_116\"\u003e[Pg 116]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XII: SELECTIONS AND THE MULTIPLICATIVE AXIOM\u0027\u003e\u003ca id=\"chap12\"\u003e\u003c/a\u003eCHAPTER XII\r\n\u003cbr\u003e\u003cbr\u003e\r\nSELECTIONS AND THE MULTIPLICATIVE AXIOM\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nIN this chapter we have to consider an axiom which can be\r\nenunciated, but not proved, in terms of logic, and which is convenient,\r\nthough not indispensable, in certain portions of mathematics.\r\nIt is convenient, in the sense that many interesting\r\npropositions, which it seems natural to suppose true, cannot\r\nbe proved without its help; but it is not indispensable, because\r\neven without those propositions the subjects in which they\r\noccur still exist, though in a somewhat mutilated form.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBefore enunciating the multiplicative axiom, we must first\r\nexplain the theory of selections, and the definition of multiplication\r\nwhen the number of factors may be infinite.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn defining the arithmetical operations, the only correct procedure\r\nis to construct an actual class (or relation, in the case\r\nof relation-numbers) having the required number of terms.\r\nThis sometimes demands a certain amount of ingenuity, but\r\nit is essential in order to prove the existence of the number\r\ndefined. Take, as the simplest example, the case of addition.\r\nSuppose we are given a cardinal number \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e,\u003c/span\u003e and a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e which\r\nhas \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms. How shall we define \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 5.494ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-233.png\" alt=\"\" data-tex=\"\\mu + \\mu\"\u003e?\u003c/span\u003e For this purpose\r\nwe must have \u003ci\u003etwo\u003c/i\u003e classes having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms, and they must not\r\noverlap. We can construct such classes from \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e in various ways,\r\nof which the following is perhaps the simplest: Form first all\r\nthe ordered couples whose first term is a class consisting of a\r\nsingle member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e and whose second term is the null-class;\r\nthen, secondly, form all the ordered couples whose first term is\r\n\u003cspan class=\"pagenum\" id=\"Page_117\"\u003e[Pg 117]\u003c/span\u003e\r\nthe null-class and whose second term is a class consisting of a\r\nsingle member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e These two classes of couples have no\r\nmember in common, and the logical sum of the two classes will\r\nhave \u003cimg style=\"vertical-align: -0.489ex; width: 5.494ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-233.png\" alt=\"\" data-tex=\"\\mu + \\mu\"\u003e terms. Exactly analogously we can define \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 5.329ex; height: 1.808ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-181.png\" alt=\"\" data-tex=\"\\mu + \\nu\"\u003e,\u003c/span\u003e\r\ngiven that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is the number of some class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is the number\r\nof some class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSuch definitions, as a rule, are merely a question of a suitable\r\ntechnical device. But in the case of multiplication, where the\r\nnumber of factors may be infinite, important problems arise out\r\nof the definition.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMultiplication when the number of factors is finite offers no\r\ndifficulty. Given two classes \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e of which the first has\r\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms and the second \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e terms, we can define \u003cimg style=\"vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-234.png\" alt=\"\" data-tex=\"\\mu × \\nu\"\u003e as the number\r\nof ordered couples that can be formed by choosing the first term\r\nout of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and the second out of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e It will be seen that this definition\r\ndoes not require that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e should not overlap; it\r\neven remains adequate when \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e are identical. For example,\r\nlet \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e be the class whose members are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e.\u003c/span\u003e Then the class\r\nwhich is used to define the product \u003cimg style=\"vertical-align: -0.489ex; width: 4.489ex; height: 1.6ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-238.png\" alt=\"\" data-tex=\"\\mu × \\mu\"\u003e is the class of couples:\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.507ex; width: 25.762ex; height: 8.145ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-41.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n(x_{1}, x_{1}),\\ (x_{1}, x_{2}),\\ (x_{1}, x_{3}); \\\\\r\n(x_{2}, x_{1}),\\ (x_{2}, x_{2}),\\ (x_{2}, x_{3}); \\\\\r\n(x_{3}, x_{1}),\\ (x_{3}, x_{2}),\\ (x_{3}, x_{3}).\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nThis definition remains applicable when \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e or \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e or both are\r\ninfinite, and it can be extended step by step to three or four or\r\nany finite number of factors. No difficulty arises as regards\r\nthis definition, except that it cannot be extended to an \u003ci\u003einfinite\u003c/i\u003e\r\nnumber of factors.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe problem of multiplication when the number of factors\r\nmay be infinite arises in this way: Suppose we have a class \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e\r\nconsisting of classes; suppose the number of terms in each of\r\nthese classes is given. How shall we define the product of all\r\nthese numbers? If we can frame our definition generally, it\r\nwill be applicable whether \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is finite or infinite. It is to be\r\nobserved that the problem is to be able to deal with the case\r\nwhen \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is infinite, not with the case when its members are. If\r\n\u003cspan class=\"pagenum\" id=\"Page_118\"\u003e[Pg 118]\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is not infinite, the method defined above is just as applicable\r\nwhen its members are infinite as when they are finite. It is\r\nthe case when \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is infinite, even though its members may be\r\nfinite, that we have to find a way of dealing with.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe following method of defining multiplication generally is\r\ndue to Dr Whitehead. It is explained and treated at length in\r\n\u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. I. * 80 ff., and vol. II. * 114.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us suppose to begin with that \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is a class of classes no two\r\nof which overlap\u0026mdash;say the constituencies in a country where\r\nthere is no plural voting, each constituency being considered\r\nas a class of voters. Let us now set to work to choose one term\r\nout of each class to be its \u003ci\u003erepresentative\u003c/i\u003e, as constituencies do\r\nwhen they elect members of Parliament, assuming that by law\r\neach constituency has to elect a man who is a voter in that\r\nconstituency. We thus arrive at a class of representatives, who\r\nmake up our Parliament, one being selected out of each constituency.\r\nHow many different possible ways of choosing a\r\nParliament are there? Each constituency can select any one\r\nof its voters, and therefore if there are \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e voters in a constituency,\r\nit can make \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e choices. The choices of the different constituencies\r\nare independent; thus it is obvious that, when the total number\r\nof constituencies is finite, the number of possible Parliaments\r\nis obtained by multiplying together the numbers of voters in the\r\nvarious constituencies. When we do not know whether the\r\nnumber of constituencies is finite or infinite, we may take the\r\nnumber of possible Parliaments as \u003ci\u003edefining\u003c/i\u003e the product of the\r\nnumbers of the separate constituencies. This is the method\r\nby which infinite products are defined. We must now drop our\r\nillustration, and proceed to exact statements.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e be a class of classes, and let us assume to begin with that\r\nno two members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e overlap, \u003ci\u003ei.e.\u003c/i\u003e that if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e are two different\r\nmembers of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e then no member of the one is a member of the\r\nother. We shall call a class a \"selection\" from \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e when it consists\r\nof just one term from each member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e;\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is a \"selection\"\r\nfrom \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e if every member of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e belongs to some member\r\n\u003cspan class=\"pagenum\" id=\"Page_119\"\u003e[Pg 119]\u003c/span\u003e\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e and if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e be any member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e have exactly one term\r\nin common. The class of all \"selections\" from \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e we shall call\r\nthe \"multiplicative class\" of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e.\u003c/span\u003e The number of terms in the\r\nmultiplicative class of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e the number of possible selections\r\nfrom \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e is defined as the product of the numbers of the members\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e.\u003c/span\u003e This definition is equally applicable whether \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is finite\r\nor infinite.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBefore we can be wholly satisfied with these definitions, we\r\nmust remove the restriction that no two members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e are to\r\noverlap. For this purpose, instead of defining first a class\r\ncalled a \"selection,\" we will define first a relation which we will\r\ncall a \"selector.\" A relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e will be called a \"selector\"\r\nfrom \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e if, from every member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e it picks out one term as the\r\nrepresentative of that member, \u003ci\u003ei.e.\u003c/i\u003e if, given any member \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e\r\nthere is just one term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e which is a member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and has the\r\nrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e;\u003c/span\u003e and this is to be all that \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e does. The formal\r\ndefinition is:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"selector\" from a class of classes \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e is a one-many relation,\r\nhaving \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e for its converse domain, and such that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the\r\nrelation to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e is a selector from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a member\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the\r\nterm which has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e we call \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e the \"representative\"\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e in respect of the relation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"selection\" from \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e will now be defined as the domain of a\r\nselector; and the multiplicative class, as before, will be the class\r\nof selections.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut when the members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e overlap, there may be more selectors\r\nthan selections, since a term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e which belongs to two classes \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\r\nand \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e may be selected once to represent \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and once\r\nto represent \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e\r\ngiving rise to different selectors in the two cases, but to the same\r\nselection. For purposes of defining multiplication, it is the\r\nselectors we require rather than the selections. Thus we define:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"The product of the numbers of the members of a class of\r\nclasses \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e\" is the number of selectors from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe can define exponentiation by an adaptation of the above\r\n\u003cspan class=\"pagenum\" id=\"Page_120\"\u003e[Pg 120]\u003c/span\u003e\r\nplan. We might, of course, define \u003cimg style=\"vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-240.png\" alt=\"\" data-tex=\"\\mu^{\\nu}\"\u003e as the number of selectors\r\nfrom \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes, each of which has \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms. But there are\r\nobjections to this definition, derived from the fact that the\r\nmultiplicative axiom (of which we shall speak shortly) is unnecessarily\r\ninvolved if it is adopted. We adopt instead the following\r\nconstruction:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e be a class having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e a\r\nclass having \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e terms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e be a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e and form the class of all ordered\r\ncouples that have \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e for their second term and a member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e for\r\ntheir first term. There will be \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e such couples for a given \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e since\r\nany member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e may be chosen for the first term, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e members.\r\nIf we now form all the classes of this sort that result\r\nfrom varying \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e we obtain altogether \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes, since \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e may be\r\nany member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e has \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e members. These \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes are each\r\nof them a class of couples, namely, all the couples that can be\r\nformed of a variable member of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and a fixed member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e We\r\ndefine \u003cimg style=\"vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-240.png\" alt=\"\" data-tex=\"\\mu^{\\nu}\"\u003e as the number of selectors from the class consisting of\r\nthese \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes. Or we may equally well define \u003cimg style=\"vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-240.png\" alt=\"\" data-tex=\"\\mu^{\\nu}\"\u003e as the number of\r\nselections, for, since our classes of couples are mutually exclusive,\r\nthe number of selectors is the same as the number of selections.\r\nA selection from our class of classes will be a set of ordered couples,\r\nof which there will be exactly one having any given member of \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\r\nfor its second term, and the first term may be any member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e\r\nThus \u003cimg style=\"vertical-align: -0.489ex; width: 2.4ex; height: 2.017ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-240.png\" alt=\"\" data-tex=\"\\mu^{\\nu}\"\u003e is defined by the selectors from a certain set of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes\r\neach having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms, but the set is one having a certain structure\r\nand a more manageable composition than is the case in general.\r\nThe relevance of this to the multiplicative axiom will appear\r\nshortly.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhat applies to exponentiation applies also to the product of\r\ntwo cardinals. We might define \"\u003cimg style=\"vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-234.png\" alt=\"\" data-tex=\"\\mu × \\nu\"\u003e\" as the sum of the\r\nnumbers of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes each having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms, but we prefer to define\r\nit as the number of ordered couples to be formed consisting of a\r\nmember of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e followed by a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e,\u003c/span\u003e where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms\r\nand \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e has \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e terms. This definition, also, is designed to evade the\r\nnecessity of assuming the multiplicative axiom.\r\n\u003cspan class=\"pagenum\" id=\"Page_121\"\u003e[Pg 121]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWith our definitions, we can prove the usual formal laws of\r\nmultiplication and exponentiation. But there is one thing we\r\ncannot prove: we cannot prove that a product is only zero when\r\none of its factors is zero. We can prove this when the number\r\nof factors is finite, but not when it is infinite. In other words,\r\nwe cannot prove that, given a class of classes none of which is\r\nnull, there must be selectors from them; or that, given a class\r\nof mutually exclusive classes, there must be at least one class\r\nconsisting of one term out of each of the given classes. These\r\nthings cannot be proved; and although, at first sight, they seem\r\nobviously true, yet reflection brings gradually increasing doubt,\r\nuntil at last we become content to register the assumption and\r\nits consequences, as we register the axiom of parallels, without\r\nassuming that we can know whether it is true or false. The\r\nassumption, loosely worded, is that selectors and selections exist\r\nwhen we should expect them. There are many equivalent ways\r\nof stating it precisely. We may begin with the following:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"Given any class of mutually exclusive classes, of which none\r\nis null, there is at least one class which has exactly one term in\r\ncommon with each of the given classes.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis proposition we will call the \"multiplicative axiom.\"\u003ca id=\"FNanchor_24_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_24_1\" class=\"fnanchor\"\u003e[24]\u003c/a\u003e\r\nWe will first give various equivalent forms of the proposition,\r\nand then consider certain ways in which its truth or falsehood\r\nis of interest to mathematics.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_24_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_24_1\"\u003e\u003cspan class=\"label\"\u003e[24]\u003c/span\u003e\u003c/a\u003e\u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. I. * 88. Also vol. III. * 257-258.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe multiplicative axiom is equivalent to the proposition that\r\na product is only zero when at least one of its factors is zero;\r\n\u003ci\u003ei.e.\u003c/i\u003e that, if any number of cardinal numbers be multiplied together,\r\nthe result cannot be 0 unless one of the numbers concerned is 0.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe multiplicative axiom is equivalent to the proposition that,\r\nif \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e be any relation, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e any class contained in the converse\r\ndomain of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e,\u003c/span\u003e then there is at least one one-many relation implying \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nand having \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e for its converse domain.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe multiplicative axiom is equivalent to the assumption that\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e be any class, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e all the sub-classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with the exception\r\n\u003cspan class=\"pagenum\" id=\"Page_122\"\u003e[Pg 122]\u003c/span\u003e\r\nof the null-class, then there is at least one selector from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e.\u003c/span\u003e This\r\nis the form in which the axiom was first brought to the notice of\r\nthe learned world by Zermelo, in his \"Beweis, dass jede Menge\r\nwohlgeordnet werden kann.\"\u003ca id=\"FNanchor_25_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_25_1\" class=\"fnanchor\"\u003e[25]\u003c/a\u003e\r\nZermelo regards the axiom as an\r\nunquestionable truth. It must be confessed that, until he made\r\nit explicit, mathematicians had used it without a qualm; but it\r\nwould seem that they had done so unconsciously. And the credit\r\ndue to Zermelo for having made it explicit is entirely independent\r\nof the question whether it is true or false.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_25_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_25_1\"\u003e\u003cspan class=\"label\"\u003e[25]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eMathematische Annalen\u003c/i\u003e, vol. LIX. pp. 514-6. In this form we shall\r\nspeak of it as Zermelo\u0027s axiom.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe multiplicative axiom has been shown by Zermelo, in the\r\nabove-mentioned proof, to be equivalent to the proposition that\r\nevery class can be well-ordered, \u003ci\u003ei.e.\u003c/i\u003e can be arranged in a series in\r\nwhich every sub-class has a first term (except, of course, the null-class).\r\nThe full proof of this proposition is difficult, but it is not\r\ndifficult to see the general principle upon which it proceeds. It\r\nuses the form which we call \"Zermelo\u0027s axiom,\" \u003ci\u003ei.e.\u003c/i\u003e it assumes\r\nthat, given any class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e there is at least one one-many relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\r\nwhose converse domain consists of all existent sub-classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\r\nand which is such that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-241.png\" alt=\"\" data-tex=\"\\xi\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a\r\nmember of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 0.991ex; height: 2.057ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-241.png\" alt=\"\" data-tex=\"\\xi\"\u003e.\u003c/span\u003e Such a relation picks out a \"representative\"\r\nfrom each sub-class; of course, it will often happen that two\r\nsub-classes have the same representative. What Zermelo does,\r\nin effect, is to count off the members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e one by one, by means\r\nof \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e and transfinite induction. We put first the representative\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e;\u003c/span\u003e call it \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e.\u003c/span\u003e Then take the representative of the class consisting\r\nof all of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e except \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e;\u003c/span\u003e call it \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e.\u003c/span\u003e It must be different from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e\r\nbecause every representative is a member of its class, and \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003eis\r\nshut out from this class. Proceed similarly to take away \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e and\r\nlet \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e be the representative of what is left. In this way we first\r\nobtain a progression \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e … \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.357ex; width: 2.442ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-52.png\" alt=\"\" data-tex=\"x_{n}\"\u003e,\u003c/span\u003e …, assuming\r\nthat \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is not\r\nfinite. We then take away the whole progression; let \u003cimg style=\"vertical-align: -0.357ex; width: 2.477ex; height: 1.357ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-242.png\" alt=\"\" data-tex=\"x_{\\omega}\"\u003e be the\r\nrepresentative of what is left of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e In this way we can go on\r\nuntil nothing is left. The successive representatives will form a\r\n\u003cspan class=\"pagenum\" id=\"Page_123\"\u003e[Pg 123]\u003c/span\u003e\r\nwell-ordered series containing all the members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e (The above\r\nis, of course, only a hint of the general lines of the proof.) This\r\nproposition is called \"Zermelo\u0027s theorem.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe multiplicative axiom is also equivalent to the assumption\r\nthat of any two cardinals which are not equal, one must be the\r\ngreater. If the axiom is false, there will be cardinals \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e\r\nsuch that \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is neither less than, equal to, nor greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e.\u003c/span\u003e We\r\nhave seen that \u003cimg style=\"vertical-align: -0.339ex; width: 2.37ex; height: 1.91ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-191.png\" alt=\"\" data-tex=\"\\aleph_{1}\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.995ex; height: 1.932ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-173.png\" alt=\"\" data-tex=\"2^{\\aleph_{0}}\"\u003e possibly form an instance of such a pair.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMany other forms of the axiom might be given, but the above\r\nare the most important of the forms known at present. As to\r\nthe truth or falsehood of the axiom in any of its forms, nothing\r\nis known at present.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe propositions that depend upon the axiom, without being\r\nknown to be equivalent to it, are numerous and important. Take\r\nfirst the connection of addition and multiplication. We naturally\r\nthink that the sum of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e mutually exclusive classes, each having\r\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms, must have \u003cimg style=\"vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-234.png\" alt=\"\" data-tex=\"\\mu × \\nu\"\u003e terms. When \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is finite, this can be\r\nproved. But when \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is infinite, it cannot be proved without the\r\nmultiplicative axiom, except where, owing to some special circumstance,\r\nthe existence of certain selectors can be proved. The\r\nway the multiplicative axiom enters in is as follows: Suppose\r\nwe have two sets of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e mutually exclusive classes, each having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms,\r\nand we wish to prove that the sum of one set has as many\r\nterms as the sum of the other. In order to prove this, we must\r\nestablish a one-one relation. Now, since there are in each case\r\n\u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes, there is some one-one relation between the two sets of\r\nclasses; but what we want is a one-one relation between their\r\nterms. Let us consider some one-one relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e between the\r\nclasses. Then if \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e and \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-243.png\" alt=\"\" data-tex=\"\\lambda\"\u003e are the two sets of classes, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is some\r\nmember of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e,\u003c/span\u003e there will be a member \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e of \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-243.png\" alt=\"\" data-tex=\"\\lambda\"\u003e which will be the\r\ncorrelate of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with respect to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e.\u003c/span\u003e Now \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e each have \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms,\r\nand are therefore similar. There are, accordingly, one-one correlations\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e The trouble is that there are so many. In\r\norder to obtain a one-one correlation of the sum of \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e with the\r\nsum of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-243.png\" alt=\"\" data-tex=\"\\lambda\"\u003e,\u003c/span\u003e we have to pick out \u003ci\u003eone selection\u003c/i\u003e from a set of classes\r\n\u003cspan class=\"pagenum\" id=\"Page_124\"\u003e[Pg 124]\u003c/span\u003e\r\nof correlators, one class of the set being all the one-one correlators\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e If \u003cimg style=\"vertical-align: -0.025ex; width: 1.303ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-239.png\" alt=\"\" data-tex=\"\\kappa\"\u003e and \u003cimg style=\"vertical-align: -0.027ex; width: 1.319ex; height: 1.597ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-243.png\" alt=\"\" data-tex=\"\\lambda\"\u003e are infinite, we cannot in general know\r\nthat such a selection exists, unless we can know that the multiplicative\r\naxiom is true. Hence we cannot establish the usual\r\nkind of connection between addition and multiplication.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis fact has various curious consequences. To begin with,\r\nwe know that \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.687ex; width: 17.274ex; height: 2.573ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-244.png\" alt=\"\" data-tex=\"\\aleph_{0}^{2} = \\aleph_{0} × \\aleph_{0} = \\aleph_{0}\"\u003e.\u003c/span\u003e It is commonly inferred from\r\nthis that the sum of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e classes each having \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e members must\r\nitself have \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e members, but this inference is fallacious, since we\r\ndo not know that the number of terms in such a sum is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 6.5ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-245.png\" alt=\"\" data-tex=\"\\aleph_{0} × \\aleph_{0}\"\u003e,\u003c/span\u003e\r\nnor consequently that it is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e This has a bearing upon the theory\r\nof transfinite ordinals. It is easy to prove that an ordinal which\r\nhas \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e predecessors must be one of what Cantor calls the \"second\r\nclass,\" \u003ci\u003ei.e.\u003c/i\u003e such that a series having this ordinal number will have\r\n\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms in its field. It is also easy to see that, if we take any\r\nprogression of ordinals of the second class, the predecessors of\r\ntheir limit form at most the sum of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e classes each having \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms.\r\nIt is inferred thence\u0026mdash;fallaciously, unless the multiplicative\r\naxiom is true\u0026mdash;that the predecessors of the limit are \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e in\r\nnumber, and therefore that the limit is a number of the \"second\r\nclass.\" That is to say, it is supposed to be proved that any progression\r\nof ordinals of the second class has a limit which is again\r\nan ordinal of the second class. This proposition, with the corollary\r\nthat \u003cimg style=\"vertical-align: -0.339ex; width: 2.395ex; height: 1.342ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-192.png\" alt=\"\" data-tex=\"\\omega_{1}\"\u003e (the smallest ordinal of the third class) is not the\r\nlimit of any progression, is involved in most of the recognised\r\ntheory of ordinals of the second class. In view of the way in\r\nwhich the multiplicative axiom is involved, the proposition and\r\nits corollary cannot be regarded as proved. They may be true,\r\nor they may not. All that can be said at present is that we do\r\nnot know. Thus the greater part of the theory of ordinals of\r\nthe second class must be regarded as unproved.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother illustration may help to make the point clearer. We\r\nknow that \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 10.649ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-246.png\" alt=\"\" data-tex=\"2 × \\aleph_{0} = \\aleph_{0}\"\u003e.\u003c/span\u003e Hence we might suppose that the sum\r\nof \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e pairs must have \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms. But this, though we can prove\r\nthat it is sometimes the case, cannot be proved to happen \u003ci\u003ealways\u003c/i\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_125\"\u003e[Pg 125]\u003c/span\u003e\r\nunless we assume the multiplicative axiom. This is illustrated\r\nby the millionaire who bought a pair of socks whenever he bought\r\na pair of boots, and never at any other time, and who had such\r\na passion for buying both that at last he had \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e pairs of boots\r\nand \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e pairs of socks. The problem is: How many boots had\r\nhe, and how many socks? One would naturally suppose that\r\nhe had twice as many boots and twice as many socks as he had\r\npairs of each, and that therefore he had \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e of each, since that\r\nnumber is not increased by doubling. But this is an instance of\r\nthe difficulty, already noted, of connecting the sum of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes\r\neach having \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e terms with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 4.324ex; height: 1.6ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-234.png\" alt=\"\" data-tex=\"\\mu × \\nu\"\u003e.\u003c/span\u003e Sometimes this can be done,\r\nsometimes it cannot. In our case it can be done with the boots,\r\nbut not with the socks, except by some very artificial device.\r\nThe reason for the difference is this: Among boots we can distinguish\r\nright and left, and therefore we can make a selection of\r\none out of each pair, namely, we can choose all the right boots or\r\nall the left boots; but with socks no such principle of selection\r\nsuggests itself, and we cannot be sure, unless we assume the\r\nmultiplicative axiom, that there is any class consisting of one\r\nsock out of each pair. Hence the problem.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may put the matter in another way. To prove that a\r\nclass has \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms, it is necessary and sufficient to find some way\r\nof arranging its terms in a progression. There is no difficulty in\r\ndoing this with the boots. The \u003ci\u003epairs\u003c/i\u003e are given as forming an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e\r\nand therefore as the field of a progression. Within each pair,\r\ntake the left boot first and the right second, keeping the order\r\nof the pairs unchanged; in this way we obtain a progression of\r\nall the boots. But with the socks we shall have to choose arbitrarily,\r\nwith each pair, which to put first; and an infinite number\r\nof arbitrary choices is an impossibility. Unless we can find a\r\n\u003ci\u003erule\u003c/i\u003e for selecting, \u003ci\u003ei.e.\u003c/i\u003e a relation which is a selector, we do not know\r\nthat a selection is even theoretically possible. Of course, in the\r\ncase of objects in space, like socks, we always can find some\r\nprinciple of selection. For example, take the centres of mass\r\nof the socks: there will be points \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e in space such that, with any\r\n\u003cspan class=\"pagenum\" id=\"Page_126\"\u003e[Pg 126]\u003c/span\u003e\r\npair, the centres of mass of the two socks are not both at exactly\r\nthe same distance from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e;\u003c/span\u003e thus we can choose, from each pair,\r\nthat sock which has its centre of mass nearer to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e But there is\r\nno theoretical reason why a method of selection such as this\r\nshould always be possible, and the case of the socks, with a little\r\ngoodwill on the part of the reader, may serve to show how a\r\nselection might be impossible.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is to be observed that, if it \u003ci\u003ewere\u003c/i\u003e impossible to select one out\r\nof each pair of socks, it would follow that the socks \u003ci\u003ecould\u003c/i\u003e not be\r\narranged in a progression, and therefore that there were not \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e of\r\nthem. This case illustrates that, if \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is an infinite number,\r\none set of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e pairs may not contain the same number of terms as\r\nanother set of \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e pairs; for, given \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e pairs of boots, there are\r\ncertainly \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e boots, but we cannot be sure of this in the case of\r\nthe socks unless we assume the multiplicative axiom or fall back\r\nupon some fortuitous geometrical method of selection such as\r\nthe above.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother important problem involving the multiplicative\r\naxiom is the relation of reflexiveness to non-inductiveness. It\r\nwill be remembered that in Chapter VIII. we pointed out that a\r\nreflexive number must be non-inductive, but that the converse\r\n(so far as is known at present) can only be proved if we assume\r\nthe multiplicative axiom. The way in which this comes about\r\nis as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is easy to prove that a reflexive class is one which contains\r\nsub-classes having \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms. (The class may, of course, itself\r\nhave \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms.) Thus we have to prove, if we can, that, given\r\nany non-inductive class, it is possible to choose a progression\r\nout of its terms. Now there is no difficulty in showing that\r\na non-inductive class must contain more terms than any inductive\r\nclass, or, what comes to the same thing, that if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a non-inductive\r\nclass and \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is any inductive number, there are sub-classes\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e that have \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e terms. Thus we can form sets of finite sub-classes\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e:\u003c/span\u003e First one class having no terms, then classes having\r\n1 term (as many as there are members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e)\u003c/span\u003e, then classes having\r\n\u003cspan class=\"pagenum\" id=\"Page_127\"\u003e[Pg 127]\u003c/span\u003e\r\n2 terms, and so on. We thus get a progression of sets of sub-classes,\r\neach set consisting of all those that have a certain given\r\nfinite number of terms. So far we have not used the multiplicative\r\naxiom, but we have only proved that the number of collections\r\nof sub-classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a reflexive number, \u003ci\u003ei.e.\u003c/i\u003e that, if \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is\r\nthe number of members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e so that \u003cimg style=\"vertical-align: 0; width: 2.284ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-247.png\" alt=\"\" data-tex=\"2^{\\mu}\"\u003e is the number of sub-classes\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: 0; width: 2.934ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-248.png\" alt=\"\" data-tex=\"2^{2^{\\mu}}\"\u003e is the number of collections of sub-classes,\r\nthen, provided \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is not inductive, \u003cimg style=\"vertical-align: 0; width: 2.934ex; height: 1.902ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-248.png\" alt=\"\" data-tex=\"2^{2^{\\mu}}\"\u003e must be reflexive. But\r\nthis is a long way from what we set out to prove.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn order to advance beyond this point, we must employ the\r\nmultiplicative axiom. From each set of sub-classes let us\r\nchoose out one, omitting the sub-class consisting of the null-class\r\nalone. That is to say, we select one sub-class containing\r\none term, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e say; one containing two terms, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e say; one containing\r\nthree, \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e,\u003c/span\u003e say; and so on. (We can do this if the multiplicative\r\naxiom is assumed; otherwise, we do not know whether\r\nwe can always do it or not.) We have now a progression\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e,\u003c/span\u003e … sub-classes of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e instead of a progression of\r\ncollections of sub-classes; thus we are one step nearer to our\r\ngoal. We now know that, assuming the multiplicative axiom,\r\nif \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e is a non-inductive number, \u003cimg style=\"vertical-align: 0; width: 2.284ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-247.png\" alt=\"\" data-tex=\"2^{\\mu}\"\u003e must be a reflexive number.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe next step is to notice that, although we cannot be sure\r\nthat new members of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e come in at any one specified stage in the\r\nprogression \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e,\u003c/span\u003e … we can be sure that new members\r\nkeep on coming in from time to time. Let us illustrate.\r\nThe class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e which consists of one term, is a new beginning;\r\nlet the one term be \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e.\u003c/span\u003e The class \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e consisting of two terms,\r\nmay or may not contain \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e;\u003c/span\u003e if it does, it introduces one new\r\nterm; and if it does not, it must introduce two new terms, say\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e.\u003c/span\u003e In this case it is possible that \u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e consists\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e,\u003c/span\u003e\r\nand so introduces no new terms, but in that case \u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-252.png\" alt=\"\" data-tex=\"\\alpha_{4}\"\u003e must introduce\r\na new term. The first \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e,\u003c/span\u003e … \u003cimg style=\"vertical-align: -0.339ex; width: 2.484ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-253.png\" alt=\"\" data-tex=\"\\alpha_{\\nu}\"\u003e contain, at\r\nthe very most, \u003cimg style=\"vertical-align: -0.186ex; width: 18.307ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-254.png\" alt=\"\" data-tex=\"1 + 2 + 3 + \\dots + \\nu\"\u003e terms, \u003ci\u003ei.e.\u003c/i\u003e\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 11.449ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-255.png\" alt=\"\" data-tex=\"\\nu/(\\nu + 1)/2\"\u003e terms;\r\nthus it would be possible, if there were no repetitions in the\r\nfirst \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e classes, to go on with only repetitions from the \u003cimg style=\"vertical-align: -0.566ex; width: 8.543ex; height: 2.497ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-256.png\" alt=\"\" data-tex=\"(\\nu + 1)^{th}\"\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_128\"\u003e[Pg 128]\u003c/span\u003e\r\nclass to the \u003cimg style=\"vertical-align: -0.566ex; width: 12.005ex; height: 2.497ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-257.png\" alt=\"\" data-tex=\"\\nu(\\nu + 1)/2^{th}\"\u003e class. But by that time the old terms\r\nwould no longer be sufficiently numerous to form a next class\r\nwith the right number of members, \u003ci\u003ei.e.\u003c/i\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 14.215ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-258.png\" alt=\"\" data-tex=\"\\nu(\\nu + 1)/2 + 1\"\u003e,\u003c/span\u003e therefore\r\nnew terms must come in at this point if not sooner. It\r\nfollows that, if we omit from our progression\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.436ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-251.png\" alt=\"\" data-tex=\"\\alpha_{3}\"\u003e,\u003c/span\u003e… all\r\nthose classes that are composed entirely of members that have\r\noccurred in previous classes, we shall still have a progression.\r\nLet our new progression be called \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-259.png\" alt=\"\" data-tex=\"\\beta_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-260.png\" alt=\"\" data-tex=\"\\beta_{2}\"\u003e,\u003c/span\u003e\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-261.png\" alt=\"\" data-tex=\"\\beta_{3}\"\u003e.\u003c/span\u003e… (We shall\r\nhave \u003cimg style=\"vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-262.png\" alt=\"\" data-tex=\"\\alpha_{1} = \\beta_{1}\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-263.png\" alt=\"\" data-tex=\"\\alpha_{2} = \\beta_{2}\"\u003e,\u003c/span\u003e\r\nbecause \u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e and \u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e \u003ci\u003emust\u003c/i\u003e introduce new\r\nterms. We may or may not have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 7.721ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-264.png\" alt=\"\" data-tex=\"\\alpha_{3} = \\beta_{3}\"\u003e,\u003c/span\u003e but, speaking generally,\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.316ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-265.png\" alt=\"\" data-tex=\"\\beta_{\\nu}\"\u003e will be \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.484ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-253.png\" alt=\"\" data-tex=\"\\alpha_{\\nu}\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is some number greater\r\nthan \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e;\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e the\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003es are \u003ci\u003esome\u003c/i\u003e of the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es.) Now these \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003es are such that any one\r\nof them, say \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.685ex; width: 2.433ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-266.png\" alt=\"\" data-tex=\"\\beta_{\\mu}\"\u003e,\u003c/span\u003e contains members which have not occurred in\r\nany of the previous \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003es. Let \u003cimg style=\"vertical-align: -0.685ex; width: 2.324ex; height: 1.683ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-267.png\" alt=\"\" data-tex=\"\\gamma_{\\mu}\"\u003e be the part of \u003cimg style=\"vertical-align: -0.685ex; width: 2.433ex; height: 2.28ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-266.png\" alt=\"\" data-tex=\"\\beta_{\\mu}\"\u003e which consists\r\nof new members. Thus we get a new progression \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-268.png\" alt=\"\" data-tex=\"\\gamma_{1}\"\u003e,\u003c/span\u003e\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-269.png\" alt=\"\" data-tex=\"\\gamma_{2}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-270.png\" alt=\"\" data-tex=\"\\gamma_{3}\"\u003e,\u003c/span\u003e…\r\n(Again \u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-268.png\" alt=\"\" data-tex=\"\\gamma_{1}\"\u003e will be identical with \u003cimg style=\"vertical-align: -0.439ex; width: 2.268ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-259.png\" alt=\"\" data-tex=\"\\beta_{1}\"\u003e and with \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e;\u003c/span\u003e\r\nif \u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e does not\r\ncontain the one member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-249.png\" alt=\"\" data-tex=\"\\alpha_{1}\"\u003e,\u003c/span\u003e we shall have \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 12.898ex; height: 2.084ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-271.png\" alt=\"\" data-tex=\"\\gamma_{2} = \\beta_{2} = \\alpha_{2}\"\u003e,\u003c/span\u003e but if\r\n\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e does contain this one member, \u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-269.png\" alt=\"\" data-tex=\"\\gamma_{2}\"\u003e will consist of the other\r\nmember of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.436ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-250.png\" alt=\"\" data-tex=\"\\alpha_{2}\"\u003e)\u003c/span\u003e. This new progression of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 1.229ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-65.png\" alt=\"\" data-tex=\"\\gamma\"\u003e\u0027\u003c/span\u003es consists of mutually\r\nexclusive classes. Hence a selection from them will be a progression;\r\n\u003ci\u003ei.e.\u003c/i\u003e if \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e is the member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-268.png\" alt=\"\" data-tex=\"\\gamma_{1}\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e is\r\na member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-269.png\" alt=\"\" data-tex=\"\\gamma_{2}\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e is\r\na member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.489ex; width: 2.16ex; height: 1.486ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-270.png\" alt=\"\" data-tex=\"\\gamma_{3}\"\u003e,\u003c/span\u003e and so on; then \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e,\u003c/span\u003e … is a progression,\r\nand is a sub-class of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e Assuming the multiplicative axiom,\r\nsuch a selection can be made. Thus by twice using this axiom\r\nwe can prove that, if the axiom is true, every non-inductive\r\ncardinal must be reflexive. This could also be deduced from\r\nZermelo\u0027s theorem, that, if the axiom is true, every class can be\r\nwell ordered; for a well-ordered series must have either a finite\r\nor a reflexive number of terms in its field.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is one advantage in the above direct argument, as\r\nagainst deduction from Zermelo\u0027s theorem, that the above\r\nargument does not demand the universal truth of the multiplicative\r\naxiom, but only its truth as applied to a set of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e classes.\r\nIt may happen that the axiom holds for \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e classes, though not\r\nfor larger numbers of classes. For this reason it is better, when\r\n\u003cspan class=\"pagenum\" id=\"Page_129\"\u003e[Pg 129]\u003c/span\u003e\r\nit is possible, to content ourselves with the more restricted\r\nassumption. The assumption made in the above direct argument\r\nis that a product of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e factors is never zero unless one of\r\nthe factors is zero. We may state this assumption in the form:\r\n\"\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e is a \u003ci\u003emultipliable\u003c/i\u003e number,\" where a number \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e is defined as\r\n\"multipliable\" when a product of \u003cimg style=\"vertical-align: 0; width: 1.199ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-143.png\" alt=\"\" data-tex=\"\\nu\"\u003e factors is never zero unless\r\none of the factors is zero. We can \u003ci\u003eprove\u003c/i\u003e that a \u003ci\u003efinite\u003c/i\u003e number is\r\nalways multipliable, but we cannot prove that any infinite number\r\nis so. The multiplicative axiom is equivalent to the assumption\r\nthat \u003ci\u003eall\u003c/i\u003e cardinal numbers are multipliable. But in order to\r\nidentify the reflexive with the non-inductive, or to deal with the\r\nproblem of the boots and socks, or to show that any progression\r\nof numbers of the second class is of the second class, we only\r\nneed the very much smaller assumption that \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e is multipliable.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is not improbable that there is much to be discovered\r\nin regard to the topics discussed in the present chapter. Cases\r\nmay be found where propositions which seem to involve the\r\nmultiplicative axiom can be proved without it. It is conceivable\r\nthat the multiplicative axiom in its general form may be shown\r\nto be false. From this point of view, Zermelo\u0027s theorem offers\r\nthe best hope: the continuum or some still more dense series\r\n\u003ci\u003emight\u003c/i\u003e be proved to be incapable of having its terms well ordered,\r\nwhich would prove the multiplicative axiom false, in virtue of\r\nZermelo\u0027s theorem. But so far, no method of obtaining such\r\nresults has been discovered, and the subject remains wrapped in\r\nobscurity.\r\n\u003cspan class=\"pagenum\" id=\"Page_130\"\u003e[Pg 130]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XIII: THE AXIOM OF INFINITY AND LOGICAL TYPES\u0027\u003e\u003ca id=\"chap13\"\u003e\u003c/a\u003eCHAPTER XIII\r\n\u003cbr\u003e\u003cbr\u003e\r\nTHE AXIOM OF INFINITY AND LOGICAL TYPES\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nTHE axiom of infinity is an assumption which may be enunciated\r\nas follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"If \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e be any inductive cardinal number, there is at least one\r\nclass of individuals having \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf this is true, it follows, of course, that there are many classes\r\nof individuals having \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms, and that the total number of\r\nindividuals in the world is not an inductive number. For, by\r\nthe axiom, there is at least one class having \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e terms, from which\r\nit follows that there are many classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms and that \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is\r\nnot the number of individuals in the world. Since \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is \u003ci\u003eany\u003c/i\u003e\r\ninductive number, it follows that the number of individuals\r\nin the world must (if our axiom be true) exceed any inductive\r\nnumber. In view of what we found in the preceding chapter,\r\nabout the possibility of cardinals which are neither inductive\r\nnor reflexive, we cannot infer from our axiom that there are at\r\nleast \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e individuals, unless we assume the multiplicative axiom.\r\nBut we do know that there are at least \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e classes of classes,\r\nsince the inductive cardinals are classes of classes, and form a\r\nprogression if our axiom is true. The way in which the need\r\nfor this axiom arises may be explained as follows:\u0026mdash;One of\r\nPeano\u0027s assumptions is that no two inductive cardinals have the\r\nsame successor, \u003ci\u003ei.e.\u003c/i\u003e that we shall not have \u003cimg style=\"vertical-align: -0.186ex; width: 14.155ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-68.png\" alt=\"\" data-tex=\"m + 1 = n + 1\"\u003e unless\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 6.361ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-69.png\" alt=\"\" data-tex=\"m = n\"\u003e,\u003c/span\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e are inductive cardinals. In Chapter VIII. we\r\nhad occasion to use what is virtually the same as the above\r\nassumption of Peano\u0027s, namely, that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive cardinal,\r\n\u003cspan class=\"pagenum\" id=\"Page_131\"\u003e[Pg 131]\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e It might be thought that this could be\r\nproved. We can prove that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is an inductive class, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is\r\nthe number of members of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e\r\nThis proposition is easily proved by induction, and might be\r\nthought to imply the other. But in fact it does not, since there\r\nmight be no such class as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e What it does imply is this: If\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive cardinal such that there is at least one class\r\nhaving \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e members, then \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e The axiom of\r\ninfinity assures us (whether truly or falsely) that there are classes\r\nhaving \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e members, and thus enables us to assert that \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not\r\nequal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e.\u003c/span\u003e But without this axiom we should be left with\r\nthe possibility that \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e and \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e might both be the null-class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us illustrate this possibility by an example: Suppose\r\nthere were exactly nine individuals in the world. (As to what\r\nis meant by the word \"individual,\" I must ask the reader to\r\nbe patient.) Then the inductive cardinals from 0 up to 9 would\r\nbe such as we expect, but 10 (defined as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-272.png\" alt=\"\" data-tex=\"9 + 1\"\u003e)\u003c/span\u003e would be the\r\nnull-class. It will be remembered that \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e may be defined as\r\nfollows: \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e is the collection of all those classes which have a\r\nterm \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e such that, when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is taken away, there remains a class\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms. Now applying this definition, we see that, in the\r\ncase supposed, \u003cimg style=\"vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-272.png\" alt=\"\" data-tex=\"9 + 1\"\u003e is a class consisting of no classes, \u003ci\u003ei.e.\u003c/i\u003e it is\r\nthe null-class. The same will be true of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.028ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-273.png\" alt=\"\" data-tex=\"9 + 2\"\u003e,\u003c/span\u003e or generally of\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-274.png\" alt=\"\" data-tex=\"9 + n\"\u003e,\u003c/span\u003e unless \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is zero. Thus 10 and all subsequent inductive\r\ncardinals will all be identical, since they will all be the null-class.\r\nIn such a case the inductive cardinals will not form a progression,\r\nnor will it be true that no two have the same successor, for 9\r\nand 10 will both be succeeded by the null-class (10 being itself\r\nthe null-class). It is in order to prevent such arithmetical\r\ncatastrophes that we require the axiom of infinity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAs a matter of fact, so long as we are content with the arithmetic\r\nof finite integers, and do not introduce either infinite\r\nintegers or infinite classes or series of finite integers or ratios,\r\nit is possible to obtain all desired results without the axiom of\r\ninfinity. That is to say, we can deal with the addition, multiplication,\r\n\u003cspan class=\"pagenum\" id=\"Page_132\"\u003e[Pg 132]\u003c/span\u003e\r\nand exponentiation of finite integers and of ratios,\r\nbut we cannot deal with infinite integers or with irrationals.\r\nThus the theory of the transfinite and the theory of real numbers\r\nfails us. How these various results come about must now be\r\nexplained.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAssuming that the number of individuals in the world is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e\r\nthe number of classes of individuals will be \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e.\u003c/span\u003e This is in virtue\r\nof the general proposition mentioned in Chapter VIII. that the\r\nnumber of classes contained in a class which has \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e members\r\nis \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e.\u003c/span\u003e Now \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e is always greater than \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e Hence the number\r\nof classes in the world is greater than the number of individuals.\r\nIf, now, we suppose the number of individuals to be 9, as we did\r\njust now, the number of classes will be \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 2.119ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-275.png\" alt=\"\" data-tex=\"2^{9}\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e 512. Thus if we\r\ntake our numbers as being applied to the counting of classes\r\ninstead of to the counting of individuals, our arithmetic will\r\nbe normal until we reach 512: the first number to be null will\r\nbe 513. And if we advance to classes of classes we shall do still\r\nbetter: the number of them will be \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 3.719ex; height: 1.887ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-276.png\" alt=\"\" data-tex=\"2^{512}\"\u003e,\u003c/span\u003e a number which is so\r\nlarge as to stagger imagination, since it has about 153 digits.\r\nAnd if we advance to classes of classes of classes, we shall obtain\r\na number represented by 2 raised to a power which has about\r\n153 digits; the number of digits in this number will be about\r\nthree times \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 4.85ex; height: 2.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-277.png\" alt=\"\" data-tex=\"10^{152}\"\u003e.\u003c/span\u003e In a time of paper shortage it is undesirable\r\nto write out this number, and if we want larger ones we can\r\nobtain them by travelling further along the logical hierarchy.\r\nIn this way any assigned inductive cardinal can be made to\r\nfind its place among numbers which are not null, merely by\r\ntravelling along the hierarchy for a sufficient distance.\u003ca id=\"FNanchor_26_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_26_1\" class=\"fnanchor\"\u003e[26]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_26_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_26_1\"\u003e\u003cspan class=\"label\"\u003e[26]\u003c/span\u003e\u003c/a\u003eOn this subject see \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. II. * 120 ff. On the\r\ncorresponding problems as regards ratio, see \u003ci\u003eibid.\u003c/i\u003e, vol. III. * 303 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nAs regards ratios, we have a very similar state of affairs.\r\nIf a ratio \u003cimg style=\"vertical-align: -0.566ex; width: 3.695ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-278.png\" alt=\"\" data-tex=\"\\mu/\\nu\"\u003e is to have the expected properties, there must\r\nbe enough objects of whatever sort is being counted to insure\r\nthat the null-class does not suddenly obtrude itself. But this\r\ncan be insured, for any given ratio \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.695ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-278.png\" alt=\"\" data-tex=\"\\mu/\\nu\"\u003e,\u003c/span\u003e without the axiom of\r\n\u003cspan class=\"pagenum\" id=\"Page_133\"\u003e[Pg 133]\u003c/span\u003e\r\ninfinity, by merely travelling up the hierarchy a sufficient distance.\r\nIf we cannot succeed by counting individuals, we can try counting\r\nclasses of individuals; if we still do not succeed, we can try\r\nclasses of classes, and so on. Ultimately, however few individuals\r\nthere may be in the world, we shall reach a stage where\r\nthere are many more than \u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e objects, whatever inductive number\r\n\u003cimg style=\"vertical-align: -0.489ex; width: 1.364ex; height: 1.489ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-142.png\" alt=\"\" data-tex=\"\\mu\"\u003e may be. Even if there were no individuals at all, this would\r\nstill be true, for there would then be one class, namely, the null-class,\r\n2 classes of classes (namely, the null-class of classes and the\r\nclass whose only member is the null-class of individuals), 4 classes\r\nof classes of classes, 16 at the next stage, 65,536 at the next\r\nstage, and so on. Thus no such assumption as the axiom of\r\ninfinity is required in order to reach any given ratio or any given\r\ninductive cardinal.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is when we wish to deal with the whole class or series of\r\ninductive cardinals or of ratios that the axiom is required. We\r\nneed the whole class of inductive cardinals in order to establish\r\nthe existence of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e,\u003c/span\u003e and the whole series in order to establish\r\nthe existence of progressions: for these results, it is necessary\r\nthat we should be able to make a single class or series in which\r\nno inductive cardinal is null. We need the whole series of ratios\r\nin order of magnitude in order to define real numbers as segments:\r\nthis definition will not give the desired result unless the series\r\nof ratios is compact, which it cannot be if the total number of\r\nratios, at the stage concerned, is finite.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt would be natural to suppose\u0026mdash;as I supposed myself in former\r\ndays\u0026mdash;that, by means of constructions such as we have been\r\nconsidering, the axiom of infinity could be \u003ci\u003eproved\u003c/i\u003e. It may be\r\nsaid: Let us assume that the number of individuals is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be 0 without spoiling our argument; then if we form the\r\ncomplete set of individuals, classes, classes of classes, etc., all\r\ntaken together, the number of terms in our whole set will be\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.462ex; width: 25.132ex; height: 2.477ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-42.png\" alt=\"\" data-tex=\"\r\nn + 2^{n} + 2^{2^{n}} + \\dots\\ \\mathit{ad\\,inf.},\r\n\"\u003e\u003c/span\u003e\r\nwhich is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e.\u003c/span\u003e Thus taking all kinds of objects together, and not\r\n\u003cspan class=\"pagenum\" id=\"Page_134\"\u003e[Pg 134]\u003c/span\u003e\r\nconfining ourselves to objects of any one type, we shall certainly\r\nobtain an infinite class, and shall therefore not need the axiom\r\nof infinity. So it might be said.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nNow, before going into this argument, the first thing to observe\r\nis that there is an air of hocus-pocus about it: something reminds\r\none of the conjurer who brings things out of the hat. The man\r\nwho has lent his hat is quite sure there wasn\u0027t a live rabbit in it\r\nbefore, but he is at a loss to say how the rabbit got there. So\r\nthe reader, if he has a robust sense of reality, will feel convinced\r\nthat it is impossible to manufacture an infinite collection out of\r\na finite collection of individuals, though he may be unable to\r\nsay where the flaw is in the above construction. It would be a\r\nmistake to lay too much stress on such feelings of hocus-pocus;\r\nlike other emotions, they may easily lead us astray. But they\r\nafford a \u003ci\u003eprima facie\u003c/i\u003e ground for scrutinising very closely any\r\nargument which arouses them. And when the above argument\r\nis scrutinised it will, in my opinion, be found to be fallacious,\r\nthough the fallacy is a subtle one and by no means easy to avoid\r\nconsistently.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe fallacy involved is the fallacy which may be called \"confusion\r\nof types.\" To explain the subject of \"types\" fully would\r\nrequire a whole volume; moreover, it is the purpose of this book\r\nto avoid those parts of the subjects which are still obscure and\r\ncontroversial, isolating, for the convenience of beginners, those\r\nparts which can be accepted as embodying mathematically ascertained\r\ntruths. Now the theory of types emphatically does not\r\nbelong to the finished and certain part of our subject: much of\r\nthis theory is still inchoate, confused, and obscure. But the need\r\nof \u003ci\u003esome\u003c/i\u003e doctrine of types is less doubtful than the precise form\r\nthe doctrine should take; and in connection with the axiom of\r\ninfinity it is particularly easy to see the necessity of some such\r\ndoctrine.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis necessity results, for example, from the \"contradiction of\r\nthe greatest cardinal.\" We saw in Chapter VIII. that the number\r\nof classes contained in a given class is always greater than the\r\n\u003cspan class=\"pagenum\" id=\"Page_135\"\u003e[Pg 135]\u003c/span\u003e\r\nnumber of members of the class, and we inferred that there is\r\nno greatest cardinal number. But if we could, as we suggested\r\na moment ago, add together into one class the individuals, classes\r\nof individuals, classes of classes of individuals, etc., we should\r\nobtain a class of which its own sub-classes would be members.\r\nThe class consisting of all objects that can be counted, of whatever\r\nsort, must, if there be such a class, have a cardinal number which\r\nis the greatest possible. Since all its sub-classes will be members\r\nof it, there cannot be more of them than there are members.\r\nHence we arrive at a contradiction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen I first came upon this contradiction, in the year 1901,\r\nI attempted to discover some flaw in Cantor\u0027s proof that there is\r\nno greatest cardinal, which we gave in Chapter VIII. Applying\r\nthis proof to the supposed class of all imaginable objects,\r\nI was led to a new and simpler contradiction, namely, the\r\nfollowing:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe comprehensive class we are considering, which is to embrace\r\neverything, must embrace itself as one of its members. In other\r\nwords, if there is such a thing as \"everything,\" then \"everything\"\r\nis something, and is a member of the class \"everything.\"\r\nBut normally a class is not a member of itself. Mankind, for\r\nexample, is not a man. Form now the assemblage of all classes\r\nwhich are not members of themselves. This is a class: is it a\r\nmember of itself or not? If it is, it is one of those classes that\r\nare not members of themselves, \u003ci\u003ei.e.\u003c/i\u003e it is not a member of itself.\r\nIf it is not, it is not one of those classes that are not members of\r\nthemselves, \u003ci\u003ei.e.\u003c/i\u003e it is a member of itself. Thus of the two hypotheses\u0026mdash;that\r\nit is, and that it is not, a member of itself\u0026mdash;each\r\nimplies its contradictory. This is a contradiction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is no difficulty in manufacturing similar contradictions\r\n\u003ci\u003ead lib\u003c/i\u003e. The solution of such contradictions by the theory of\r\ntypes is set forth fully in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e,\u003ca id=\"FNanchor_27_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_27_1\" class=\"fnanchor\"\u003e[27]\u003c/a\u003e\r\nand also, more\r\nbriefly, in articles by the present author in the \u003ci\u003eAmerican Journal\r\n\u003cspan class=\"pagenum\" id=\"Page_136\"\u003e[Pg 136]\u003c/span\u003e\r\nof Mathematics\u003c/i\u003e,\u003ca id=\"FNanchor_28_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_28_1\" class=\"fnanchor\"\u003e[28]\u003c/a\u003e\r\nand in the \u003ci\u003eRevue de Metaphysique et de Morale\u003c/i\u003e.\u003ca id=\"FNanchor_29_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_29_1\" class=\"fnanchor\"\u003e[29]\u003c/a\u003e\r\nFor the present an outline of the solution must suffice.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_27_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_27_1\"\u003e\u003cspan class=\"label\"\u003e[27]\u003c/span\u003e\u003c/a\u003eVol. I., Introduction, chap. II., * 12 and * 20; vol. II., Prefatory\r\nStatement\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_28_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_28_1\"\u003e\u003cspan class=\"label\"\u003e[28]\u003c/span\u003e\u003c/a\u003e\"Mathematical Logic as based on the Theory of Types,\" vol. XXX.,\r\n1908, pp. 222-262.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_29_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_29_1\"\u003e\u003cspan class=\"label\"\u003e[29]\u003c/span\u003e\u003c/a\u003e\"Les paradoxes de la logique,\" 1906, pp. 627-650.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe fallacy consists in the formation of what we may call\r\n\"impure\" classes, \u003ci\u003ei.e.\u003c/i\u003e classes which are not pure as to \"type.\"\r\nAs we shall see in a later chapter, classes are logical fictions, and\r\na statement which appears to be about a class will only be significant\r\nif it is capable of translation into a form in which no mention\r\nis made of the class. This places a limitation upon the ways in\r\nwhich what are nominally, though not really, names for classes\r\ncan occur significantly: a sentence or set of symbols in which\r\nsuch pseudo-names occur in wrong ways is not false, but strictly\r\ndevoid of meaning. The supposition that a class is, or that it\r\nis not, a member of itself is meaningless in just this way. And\r\nmore generally, to suppose that one class of individuals is a\r\nmember, or is not a member, of another class of individuals\r\nwill be to suppose nonsense; and to construct symbolically any\r\nclass whose members are not all of the same grade in the logical\r\nhierarchy is to use symbols in a way which makes them no\r\nlonger symbolise anything.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus if there are \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals in the world, and \u003cimg style=\"vertical-align: 0; width: 2.279ex; height: 1.528ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-174.png\" alt=\"\" data-tex=\"2^{n}\"\u003e classes of\r\nindividuals, we cannot form a new class, consisting of both\r\nindividuals and classes and having \u003cimg style=\"vertical-align: -0.186ex; width: 6.402ex; height: 1.714ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-279.png\" alt=\"\" data-tex=\"n + 2^{n}\"\u003e members. In this way\r\nthe attempt to escape from the need for the axiom of infinity\r\nbreaks down. I do not pretend to have explained the doctrine\r\nof types, or done more than indicate, in rough outline, why there\r\nis need of such a doctrine. I have aimed only at saying just\r\nso much as was required in order to show that we cannot \u003ci\u003eprove\u003c/i\u003e\r\nthe existence of infinite numbers and classes by such conjurer\u0027s\r\nmethods as we have been examining. There remain, however,\r\ncertain other possible methods which must be considered.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nVarious arguments professing to prove the existence of infinite\r\nclasses are given in the \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e, \u0026sect; 339 (p. 357).\r\n\u003cspan class=\"pagenum\" id=\"Page_137\"\u003e[Pg 137]\u003c/span\u003e\r\nIn so far as these arguments assume that, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive\r\ncardinal, \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is not equal to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e,\u003c/span\u003e they have been already dealt\r\nwith. There is an argument, suggested by a passage in Plato\u0027s\r\n\u003ci\u003eParmenides\u003c/i\u003e, to the effect that, if there is such a number as 1,\r\nthen 1 has being; but 1 is not identical with being, and therefore\r\n1 and being are two, and therefore there is such a number as 2,\r\nand 2 together with 1 and being gives a class of three terms, and\r\nso on. This argument is fallacious, partly because \"being\" is\r\nnot a term having any definite meaning, and still more because,\r\nif a definite meaning were invented for it, it would be found that\r\nnumbers do not have being\u0026mdash;they are, in fact, what are called\r\n\"logical fictions,\" as we shall see when we come to consider\r\nthe definition of classes.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe argument that the number of numbers from 0 to \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e (both\r\ninclusive) is \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e depends upon the assumption that up to and\r\nincluding \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e no number is equal to its successor, which, as we have\r\nseen, will not be always true if the axiom of infinity is false. It\r\nmust be understood that the equation \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 9.629ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-166.png\" alt=\"\" data-tex=\"n = n + 1\"\u003e,\u003c/span\u003e which might be\r\ntrue for a finite \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e exceeded the total number of individuals\r\nin the world, is quite different from the same equation as applied\r\nto a reflexive number. As applied to a reflexive number, it\r\nmeans that, given a class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms, this class is \"similar\" to\r\nthat obtained by adding another term. But as applied to a\r\nnumber which is too great for the actual world, it merely means\r\nthat there is no class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals, and no class of \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e individuals;\r\nit does not mean that, if we mount the hierarchy of\r\ntypes sufficiently far to secure the existence of a class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms,\r\nwe shall then find this class \"similar\" to one of \u003cimg style=\"vertical-align: -0.186ex; width: 5.254ex; height: 1.692ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-58.png\" alt=\"\" data-tex=\"n + 1\"\u003e terms, for\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is inductive this will not be the case, quite independently of\r\nthe truth or falsehood of the axiom of infinity.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is an argument employed by both Bolzano\u003ca id=\"FNanchor_30_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_30_1\" class=\"fnanchor\"\u003e[30]\u003c/a\u003e\r\nand Dedekind\u003ca id=\"FNanchor_31_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_31_1\" class=\"fnanchor\"\u003e[31]\u003c/a\u003e\r\nto prove the existence of reflexive classes. The argument,\r\nin brief, is this: An object is not identical with the idea of the\r\n\u003cspan class=\"pagenum\" id=\"Page_138\"\u003e[Pg 138]\u003c/span\u003e\r\nobject, but there is (at least in the realm of being) an idea of any\r\nobject. The relation of an object to the idea of it is one-one, and\r\nideas are only some among objects. Hence the relation \"idea\r\nof\" constitutes a reflexion of the whole class of objects into a\r\npart of itself, namely, into that part which consists of ideas.\r\nAccordingly, the class of objects and the class of ideas are both\r\ninfinite. This argument is interesting, not only on its own\r\naccount, but because the mistakes in it (or what I judge to be\r\nmistakes) are of a kind which it is instructive to note. The\r\nmain error consists in assuming that there is an idea of every\r\nobject. It is, of course, exceedingly difficult to decide what is\r\nmeant by an \"idea\"; but let us assume that we know. We are\r\nthen to suppose that, starting (say) with Socrates, there is the\r\nidea of Socrates, and so on \u003ci\u003ead inf\u003c/i\u003e. Now it is plain that this is not\r\nthe case in the sense that all these ideas have actual empirical\r\nexistence in people\u0027s minds. Beyond the third or fourth stage\r\nthey become mythical. If the argument is to be upheld, the\r\n\"ideas\" intended must be Platonic ideas laid up in heaven, for\r\ncertainly they are not on earth. But then it at once becomes\r\ndoubtful whether there are such ideas. If we are to know that\r\nthere are, it must be on the basis of some logical theory, proving\r\nthat it is necessary to a thing that there should be an idea of it.\r\nWe certainly cannot obtain this result empirically, or apply it,\r\nas Dedekind does, to \"meine Gedankenwelt\"\u0026mdash;the world of my\r\nthoughts.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_30_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_30_1\"\u003e\u003cspan class=\"label\"\u003e[30]\u003c/span\u003e\u003c/a\u003eBolzano, \u003ci\u003eParadoxien des Unendlichen\u003c/i\u003e, 13.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_31_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_31_1\"\u003e\u003cspan class=\"label\"\u003e[31]\u003c/span\u003e\u003c/a\u003eDedekind, \u003ci\u003eWas sind und was sollen die Zahlen?\u003c/i\u003e No. 66.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIf we were concerned to examine fully the relation of idea and\r\nobject, we should have to enter upon a number of psychological\r\nand logical inquiries, which are not relevant to our main purpose.\r\nBut a few further points should be noted. If \"idea\" is to be\r\nunderstood logically, it may be \u003ci\u003eidentical\u003c/i\u003e with the object, or it\r\nmay stand for a \u003ci\u003edescription\u003c/i\u003e (in the sense to be explained in a\r\nsubsequent chapter). In the former case the argument fails,\r\nbecause it was essential to the proof of reflexiveness that object\r\nand idea should be distinct. In the second case the argument\r\nalso fails, because the relation of object and description is not\r\n\u003cspan class=\"pagenum\" id=\"Page_139\"\u003e[Pg 139]\u003c/span\u003e\r\none-one: there are innumerable correct descriptions of any given\r\nobject. Socrates (\u003ci\u003ee.g.\u003c/i\u003e) may be described as \"the master of\r\nPlato,\" or as \"the philosopher who drank the hemlock,\" or as\r\n\"the husband of Xantippe.\" If\u0026mdash;to take up the remaining\r\nhypothesis\u0026mdash;\"idea\" is to be interpreted psychologically, it must\r\nbe maintained that there is not any one definite psychological\r\nentity which could be called \u003ci\u003ethe\u003c/i\u003e idea of the object: there are innumerable\r\nbeliefs and attitudes, each of which could be called \u003ci\u003ean\u003c/i\u003e\r\nidea of the object in the sense in which we might say \"my idea\r\nof Socrates is quite different from yours,\" but there is not any\r\ncentral entity (except Socrates himself) to bind together various\r\n\"ideas of Socrates,\" and thus there is not any such one-one relation\r\nof idea and object as the argument supposes. Nor, of course,\r\nas we have already noted, is it true psychologically that there are\r\nideas (in however extended a sense) of more than a tiny proportion\r\nof the things in the world. For all these reasons, the above\r\nargument in favour of the logical existence of reflexive classes\r\nmust be rejected.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt might be thought that, whatever may be said of \u003ci\u003elogical\u003c/i\u003e\r\narguments, the \u003ci\u003eempirical\u003c/i\u003e arguments derivable from space and\r\ntime, the diversity of colours, etc., are quite sufficient to prove\r\nthe actual existence of an infinite number of particulars. I do\r\nnot believe this. We have no reason except prejudice for believing\r\nin the infinite extent of space and time, at any rate in the sense\r\nin which space and time are physical facts, not mathematical\r\nfictions. We naturally regard space and time as continuous, or,\r\nat least, as compact; but this again is mainly prejudice. The\r\ntheory of \"quanta\" in physics, whether true or false, illustrates\r\nthe fact that physics can never afford proof of continuity, though\r\nit might quite possibly afford disproof. The senses are not\r\nsufficiently exact to distinguish between continuous motion and\r\nrapid discrete succession, as anyone may discover in a cinema.\r\nA world in which all motion consisted of a series of small finite\r\njerks would be empirically indistinguishable from one in which\r\nmotion was continuous. It would take up too much space to\r\n\u003cspan class=\"pagenum\" id=\"Page_140\"\u003e[Pg 140]\u003c/span\u003e\r\ndefend these theses adequately; for the present I am merely\r\nsuggesting them for the reader\u0027s consideration. If they are valid,\r\nit follows that there is no empirical reason for believing the\r\nnumber of particulars in the world to be infinite, and that there\r\nnever can be; also that there is at present no empirical reason\r\nto believe the number to be finite, though it is theoretically\r\nconceivable that some day there might be evidence pointing,\r\nthough not conclusively, in that direction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFrom the fact that the infinite is not self-contradictory, but is\r\nalso not demonstrable logically, we must conclude that nothing\r\ncan be known \u003ci\u003ea priori\u003c/i\u003e as to whether the number of things\r\nin the world is finite or infinite. The conclusion is, therefore,\r\nto adopt a Leibnizian phraseology, that some of the possible\r\nworlds are finite, some infinite, and we have no means of\r\nknowing to which of these two kinds our actual world belongs.\r\nThe axiom of infinity will be true in some possible worlds\r\nand false in others; whether it is true or false in this world,\r\nwe cannot tell.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThroughout this chapter the synonyms \"individual\" and\r\n\"particular\" have been used without explanation. It would be\r\nimpossible to explain them adequately without a longer disquisition\r\non the theory of types than would be appropriate to the\r\npresent work, but a few words before we leave this topic may\r\ndo something to diminish the obscurity which would otherwise\r\nenvelop the meaning of these words.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn an ordinary statement we can distinguish a verb, expressing\r\nan attribute or relation, from the substantives which express the\r\nsubject of the attribute or the terms of the relation. \"Cæsar\r\nlived\" ascribes an attribute to Cæsar; \"Brutus killed Cæsar\"\r\nexpresses a relation between Brutus and Cæsar. Using the word\r\n\"subject\" in a generalised sense, we may call both Brutus and\r\nCæsar subjects of this proposition: the fact that Brutus is grammatically\r\nsubject and Cæsar object is logically irrelevant, since\r\nthe same occurrence may be expressed in the words \"Cæsar was\r\nkilled by Brutus,\" where Cæsar is the grammatical subject.\r\n\u003cspan class=\"pagenum\" id=\"Page_141\"\u003e[Pg 141]\u003c/span\u003e\r\nThus in the simpler sort of proposition we shall have an attribute\r\nor relation holding of or between one, two or more \"subjects\"\r\nin the extended sense. (A relation may have more than two\r\nterms: \u003ci\u003ee.g.\u003c/i\u003e \"\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e gives \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e to \u003cimg style=\"vertical-align: -0.048ex; width: 1.633ex; height: 1.643ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-76.png\" alt=\"\" data-tex=\"\\mathrm C\"\u003e\"\r\nis a relation of \u003ci\u003ethree\u003c/i\u003e terms.) Now\r\nit often happens that, on a closer scrutiny, the apparent subjects\r\nare found to be not really subjects, but to be capable of analysis;\r\nthe only result of this, however, is that new subjects take their\r\nplaces. It also happens that the verb may grammatically be\r\nmade subject: \u003ci\u003ee.g.\u003c/i\u003e we may say, \"Killing is a relation which\r\nholds between Brutus and Cæsar.\" But in such cases the\r\ngrammar is misleading, and in a straightforward statement,\r\nfollowing the rules that should guide philosophical grammar,\r\nBrutus and Cæsar will appear as the subjects and killing\r\nas the verb.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe are thus led to the conception of terms which, when they\r\noccur in propositions, can \u003ci\u003eonly\u003c/i\u003e occur as subjects, and never in\r\nany other way. This is part of the old scholastic definition\r\nof \u003ci\u003esubstance\u003c/i\u003e; but persistence through time, which belonged to\r\nthat notion, forms no part of the notion with which we are concerned.\r\nWe shall define \"proper names\" as those terms which\r\ncan only occur as \u003ci\u003esubjects\u003c/i\u003e in propositions (using \"subject\"\r\nin the extended sense just explained). We shall further define\r\n\"individuals\" or \"particulars\" as the objects that can be\r\nnamed by proper names. (It would be better to define them\r\ndirectly, rather than by means of the kind of symbols by which\r\nthey are symbolised; but in order to do that we should have\r\nto plunge deeper into metaphysics than is desirable here.) It\r\nis, of course, possible that there is an endless regress: that\r\nwhatever appears as a particular is really, on closer scrutiny,\r\na class or some kind of complex. If this be the case, the axiom\r\nof infinity must of course be true. But if it be not the case,\r\nit must be theoretically possible for analysis to reach ultimate\r\nsubjects, and it is these that give the meaning of \"particulars\"\r\nor \"individuals.\" It is to the number of these that the axiom\r\nof infinity is assumed to apply. If it is true of them, it is true\r\n\u003cspan class=\"pagenum\" id=\"Page_142\"\u003e[Pg 142]\u003c/span\u003e\r\nof classes of them, and classes of classes of them, and so on;\r\nsimilarly if it is false of them, it is false throughout this hierarchy.\r\nHence it is natural to enunciate the axiom concerning them rather\r\nthan concerning any other stage in the hierarchy. But whether\r\nthe axiom is true or false, there seems no known method of\r\ndiscovering.\r\n\u003cspan class=\"pagenum\" id=\"Page_143\"\u003e[Pg 143]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XIV: INCOMPATIBILITY AND THE THEORY OF DEDUCTION\u0027\u003e\u003ca id=\"chap14\"\u003e\u003c/a\u003eCHAPTER XIV\r\n\u003cbr\u003e\u003cbr\u003e\r\nINCOMPATIBILITY AND THE THEORY OF DEDUCTION\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWE have now explored, somewhat hastily it is true, that part\r\nof the philosophy of mathematics which does not demand a\r\ncritical examination of the idea of \u003ci\u003eclass\u003c/i\u003e. In the preceding\r\nchapter, however, we found ourselves confronted by problems\r\nwhich make such an examination imperative. Before we can\r\nundertake it, we must consider certain other parts of the philosophy\r\nof mathematics, which we have hitherto ignored. In a\r\nsynthetic treatment, the parts which we shall now be concerned\r\nwith come first: they are more fundamental than anything\r\nthat we have discussed hitherto. Three topics will concern us\r\nbefore we reach the theory of classes, namely: (1) the theory\r\nof deduction, (2) propositional functions, (3) descriptions. Of\r\nthese, the third is not logically presupposed in the theory of\r\nclasses, but it is a simpler example of the \u003ci\u003ekind\u003c/i\u003e of theory that\r\nis needed in dealing with classes. It is the first topic, the theory\r\nof deduction, that will concern us in the present chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nMathematics is a deductive science: starting from certain\r\npremisses, it arrives, by a strict process of deduction, at the\r\nvarious theorems which constitute it. It is true that, in the past,\r\nmathematical deductions were often greatly lacking in rigour;\r\nit is true also that perfect rigour is a scarcely attainable ideal.\r\nNevertheless, in so far as rigour is lacking in a mathematical\r\nproof, the proof is defective; it is no defence to urge that common\r\nsense shows the result to be correct, for if we were to rely upon\r\nthat, it would be better to dispense with argument altogether,\r\n\u003cspan class=\"pagenum\" id=\"Page_144\"\u003e[Pg 144]\u003c/span\u003e\r\nrather than bring fallacy to the rescue of common sense. No\r\nappeal to common sense, or \"intuition,\" or anything except strict\r\ndeductive logic, ought to be needed in mathematics after the\r\npremisses have been laid down.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nKant, having observed that the geometers of his day could\r\nnot prove their theorems by unaided argument, but required\r\nan appeal to the figure, invented a theory of mathematical\r\nreasoning according to which the inference is never strictly\r\nlogical, but always requires the support of what is called\r\n\"intuition.\" The whole trend of modern mathematics, with\r\nits increased pursuit of rigour, has been against this Kantian\r\ntheory. The things in the mathematics of Kant\u0027s day which\r\ncannot be \u003ci\u003eproved\u003c/i\u003e, cannot be \u003ci\u003eknown\u003c/i\u003e\u0026mdash;for example, the axiom of\r\nparallels. What can be known, in mathematics and by mathematical\r\nmethods, is what can be deduced from pure logic. What\r\nelse is to belong to human knowledge must be ascertained otherwise\u0026mdash;empirically,\r\nthrough the senses or through experience in\r\nsome form, but not \u003ci\u003ea priori\u003c/i\u003e. The positive grounds for this\r\nthesis are to be found in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, \u003ci\u003epassim\u003c/i\u003e; a\r\ncontroversial defence of it is given in the \u003ci\u003ePrinciples of Mathematics\u003c/i\u003e.\r\nWe cannot here do more than refer the reader to those\r\nworks, since the subject is too vast for hasty treatment. Meanwhile,\r\nwe shall assume that all mathematics is deductive, and\r\nproceed to inquire as to what is involved in deduction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn deduction, we have one or more propositions called \u003ci\u003epremisses\u003c/i\u003e,\r\nfrom which we infer a proposition called the \u003ci\u003econclusion\u003c/i\u003e.\r\nFor our purposes, it will be convenient, when there are originally\r\nseveral premisses, to amalgamate them into a single proposition,\r\nso as to be able to speak of \u003ci\u003ethe\u003c/i\u003e premiss as well as of \u003ci\u003ethe\u003c/i\u003e conclusion.\r\nThus we may regard deduction as a process by which\r\nwe pass from knowledge of a certain proposition, the premiss,\r\nto knowledge of a certain other proposition, the conclusion.\r\nBut we shall not regard such a process as \u003ci\u003elogical\u003c/i\u003e deduction unless\r\nit is \u003ci\u003ecorrect\u003c/i\u003e, \u003ci\u003ei.e.\u003c/i\u003e unless there is such a relation between premiss\r\nand conclusion that we have a right to believe the conclusion\r\n\u003cspan class=\"pagenum\" id=\"Page_145\"\u003e[Pg 145]\u003c/span\u003e\r\nif we know the premiss to be true. It is this relation that is\r\nchiefly of interest in the logical theory of deduction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn order to be able validly to infer the truth of a proposition,\r\nwe must know that some other proposition is true, and that\r\nthere is between the two a relation of the sort called \"implication,\"\r\n\u003ci\u003ei.e.\u003c/i\u003e that (as we say) the premiss \"implies\" the conclusion. (We\r\nshall define this relation shortly.) Or we may know that a certain\r\nother proposition is false, and that there is a relation between\r\nthe two of the sort called \"disjunction,\" expressed by \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\"\u003ca id=\"FNanchor_32_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_32_1\" class=\"fnanchor\"\u003e[32]\u003c/a\u003e\r\nso that the knowledge that the one is false allows us to infer\r\nthat the other is true. Again, what we wish to infer may be\r\nthe \u003ci\u003efalsehood\u003c/i\u003e of some proposition, not its truth. This may be\r\ninferred from the truth of another proposition, provided we know\r\nthat the two are \"incompatible,\" \u003ci\u003ei.e.\u003c/i\u003e that if one is true, the other\r\nis false. It may also be inferred from the falsehood of another\r\nproposition, in just the same circumstances in which the truth\r\nof the other might have been inferred from the truth of the one;\r\n\u003ci\u003ei.e.\u003c/i\u003e from the falsehood of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e we may infer the falsehood of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e when\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e All these four are cases of inference. When our\r\nminds are fixed upon inference, it seems natural to take \"implication\"\r\nas the primitive fundamental relation, since this is the\r\nrelation which must hold between \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e if we are to be able\r\nto infer the \u003ci\u003etruth\u003c/i\u003e of \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e from the \u003ci\u003etruth\u003c/i\u003e of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e But for technical\r\nreasons this is not the best primitive idea to choose. Before\r\nproceeding to primitive ideas and definitions, let us consider\r\nfurther the various functions of propositions suggested by the\r\nabove-mentioned relations of propositions.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_32_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_32_1\"\u003e\u003cspan class=\"label\"\u003e[32]\u003c/span\u003e\u003c/a\u003eWe shall use the letters \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e to denote variable propositions.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe simplest of such functions is the negative, \"not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e\"\r\nThis is that function of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e which is true when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false, and false\r\nwhen \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true. It is convenient to speak of the truth of a proposition,\r\nor its falsehood, as its \"truth-value\"\u003ca id=\"FNanchor_33_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_33_1\" class=\"fnanchor\"\u003e[33]\u003c/a\u003e;\r\n\u003ci\u003ei.e.\u003c/i\u003e \u003ci\u003etruth\u003c/i\u003e is\r\nthe \"truth-value\" of a true proposition, and \u003ci\u003efalsehood\u003c/i\u003e of a false\r\none. Thus not \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e has the opposite truth-value to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_146\"\u003e[Pg 146]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_33_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_33_1\"\u003e\u003cspan class=\"label\"\u003e[33]\u003c/span\u003e\u003c/a\u003eThis term is due to Frege.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe may take next \u003ci\u003edisjunction\u003c/i\u003e, \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\" This is a function\r\nwhose truth-value is truth when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true and also when \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true,\r\nbut is falsehood when both \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e are false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nNext we may take \u003ci\u003econjunction\u003c/i\u003e, \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" This has truth\r\nfor its truth-value when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e are both true; otherwise it\r\nhas falsehood for its truth-value.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTake next \u003ci\u003eincompatibility\u003c/i\u003e, \u003ci\u003ei.e.\u003c/i\u003e \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e are not both true.\"\r\nThis is the negation of conjunction; it is also the disjunction\r\nof the negations of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e it is \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\" Its truth-value\r\nis truth when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false and likewise when \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is false; its\r\ntruth-value is falsehood when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e are both true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLast take \u003ci\u003eimplication\u003c/i\u003e, \u003ci\u003ei.e.\u003c/i\u003e \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\" or \"if \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e then \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\"\r\nThis is to be understood in the widest sense that will allow us\r\nto infer the truth of \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e if we know the truth of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e . Thus we interpret\r\nit as meaning: \"Unless \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false, \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true,\" or \"either\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true.\" (The fact that \"implies\" is capable\r\nof other meanings does not concern us; this is the meaning which\r\nis convenient for us.) That is to say, \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is to mean\r\n\"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\": its truth-value is to be truth if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false, likewise\r\nif \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true, and is to be falsehood if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have thus five functions: negation, disjunction, conjunction,\r\nincompatibility, and implication. We might have added others,\r\nfor example, joint falsehood, \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\" but the above\r\nfive will suffice. Negation differs from the other four in being\r\na function of \u003ci\u003eone\u003c/i\u003e proposition, whereas the others are functions\r\nof \u003ci\u003etwo\u003c/i\u003e. But all five agree in this, that their truth-value depends\r\nonly upon that of the propositions which are their arguments.\r\nGiven the truth or falsehood of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e or of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e (as the case may\r\nbe), we are given the truth or falsehood of the negation, disjunction,\r\nconjunction, incompatibility, or implication. A function of\r\npropositions which has this property is called a \"truth-function.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe whole meaning of a truth-function is exhausted by the\r\nstatement of the circumstances under which it is true or false.\r\n\"Not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e\" for example, is simply that function of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e which is true\r\nwhen \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false, and false when \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true: there is no further\r\n\u003cspan class=\"pagenum\" id=\"Page_147\"\u003e[Pg 147]\u003c/span\u003e\r\nmeaning to be assigned to it. The same applies to \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"\r\nand the rest. It follows that two truth-functions which have\r\nthe same truth-value for all values of the argument are indistinguishable.\r\nFor example, \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is the negation of\r\n\"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" and \u003ci\u003evice versa\u003c/i\u003e; thus either of these may be\r\n\u003ci\u003edefined\u003c/i\u003e as the negation of the other. There is no further meaning\r\nin a truth-function over and above the conditions under which\r\nit is true or false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that the above five truth-functions are not all independent.\r\nWe can define some of them in terms of others. There\r\nis no great difficulty in reducing the number to two; the two\r\nchosen in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e are negation and disjunction.\r\nImplication is then defined as \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"; incompatibility\r\nas \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"; conjunction as the negation of incompatibility.\r\nBut it has been shown by Sheffer\u003ca id=\"FNanchor_34_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_34_1\" class=\"fnanchor\"\u003e[34]\u003c/a\u003e\r\nthat we can be content\r\nwith \u003ci\u003eone\u003c/i\u003e primitive idea for all five, and by Nicod\u003ca id=\"FNanchor_35_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_35_1\" class=\"fnanchor\"\u003e[35]\u003c/a\u003e\r\nthat this enables\r\nus to reduce the primitive propositions required in the theory\r\nof deduction to two non-formal principles and one formal one.\r\nFor this purpose, we may take as our one indefinable either\r\nincompatibility or joint falsehood. We will choose the former.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_34_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_34_1\"\u003e\u003cspan class=\"label\"\u003e[34]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eTrans. Am. Math. Soc.\u003c/i\u003e, vol. XIV. pp. 481-488.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_35_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_35_1\"\u003e\u003cspan class=\"label\"\u003e[35]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eProc. Camb. Phil. Soc.\u003c/i\u003e, vol. XIX., i., January 1917.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nOur primitive idea, now, is a certain truth-function called\r\n\"incompatibility,\" which we will denote by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.31ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-120.png\" alt=\"\" data-tex=\"p/q\"\u003e.\u003c/span\u003e Negation\r\ncan be at once defined as the incompatibility of a proposition\r\nwith itself, \u003ci\u003ei.e.\u003c/i\u003e \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e\" is defined as \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.407ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-281.png\" alt=\"\" data-tex=\"p/p\"\u003e.\u003c/span\u003e\" Disjunction is\r\nthe incompatibility of not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e it is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 10.769ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-282.png\" alt=\"\" data-tex=\"(p/p) | (q/q)\"\u003e.\u003c/span\u003e\r\nImplication is the incompatibility of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-283.png\" alt=\"\" data-tex=\"p | (q/q)\"\u003e.\u003c/span\u003e\r\nConjunction is the negation of incompatibility, \u003ci\u003ei.e.\u003c/i\u003e it is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 10.769ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-284.png\" alt=\"\" data-tex=\"(p/q) | (p/q)\"\u003e.\u003c/span\u003e\r\nThus all our four other functions are defined in terms\r\nof incompatibility.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is obvious that there is no limit to the manufacture of truth-functions,\r\neither by introducing more arguments or by repeating\r\narguments. What we are concerned with is the connection of\r\nthis subject with inference.\r\n\u003cspan class=\"pagenum\" id=\"Page_148\"\u003e[Pg 148]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf we know that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true and that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e we can proceed\r\nto assert \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e There is always unavoidably \u003ci\u003esomething\u003c/i\u003e psychological\r\nabout inference: inference is a method by which we arrive\r\nat new knowledge, and what is not psychological about it is the\r\nrelation which allows us to infer correctly; but the actual passage\r\nfrom the assertion of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e to the assertion of \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is a psychological\r\nprocess, and we must not seek to represent it in purely logical\r\nterms.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn mathematical practice, when we infer, we have always\r\nsome expression containing variable propositions, say \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\r\nwhich is known, in virtue of its form, to be true for all values\r\nof \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e;\u003c/span\u003e we have also some other expression, part of the former,\r\nwhich is also known to be true for all values of \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e;\u003c/span\u003e and in\r\nvirtue of the principles of inference, we are able to drop this part\r\nof our original expression, and assert what is left. This somewhat\r\nabstract account may be made clearer by a few examples.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us assume that we know the five formal principles of\r\ndeduction enumerated in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e. (M. Nicod has\r\nreduced these to one, but as it is a complicated proposition,\r\nwe will begin with the five.) These five propositions are as\r\nfollows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e\" implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e if either \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true\r\nor \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true, then \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e the disjunction \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"\r\nis true when one of its alternatives is true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" implies \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e.\u003c/span\u003e\" This would not be required\r\nif we had a theoretically more perfect notation, since in the\r\nconception of disjunction there is no order involved, so that\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" and \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e\" should be identical. But since our\r\nsymbols, in any convenient form, inevitably introduce an order,\r\nwe need suitable assumptions for showing that the order is\r\nirrelevant.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(4) If either \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true or \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e\" is true, then either \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true\r\nor \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e\" is true. (The twist in this proposition serves to\r\nincrease its deductive power.)\r\n\u003cspan class=\"pagenum\" id=\"Page_149\"\u003e[Pg 149]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(5) If \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e then \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" implies \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese are the \u003ci\u003eformal\u003c/i\u003e principles of deduction employed in\r\n\u003ci\u003ePrincipia Mathematica\u003c/i\u003e. A formal principle of deduction has a\r\ndouble use, and it is in order to make this clear that we have\r\ncited the above five propositions. It has a use as the premiss\r\nof an inference, and a use as establishing the fact that the premiss\r\nimplies the conclusion. In the schema of an inference\r\nwe have a proposition \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e and a proposition \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\" from\r\nwhich we infer \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e Now when we are concerned with the principles\r\nof deduction, our apparatus of primitive propositions has\r\nto yield both the \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and the \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" of our inferences.\r\nThat is to say, our rules of deduction are to be used, not \u003ci\u003eonly\u003c/i\u003e as\r\n\u003ci\u003erules\u003c/i\u003e, which is their use for establishing \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" but \u003ci\u003ealso\u003c/i\u003e\r\nas substantive premisses, \u003ci\u003ei.e.\u003c/i\u003e as the \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e of our schema. Suppose,\r\nfor example, we wish to prove that if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e then if \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e\r\nit follows that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e We have here a relation of\r\nthree propositions which state implications. Put\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 6.281ex; height: 1.758ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-285.png\" alt=\"\" data-tex=\"p_{1} = p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 6.184ex; height: 1.758ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-286.png\" alt=\"\" data-tex=\"p_{2} = q\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e and\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 6.281ex; height: 1.758ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-287.png\" alt=\"\" data-tex=\"p_{3} = p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nThen we have to prove that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e.\u003c/span\u003e Now\r\ntake the fifth of our above principles, substitute not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e for \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e\r\nand remember that \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is by definition the same as\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\" Thus our fifth principle yields:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"If \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e then \u0027\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\u0027\u003c/span\u003e implies \u0027\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003eimplies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e\u0027\"\r\n\u003ci\u003ei.e.\u003c/i\u003e \"\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003eimplies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e.\u003c/span\u003e\" Call this\r\nproposition \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nBut the fourth of our principles, when we substitute not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e\r\nnot-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e for \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e and remember the definition of implication,\r\nbecomes:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"If \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies\r\nthat \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWriting \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e in place of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e in place of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e and\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e in place of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e this\r\nbecomes:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"If \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e,\u003c/span\u003e then\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e.\u003c/span\u003e\"\r\nCall this \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_150\"\u003e[Pg 150]\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nNow we proved by means of our fifth principle that\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e,\u003c/span\u003e\" which was\r\nwhat we called \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nThus we have here an instance of the schema of inference,\r\nsince \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e represents the \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e of our scheme, and \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e represents the\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\" Hence we arrive at \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e namely,\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-288.png\" alt=\"\" data-tex=\"p_{1}\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-289.png\" alt=\"\" data-tex=\"p_{2}\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 2.126ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-290.png\" alt=\"\" data-tex=\"p_{3}\"\u003e,\u003c/span\u003e\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nwhich was the proposition to be proved. In this proof, the\r\nadaptation of our fifth principle, which yields \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e occurs as a\r\nsubstantive premiss; while the adaptation of our fourth principle,\r\nwhich yields \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e,\u003c/span\u003e is used to give the \u003ci\u003eform\u003c/i\u003e of the inference. The\r\nformal and material employments of premisses in the theory\r\nof deduction are closely intertwined, and it is not very important\r\nto keep them separated, provided we realise that they are in\r\ntheory distinct.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe earliest method of arriving at new results from a premiss\r\nis one which is illustrated in the above deduction, but which\r\nitself can hardly be called deduction. The primitive propositions,\r\nwhatever they may be, are to be regarded as asserted for all\r\npossible values of the variable propositions \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e which occur\r\nin them. We may therefore substitute for (say) \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e any expression\r\nwhose value is always a proposition, \u003ci\u003ee.g.\u003c/i\u003e not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e\r\n\"\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e,\u003c/span\u003e\"\r\nand so on. By means of such substitutions we really obtain\r\nsets of special cases of our original proposition, but from a practical\r\npoint of view we obtain what are virtually new propositions.\r\nThe legitimacy of substitutions of this kind has to be insured by\r\nmeans of a non-formal principle of inference.\u003ca id=\"FNanchor_36_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_36_1\" class=\"fnanchor\"\u003e[36]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_36_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_36_1\"\u003e\u003cspan class=\"label\"\u003e[36]\u003c/span\u003e\u003c/a\u003eNo such principle is enunciated in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, or in M. Nicod\u0027s\r\narticle mentioned above. But this would seem to be an omission.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe may now state the one formal principle of inference to\r\nwhich M. Nicod has reduced the five given above. For this\r\npurpose we will first show how certain truth-functions can be\r\ndefined in terms of incompatibility. We saw already that\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-283.png\" alt=\"\" data-tex=\"p | (q/q)\"\u003e means \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e\"\r\n\u003cspan class=\"pagenum\" id=\"Page_151\"\u003e[Pg 151]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nWe now observe that\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 6.719ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-291.png\" alt=\"\" data-tex=\"p | (q/r)\"\u003e means \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies both \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e\"\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nFor this expression means \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is incompatible with the incompatibility\r\nof \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e\" \u003ci\u003ei.e.\u003c/i\u003e \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e are not incompatible,\"\r\n\u003ci\u003ei.e.\u003c/i\u003e \"\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies that \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e are both true\"\u0026mdash;for, as\r\nwe saw, the conjunction of \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e is the negation of their\r\nincompatibility.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nObserve next that \u003cimg style=\"vertical-align: -0.566ex; width: 5.971ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-292.png\" alt=\"\" data-tex=\"t | (t/t)\"\u003e means \"\u003cimg style=\"vertical-align: -0.025ex; width: 0.817ex; height: 1.441ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-202.png\" alt=\"\" data-tex=\"t\"\u003e implies itself.\" This is a\r\nparticular case of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 6.74ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-283.png\" alt=\"\" data-tex=\"p | (q/q)\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us write \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 2.156ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-293.png\" alt=\"\" data-tex=\"\\overline{p}\"\u003e for the negation of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e;\u003c/span\u003e thus\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 3.33ex; height: 2.98ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-294.png\" alt=\"\" data-tex=\"\\overline{p/s}\"\u003e will mean the\r\nnegation of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 3.33ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-295.png\" alt=\"\" data-tex=\"p/s\"\u003e,\u003c/span\u003e \u003ci\u003ei.e.\u003c/i\u003e it will mean the conjunction of\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e.\u003c/span\u003e It follows that\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 8.952ex; height: 2.98ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-43.png\" alt=\"\" data-tex=\"\r\n(s/q) | \\overline{p/s}\r\n\"\u003e\u003c/span\u003e\r\nexpresses the incompatibility of \u003cimg style=\"vertical-align: -0.566ex; width: 3.233ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-296.png\" alt=\"\" data-tex=\"s/q\"\u003e with the conjunction of\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e;\u003c/span\u003e in other words, it states that if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e are both true,\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 3.233ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-296.png\" alt=\"\" data-tex=\"s/q\"\u003e is false, \u003ci\u003ei.e.\u003c/i\u003e \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e are both true; in still simpler words,\r\nit states that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e jointly imply \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e jointly.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nNow, put\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -3.865ex; width: 14.388ex; height: 8.862ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-44.png\" alt=\"\" data-tex=\"\r\n\\begin{align*}\r\n P \u0026= p | (q/r), \\\\\r\n \\pi \u0026= t | (t/t), \\\\\r\n Q \u0026= (s/q) | \\overline{p/s}.\r\n\\end{align*}\r\n\"\u003e\u003c/span\u003e\r\nThen M. Nicod\u0027s sole formal principle of deduction is\r\n\u003cspan class=\"align-center\"\u003e\u003cimg style=\"vertical-align: -0.566ex; width: 7.167ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-45.png\" alt=\"\" data-tex=\"\r\nP | \\pi/Q,\r\n\"\u003e\u003c/span\u003e\r\nin other words, \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e implies both \u003cimg style=\"vertical-align: -0.025ex; width: 1.29ex; height: 1ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-297.png\" alt=\"\" data-tex=\"\\pi\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.437ex; width: 1.76ex; height: 2.032ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-95.png\" alt=\"\" data-tex=\"\\mathrm Q\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nHe employs in addition one non-formal principle belonging\r\nto the theory of types (which need not concern us), and one\r\ncorresponding to the principle that, given \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e and given that\r\n\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e we can assert \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e This principle is:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\n\"If \u003cimg style=\"vertical-align: -0.566ex; width: 6.719ex; height: 2.262ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-298.png\" alt=\"\" data-tex=\"p | (r/q)\"\u003e is true, and \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is true, then \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is true.\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nFrom this apparatus the whole theory of deduction follows, except\r\nin so far as we are concerned with deduction from or to the\r\nexistence or the universal truth of \"propositional functions,\"\r\nwhich we shall consider in the next chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere is? if I am not mistaken, a certain confusion in the\r\n\u003cspan class=\"pagenum\" id=\"Page_152\"\u003e[Pg 152]\u003c/span\u003e\r\nminds of some authors as to the relation, between propositions,\r\nin virtue of which an inference is valid. In order that it may\r\nbe \u003ci\u003evalid\u003c/i\u003e to infer \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e it is only necessary that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e should be\r\ntrue and that the proposition \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" should be true.\r\nWhenever this is the case, it is clear that \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e must be true. But\r\ninference will only in fact take place when the proposition \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e\r\nor \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is \u003ci\u003eknown\u003c/i\u003e otherwise than through knowledge of not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or\r\nknowledge of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e Whenever \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e is false, \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is true,\r\nbut is useless for inference, which requires that \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e should be true.\r\nWhenever \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is already known to be true, \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" is of\r\ncourse also known to be true, but is again useless for inference,\r\nsince \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is already known, and therefore does not need to be\r\ninferred. In fact, inference only arises when \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\"\r\ncan be known without our knowing already which of the two\r\nalternatives it is that makes the disjunction true. Now, the\r\ncircumstances under which this occurs are those in which certain\r\nrelations of form exist between \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e For example, we know\r\nthat if \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e implies the negation of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e implies the negation\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e.\u003c/span\u003e Between \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e implies not-\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e\" and \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.061ex; height: 1.023ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-165.png\" alt=\"\" data-tex=\"s\"\u003e implies not-\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e\" there\r\nis a formal relation which enables us to \u003ci\u003eknow\u003c/i\u003e that the first implies\r\nthe second, without having first to know that the first is false\r\nor to know that the second is true. It is under such circumstances\r\nthat the relation of implication is practically useful for\r\ndrawing inferences.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut this formal relation is only required in order that we may\r\nbe able to \u003ci\u003eknow\u003c/i\u003e that either the premiss is false or the conclusion\r\nis true. It is the truth of \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" that is required for\r\nthe \u003ci\u003evalidity\u003c/i\u003e of the inference; what is required further is only\r\nrequired for the practical feasibility of the inference. Professor\r\nC. I. Lewis\u003ca id=\"FNanchor_37_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_37_1\" class=\"fnanchor\"\u003e[37]\u003c/a\u003e\r\nhas especially studied the narrower, formal relation\r\nwhich we may call \"formal deducibility.\" He urges that the\r\nwider relation, that expressed by \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" should not be\r\ncalled \"implication.\" That is, however, a matter of words.\r\n\u003cspan class=\"pagenum\" id=\"Page_153\"\u003e[Pg 153]\u003c/span\u003e\r\nProvided our use of words is consistent, it matters little how we\r\ndefine them. The essential point of difference between the\r\ntheory which I advocate and the theory advocated by Professor\r\nLewis is this: He maintains that, when one proposition \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e is\r\n\"formally deducible\" from another \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e the relation which we\r\nperceive between them is one which he calls \"strict implication,\"\r\nwhich is not the relation expressed by \"not-\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\" but a narrower\r\nrelation, holding only when there are certain formal connections\r\nbetween \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e.\u003c/span\u003e I maintain that, whether or not there be\r\nsuch a relation as he speaks of, it is in any case one that mathematics\r\ndoes not need, and therefore one that, on general grounds\r\nof economy, ought not to be admitted into our apparatus of\r\nfundamental notions; that, whenever the relation of \"formal\r\ndeducibility\" holds between two propositions, it is the case that\r\nwe can see that either the first is false or the second true, and that\r\nnothing beyond this fact is necessary to be admitted into our\r\npremisses; and that, finally, the reasons of detail which Professor\r\nLewis adduces against the view which I advocate can all be met\r\nin detail, and depend for their plausibility upon a covert and\r\nunconscious assumption of the point of view which I reject.\r\nI conclude, therefore, that there is no need to admit as a fundamental\r\nnotion any form of implication not expressible as a\r\ntruth-function.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_37_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_37_1\"\u003e\u003cspan class=\"label\"\u003e[37]\u003c/span\u003e\u003c/a\u003eSee \u003ci\u003eMind\u003c/i\u003e, vol. XXI., 1912, pp. 522-531; and vol. XXIII., 1914, pp. 240-247.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_154\"\u003e[Pg 154]\u003c/span\u003e\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XV: PROPOSITIONAL FUNCTIONS\u0027\u003e\u003ca id=\"chap15\"\u003e\u003c/a\u003eCHAPTER XV\r\n\u003cbr\u003e\u003cbr\u003e\r\nPROPOSITIONAL FUNCTIONS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWHEN, in the preceding chapter, we were discussing propositions,\r\nwe did not attempt to give a definition of the word \"proposition.\"\r\nBut although the word cannot be formally defined, it is necessary\r\nto say something as to its meaning, in order to avoid the very\r\ncommon confusion with \"propositional functions,\" which are to\r\nbe the topic of the present chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe mean by a \"proposition\" primarily a form of words which\r\nexpresses what is either true or false. I say \"primarily,\"\r\nbecause I do not wish to exclude other than verbal symbols, or\r\neven mere thoughts if they have a symbolic character. But I\r\nthink the word \"proposition\" should be limited to what may,\r\nin some sense, be called \"symbols,\" and further to such symbols\r\nas give expression to truth and falsehood. Thus \"two and two\r\nare four\" and \"two and two are five\" will be propositions,\r\nand so will \"Socrates is a man\" and \"Socrates is not a man.\"\r\nThe statement: \"Whatever numbers \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e may be,\r\n\u003cimg style=\"vertical-align: -0.566ex; width: 23.671ex; height: 2.452ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-299.png\" alt=\"\" data-tex=\"(a + b)^{2} = a^{2} + 2ab + b^{2}\"\u003e\"\r\nis a proposition; but the bare formula \"\u003cimg style=\"vertical-align: -0.566ex; width: 23.671ex; height: 2.452ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-299.png\" alt=\"\" data-tex=\"(a + b)^{2} = a^{2} + 2ab + b^{2}\"\u003e\"\r\nalone is not, since it asserts nothing definite unless\r\nwe are further told, or led to suppose, that \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 0.971ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-86.png\" alt=\"\" data-tex=\"b\"\u003e are to have\r\nall possible values, or are to have such-and-such values. The\r\nformer of these is tacitly assumed, as a rule, in the enunciation\r\nof mathematical formulæ, which thus become propositions;\r\nbut if no such assumption were made, they would be \"propositional\r\nfunctions.\" A \"propositional function,\" in fact, is an\r\nexpression containing one or more undetermined constituents,\r\n\u003cspan class=\"pagenum\" id=\"Page_155\"\u003e[Pg 155]\u003c/span\u003e\r\nsuch that, when values are assigned to these constituents, the\r\nexpression becomes a proposition. In other words, it is a function\r\nwhose values are propositions. But this latter definition must\r\nbe used with caution. A descriptive function, \u003ci\u003ee.g.\u003c/i\u003e \"the hardest\r\nproposition in \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e\u0027\u003c/span\u003es mathematical treatise,\" will not be a propositional\r\nfunction, although its values are propositions. But in\r\nsuch a case the propositions are only described: in a propositional\r\nfunction, the values must actually \u003ci\u003eenunciate\u003c/i\u003e propositions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nExamples of propositional functions are easy to give: \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is\r\nhuman\" is a propositional function; so long as \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e remains\r\nundetermined, it is neither true nor false, but when a value\r\nis assigned to \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e it becomes a true or false proposition. Any\r\nmathematical equation is a propositional function. So long as\r\nthe variables have no definite value, the equation is merely an\r\nexpression awaiting determination in order to become a true or\r\nfalse proposition. If it is an equation containing one variable,\r\nit becomes true when the variable is made equal to a root\r\nof the equation, otherwise it becomes false; but if it is an\r\n\"identity\" it will be true when the variable is any number.\r\nThe equation to a curve in a plane or to a surface in space is a\r\npropositional function, true for values of the co-ordinates belonging\r\nto points on the curve or surface, false for other values.\r\nExpressions of traditional logic such as \"all \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e\" are propositional\r\nfunctions: \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e and \u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e have to be determined as definite\r\nclasses before such expressions become true or false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe notion of \"cases\" or \"instances\" depends upon propositional\r\nfunctions. Consider, for example, the kind of process\r\nsuggested by what is called \"generalisation,\" and let us take\r\nsome very primitive example, say, \"lightning is followed by\r\nthunder.\" We have a number of \"instances\" of this, \u003ci\u003ei.e.\u003c/i\u003e a\r\nnumber of propositions such as: \"this is a flash of lightning\r\nand is followed by thunder.\" What are these occurrences\r\n\"instances\" of? They are instances of the propositional\r\nfunction: \"If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a flash of lightning, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is followed by thunder.\"\r\nThe process of generalisation (with whose validity we are fortunately\r\n\u003cspan class=\"pagenum\" id=\"Page_156\"\u003e[Pg 156]\u003c/span\u003e\r\nnot concerned) consists in passing from a number of such\r\ninstances to the \u003ci\u003euniversal\u003c/i\u003e truth of the propositional function:\r\n\"If \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a flash of lightning, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is followed by thunder.\" It will\r\nbe found that, in an analogous way, propositional functions\r\nare always involved whenever we talk of instances or cases or\r\nexamples.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe do not need to ask, or attempt to answer, the question:\r\n\"What \u003ci\u003eis\u003c/i\u003e a propositional function?\" A propositional function\r\nstanding all alone may be taken to be a mere schema, a mere\r\nshell, an empty receptacle for meaning, not something already\r\nsignificant. We are concerned with propositional functions,\r\nbroadly speaking, in two ways: first, as involved in the notions\r\n\"true in all cases\" and \"true in some cases\"; secondly, as\r\ninvolved in the theory of classes and relations. The second of\r\nthese topics we will postpone to a later chapter; the first must\r\noccupy us now.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen we say that something is \"always true\" or \"true in\r\nall cases,\" it is clear that the \"something\" involved cannot be\r\na proposition. A proposition is just true or false, and there\r\nis an end of the matter. There are no instances or cases of\r\n\"Socrates is a man\" or \"Napoleon died at St Helena.\" These\r\nare propositions, and it would be meaningless to speak of their\r\nbeing true \"in all cases.\" This phrase is only applicable to\r\npropositional \u003ci\u003efunctions\u003c/i\u003e. Take, for example, the sort of thing\r\nthat is often said when causation is being discussed. (We are\r\nnet concerned with the truth or falsehood of what is said, but\r\nonly with its logical analysis.) We are told that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e is, in every\r\ninstance, followed by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e.\u003c/span\u003e Now if there are \"instances\" of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e must be some general concept of which it is significant to say\r\n\"\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e\" \"\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e\" \"\u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e\"\r\nand so on, where \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-235.png\" alt=\"\" data-tex=\"x_{1}\"\u003e,\u003c/span\u003e \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.339ex; width: 2.282ex; height: 1.339ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-236.png\" alt=\"\" data-tex=\"x_{2}\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.375ex; width: 2.282ex; height: 1.375ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-237.png\" alt=\"\" data-tex=\"x_{3}\"\u003e are\r\nparticulars which are not identical one with another. This\r\napplies, \u003ci\u003ee.g.\u003c/i\u003e, to our previous case of lightning. We say that\r\nlightning (\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e)\u003c/span\u003e is followed by thunder (\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e)\u003c/span\u003e. But the separate\r\nflashes are particulars, not identical, but sharing the common\r\nproperty of being lightning. The only way of expressing a\r\n\u003cspan class=\"pagenum\" id=\"Page_157\"\u003e[Pg 157]\u003c/span\u003e\r\ncommon property generally is to say that a common property\r\nof a number of objects is a propositional function which becomes\r\ntrue when any one of these objects is taken as the value of the\r\nvariable. In this case all the objects are \"instances\" of the\r\ntruth of the propositional function\u0026mdash;for a propositional function,\r\nthough it cannot itself be true or false, is true in certain instances\r\nand false in certain others, unless it is \"always true\" or \"always\r\nfalse.\" When, to return to our example, we say that \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e is in\r\nevery instance followed by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e,\u003c/span\u003e we mean that, whatever \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e may be,\r\nif \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e,\u003c/span\u003e it is followed by a \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.602ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-75.png\" alt=\"\" data-tex=\"\\mathrm B\"\u003e;\u003c/span\u003e that is,\r\nwe are asserting that\r\na certain propositional function is \"always true.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSentences involving such words as \"all,\" \"every,\" \"a,\"\r\n\"the,\" \"some\" require propositional functions for their interpretation.\r\nThe way in which propositional functions occur\r\ncan be explained by means of two of the above words, namely,\r\n\"all\" and \"some.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere are, in the last analysis, only two things that can be\r\ndone with a propositional function: one is to assert that it is\r\ntrue in \u003ci\u003eall\u003c/i\u003e cases, the other to assert that it is true in at least one\r\ncase, or in \u003ci\u003esome\u003c/i\u003e cases (as we shall say, assuming that there is\r\nto be no necessary implication of a plurality of cases). All the\r\nother uses of propositional functions can be reduced to these two.\r\nWhen we say that a propositional function is true \"in all cases,\"\r\nor \"always\" (as we shall also say, without any temporal suggestion),\r\nwe mean that all its values are true. If \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\" is the\r\nfunction, and \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is the right sort of object to be an argument to \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e\"\r\nthen \u003cimg style=\"vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-301.png\" alt=\"\" data-tex=\"\\phi a\"\u003e is to be true, however \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e may have been chosen.\r\nFor example, \"if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is human, \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is mortal\" is true whether \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is\r\nhuman or not; in fact, every proposition of this form is true.\r\nThus the propositional function \"if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is mortal\"\r\nis \"always true,\" or \"true in all cases.\" Or, again, the statement\r\n\"there are no unicorns\" is the same as the statement\r\n\"the propositional function \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not a unicorn\u0027 is true in all\r\ncases.\" The assertions in the preceding chapter about propositions,\r\n\u003ci\u003ee.g.\u003c/i\u003e \"\u0027\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\u0027\u003c/span\u003e implies \u0027\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e,\u003c/span\u003e\u0027\" are really assertions\r\n\u003cspan class=\"pagenum\" id=\"Page_158\"\u003e[Pg 158]\u003c/span\u003e\r\nthat certain propositional functions are true in all cases. We do\r\nnot assert the above principle, for example, as being true only\r\nof this or that particular \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e,\u003c/span\u003e but as being true of\r\n\u003ci\u003eany\u003c/i\u003e \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e or \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e\r\nconcerning which it can be made significantly. The condition\r\nthat a function is to be \u003ci\u003esignificant\u003c/i\u003e for a given argument is the same\r\nas the condition that it shall have a value for that argument,\r\neither true or false. The study of the conditions of significance\r\nbelongs to the doctrine of types, which we shall not pursue\r\nbeyond the sketch given in the preceding chapter.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nNot only the principles of deduction, but all the primitive\r\npropositions of logic, consist of assertions that certain propositional\r\nfunctions are always true. If this were not the case, they\r\nwould have to mention particular things or concepts\u0026mdash;Socrates,\r\nor redness, or east and west, or what not,\u0026mdash;and clearly it is not\r\nthe province of logic to make assertions which are true concerning\r\none such thing or concept but not concerning another. It is\r\npart of the definition of logic (but not the whole of its definition)\r\nthat all its propositions are completely general, \u003ci\u003ei.e.\u003c/i\u003e they all\r\nconsist of the assertion that some propositional function containing\r\nno constant terms is always true. We shall return in\r\nour final chapter to the discussion of propositional functions\r\ncontaining no constant terms. For the present we will proceed\r\nto the other thing that is to be done with a propositional function,\r\nnamely, the assertion that it is \"sometimes true,\" \u003ci\u003ei.e.\u003c/i\u003e true in at\r\nleast one instance.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen we say \"there are men,\" that means that the propositional\r\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a man\" is sometimes true. When we\r\nsay \"some men are Greeks,\" that means that the propositional\r\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a man and a Greek\" is sometimes true. When we\r\nsay \"cannibals still exist in Africa,\" that means that the propositional\r\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a cannibal now in Africa\" is sometimes\r\ntrue, \u003ci\u003ei.e.\u003c/i\u003e is true for some values of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e To say \"there are at least\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals in the world\" is to say that the propositional\r\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a class of individuals and a member of the cardinal\r\nnumber \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e\" is sometimes true, or, as we may say, is true for certain\r\n\u003cspan class=\"pagenum\" id=\"Page_159\"\u003e[Pg 159]\u003c/span\u003e\r\nvalues of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e.\u003c/span\u003e This form of expression is more convenient when it\r\nis necessary to indicate which is the variable constituent which\r\nwe are taking as the argument to our propositional function.\r\nFor example, the above propositional function, which we may\r\nshorten to \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals,\" contains two variables,\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e The axiom of infinity, in the language of propositional\r\nfunctions, is: \"The propositional function \u0027if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive\r\nnumber, it is true for some values of \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e that \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a\r\nclass of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals\u0027 is true for all possible values of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e.\u003c/span\u003e\"\r\nHere there is a\r\nsubordinate function, \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is a class of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals,\" which is\r\nsaid to be, in respect of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e \u003ci\u003esometimes\u003c/i\u003e true; and the assertion\r\nthat this happens if \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is an inductive number is said to be, in\r\nrespect of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e \u003ci\u003ealways\u003c/i\u003e true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe statement that a function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always true is the negation\r\nof the statement that not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true, and the statement\r\nthat \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true is the negation of the statement\r\nthat not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always true. Thus the statement \"all\r\nmen are mortals\" is the negation of the statement that the\r\nfunction \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an immortal man\" is sometimes true. And the\r\nstatement \"there are unicorns\" is the negation of the statement\r\nthat the function \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not a unicorn\" is always true.\u003ca id=\"FNanchor_38_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_38_1\" class=\"fnanchor\"\u003e[38]\u003c/a\u003e\r\nWe say that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is \"never true\" or \"always false\" if not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is\r\nalways true. We can, if we choose, take one of the pair \"always,\"\r\n\"sometimes\" as a primitive idea, and define the other by means\r\nof the one and negation. Thus if we choose \"sometimes\" as\r\nour primitive idea, we can define: \"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always true\u0027 is to\r\nmean \u0027it is false that not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true.\u0027\"\u003ca id=\"FNanchor_39_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_39_1\" class=\"fnanchor\"\u003e[39]\u003c/a\u003e\r\nBut for\r\nreasons connected with the theory of types it seems more correct\r\nto take both \"always\" and \"sometimes\" as primitive ideas,\r\nand define by their means the negation of propositions in which\r\nthey occur. That is to say, assuming that we have already\r\n\u003cspan class=\"pagenum\" id=\"Page_160\"\u003e[Pg 160]\u003c/span\u003e\r\ndefined (or adopted as a primitive idea) the negation of propositions\r\nof the type to which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e belongs, we define: \"The\r\nnegation of \u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always\u0027 is \u0027not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e sometimes\u0027; and the negation\r\nof \u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e sometimes\u0027 is \u0027not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always.\u0027\" In like manner\r\nwe can re-define disjunction and the other truth-functions,\r\nas applied to propositions containing apparent variables, in\r\nterms of the definitions and primitive ideas for propositions\r\ncontaining no apparent variables. Propositions containing no\r\napparent variables are called \"elementary propositions.\" From\r\nthese we can mount up step by step, using such methods as have\r\njust been indicated, to the theory of truth-functions as applied\r\nto propositions containing one, two, three, … variables, or any\r\nnumber up to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any assigned finite number.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_38_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_38_1\"\u003e\u003cspan class=\"label\"\u003e[38]\u003c/span\u003e\u003c/a\u003eThe method of deduction is given in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e,\r\nvol. I. * 9.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_39_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_39_1\"\u003e\u003cspan class=\"label\"\u003e[39]\u003c/span\u003e\u003c/a\u003eFor linguistic reasons, to avoid suggesting either the plural or the\r\nsingular, it is often convenient to say \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003eis not always false\" rather\r\nthan \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e sometimes\" or \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true.\"\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nThe forms which are taken as simplest in traditional formal\r\nlogic are really far from being so, and all involve the assertion\r\nof all values or some values of a compound propositional function.\r\nTake, to begin with, \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\" We will take it that \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is\r\ndefined by a propositional function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e by a propositional\r\nfunction \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e.\u003c/span\u003e \u003ci\u003eE.g.\u003c/i\u003e, if \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003ci\u003emen\u003c/i\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nwill be \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\"; if \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is\r\n\u003ci\u003emortals\u003c/i\u003e, \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e will be \"there is a time at which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e dies.\" Then\r\n\"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" means: \"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\u0027\u003c/span\u003e is always true.\" It is\r\nto be observed that \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" does not apply only to those\r\nterms that actually are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es; it says something equally about\r\nterms which are not \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es. Suppose we come across an \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e of which\r\nwe do not know whether it is an \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e or not; still, our statement\r\n\"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" tells us something about \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e namely,\r\nthat if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e,\u003c/span\u003e\r\nthen \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e And this is every bit as true when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not an \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e as\r\nwhen \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e.\u003c/span\u003e If it were not equally true in both cases, the\r\n\u003ci\u003ereductio ad absurdum\u003c/i\u003e would not be a valid method; for the\r\nessence of this method consists in using implications in cases\r\nwhere (as it afterwards turns out) the hypothesis is false. We may\r\nput the matter another way. In order to understand \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\"\r\nit is not necessary to be able to enumerate what terms are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es;\r\nprovided we know what is meant by being an \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e and what by\r\nbeing a \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e we can understand completely what is actually affirmed\r\n\u003cspan class=\"pagenum\" id=\"Page_161\"\u003e[Pg 161]\u003c/span\u003e\r\nby \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\" however little we may know of actual instances\r\nof either. This shows that it is not merely the actual terms that\r\nare \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es that are relevant in the statement \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\" but all the\r\nterms concerning which the supposition that they are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es is\r\nsignificant, \u003ci\u003ei.e.\u003c/i\u003e all the terms that are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es, together with all the\r\nterms that are not \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e the whole of the appropriate logical\r\n\"type.\" What applies to statements about \u003ci\u003eall\u003c/i\u003e applies also to\r\nstatements about \u003ci\u003esome\u003c/i\u003e. \"There are men,\" \u003ci\u003ee.g.\u003c/i\u003e, means that\r\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\" is true for \u003ci\u003esome\u003c/i\u003e values of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e Here \u003ci\u003eall\u003c/i\u003e values of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\n(\u003ci\u003ei.e.\u003c/i\u003e all values for which \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\" is significant, whether\r\ntrue or false) are relevant, and not only those that in fact are\r\nhuman. (This becomes obvious if we consider how we could\r\nprove such a statement to be \u003ci\u003efalse\u003c/i\u003e.) Every assertion about\r\n\"all\" or \"some\" thus involves not only the arguments that\r\nmake a certain function true, but all that make it significant,\r\n\u003ci\u003ei.e.\u003c/i\u003e all for which it has a value at all, whether true or false.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may now proceed with our interpretation of the traditional\r\nforms of the old-fashioned formal logic. We assume that \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is\r\nthose terms \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for which \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is true, and \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e is\r\nthose for which \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e is\r\ntrue. (As we shall see in a later chapter, all classes are derived\r\nin this way from propositional functions.) Then:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"All \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" means \"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e implies\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\u0027\u003c/span\u003e is always true.\"\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\"Some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" means \"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\u0027\u003c/span\u003e is sometimes true.\"\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\"No \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" means \"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nimplies not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\u0027\u003c/span\u003e is always true.\"\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\"Some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is not \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" means\r\n\"\u0027\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and not-\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\u0027\u003c/span\u003e is sometimes true.\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nIt will be observed that the propositional functions which are\r\nhere asserted for all or some values are not \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e themselves,\r\nbut truth-functions of \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e for the \u003ci\u003esame\u003c/i\u003e argument \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\r\nThe easiest way to conceive of the sort of thing that is\r\nintended is to start not from \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e in general, but from\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-301.png\" alt=\"\" data-tex=\"\\phi a\"\u003e and \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.67ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-303.png\" alt=\"\" data-tex=\"\\psi a\"\u003e,\u003c/span\u003e where \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is some constant. Suppose we are considering\r\n\"all men are mortal\": we will begin with\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"If Socrates is human, Socrates is mortal,\"\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\" id=\"Page_162\"\u003e[Pg 162]\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nand then we will regard \"Socrates\" as replaced by a variable \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\r\nwherever \"Socrates\" occurs. The object to be secured is that,\r\nalthough \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e remains a variable, without any definite value, yet\r\nit is to have the same value in \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\" as in \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" when we are\r\nasserting that \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e implies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" is always true. This requires\r\nthat we shall start with a function whose values are such as\r\n\"\u003cimg style=\"vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-301.png\" alt=\"\" data-tex=\"\\phi a\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.67ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-303.png\" alt=\"\" data-tex=\"\\psi a\"\u003e,\u003c/span\u003e\" rather than with two separate functions \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nand \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e;\u003c/span\u003e for if we start with two separate functions we can\r\nnever secure that the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e while remaining undetermined, shall\r\nhave the same value in both.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFor brevity we say \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always implies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" when we\r\nmean that \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e implies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" is always true. Propositions\r\nof the form \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always implies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" are called \"formal\r\nimplications\"; this name is given equally if there are several\r\nvariables.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe above definitions show how far removed from the simplest\r\nforms are such propositions as \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\" with which traditional\r\nlogic begins. It is typical of the lack of analysis involved\r\nthat traditional logic treats \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" as a proposition of\r\nthe same form as \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\"\u0026mdash;\u003ci\u003ee.g.\u003c/i\u003e, it treats \"all men are mortal\"\r\nas of the same form as \"Socrates is mortal.\" As we have just\r\nseen, the first is of the form \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e,\u003c/span\u003e\" while the\r\nsecond is of the form \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e.\u003c/span\u003e\" The emphatic separation of these\r\ntwo forms, which was effected by Peano and Frege, was a very\r\nvital advance in symbolic logic.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt will be seen that \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" and\r\n\"no \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" do not\r\nreally differ in form, except by the substitution of not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e for \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e,\u003c/span\u003e\r\nand that the same applies to \"some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" and \"some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is\r\nnot \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e.\u003c/span\u003e\" It should also be observed that the traditional rules\r\nof conversion are faulty, if we adopt the view, which is the only\r\ntechnically tolerable one, that such propositions as \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\"\r\ndo not involve the \"existence\" of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es, \u003ci\u003ei.e.\u003c/i\u003e do not require that\r\nthere should be terms which are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es. The above definitions\r\nlead to the result that, if \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always false, \u003ci\u003ei.e.\u003c/i\u003e if there are no \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e\u0027\u003c/span\u003es,\r\nthen \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" and \"no \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" will both be true, whatever\r\n\u003cspan class=\"pagenum\" id=\"Page_163\"\u003e[Pg 163]\u003c/span\u003e\r\n\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e may be. For, according to the definition in the last\r\nchapter, \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e implies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" means \"not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e or \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" which is\r\nalways true if not-\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always true. At the first moment,\r\nthis result might lead the reader to desire different definitions,\r\nbut a little practical experience soon shows that any different\r\ndefinitions would be inconvenient and would conceal the important\r\nideas. The proposition \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e,\u003c/span\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is\r\nsometimes true\" is essentially composite, and it would be\r\nvery awkward to give this as the definition of \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\"\r\nfor then we should have no language left for \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e always implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e,\u003c/span\u003e\"\r\nwhich is needed a hundred times for once that the other is\r\nneeded. But, with our definitions, \"all \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e\" does not imply\r\n\"some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\" since the first allows the non-existence of \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e and\r\nthe second does not; thus conversion \u003ci\u003eper accidens\u003c/i\u003e becomes\r\ninvalid, and some moods of the syllogism are fallacious, \u003ci\u003ee.g.\u003c/i\u003e\r\nDarapti: \"All \u003cimg style=\"vertical-align: 0; width: 2.075ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-304.png\" alt=\"\" data-tex=\"\\mathrm M\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e,\u003c/span\u003e all \u003cimg style=\"vertical-align: 0; width: 2.075ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-304.png\" alt=\"\" data-tex=\"\\mathrm M\"\u003e is\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e therefore some \u003cimg style=\"vertical-align: -0.05ex; width: 1.258ex; height: 1.645ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-90.png\" alt=\"\" data-tex=\"\\mathrm S\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 1.541ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-80.png\" alt=\"\" data-tex=\"\\mathrm P\"\u003e,\u003c/span\u003e\" which\r\nfails if there is no \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: 0; width: 2.075ex; height: 1.545ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-304.png\" alt=\"\" data-tex=\"\\mathrm M\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe notion of \"existence\" has several forms, one of which\r\nwill occupy us in the next chapter; but the fundamental form\r\nis that which is derived immediately from the notion of \"sometimes\r\ntrue.\" We say that an argument \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e \"satisfies\" a function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nif \u003cimg style=\"vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-301.png\" alt=\"\" data-tex=\"\\phi a\"\u003e is true; this is the same sense in which the roots of an\r\nequation are said to satisfy the equation. Now if \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes\r\ntrue, we may say there are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\u0027\u003c/span\u003es for which it is true, or we may say\r\n\"arguments satisfying \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e \u003ci\u003eexist\u003c/i\u003e\" This is the fundamental meaning\r\nof the word \"existence.\" Other meanings are either derived\r\nfrom this, or embody mere confusion of thought. We may\r\ncorrectly say \"men exist,\" meaning that \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a man\" is sometimes\r\ntrue. But if we make a pseudo-syllogism: \"Men exist,\r\nSocrates is a man, therefore Socrates exists,\" we are talking\r\nnonsense, since \"Socrates\" is not, like \"men,\" merely an undetermined\r\nargument to a given propositional function. The\r\nfallacy is closely analogous to that of the argument: \"Men are\r\nnumerous, Socrates is a man, therefore Socrates is numerous.\"\r\nIn this case it is obvious that the conclusion is nonsensical, but\r\n\u003cspan class=\"pagenum\" id=\"Page_164\"\u003e[Pg 164]\u003c/span\u003e\r\nin the case of existence it is not obvious, for reasons which will\r\nappear more fully in the next chapter. For the present let us\r\nmerely note the fact that, though it is correct to say \"men exist,\"\r\nit is incorrect, or rather meaningless, to ascribe existence to a\r\ngiven particular \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e who happens to be a man. Generally, \"terms\r\nsatisfying \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e exist\" means \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true\"; but \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e exists\"\r\n(where \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is a term satisfying \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e)\u003c/span\u003e is a mere noise or shape,\r\ndevoid of significance. It will be found that by bearing in mind\r\nthis simple fallacy we can solve many ancient philosophical\r\npuzzles concerning the meaning of existence.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAnother set of notions as to which philosophy has allowed\r\nitself to fall into hopeless confusions through not sufficiently\r\nseparating propositions and propositional functions are the\r\nnotions of \"modality\": \u003ci\u003enecessary\u003c/i\u003e, \u003ci\u003epossible\u003c/i\u003e, and \u003ci\u003eimpossible\u003c/i\u003e.\r\n(Sometimes \u003ci\u003econtingent\u003c/i\u003e or \u003ci\u003eassertoric\u003c/i\u003e is used instead of \u003ci\u003epossible\u003c/i\u003e.)\r\nThe traditional view was that, among true propositions, some\r\nwere necessary, while others were merely contingent or assertoric;\r\nwhile among false propositions some were impossible, namely,\r\nthose whose contradictories were necessary, while others merely\r\nhappened not to be true. In fact, however, there was never\r\nany clear account of what was added to truth by the conception\r\nof necessity. In the case of propositional functions, the three-fold\r\ndivision is obvious. If \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\" is an undetermined value of a\r\ncertain propositional function, it will be \u003ci\u003enecessary\u003c/i\u003e if the function\r\nis always true, \u003ci\u003epossible\u003c/i\u003e if it is sometimes true, and \u003ci\u003eimpossible\u003c/i\u003e if\r\nit is never true. This sort of situation arises in regard to probability,\r\nfor example. Suppose a ball \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is drawn from a bag\r\nwhich contains a number of balls: if all the balls are white,\r\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is white\" is necessary; if some are white, it is possible;\r\nif none, it is impossible. Here all that is \u003ci\u003eknown\u003c/i\u003e about \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is that\r\nit satisfies a certain propositional function, namely, \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e was a\r\nball in the bag.\" This is a situation which is general in probability\r\nproblems and not uncommon in practical life\u0026mdash;\u003ci\u003ee.g.\u003c/i\u003e when\r\na person calls of whom we know nothing except that he brings\r\na letter of introduction from our friend so-and-so. In all such\r\n\u003cspan class=\"pagenum\" id=\"Page_165\"\u003e[Pg 165]\u003c/span\u003e\r\ncases, as in regard to modality in general, the propositional\r\nfunction is relevant. For clear thinking, in many very diverse\r\ndirections, the habit of keeping propositional functions sharply\r\nseparated from propositions is of the utmost importance, and\r\nthe failure to do so in the past has been a disgrace to\r\nphilosophy.\r\n\u003cspan class=\"pagenum\" id=\"Page_166\"\u003e[Pg 166]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XVI: DESCRIPTIONS\u0027\u003e\u003ca id=\"chap16\"\u003e\u003c/a\u003eCHAPTER XVI\r\n\u003cbr\u003e\u003cbr\u003e\r\nDESCRIPTIONS\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nWe dealt in the preceding chapter with the words \u003ci\u003eall\u003c/i\u003e and \u003ci\u003esome\u003c/i\u003e;\r\nin this chapter we shall consider the word \u003ci\u003ethe\u003c/i\u003e in the singular,\r\nand in the next chapter we shall consider the word \u003ci\u003ethe\u003c/i\u003e in the\r\nplural. It may be thought excessive to devote two chapters\r\nto one word, but to the philosophical mathematician it is a\r\nword of very great importance: like Browning\u0027s Grammarian\r\nwith the enclitic \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.923ex; height: 1.647ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-305.png\" alt=\"\" data-tex=\"\\delta\\epsilon\"\u003e,\u003c/span\u003e I would give the doctrine of this word if I\r\nwere \"dead from the waist down\" and not merely in a prison.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have already had occasion to mention \"descriptive\r\nfunctions,\" \u003ci\u003ei.e.\u003c/i\u003e such expressions as \"the father of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" or \"the sine\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\" These are to be defined by first defining \"descriptions.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA \"description\" may be of two sorts, definite and indefinite\r\n(or ambiguous). An indefinite description is a phrase of the\r\nform \"a so-and-so,\" and a definite description is a phrase of\r\nthe form \"the so-and-so\" (in the singular). Let us begin with\r\nthe former.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"Who did you meet?\" \"I met a man.\" \"That is a very\r\nindefinite description.\" We are therefore not departing from\r\nusage in our terminology. Our question is: What do I really\r\nassert when I assert \"I met a man\"? Let us assume, for the\r\nmoment, that my assertion is true, and that in fact I met Jones.\r\nIt is clear that what I assert is \u003ci\u003enot\u003c/i\u003e \"I met Jones.\" I may say\r\n\"I met a man, but it was not Jones\"; in that case, though I lie,\r\nI do not contradict myself, as I should do if when I say I met a\r\n\u003cspan class=\"pagenum\" id=\"Page_167\"\u003e[Pg 167]\u003c/span\u003e\r\nman I really mean that I met Jones. It is clear also that the\r\nperson to whom I am speaking can understand what I say, even\r\nif he is a foreigner and has never heard of Jones.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut we may go further: not only Jones, but no actual man,\r\nenters into my statement. This becomes obvious when the statement\r\nis false, since then there is no more reason why Jones\r\nshould be supposed to enter into the proposition than why anyone\r\nelse should. Indeed the statement would remain significant,\r\nthough it could not possibly be true, even if there were no man\r\nat all. \"I met a unicorn\" or \"I met a sea-serpent\" is a\r\nperfectly significant assertion, if we know what it would be to\r\nbe a unicorn or a sea-serpent, \u003ci\u003ei.e.\u003c/i\u003e what is the definition of these\r\nfabulous monsters. Thus it is only what we may call the \u003ci\u003econcept\u003c/i\u003e\r\nthat enters into the proposition. In the case of \"unicorn,\"\r\nfor example, there is only the concept: there is not also, somewhere\r\namong the shades, something unreal which may be called\r\n\"a unicorn.\" Therefore, since it is significant (though false)\r\nto say \"I met a unicorn,\" it is clear that this proposition, rightly\r\nanalysed, does not contain a constituent \"a unicorn,\" though\r\nit does contain the concept \"unicorn.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe question of \"unreality,\" which confronts us at this\r\npoint, is a very important one. Misled by grammar, the great\r\nmajority of those logicians who have dealt with this question\r\nhave dealt with it on mistaken lines. They have regarded\r\ngrammatical form as a surer guide in analysis than, in fact,\r\nit is. And they have not known what differences in grammatical\r\nform are important. \"I met Jones\" and \"I met a\r\nman\" would count traditionally as propositions of the same form,\r\nbut in actual fact they are of quite different forms: the first\r\nnames an actual person, Jones; while the second involves a\r\npropositional function, and becomes, when made explicit: \"The\r\nfunction \u0027I met \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\u0027 is sometimes true.\" (It\r\nwill be remembered that we adopted the convention of using\r\n\"sometimes\" as not implying more than once.) This proposition\r\nis obviously not of the form \"I met \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\" which accounts\r\n\u003cspan class=\"pagenum\" id=\"Page_168\"\u003e[Pg 168]\u003c/span\u003e\r\nfor the existence of the proposition \"I met a unicorn\" in spite\r\nof the fact that there is no such thing as \"a unicorn.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nFor want of the apparatus of propositional functions, many\r\nlogicians have been driven to the conclusion that there are\r\nunreal objects. It is argued, \u003ci\u003ee.g.\u003c/i\u003e by Meinong,\u003ca id=\"FNanchor_40_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_40_1\" class=\"fnanchor\"\u003e[40]\u003c/a\u003e\r\nthat we can\r\nspeak about \"the golden mountain,\" \"the round square,\"\r\nand so on; we can make true propositions of which these are\r\nthe subjects; hence they must have some kind of logical being,\r\nsince otherwise the propositions in which they occur would be\r\nmeaningless. In such theories, it seems to me, there is a failure\r\nof that feeling for reality which ought to be preserved even in\r\nthe most abstract studies. Logic, I should maintain, must no\r\nmore admit a unicorn than zoology can; for logic is concerned\r\nwith the real world just as truly as zoology, though with its\r\nmore abstract and general features. To say that unicorns have\r\nan existence in heraldry, or in literature, or in imagination,\r\nis a most pitiful and paltry evasion. What exists in heraldry\r\nis not an animal, made of flesh and blood, moving and breathing\r\nof its own initiative. What exists is a picture, or a description\r\nin words. Similarly, to maintain that Hamlet, for example,\r\nexists in his own world, namely, in the world of Shakespeare\u0027s\r\nimagination, just as truly as (say) Napoleon existed in the\r\nordinary world, is to say something deliberately confusing, or\r\nelse confused to a degree which is scarcely credible. There is\r\nonly one world, the \"real\" world: Shakespeare\u0027s imagination\r\nis part of it, and the thoughts that he had in writing Hamlet\r\nare real. So are the thoughts that we have in reading the play.\r\nBut it is of the very essence of fiction that only the thoughts,\r\nfeelings, etc., in Shakespeare and his readers are real, and that\r\nthere is not, in addition to them, an objective Hamlet. When\r\nyou have taken account of all the feelings roused by Napoleon\r\nin writers and readers of history, you have not touched the actual\r\nman; but in the case of Hamlet you have come to the end of\r\nhim. If no one thought about Hamlet, there would be nothing\r\n\u003cspan class=\"pagenum\" id=\"Page_169\"\u003e[Pg 169]\u003c/span\u003e\r\nleft of him; if no one had thought about Napoleon, he would\r\nhave soon seen to it that some one did. The sense of reality is\r\nvital in logic, and whoever juggles with it by pretending that\r\nHamlet has another kind of reality is doing a disservice to\r\nthought. A robust sense of reality is very necessary in framing\r\na correct analysis of propositions about unicorns, golden mountains,\r\nround squares, and other such pseudo-objects.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_40_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_40_1\"\u003e\u003cspan class=\"label\"\u003e[40]\u003c/span\u003e\u003c/a\u003e\u003ci\u003eUntersuchungen zur Gegenstandstheorie und Psychologie\u003c/i\u003e, 1904.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nIn obedience to the feeling of reality, we shall insist that,\r\nin the analysis of propositions, nothing \"unreal\" is to be\r\nadmitted. But, after all, if there \u003ci\u003eis\u003c/i\u003e nothing unreal, how, it\r\nmay be asked, \u003ci\u003ecould\u003c/i\u003e we admit anything unreal? The reply\r\nis that, in dealing with propositions, we are dealing in the first\r\ninstance with symbols, and if we attribute significance to groups\r\nof symbols which have no significance, we shall fall into the\r\nerror of admitting unrealities, in the only sense in which this is\r\npossible, namely, as objects described. In the proposition\r\n\"I met a unicorn,\" the whole four words together make a significant\r\nproposition, and the word \"unicorn\" by itself is significant,\r\nin just the same sense as the word \"man.\" But the \u003ci\u003etwo\u003c/i\u003e words\r\n\"a unicorn\" do not form a subordinate group having a meaning\r\nof its own. Thus if we falsely attribute meaning to these two\r\nwords, we find ourselves saddled with \"a unicorn,\" and with\r\nthe problem how there can be such a thing in a world where\r\nthere are no unicorns. \"A unicorn\" is an indefinite description\r\nwhich describes nothing. It is not an indefinite description\r\nwhich describes something unreal. Such a proposition as\r\n\"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is unreal\" only has meaning when \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a description,\r\ndefinite or indefinite; in that case the proposition will be true\r\nif \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a description which describes nothing. But whether\r\nthe description \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" describes something or describes nothing,\r\nit is in any case not a constituent of the proposition in which it\r\noccurs; like \"a unicorn\" just now, it is not a subordinate group\r\nhaving a meaning of its own. All this results from the fact that,\r\nwhen \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a description, \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is unreal\" or \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e does not exist\"\r\nis not nonsense, but is always significant and sometimes true.\r\n\u003cspan class=\"pagenum\" id=\"Page_170\"\u003e[Pg 170]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may now proceed to define generally the meaning of\r\npropositions which contain ambiguous descriptions. Suppose\r\nwe wish to make some statement about \"a so-and-so,\" where\r\n\"so-and-so\u0027s\" are those objects that have a certain property \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e,\u003c/span\u003e\r\n\u003ci\u003ei.e.\u003c/i\u003e those objects \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for which the propositional function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is\r\ntrue. (\u003ci\u003eE.g.\u003c/i\u003e if we take \"a man\" as our instance of \"a so-and-so,\"\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e will be \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human.\") Let us now wish to assert the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e\r\nof \"a so-and-so,\" \u003ci\u003ei.e.\u003c/i\u003e we wish to assert that \"a so-and-so\" has\r\nthat property which \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has when \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e is true. (\u003ci\u003eE.g.\u003c/i\u003e in the case\r\nof \"I met a man,\" \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e will be \"I met \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\") Now the proposition\r\nthat \"a so-and-so\" has the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e is \u003ci\u003enot\u003c/i\u003e a proposition of\r\nthe form \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e.\u003c/span\u003e\" If it were, \"a so-and-so\" would have to be\r\nidentical with \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for a suitable \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e;\u003c/span\u003e and although (in a sense) this\r\nmay be true in some cases, it is certainly not true in such a case\r\nas \"a unicorn.\" It is just this fact, that the statement that a\r\nso-and-so has the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e is not of the form \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e,\u003c/span\u003e which makes\r\nit possible for \"a so-and-so\" to be, in a certain clearly definable\r\nsense, \"unreal.\" The definition is as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe statement that \"an object having the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e has\r\nthe property \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e\"\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nmeans:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"The joint assertion of \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e is not always false.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSo far as logic goes, this is the same proposition as might\r\nbe expressed by \"some \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e\u0027\u003c/span\u003es are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e\u0027\u003c/span\u003es\"; but rhetorically there is\r\na difference, because in the one case there is a suggestion of\r\nsingularity, and in the other case of plurality. This, however,\r\nis not the important point. The important point is that, when\r\nrightly analysed, propositions verbally about \"a so-and-so\"\r\nare found to contain no constituent represented by this phrase.\r\nAnd that is why such propositions can be significant even when\r\nthere is no such thing as a so-and-so.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe definition of \u003ci\u003eexistence\u003c/i\u003e, as applied to ambiguous descriptions,\r\nresults from what was said at the end of the preceding\r\nchapter. We say that \"men exist\" or \"a man exists\" if the\r\n\u003cspan class=\"pagenum\" id=\"Page_171\"\u003e[Pg 171]\u003c/span\u003e\r\npropositional function \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\" is sometimes true; and\r\ngenerally \"a so-and-so\" exists if \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is so-and-so\" is sometimes\r\ntrue. We may put this in other language. The proposition\r\n\"Socrates is a man\" is no doubt \u003ci\u003eequivalent\u003c/i\u003e to \"Socrates is\r\nhuman,\" but it is not the very same proposition. The \u003ci\u003eis\u003c/i\u003e of\r\n\"Socrates is human\" expresses the relation of subject and\r\npredicate; the \u003ci\u003eis\u003c/i\u003e of \"Socrates is a man\" expresses identity.\r\nIt is a disgrace to the human race that it has chosen to employ\r\nthe same word \"is\" for these two entirely different ideas\u0026mdash;a\r\ndisgrace which a symbolic logical language of course remedies.\r\nThe identity in \"Socrates is a man\" is identity between an\r\nobject named (accepting \"Socrates\" as a name, subject to\r\nqualifications explained later) and an object ambiguously\r\ndescribed. An object ambiguously described will \"exist\" when\r\nat least one such proposition is true, \u003ci\u003ei.e.\u003c/i\u003e when there is at least\r\none true proposition of the form \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a so-and-so,\" where \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is\r\na name. It is characteristic of ambiguous (as opposed to\r\ndefinite) descriptions that there may be any number of true\r\npropositions of the above form\u0026mdash;Socrates is a man, Plato is a\r\nman, etc. Thus \"a man exists\" follows from Socrates, or\r\nPlato, or anyone else. With definite descriptions, on the other\r\nhand, the corresponding form of proposition, namely, \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the\r\nso-and-so\" (where \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a name), can only be true for one\r\nvalue of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e at most. This brings us to the subject of definite\r\ndescriptions, which are to be defined in a way analogous to\r\nthat employed for ambiguous descriptions, but rather more\r\ncomplicated.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe come now to the main subject of the present chapter,\r\nnamely, the definition of the word \u003ci\u003ethe\u003c/i\u003e (in the singular). One\r\nvery important point about the definition of \"a so-and-so\"\r\napplies equally to \"the so-and-so\"; the definition to be sought\r\nis a definition of propositions in which this phrase occurs, not a\r\ndefinition of the phrase itself in isolation. In the case of \"a\r\nso-and-so,\" this is fairly obvious: no one could suppose that\r\n\"a man\" was a definite object, which could be defined by itself.\r\n\u003cspan class=\"pagenum\" id=\"Page_172\"\u003e[Pg 172]\u003c/span\u003e\r\nSocrates is a man, Plato is a man, Aristotle is a man, but we\r\ncannot infer that \"a man\" means the same as \"Socrates\"\r\nmeans and also the same as \"Plato\" means and also the same\r\nas \"Aristotle\" means, since these three names have different\r\nmeanings. Nevertheless, when we have enumerated all the\r\nmen in the world, there is nothing left of which we can say,\r\n\"This is a man, and not only so, but it is \u003ci\u003ethe\u003c/i\u003e \u0027a man,\u0027 the quintessential\r\nentity that is just an indefinite man without being anybody\r\nin particular.\" It is of course quite clear that whatever\r\nthere is in the world is definite: if it is a man it is one definite\r\nman and not any other. Thus there cannot be such an entity\r\nas \"a man\" to be found in the world, as opposed to specific\r\nman. And accordingly it is natural that we do not define \"a\r\nman\" itself, but only the propositions in which it occurs.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the case of \"the so-and-so\" this is equally true, though\r\nat first sight less obvious. We may demonstrate that this must\r\nbe the case, by a consideration of the difference between a \u003ci\u003ename\u003c/i\u003e\r\nand a \u003ci\u003edefinite description\u003c/i\u003e. Take the proposition, \"Scott is the\r\nauthor of \u003ci\u003eWaverley\u003c/i\u003e.\" We have here a name, \"Scott,\" and a\r\ndescription, \"the author of \u003ci\u003eWaverley\u003c/i\u003e,\" which are asserted to\r\napply to the same person. The distinction between a name and\r\nall other symbols may be explained as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA name is a simple symbol whose meaning is something that\r\ncan only occur as subject, \u003ci\u003ei.e.\u003c/i\u003e something of the kind that, in\r\nChapter XIII., we defined as an \"individual\" or a \"particular.\"\r\nAnd a \"simple\" symbol is one which has no parts that are\r\nsymbols. Thus \"Scott\" is a simple symbol, because, though it\r\nhas parts (namely, separate letters), these parts are not symbols.\r\nOn the other hand, \"the author of \u003ci\u003eWaverley\u003c/i\u003e\" is not a simple\r\nsymbol, because the separate words that compose the phrase\r\nare parts which are symbols. If, as may be the case, whatever\r\n\u003ci\u003eseems\u003c/i\u003e to be an \"individual\" is really capable of further analysis,\r\nwe shall have to content ourselves with what may be called\r\n\"relative individuals,\" which will be terms that, throughout\r\nthe context in question, are never analysed and never occur\r\n\u003cspan class=\"pagenum\" id=\"Page_173\"\u003e[Pg 173]\u003c/span\u003e\r\notherwise than as subjects. And in that case we shall have\r\ncorrespondingly to content ourselves with \"relative names.\"\r\nFrom the standpoint of our present problem, namely, the definition\r\nof descriptions, this problem, whether these are absolute\r\nnames or only relative names, may be ignored, since it concerns\r\ndifferent stages in the hierarchy of \"types,\" whereas we\r\nhave to compare such couples as \"Scott\" and \"the author of\r\n\u003ci\u003eWaverley\u003c/i\u003e,\" which both apply to the same object, and do not\r\nraise the problem of types. We may, therefore, for the moment,\r\ntreat names as capable of being absolute; nothing that we shall\r\nhave to say will depend upon this assumption, but the wording\r\nmay be a little shortened by it.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have, then, two things to compare: (1) a \u003ci\u003ename\u003c/i\u003e, which\r\nis a simple symbol, directly designating an individual which\r\nis its meaning, and having this meaning in its own right, independently\r\nof the meanings of all other words; (2) a \u003ci\u003edescription\u003c/i\u003e,\r\nwhich consists of several words, whose meanings are already\r\nfixed, and from which results whatever is to be taken as the\r\n\"meaning\" of the description.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nA proposition containing a description is not identical with\r\nwhat that proposition becomes when a name is substituted,\r\neven if the name names the same object as the description\r\ndescribes. \"Scott is the author of \u003ci\u003eWaverley\u003c/i\u003e\" is obviously a\r\ndifferent proposition from \"Scott is Scott\": the first is a fact\r\nin literary history, the second a trivial truism. And if we put\r\nanyone other than Scott in place of \"the author of \u003ci\u003eWaverley\u003c/i\u003e,\"\r\nour proposition would become false, and would therefore certainly\r\nno longer be the same proposition. But, it may be said, our\r\nproposition is essentially of the same form as (say) \"Scott is\r\nSir Walter,\" in which two names are said to apply to the same\r\nperson. The reply is that, if \"Scott is Sir Walter\" really means\r\n\"the person named \u0027Scott\u0027 is the person named \u0027Sir Walter,\u0027\"\r\nthen the names are being used as descriptions: \u003ci\u003ei.e.\u003c/i\u003e the individual,\r\ninstead of being named, is being described as the person having\r\nthat name. This is a way in which names are frequently used\r\n\u003cspan class=\"pagenum\" id=\"Page_174\"\u003e[Pg 174]\u003c/span\u003e\r\nin practice, and there will, as a rule, be nothing in the phraseology\r\nto show whether they are being used in this way or \u003ci\u003eas\u003c/i\u003e names.\r\nWhen a name is used directly, merely to indicate what we are\r\nspeaking about, it is no part of the \u003ci\u003efact\u003c/i\u003e asserted, or of the falsehood\r\nif our assertion happens to be false: it is merely part of the\r\nsymbolism by which we express our thought. What we want\r\nto express is something which might (for example) be translated\r\ninto a foreign language; it is something for which the actual\r\nwords are a vehicle, but of which they are no part. On the other\r\nhand, when we make a proposition about \"the person called\r\n\u0027Scott,\u0027\" the actual name \"Scott\" enters into what we are\r\nasserting, and not merely into the language used in making the\r\nassertion. Our proposition will now be a different one if we\r\nsubstitute \"the person called \u0027Sir Walter.\u0027\" But so long as\r\nwe are using names \u003ci\u003eas\u003c/i\u003e names, whether we say \"Scott\" or whether\r\nwe say \"Sir Walter\" is as irrelevant to what we are asserting\r\nas whether we speak English or French. Thus so long as names\r\nare used as names, \"Scott is Sir Walter\" is the same trivial\r\nproposition as \"Scott is Scott.\" This completes the proof that\r\n\"Scott is the author of \u003ci\u003eWaverley\u003c/i\u003e\" is not the same proposition\r\nas results from substituting a name for \"the author of \u003ci\u003eWaverley\u003c/i\u003e,\"\r\nno matter what name may be substituted.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen we use a variable, and speak of a propositional function,\r\n\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e say, the process of applying general statements about \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to\r\nparticular cases will consist in substituting a name for the letter \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\"\r\nassuming that \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is a function which has individuals for its\r\narguments. Suppose, for example, that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is \"always true\";\r\nlet it be, say, the \"law of identity,\" \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-308.png\" alt=\"\" data-tex=\"x = x\"\u003e.\u003c/span\u003e Then we may substitute\r\nfor \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" any name we choose, and we shall obtain a true\r\nproposition. Assuming for the moment that \"Socrates,\"\r\n\"Plato,\" and \"Aristotle\" are names (a very rash assumption),\r\nwe can infer from the law of identity that Socrates is Socrates,\r\nPlato is Plato, and Aristotle is Aristotle. But we shall commit\r\na fallacy if we attempt to infer, without further premisses, that\r\nthe author of \u003ci\u003eWaverley\u003c/i\u003e is the author of \u003ci\u003eWaverley\u003c/i\u003e. This results\r\n\u003cspan class=\"pagenum\" id=\"Page_175\"\u003e[Pg 175]\u003c/span\u003e\r\nfrom what we have just proved, that, if we substitute a name for\r\n\"the author of \u003ci\u003eWaverley\u003c/i\u003e\" in a proposition, the proposition\r\nwe obtain is a different one. That is to say, applying the result\r\nto our present case: If \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a name, \"\u003cimg style=\"vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-308.png\" alt=\"\" data-tex=\"x = x\"\u003e\" is not the same\r\nproposition as \"the author of \u003ci\u003eWaverley\u003c/i\u003e is the author of \u003ci\u003eWaverley\u003c/i\u003e,\"\r\nno matter what name \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" may be. Thus from the fact that\r\nall propositions of the form \"\u003cimg style=\"vertical-align: -0.186ex; width: 5.605ex; height: 1.505ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-308.png\" alt=\"\" data-tex=\"x = x\"\u003e\" are true we cannot infer,\r\nwithout more ado, that the author of \u003ci\u003eWaverley\u003c/i\u003e is the author of\r\n\u003ci\u003eWaverley\u003c/i\u003e. In fact, propositions of the form \"the so-and-so\r\nis the so-and-so\" are not always true: it is necessary that the\r\nso-and-so should \u003ci\u003eexist\u003c/i\u003e (a term which will be explained shortly).\r\nIt is false that the present King of France is the present King of\r\nFrance, or that the round square is the round square. When we\r\nsubstitute a description for a name, propositional functions\r\nwhich are \"always true\" may become false, if the description\r\ndescribes nothing. There is no mystery in this as soon as we\r\nrealise (what was proved in the preceding paragraph) that when\r\nwe substitute a description the result is not a value of the\r\npropositional function in question.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe are now in a position to define propositions in which a\r\ndefinite description occurs. The only thing that distinguishes\r\n\"the so-and-so\" from \"a so-and-so\" is the implication of\r\nuniqueness. We cannot speak of \"\u003ci\u003ethe\u003c/i\u003e inhabitant of London,\"\r\nbecause inhabiting London is an attribute which is not unique.\r\nWe cannot speak about \"the present King of France,\" because\r\nthere is none; but we can speak about \"the present King of\r\nEngland.\" Thus propositions about \"the so-and-so\" always\r\nimply the corresponding propositions about \"a so-and-so,\"\r\nwith the addendum that there is not more than one so-and-so.\r\nSuch a proposition as \"Scott is the author of \u003ci\u003eWaverly\u003c/i\u003e\" could\r\nnot be true if \u003ci\u003eWaverly\u003c/i\u003e had never been written, or if several\r\npeople had written it; and no more could any other proposition\r\nresulting from a propositional function \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e by the substitution\r\nof \"the author of \u003ci\u003eWaverly\u003c/i\u003e\" for \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e.\u003c/span\u003e\" We may say that \"the\r\nauthor of \u003ci\u003eWaverly\u003c/i\u003e\" means \"the value of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for which \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote\r\n\u003cspan class=\"pagenum\" id=\"Page_176\"\u003e[Pg 176]\u003c/span\u003e\r\n\u003ci\u003eWaverly\u003c/i\u003e\u0027 is true.\" Thus the proposition \"the author of\r\n\u003ci\u003eWaverly\u003c/i\u003e was Scotch,\" for example, involves:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e\" is not always false;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) \"if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e are identical\" is\r\nalways true;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) \"if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e was Scotch\" is always true.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nThese three propositions, translated into ordinary language,\r\nstate:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) at least one person wrote \u003ci\u003eWaverly\u003c/i\u003e;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) at most one person wrote \u003ci\u003eWaverly\u003c/i\u003e;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) whoever wrote \u003ci\u003eWaverly\u003c/i\u003e was Scotch.\r\n\u003c/p\u003e\r\n\u003cp class=\"nind\"\u003e\r\nAll these three are implied by \"the author of \u003ci\u003eWaverly\u003c/i\u003e was\r\nScotch.\" Conversely, the three together (but no two of them)\r\nimply that the author of \u003ci\u003eWaverly\u003c/i\u003e was Scotch. Hence the\r\nthree together may be taken as defining what is meant by the\r\nproposition \"the author of \u003ci\u003eWaverly\u003c/i\u003e was Scotch.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may somewhat simplify these three propositions. The\r\nfirst and second together are equivalent to: \"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e\r\nsuch that \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e\u0027 is true when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e and is false\r\nwhen \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e\" In other words, \"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e such that\r\n\u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e\u0027 is always equivalent to \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e\u0027\" (Two\r\npropositions are \"equivalent\" when both are true or both are\r\nfalse.) We have here, to begin with, two functions of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote\r\n\u003ci\u003eWaverly\u003c/i\u003e\" and \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e\" and we form a function of \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e by\r\nconsidering the equivalence of these two functions of \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for all\r\nvalues of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e;\u003c/span\u003e we then proceed to assert that the resulting function\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e is \"sometimes true,\" \u003ci\u003ei.e.\u003c/i\u003e that it is true for at least one value\r\nof \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e (It obviously cannot be true for more than one value of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e)\r\nThese two conditions together are defined as giving the meaning\r\nof \"the author of \u003ci\u003eWaverly\u003c/i\u003e exists.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may now define \"the term satisfying the function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nexists.\" This is the general form of which the above is a particular\r\ncase. \"The author of \u003ci\u003eWaverly\u003c/i\u003e\" is \"the term satisfying\r\nthe function \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e.\u0027\" And \"the so-and-so\" will\r\n\u003cspan class=\"pagenum\" id=\"Page_177\"\u003e[Pg 177]\u003c/span\u003e\r\nalways involve reference to some propositional function, namely,\r\nthat which defines the property that makes a thing a so-and-so.\r\nOur definition is as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"The term satisfying the function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e exists\" means:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e such that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always equivalent\r\nto \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e\u0027\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn order to define \"the author of \u003ci\u003eWaverly\u003c/i\u003e was Scotch,\"\r\nwe have still to take account of the third of our three propositions,\r\nnamely, \"Whoever wrote \u003ci\u003eWaverly\u003c/i\u003e was Scotch.\" This\r\nwill be satisfied by merely adding that the \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e in question is to\r\nbe Scotch. Thus \"the author of \u003ci\u003eWaverly\u003c/i\u003e was Scotch\" is:\r\n\u003c/p\u003e\r\n\u003cp class=\"hanging2\"\u003e\r\n\"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e such that (1) \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e wrote \u003ci\u003eWaverly\u003c/i\u003e\u0027 is always\r\nequivalent to \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e\u0027 (2) \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e is Scotch.\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nAnd generally: \"the term satisfying \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e satisfies \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e\" is\r\ndefined as meaning:\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"hanging2\"\u003e\r\n\"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e such that (1) \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always equivalent to\r\n\u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e\u0027 (2) \u003cimg style=\"vertical-align: -0.464ex; width: 2.452ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-309.png\" alt=\"\" data-tex=\"\\psi c\"\u003e is true.\"\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nThis is the definition of propositions in which descriptions occur.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is possible to have much knowledge concerning a term\r\ndescribed, \u003ci\u003ei.e.\u003c/i\u003e to know many propositions concerning \"the so-and-so,\"\r\nwithout actually knowing what the so-and-so is, \u003ci\u003ei.e.\u003c/i\u003e\r\nwithout knowing any proposition of the form \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is the so-and-so,\"\r\nwhere \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" is a name. In a detective story propositions about\r\n\"the man who did the deed\" are accumulated, in the hope\r\nthat ultimately they will suffice to demonstrate that it was \u003cimg style=\"vertical-align: 0; width: 1.697ex; height: 1.62ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-73.png\" alt=\"\" data-tex=\"\\mathrm A\"\u003e\r\nwho did the deed. We may even go so far as to say that,\r\nin all such knowledge as can be expressed in words\u0026mdash;with the\r\nexception of \"this\" and \"that\" and a few other words of\r\nwhich the meaning varies on different occasions\u0026mdash;no names,\r\nin the strict sense, occur, but what seem like names are really\r\ndescriptions. We may inquire significantly whether Homer\r\nexisted, which we could not do if \"Homer\" were a name. The\r\nproposition \"the so-and-so exists\" is significant, whether\r\ntrue or false; but if \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is the so-and-so (where \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" is a name),\r\nthe words \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e exists\" are meaningless. It is only of descriptions\u0026mdash;definite\r\n\u003cspan class=\"pagenum\" id=\"Page_178\"\u003e[Pg 178]\u003c/span\u003e\r\nor indefinite\u0026mdash;that existence can be significantly\r\nasserted; for, if \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" is a name, it \u003ci\u003emust\u003c/i\u003e name something: what\r\ndoes not name anything is not a name, and therefore, if intended\r\nto be a name, is a symbol devoid of meaning, whereas a description,\r\nlike \"the present King of France,\" does not become incapable\r\nof occurring significantly merely on the ground that it\r\ndescribes nothing, the reason being that it is a \u003ci\u003ecomplex\u003c/i\u003e symbol,\r\nof which the meaning is derived from that of its constituent\r\nsymbols. And so, when we ask whether Homer existed, we are\r\nusing the word \"Homer\" as an abbreviated description: we\r\nmay replace it by (say) \"the author of the \u003ci\u003eIliad\u003c/i\u003e and the \u003ci\u003eOdyssey\u003c/i\u003e.\"\r\nThe same considerations apply to almost all uses of what look\r\nlike proper names.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen descriptions occur in propositions, it is necessary to\r\ndistinguish what may be called \"primary\" and \"secondary\"\r\noccurrences. The abstract distinction is as follows. A description\r\nhas a \"primary\" occurrence when the proposition in\r\nwhich it occurs results from substituting the description for \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\"\r\nin some propositional function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e;\u003c/span\u003e a description has a\r\n\"secondary\" occurrence when the result of substituting the\r\ndescription for \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e in \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e gives only \u003ci\u003epart\u003c/i\u003e of the proposition concerned.\r\nAn instance will make this clearer. Consider \"the\r\npresent King of France is bald.\" Here \"the present King of\r\nFrance\" has a primary occurrence, and the proposition is false.\r\nEvery proposition in which a description which describes nothing\r\nhas a primary occurrence is false. But now consider \"the\r\npresent King of France is not bald.\" This is ambiguous. If\r\nwe are first to take \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is bald,\" then substitute \"the present\r\nKing of France\" for \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" and then deny the result, the occurrence\r\nof \"the present King of France\" is secondary and our proposition\r\nis true; but if we are to take \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not bald\" and substitute\r\n\"the present King of France\" for \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" then \"the present\r\nKing of France\" has a primary occurrence and the proposition\r\nis false. Confusion of primary and secondary occurrences is a\r\nready source of fallacies where descriptions are concerned.\r\n\u003cspan class=\"pagenum\" id=\"Page_179\"\u003e[Pg 179]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nDescriptions occur in mathematics chiefly in the form of\r\n\u003ci\u003edescriptive functions\u003c/i\u003e, \u003ci\u003ei.e.\u003c/i\u003e \"the term having the\r\nrelation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\"\r\nor \"the \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\" as we may say, on the analogy of \"the\r\nfather of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\" and similar phrases. To say \"the father of \u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e is\r\nrich,\" for example, is to say that the following propositional\r\nfunction of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e:\u003c/span\u003e \"\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e is rich, and \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e begat \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e\u0027\u003c/span\u003e is always equivalent\r\nto \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003eis \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e,\u003c/span\u003e\u0027\" is \"sometimes true,\" \u003ci\u003ei.e.\u003c/i\u003e is true for at least one\r\nvalue of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e It obviously cannot be true for more than one\r\nvalue.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe theory of descriptions, briefly outlined in the present\r\nchapter, is of the utmost importance both in logic and in theory\r\nof knowledge. But for purposes of mathematics, the more\r\nphilosophical parts of the theory are not essential, and have\r\ntherefore been omitted in the above account, which has confined\r\nitself to the barest mathematical requisites.\r\n\u003cspan class=\"pagenum\" id=\"Page_180\"\u003e[Pg 180]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XVII: CLASSES\u0027\u003e\u003ca id=\"chap17\"\u003e\u003c/a\u003eCHAPTER XVII\r\n\u003cbr\u003e\u003cbr\u003e\r\nCLASSES\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nIN the present chapter we shall be concerned with \u003ci\u003ethe\u003c/i\u003e in the\r\nplural: the inhabitants of London, the sons of rich men, and\r\nso on. In other words, we shall be concerned with \u003ci\u003eclasses\u003c/i\u003e. We\r\nsaw in Chapter II. that a cardinal number is to be defined as a\r\nclass of classes, and in Chapter III. that the number 1 is to be\r\ndefined as the class of all unit classes, \u003ci\u003ei.e.\u003c/i\u003e of all that have just\r\none member, as we should say but for the vicious circle. Of\r\ncourse, when the number 1 is defined as the class of all unit\r\nclasses, \"unit classes\" must be defined so as not to assume\r\nthat we know what is meant by \"one\"; in fact, they are defined\r\nin a way closely analogous to that used for descriptions, namely:\r\nA class \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is said to be a \"unit\" class if the propositional function\r\n\"\u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003e is always equivalent to \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e\u0027\u003c/span\u003e\" (regarded as a\r\nfunction of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e)\u003c/span\u003e is not always false, \u003ci\u003ei.e.\u003c/i\u003e, in more ordinary language,\r\nif there is a term \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e such that \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e will be a member of\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e\r\nbut not otherwise. This gives us a definition of a unit class if we\r\nalready know what a class is in general. Hitherto we have, in\r\ndealing with arithmetic, treated \"class\" as a primitive idea.\r\nBut, for the reasons set forth in Chapter XIII., if for no others,\r\nwe cannot accept \"class\" as a primitive idea. We must seek a\r\ndefinition on the same lines as the definition of descriptions,\r\n\u003ci\u003ei.e.\u003c/i\u003e a definition which will assign a meaning to propositions in\r\nwhose verbal or symbolic expression words or symbols apparently\r\nrepresenting classes occur, but which will assign a meaning that\r\naltogether eliminates all mention of classes from a right analysis\r\n\u003cspan class=\"pagenum\" id=\"Page_181\"\u003e[Pg 181]\u003c/span\u003e\r\nof such propositions. We shall then be able to say that the\r\nsymbols for classes are mere conveniences, not representing\r\nobjects called \"classes,\" and that classes are in fact, like descriptions,\r\nlogical fictions, or (as we say) \"incomplete symbols.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe theory of classes is less complete than the theory of descriptions,\r\nand there are reasons (which we shall give in outline)\r\nfor regarding the definition of classes that will be suggested as\r\nnot finally satisfactory. Some further subtlety appears to be\r\nrequired; but the reasons for regarding the definition which\r\nwill be offered as being approximately correct and on the right\r\nlines are overwhelming.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe first thing is to realise why classes cannot be regarded\r\nas part of the ultimate furniture of the world. It is difficult\r\nto explain precisely what one means by this statement, but one\r\nconsequence which it implies may be used to elucidate its meaning.\r\nIf we had a complete symbolic language, with a definition for\r\neverything definable, and an undefined symbol for everything\r\nindefinable, the undefined symbols in this language would represent\r\nsymbolically what I mean by \"the ultimate furniture of\r\nthe world.\" I am maintaining that no symbols either for \"class\"\r\nin general or for particular classes would be included in this\r\napparatus of undefined symbols. On the other hand, all the\r\nparticular things there are in the world would have to have\r\nnames which would be included among undefined symbols.\r\nWe might try to avoid this conclusion by the use of descriptions.\r\nTake (say) \"the last thing Cæsar saw before he died.\" This\r\nis a description of some particular; we might use it as (in one\r\nperfectly legitimate sense) a \u003ci\u003edefinition\u003c/i\u003e of that particular. But\r\nif \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" is a \u003ci\u003ename\u003c/i\u003e for the same particular, a proposition in which\r\n\"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" occurs is not (as we saw in the preceding chapter) identical\r\nwith what this proposition becomes when for \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" we substitute\r\n\"the last thing Cæsar saw before he died.\" If our language\r\ndoes not contain the name \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" or some other name for the same\r\nparticular, we shall have no means of expressing the proposition\r\nwhich we expressed by means of \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e\" as opposed to the one that\r\n\u003cspan class=\"pagenum\" id=\"Page_182\"\u003e[Pg 182]\u003c/span\u003e\r\nwe expressed by means of the description. Thus descriptions\r\nwould not enable a perfect language to dispense with names for\r\nall particulars. In this respect, we are maintaining, classes\r\ndiffer from particulars, and need not be represented by undefined\r\nsymbols. Our first business is to give the reasons for this opinion.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe have already seen that classes cannot be regarded as a\r\nspecies of individuals, on account of the contradiction about\r\nclasses which are not members of themselves (explained in\r\nChapter XIII.), and because we can prove that the number of\r\nclasses is greater than the number of individuals.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe cannot take classes in the \u003ci\u003epure\u003c/i\u003e extensional way as simply\r\nheaps or conglomerations. If we were to attempt to do that,\r\nwe should find it impossible to understand how there can be such\r\na class as the null-class, which has no members at all and cannot\r\nbe regarded as a \"heap\"; we should also find it very hard to\r\nunderstand how it comes about that a class which has only one\r\nmember is not identical with that one member. I do not mean\r\nto assert, or to deny, that there are such entities as \"heaps.\"\r\nAs a mathematical logician, I am not called upon to have an\r\nopinion on this point. All that I am maintaining is that, if there\r\nare such things as heaps, we cannot identify them with the classes\r\ncomposed of their constituents.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe shall come much nearer to a satisfactory theory if we\r\ntry to identify classes with propositional functions. Every\r\nclass, as we explained in Chapter II., is defined by some propositional\r\nfunction which is true of the members of the class\r\nand false of other things. But if a class can be defined by one\r\npropositional function, it can equally well be defined by any\r\nother which is true whenever the first is true and false whenever\r\nthe first is false. For this reason the class cannot be identified\r\nwith any one such propositional function rather than with\r\nany other\u0026mdash;and given a propositional function, there are always\r\nmany others which are true when it is true and false when it is\r\nfalse. We say that two propositional functions are \"formally\r\nequivalent\" when this happens. Two \u003ci\u003epropositions\u003c/i\u003e are \"equivalent\"\r\n\u003cspan class=\"pagenum\" id=\"Page_183\"\u003e[Pg 183]\u003c/span\u003e\r\nwhen both are true or both false; two propositional\r\nfunctions \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e \u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e are \"formally equivalent\" when \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always\r\nequivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.767ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-302.png\" alt=\"\" data-tex=\"\\psi x\"\u003e.\u003c/span\u003e It is the fact that there are other functions\r\nformally equivalent to a given function that makes it impossible\r\nto identify a class with a function; for we wish classes to be such\r\nthat no two distinct classes have exactly the same members,\r\nand therefore two formally equivalent functions will have to\r\ndetermine the same class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhen we have decided that classes cannot be things of the\r\nsame sort as their members, that they cannot be just heaps or\r\naggregates, and also that they cannot be identified with propositional\r\nfunctions, it becomes very difficult to see what they\r\ncan be, if they are to be more than symbolic fictions. And if\r\nwe can find any way of dealing with them as symbolic fictions,\r\nwe increase the logical security of our position, since we avoid\r\nthe need of assuming that there are classes without being compelled\r\nto make the opposite assumption that there are no classes.\r\nWe merely abstain from both assumptions. This is an example\r\nof Occam\u0027s razor, namely, \"entities are not to be multiplied\r\nwithout necessity.\" But when we refuse to assert that there\r\nare classes, we must not be supposed to be asserting dogmatically\r\nthat there are none. We are merely agnostic as regards them:\r\nlike Laplace, we can say, \"\u003ci\u003eje n\u0027ai pas besoin de cette hypothèse.\u003c/i\u003e\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nLet us set forth the conditions that a symbol must fulfil if\r\nit is to serve as a class. I think the following conditions will\r\nbe found necessary and sufficient:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(1) Every propositional function must determine a class,\r\nconsisting of those arguments for which the function is true.\r\nGiven any proposition (true or false), say about Socrates, we\r\ncan imagine Socrates replaced by Plato or Aristotle or a gorilla\r\nor the man in the moon or any other individual in the world.\r\nIn general, some of these substitutions will give a true proposition\r\nand some a false one. The class determined will consist of all\r\nthose substitutions that give a true one. Of course, we have\r\nstill to decide what we mean by \"all those which, etc.\" All that\r\n\u003cspan class=\"pagenum\" id=\"Page_184\"\u003e[Pg 184]\u003c/span\u003e\r\nwe are observing at present is that a class is rendered determinate\r\nby a propositional function, and that every propositional function\r\ndetermines an appropriate class.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(2) Two formally equivalent propositional functions must\r\ndetermine the same class, and two which are not formally equivalent\r\nmust determine different classes. That is, a class is determined\r\nby its membership, and no two different classes can have\r\nthe same membership. (If a class is determined by a function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e\r\nwe say that \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is a \"member\" of the class if \u003cimg style=\"vertical-align: -0.464ex; width: 2.545ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-301.png\" alt=\"\" data-tex=\"\\phi a\"\u003e is true.)\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(3) We must find some way of defining not only classes, but\r\nclasses of classes. We saw in Chapter II. that cardinal numbers\r\nare to be defined as classes of classes. The ordinary phrase\r\nof elementary mathematics, \"The combinations of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e things\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e at a time\" represents a class of classes, namely, the class of\r\nall classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.986ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-47.png\" alt=\"\" data-tex=\"m\"\u003e terms that can be selected out of a given class\r\nof \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms. Without some symbolic method of dealing with\r\nclasses of classes, mathematical logic would break down.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(4) It must under all circumstances be meaningless (not false)\r\nto suppose a class a member of itself or not a member of itself.\r\nThis results from the contradiction which we discussed in\r\nChapter XIII.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n(5) Lastly\u0026mdash;and this is the condition which is most difficult\r\nof fulfilment,\u0026mdash;it must be possible to make propositions about\r\n\u003ci\u003eall\u003c/i\u003e the classes that are composed of individuals, or about \u003ci\u003eall\u003c/i\u003e the\r\nclasses that are composed of objects of any one logical \"type.\"\r\nIf this were not the case, many uses of classes would go astray\u0026mdash;for\r\nexample, mathematical induction. In defining the posterity\r\nof a given term, we need to be able to say that a member of the\r\nposterity belongs to \u003ci\u003eall\u003c/i\u003e hereditary classes to which the given\r\nterm belongs, and this requires the sort of totality that is in\r\nquestion. The reason there is a difficulty about this condition\r\nis that it can be proved to be impossible to speak of \u003ci\u003eall\u003c/i\u003e the propositional\r\nfunctions that can have arguments of a given type.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe will, to begin with, ignore this last condition and the\r\nproblems which it raises. The first two conditions may be\r\n\u003cspan class=\"pagenum\" id=\"Page_185\"\u003e[Pg 185]\u003c/span\u003e\r\ntaken together. They state that there is to be one class, no\r\nmore and no less, for each group of formally equivalent propositional\r\nfunctions; \u003ci\u003ee.g.\u003c/i\u003e the class of men is to be the same as\r\nthat of featherless bipeds or rational animals or Yahoos or whatever\r\nother characteristic may be preferred for defining a human\r\nbeing. Now, when we say that two formally equivalent propositional\r\nfunctions may be not identical, although they define\r\nthe same class, we may prove the truth of the assertion by pointing\r\nout that a statement may be true of the one function and\r\nfalse of the other; \u003ci\u003ee.g.\u003c/i\u003e \"I believe that all men are mortal\"\r\nmay be true, while \"I believe that all rational animals are\r\nmortal\" may be false, since I may believe falsely that the\r\nPhoenix is an immortal rational animal. Thus we are led to\r\nconsider \u003ci\u003estatements about functions\u003c/i\u003e, or (more correctly) \u003ci\u003efunctions\r\nof functions\u003c/i\u003e.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSome of the things that may be said about a function may\r\nbe regarded as said about the class defined by the function,\r\nwhereas others cannot. The statement \"all men are mortal\"\r\ninvolves the functions \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\" and \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is mortal\"; or,\r\nif we choose, we can say that it involves the classes \u003ci\u003emen\u003c/i\u003e and\r\n\u003ci\u003emortals\u003c/i\u003e. We can interpret the statement in either way, because\r\nits truth-value is unchanged if we substitute for \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\"\r\nor for \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is mortal\" any formally equivalent function. But,\r\nas we have just seen, the statement \"I believe that all men are\r\nmortal\" cannot be regarded as being about the class determined\r\nby either function, because its truth-value may be changed\r\nby the substitution of a formally equivalent function (which\r\nleaves the class unchanged). We will call a statement involving\r\na function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e an \"extensional\" function of the function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e if\r\nit is like \"all men are mortal,\" \u003ci\u003ei.e.\u003c/i\u003e if its truth-value is unchanged\r\nby the substitution of any formally equivalent function; and\r\nwhen a function of a function is not extensional, we will call it\r\n\"intensional,\" so that \"I believe that all men are mortal\"\r\nis an intensional function of \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\" or \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is mortal.\"\r\nThus \u003ci\u003eextensional\u003c/i\u003e functions of a function \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e may, for practical\r\n\u003cspan class=\"pagenum\" id=\"Page_186\"\u003e[Pg 186]\u003c/span\u003e\r\npurposes, be regarded as functions of the class determined by \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\r\nwhile \u003ci\u003eintensional\u003c/i\u003e functions cannot be so regarded.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is to be observed that all the \u003ci\u003especific\u003c/i\u003e functions of functions\r\nthat we have occasion to introduce in mathematical logic are\r\nextensional. Thus, for example, the two fundamental functions\r\nof functions are: \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always true\" and \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes\r\ntrue.\" Each of these has its truth-value unchanged if any\r\nformally equivalent function is substituted for \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e.\u003c/span\u003e In the\r\nlanguage of classes, if \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is the class determined\r\nby \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is\r\nalways true\" is equivalent to \"everything is a member of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e\"\r\nand \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is sometimes true\" is equivalent to \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has members\"\r\nor (better) \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e has at least one member.\" Take, again, the\r\ncondition, dealt with in the preceding chapter, for the existence\r\nof \"the term satisfying \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e.\u003c/span\u003e\" The condition is that there is a\r\nterm \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e such that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is always equivalent to \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e\" This\r\nis obviously extensional. It is equivalent to the assertion\r\nthat the class defined by the function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is a unit class, \u003ci\u003ei.e.\u003c/i\u003e a\r\nclass having one member; in other words, a class which is a\r\nmember of 1.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven a function of a function which may or may not be\r\nextensional, we can always derive from it a connected and\r\ncertainly extensional function of the same function, by the\r\nfollowing plan: Let our original function of a function be one\r\nwhich attributes to \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e the property \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e;\u003c/span\u003e then consider the assertion\r\n\"there is a function having the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e and formally\r\nequivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e.\u003c/span\u003e\" This is an extensional function of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e;\u003c/span\u003e it\r\nis true when our original statement is true, and it is formally\r\nequivalent to the original function of \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e if this original function\r\nis extensional; but when the original function is intensional,\r\nthe new one is more often true than the old one. For example,\r\nconsider again \"I believe that all men are mortal,\" regarded\r\nas a function of \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human.\" The derived extensional function\r\nis: \"There is a function formally equivalent to \u0027\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human\u0027\r\nand such that I believe that whatever satisfies it is mortal.\"\r\nThis remains true when we substitute \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a rational animal\"\r\n\u003cspan class=\"pagenum\" id=\"Page_187\"\u003e[Pg 187]\u003c/span\u003e\r\nfor \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is human,\" even if I believe falsely that the Phoenix is\r\nrational and immortal.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe give the name of \"derived extensional function\" to the\r\nfunction constructed as above, namely, to the function: \"There\r\nis a function having the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e and formally equivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e\"\r\nwhere the original function was \"the function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e has\r\nthe property \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e.\u003c/span\u003e\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may regard the derived extensional function as having\r\nfor its argument the class determined by the function \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e,\u003c/span\u003e and\r\nas asserting \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e of this class. This may be taken as the definition\r\nof a proposition about a class. \u003ci\u003eI.e.\u003c/i\u003e we may define:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTo assert that \"the class determined by the function \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\r\nhas the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e\" is to assert that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e satisfies the extensional\r\nfunction derived from \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis gives a meaning to any statement about a class which\r\ncan be made significantly about a function; and it will be\r\nfound that technically it yields the results which are required\r\nin order to make a theory symbolically satisfactory.\u003ca id=\"FNanchor_41_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_41_1\" class=\"fnanchor\"\u003e[41]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_41_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_41_1\"\u003e\u003cspan class=\"label\"\u003e[41]\u003c/span\u003e\u003c/a\u003eSee \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, vol. I. pp. 75-84 and * 20.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWhat we have said just now as regards the definition of\r\nclasses is sufficient to satisfy our first four conditions. The\r\nway in which it secures the third and fourth, namely, the possibility\r\nof classes of classes, and the impossibility of a class being\r\nor not being a member of itself, is somewhat technical; it is\r\nexplained in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, but may be taken for\r\ngranted here. It results that, but for our fifth condition, we\r\nmight regard our task as completed. But this condition\u0026mdash;at\r\nonce the most important and the most difficult\u0026mdash;is not fulfilled\r\nin virtue of anything we have said as yet. The difficulty is\r\nconnected with the theory of types, and must be briefly discussed.\u003ca id=\"FNanchor_42_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_42_1\" class=\"fnanchor\"\u003e[42]\u003c/a\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_42_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_42_1\"\u003e\u003cspan class=\"label\"\u003e[42]\u003c/span\u003e\u003c/a\u003eThe reader who desires a fuller discussion should consult \u003ci\u003ePrincipia\r\nMathematica\u003c/i\u003e, Introduction, chap. II.; also * 12.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe saw in Chapter XIII. that there is a hierarchy of logical\r\ntypes, and that it is a fallacy to allow an object belonging to\r\none of these to be substituted for an object belonging to another.\r\n\u003cspan class=\"pagenum\" id=\"Page_188\"\u003e[Pg 188]\u003c/span\u003e\r\nNow it is not difficult to show that the various functions which\r\ncan take a given object \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e as argument are not all of one type.\r\nLet us call them all \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions. We may take first those among\r\nthem which do not involve reference to any collection of functions;\r\nthese we will call \"predicative \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions.\" If we now proceed\r\nto functions involving reference to the totality of predicative\r\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions, we shall incur a fallacy if we regard these as of the\r\nsame type as the predicative \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions. Take such an everyday\r\nstatement as \"\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e is a typical Frenchman.\" How shall\r\nwe define a \"typical\" Frenchman? We may define him as\r\none \"possessing all qualities that are possessed by most French\r\nmen.\" But unless we confine \"all qualities\" to such as do not\r\ninvolve a reference to any totality of qualities, we shall have to\r\nobserve that most Frenchmen are \u003ci\u003enot\u003c/i\u003e typical in the above sense,\r\nand therefore the definition shows that to be not typical is\r\nessential to a typical Frenchman. This is not a logical contradiction,\r\nsince there is no reason why there should be any typical\r\nFrenchmen; but it illustrates the need for separating off\r\nqualities that involve reference to a totality of qualities from\r\nthose that do not.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWhenever, by statements about \"all\" or \"some\" of the\r\nvalues that a variable can significantly take, we generate a\r\nnew object, this new object must not be among the values which\r\nour previous variable could take, since, if it were, the totality\r\nof values over which the variable could range would only be\r\ndefinable in terms of itself, and we should be involved in a vicious\r\ncircle. For example, if I say \"Napoleon had all the qualities\r\nthat make a great general,\" I must define \"qualities\" in such a\r\nway that it will not include what I am now saying, \u003ci\u003ei.e.\u003c/i\u003e \"having\r\nall the qualities that make a great general\" must not be itself a\r\nquality in the sense supposed. This is fairly obvious, and is\r\nthe principle which leads to the theory of types by which vicious-circle\r\nparadoxes are avoided. As applied to \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions, we\r\nmay suppose that \"qualities\" is to mean \"predicative functions.\"\r\nThen when I say \"Napoleon had all the qualities, etc.,\" I mean\r\n\u003cspan class=\"pagenum\" id=\"Page_189\"\u003e[Pg 189]\u003c/span\u003e\r\n\"Napoleon satisfied all the predicative functions, etc.\" This\r\nstatement attributes a property to Napoleon, but not a predicative\r\nproperty; thus we escape the vicious circle. But\r\nwherever \"all functions which\" occurs, the functions in question\r\nmust be limited to one type if a vicious circle is to be avoided;\r\nand, as Napoleon and the typical Frenchman have shown, the\r\ntype is not rendered determinate by that of the argument. It\r\nwould require a much fuller discussion to set forth this point\r\nfully, but what has been said may suffice to make it clear that\r\nthe functions which can take a given argument are of an infinite\r\nseries of types. We could, by various technical devices, construct\r\na variable which would run through the first \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e of these\r\ntypes, where \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is finite, but we cannot construct a variable which\r\nwill run through them all, and, if we could, that mere fact would\r\nat once generate a new type of function with the same arguments,\r\nand would set the whole process going again.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe call predicative \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions the \u003ci\u003efirst\u003c/i\u003e type of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions;\r\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions involving reference to the totality of the first type\r\nwe call the \u003ci\u003esecond\u003c/i\u003e type; and so on. No variable \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function\r\ncan run through all these different types: it must stop short at\r\nsome definite one.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThese considerations are relevant to our definition of the\r\nderived extensional function. We there spoke of \"a function\r\nformally equivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e.\u003c/span\u003e\" It is necessary to decide upon\r\nthe type of our function. Any decision will do, but some decision\r\nis unavoidable. Let us call the supposed formally equivalent\r\nfunction \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e.\u003c/span\u003e Then \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e appears as a variable, and must be of\r\nsome determinate type. All that we know necessarily about\r\nthe type of \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is that it takes arguments of a given type\u0026mdash;that\r\nit is (say) an \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function. But this, as we have just seen, does\r\nnot determine its type. If we are to be able (as our fifth requisite\r\ndemands) to deal with \u003ci\u003eall\u003c/i\u003e classes whose members are of the same\r\ntype as \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e,\u003c/span\u003e we must be able to define all such classes by means of\r\nfunctions of some one type; that is to say, there must be some\r\ntype of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function, say the \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-311.png\" alt=\"\" data-tex=\"n^\\mathord{th}\"\u003e,\u003c/span\u003e such that any \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function is formally\r\n\u003cspan class=\"pagenum\" id=\"Page_190\"\u003e[Pg 190]\u003c/span\u003e\r\nequivalent to some \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function of the \u003cimg style=\"vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-311.png\" alt=\"\" data-tex=\"n^\\mathord{th}\"\u003e type. If this is the case,\r\nthen any extensional function which holds of all \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions\r\nof the \u003cimg style=\"vertical-align: -0.025ex; width: 3.044ex; height: 1.956ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-311.png\" alt=\"\" data-tex=\"n^\\mathord{th}\"\u003e type will hold of any \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function whatever. It is chiefly\r\nas a technical means of embodying an assumption leading to\r\nthis result that classes are useful. The assumption is called the\r\n\"axiom of reducibility,\" and may be stated as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"There is a type (\u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e say) of \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions such that, given any\r\n\u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-function, it is formally equivalent to some function of the type\r\nin question.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIf this axiom is assumed, we use functions of this type in\r\ndefining our associated extensional function. Statements about\r\nall \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-classes (\u003ci\u003ei.e.\u003c/i\u003e all classes defined by \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions) can be reduced\r\nto statements about all \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions of the type \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e.\u003c/span\u003e So long as\r\nonly extensional functions of functions are involved, this gives\r\nus in practice results which would otherwise have required the\r\nimpossible notion of \"all \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e-functions.\" One particular region\r\nwhere this is vital is mathematical induction.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe axiom of reducibility involves all that is really essential\r\nin the theory of classes. It is therefore worth while to ask\r\nwhether there is any reason to suppose it true.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis axiom, like the multiplicative axiom and the axiom\r\nof infinity, is necessary for certain results, but not for the bare\r\nexistence of deductive reasoning. The theory of deduction,\r\nas explained in Chapter XIV., and the laws for propositions\r\ninvolving \"all\" and \"some,\" are of the very texture of mathematical\r\nreasoning: without them, or something like them,\r\nwe should not merely not obtain the same results, but we should\r\nnot obtain any results at all. We cannot use them as hypotheses,\r\nand deduce hypothetical consequences, for they are\r\nrules of deduction as well as premisses. They must be absolutely\r\ntrue, or else what we deduce according to them does not even\r\nfollow from the premisses. On the other hand, the axiom of\r\nreducibility, like our two previous mathematical axioms, could\r\nperfectly well be stated as an hypothesis whenever it is used,\r\ninstead of being assumed to be actually true. We can deduce\r\n\u003cspan class=\"pagenum\" id=\"Page_191\"\u003e[Pg 191]\u003c/span\u003e\r\nits consequences hypothetically; we can also deduce the consequences\r\nof supposing it false. It is therefore only convenient,\r\nnot necessary. And in view of the complication of the theory\r\nof types, and of the uncertainty of all except its most general\r\nprinciples, it is impossible as yet to say whether there may\r\nnot be some way of dispensing with the axiom of reducibility\r\naltogether. However, assuming the correctness of the theory\r\noutlined above, what can we say as to the truth or falsehood of\r\nthe axiom?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe axiom, we may observe, is a generalised form of Leibniz\u0027s\r\nidentity of indiscernibles. Leibniz assumed, as a logical principle,\r\nthat two different subjects must differ as to predicates. Now\r\npredicates are only some among what we called \"predicative\r\nfunctions,\" which will include also relations to given terms,\r\nand various properties not to be reckoned as predicates. Thus\r\nLeibniz\u0027s assumption is a much stricter and narrower one than\r\nours. (Not, of course, according to \u003ci\u003ehis\u003c/i\u003e logic, which regarded\r\n\u003ci\u003eall\u003c/i\u003e propositions as reducible to the subject-predicate form.)\r\nBut there is no good reason for believing his form, so far as I can\r\nsee. There might quite well, as a matter of abstract logical\r\npossibility, be two things which had exactly the same predicates,\r\nin the narrow sense in which we have been using the word \"predicate.\"\r\nHow does our axiom look when we pass beyond predicates\r\nin this narrow sense? In the actual world there seems\r\nno way of doubting its empirical truth as regards particulars,\r\nowing to spatio-temporal differentiation: no two particulars\r\nhave exactly the same spatial and temporal relations to all other\r\nparticulars. But this is, as it were, an accident, a fact about\r\nthe world in which we happen to find ourselves. Pure logic,\r\nand pure mathematics (which is the same thing), aims at being\r\ntrue, in Leibnizian phraseology, in all possible worlds, not only\r\nin this higgledy-piggledy job-lot of a world in which chance has\r\nimprisoned us. There is a certain lordliness which the logician\r\nshould preserve: he must not condescend to derive arguments\r\nfrom the things he sees about him.\r\n\u003cspan class=\"pagenum\" id=\"Page_192\"\u003e[Pg 192]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nViewed from this strictly logical point of view, I do not see\r\nany reason to believe that the axiom of reducibility is logically\r\nnecessary, which is what would be meant by saying that it is\r\ntrue in all possible worlds. The admission of this axiom into\r\na system of logic is therefore a defect, even if the axiom is empirically\r\ntrue. It is for this reason that the theory of classes cannot\r\nbe regarded as being as complete as the theory of descriptions.\r\nThere is need of further work on the theory of types, in the hope\r\nof arriving at a doctrine of classes which does not require such a\r\ndubious assumption. But it is reasonable to regard the theory\r\noutlined in the present chapter as right in its main lines, \u003ci\u003ei.e.\u003c/i\u003e in\r\nits reduction of propositions nominally about classes to propositions\r\nabout their defining functions. The avoidance of\r\nclasses as entities by this method must, it would seem, be sound\r\nin principle, however the detail may still require adjustment.\r\nIt is because this seems indubitable that we have included the\r\ntheory of classes, in spite of our desire to exclude, as far as possible,\r\nwhatever seemed open to serious doubt.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe theory of classes, as above outlined, reduces itself to one\r\naxiom and one definition. For the sake of definiteness, we will\r\nhere repeat them. The axiom is:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eThere is a type \u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e such that if \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is a function which can take a\r\ngiven object\u003c/i\u003e \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e \u003ci\u003eas argument, then there is a function \u003cimg style=\"vertical-align: -0.464ex; width: 1.473ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-307.png\" alt=\"\" data-tex=\"\\psi\"\u003e\r\nof the type \u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e which is formally equivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e.\u003c/span\u003e\u003c/i\u003e\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe definition is:\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\u003ci\u003eIf \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is a function which can take a given object\u003c/i\u003e \u003cimg style=\"vertical-align: -0.023ex; width: 1.197ex; height: 1.02ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-85.png\" alt=\"\" data-tex=\"a\"\u003e \u003ci\u003eas argument,\r\nand \u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e the type mentioned in the above axiom, then to say that\r\nthe class determined by \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e has the property \u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e is to say that there\r\nis a function of type \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.029ex; width: 1.17ex; height: 1.005ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-312.png\" alt=\"\" data-tex=\"\\tau\"\u003e,\u003c/span\u003e formally equivalent to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e,\u003c/span\u003e and having the\r\nproperty \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.244ex; height: 2.059ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-310.png\" alt=\"\" data-tex=\"f\"\u003e.\u003c/span\u003e\u003c/i\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_193\"\u003e[Pg 193]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027XVIII: MATHEMATICS AND LOGIC\u0027\u003e\u003ca id=\"chap18\"\u003e\u003c/a\u003eCHAPTER XVIII\r\n\u003cbr\u003e\u003cbr\u003e\r\nMATHEMATICS AND LOGIC\u003c/h2\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\r\nMATHEMATICS and logic, historically speaking, have been entirely\r\ndistinct studies. Mathematics has been connected with science,\r\nlogic with Greek. But both have developed in modern times:\r\nlogic has become more mathematical and mathematics has\r\nbecome more logical. The consequence is that it has now become\r\nwholly impossible to draw a line between the two; in fact, the\r\ntwo are one. They differ as boy and man: logic is the youth\r\nof mathematics and mathematics is the manhood of logic. This\r\nview is resented by logicians who, having spent their time in\r\nthe study of classical texts, are incapable of following a piece\r\nof symbolic reasoning, and by mathematicians who have learnt\r\na technique without troubling to inquire into its meaning or\r\njustification. Both types are now fortunately growing rarer.\r\nSo much of modern mathematical work is obviously on the\r\nborder-line of logic, so much of modern logic is symbolic and\r\nformal, that the very close relationship of logic and mathematics\r\nhas become obvious to every instructed student. The proof\r\nof their identity is, of course, a matter of detail: starting with\r\npremisses which would be universally admitted to belong to\r\nlogic, and arriving by deduction at results which as obviously\r\nbelong to mathematics, we find that there is no point at which\r\na sharp line can be drawn, with logic to the left and mathematics\r\nto the right. If there are still those who do not admit\r\nthe identity of logic and mathematics, we may challenge them\r\nto indicate at what point, in the successive definitions and\r\n\u003cspan class=\"pagenum\" id=\"Page_194\"\u003e[Pg 194]\u003c/span\u003e\r\ndeductions of \u003ci\u003ePrincipia Mathematica\u003c/i\u003e, they consider that logic\r\nends and mathematics begins. It will then be obvious that any\r\nanswer must be quite arbitrary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn the earlier chapters of this book, starting from the natural\r\nnumbers, we have first defined \"cardinal number\" and shown\r\nhow to generalise the conception of number, and have then\r\nanalysed the conceptions involved in the definition, until we found\r\nourselves dealing with the fundamentals of logic. In a synthetic,\r\ndeductive treatment these fundamentals come first, and the\r\nnatural numbers are only reached after a long journey. Such\r\ntreatment, though formally more correct than that which we\r\nhave adopted, is more difficult for the reader, because the ultimate\r\nlogical concepts and propositions with which it starts are remote\r\nand unfamiliar as compared with the natural numbers. Also\r\nthey represent the present frontier of knowledge, beyond which\r\nis the still unknown; and the dominion of knowledge over them\r\nis not as yet very secure.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt used to be said that mathematics is the science of \"quantity.\"\r\n\"Quantity\" is a vague word, but for the sake of argument\r\nwe may replace it by the word \"number.\" The statement\r\nthat mathematics is the science of number would be untrue\r\nin two different ways. On the one hand, there are recognised\r\nbranches of mathematics which have nothing to do with number\u0026mdash;all\r\ngeometry that does not use co-ordinates or measurement,\r\nfor example: projective and descriptive geometry, down to\r\nthe point at which co-ordinates are introduced, does not have\r\nto do with number, or even with quantity in the sense of \u003ci\u003egreater\u003c/i\u003e\r\nand \u003ci\u003eless\u003c/i\u003e. On the other hand, through the definition of cardinals,\r\nthrough the theory of induction and ancestral relations, through\r\nthe general theory of series, and through the definitions of the\r\narithmetical operations, it has become possible to generalise much\r\nthat used to be proved only in connection with numbers. The\r\nresult is that what was formerly the single study of Arithmetic\r\nhas now become divided into numbers of separate studies, no\r\none of which is specially concerned with numbers. The most\r\n\u003cspan class=\"pagenum\" id=\"Page_195\"\u003e[Pg 195]\u003c/span\u003e\r\nelementary properties of numbers are concerned with one-one\r\nrelations, and similarity between classes. Addition is concerned\r\nwith the construction of mutually exclusive classes respectively\r\nsimilar to a set of classes which are not known to be mutually\r\nexclusive. Multiplication is merged in the theory of \"selections,\"\r\n\u003ci\u003ei.e.\u003c/i\u003e of a certain kind of one-many relations. Finitude\r\nis merged in the general study of ancestral relations, which yields\r\nthe whole theory of mathematical induction. The ordinal\r\nproperties of the various kinds of number-series, and the elements\r\nof the theory of continuity of functions and the limits of functions,\r\ncan be generalised so as no longer to involve any essential reference\r\nto numbers. It is a principle, in all formal reasoning, to generalise\r\nto the utmost, since we thereby secure that a given process of\r\ndeduction shall have more widely applicable results; we are,\r\ntherefore, in thus generalising the reasoning of arithmetic,\r\nmerely following a precept which is universally admitted in\r\nmathematics. And in thus generalising we have, in effect,\r\ncreated a set of new deductive systems, in which traditional\r\narithmetic is at once dissolved and enlarged; but whether any\r\none of these new deductive systems\u0026mdash;for example, the theory of\r\nselections\u0026mdash;is to be said to belong to logic or to arithmetic is\r\nentirely arbitrary, and incapable of being decided rationally.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe are thus brought face to face with the question: What\r\nis this subject, which may be called indifferently either mathematics\r\nor logic? Is there any way in which we can define it?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nCertain characteristics of the subject are clear. To begin\r\nwith, we do not, in this subject, deal with particular things or\r\nparticular properties: we deal formally with what can be said\r\nabout \u003ci\u003eany\u003c/i\u003e thing or \u003ci\u003eany\u003c/i\u003e property. We are prepared to say that\r\none and one are two, but not that Socrates and Plato are two,\r\nbecause, in our capacity of logicians or pure mathematicians,\r\nwe have never heard of Socrates and Plato. A world in which\r\nthere were no such individuals would still be a world in which\r\none and one are two. It is not open to us, as pure mathematicians\r\nor logicians, to mention anything at all, because, if we do so,\r\n\u003cspan class=\"pagenum\" id=\"Page_196\"\u003e[Pg 196]\u003c/span\u003e\r\nwe introduce something irrelevant and not formal. We may\r\nmake this clear by applying it to the case of the syllogism.\r\nTraditional logic says: \"All men are mortal, Socrates is a man,\r\ntherefore Socrates is mortal.\" Now it is clear that what we\r\n\u003ci\u003emean\u003c/i\u003e to assert, to begin with, is only that the premisses imply\r\nthe conclusion, not that premisses and conclusion are actually\r\ntrue; even the most traditional logic points out that the actual\r\ntruth of the premisses is irrelevant to logic. Thus the first\r\nchange to be made in the above traditional syllogism is to state\r\nit in the form: \"If all men are mortal and Socrates is a man,\r\nthen Socrates is mortal.\" We may now observe that it is intended\r\nto convey that this argument is valid in virtue of its \u003ci\u003eform\u003c/i\u003e, not\r\nin virtue of the particular terms occurring in it. If we had\r\nomitted \"Socrates is a man\" from our premisses, we should\r\nhave had a non-formal argument, only admissible because\r\nSocrates is in fact a man; in that case we could not have generalised\r\nthe argument. But when, as above, the argument is \u003ci\u003eformal\u003c/i\u003e,\r\nnothing depends upon the terms that occur in it. Thus we may\r\nsubstitute \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e for \u003ci\u003emen\u003c/i\u003e, \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e for \u003ci\u003emortals\u003c/i\u003e, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e for Socrates, where\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e are any classes whatever, and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is any individual. We\r\nthen arrive at the statement: \"No matter what possible values\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e may have, if all \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es are\r\n\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003es and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is\r\na \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\"; in other words, \"the propositional function \u0027if all \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es\r\nare \u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e and \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e then \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003e is always true.\" Here at last\r\nwe have a proposition of logic\u0026mdash;the one which is only \u003ci\u003esuggested\u003c/i\u003e by\r\nthe traditional statement about Socrates and men and mortals.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that, if \u003ci\u003eformal\u003c/i\u003e reasoning is what we are aiming at,\r\nwe shall always arrive ultimately at statements like the above,\r\nin which no actual things or properties are mentioned; this\r\nwill happen through the mere desire not to waste our time proving\r\nin a particular case what can be proved generally. It would be\r\nridiculous to go through a long argument about Socrates, and then\r\ngo through precisely the same argument again about Plato. If\r\nour argument is one (say) which holds of all men, we shall prove\r\nit concerning \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e,\u003c/span\u003e\" with the hypothesis \"if \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a man.\" With\r\n\u003cspan class=\"pagenum\" id=\"Page_197\"\u003e[Pg 197]\u003c/span\u003e\r\nthis hypothesis, the argument will retain its hypothetical validity\r\neven when \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is not a man. But now we shall find that our argument\r\nwould still be valid if, instead of supposing \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e to be a man,\r\nwe were to suppose him to be a monkey or a goose or a Prime\r\nMinister. We shall therefore not waste our time taking as our\r\npremiss \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a man\" but shall take \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e,\u003c/span\u003e\"\r\nwhere \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e is any\r\nclass of individuals, or \"\u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e\" where \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is any propositional\r\nfunction of some assigned type. Thus the absence of all mention\r\nof particular things or properties in logic or pure mathematics\r\nis a necessary result of the fact that this study is, as we say,\r\n\"purely formal.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAt this point we find ourselves faced with a problem which\r\nis easier to state than to solve. The problem is: \"What are\r\nthe constituents of a logical proposition?\" I do not know the\r\nanswer, but I propose to explain how the problem arises.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nTake (say) the proposition \"Socrates was before Aristotle.\"\r\nHere it seems obvious that we have a relation between two terms,\r\nand that the constituents of the proposition (as well as of the\r\ncorresponding fact) are simply the two terms and the relation,\r\n\u003ci\u003ei.e.\u003c/i\u003e Socrates, Aristotle, and \u003ci\u003ebefore\u003c/i\u003e. (I ignore the fact that\r\nSocrates and Aristotle are not simple; also the fact that what\r\nappear to be their names are really truncated descriptions.\r\nNeither of these facts is relevant to the present issue.) We may\r\nrepresent the general form of such propositions by \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-313.png\" alt=\"\" data-tex=\"x\\mathrm Ry\"\u003e,\u003c/span\u003e\"\r\nwhich may be read \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e has the relation \u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e.\u003c/span\u003e\" This general\r\nform may occur in logical propositions, but no particular instance\r\nof it can occur. Are we to infer that the general form itself is a\r\nconstituent of such logical propositions?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nGiven a proposition, such as \"Socrates is before Aristotle,\"\r\nwe have certain constituents and also a certain form. But the\r\nform is not itself a new constituent; if it were, we should need a\r\nnew form to embrace both it and the other constituents. We\r\ncan, in fact, turn \u003ci\u003eall\u003c/i\u003e the constituents of a proposition into\r\nvariables, while keeping the form unchanged. This is what we\r\ndo when we use such a schema as \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-313.png\" alt=\"\" data-tex=\"x\\mathrm Ry\"\u003e,\u003c/span\u003e\" which stands for any\r\n\u003cspan class=\"pagenum\" id=\"Page_198\"\u003e[Pg 198]\u003c/span\u003e\r\none of a certain class of propositions, namely, those asserting\r\nrelations between two terms. We can proceed to general assertions,\r\nsuch as \"\u003cimg style=\"vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-313.png\" alt=\"\" data-tex=\"x\\mathrm Ry\"\u003e is sometimes true\"\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e there are cases\r\nwhere dual relations hold. This assertion will belong to logic\r\n(or mathematics) in the sense in which we are using the word.\r\nBut in this assertion we do not mention any particular things\r\nor particular relations; no particular things or relations can\r\never enter into a proposition of pure logic. We are left with pure\r\n\u003ci\u003eforms\u003c/i\u003e as the only possible constituents of logical propositions.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nI do not wish to assert positively that pure forms\u0026mdash;\u003ci\u003ee.g.\u003c/i\u003e the\r\nform \"\u003cimg style=\"vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-313.png\" alt=\"\" data-tex=\"x\\mathrm Ry\"\u003e\"\u0026mdash;do actually enter into propositions of the kind\r\nwe are considering. The question of the analysis of such propositions\r\nis a difficult one, with conflicting considerations on the\r\none side and on the other. We cannot embark upon this question\r\nnow, but we may accept, as a first approximation, the view\r\nthat \u003ci\u003eforms\u003c/i\u003e are what enter into logical propositions as their\r\nconstituents. And we may explain (though not formally define)\r\nwhat we mean by the \"form\" of a proposition as follows:\u0026mdash;\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe \"form\" of a proposition is that, in it, that remains unchanged\r\nwhen every constituent of the proposition is replaced\r\nby another.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThus \"Socrates is earlier than Aristotle\" has the same form\r\nas \"Napoleon is greater than Wellington,\" though every constituent\r\nof the two propositions is different.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may thus lay down, as a necessary (though not sufficient)\r\ncharacteristic of logical or mathematical propositions, that they\r\nare to be such as can be obtained from a proposition containing\r\nno variables (\u003ci\u003ei.e.\u003c/i\u003e no such words as \u003ci\u003eall\u003c/i\u003e, \u003ci\u003esome\u003c/i\u003e, \u003ci\u003ea\u003c/i\u003e, \u003ci\u003ethe\u003c/i\u003e, etc.) by turning\r\nevery constituent into a variable and asserting that the result\r\nis always true or sometimes true, or that it is always true in\r\nrespect of some of the variables that the result is sometimes true\r\nin respect of the others, or any variant of these forms. And\r\nanother way of stating the same thing is to say that logic (or\r\nmathematics) is concerned only with \u003ci\u003eforms\u003c/i\u003e, and is concerned\r\nwith them only in the way of stating that they are always or\r\n\u003cspan class=\"pagenum\" id=\"Page_199\"\u003e[Pg 199]\u003c/span\u003e\r\nsometimes true\u0026mdash;with all the permutations of \"always\" and\r\n\"sometimes\" that may occur.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThere are in every language some words whose sole function is\r\nto indicate form. These words, broadly speaking, are commonest\r\nin languages having fewest inflections. Take \"Socrates is\r\nhuman.\" Here \"is\" is not a constituent of the proposition,\r\nbut merely indicates the subject-predicate form. Similarly\r\nin \"Socrates is earlier than Aristotle,\" \"is\" and \"than\"\r\nmerely indicate form; the proposition is the same as \"Socrates\r\nprecedes Aristotle,\" in which these words have disappeared\r\nand the form is otherwise indicated. Form, as a rule, \u003ci\u003ecan\u003c/i\u003e be\r\nindicated otherwise than by specific words: the order of the\r\nwords can do most of what is wanted. But this principle\r\nmust not be pressed. For example, it is difficult to see how we\r\ncould conveniently express molecular forms of propositions\r\n(\u003ci\u003ei.e.\u003c/i\u003e what we call \"truth-functions\") without any word at all.\r\nWe saw in Chapter XIV. that one word or symbol is enough for\r\nthis purpose, namely, a word or symbol expressing \u003ci\u003eincompatibility\u003c/i\u003e.\r\nBut without even one we should find ourselves in difficulties.\r\nThis, however, is not the point that is important for\r\nour present purpose. What is important for us is to observe\r\nthat form may be the one concern of a general proposition,\r\neven when no word or symbol in that proposition designates\r\nthe form. If we wish to speak about the form itself, we must\r\nhave a word for it; but if, as in mathematics, we wish to speak\r\nabout all propositions that have the form, a word for the form\r\nwill usually be found not indispensable; probably in theory it\r\nis \u003ci\u003enever\u003c/i\u003e indispensable.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nAssuming\u0026mdash;as I think we may\u0026mdash;that the forms of propositions\r\ncan be represented by the forms of the propositions in which\r\nthey are expressed without any special word for forms, we should\r\narrive at a language in which everything formal belonged to\r\nsyntax and not to vocabulary. In such a language we could\r\nexpress \u003ci\u003eall\u003c/i\u003e the propositions of mathematics even if we did not\r\nknow one single word of the language. The language of mathematical\r\n\u003cspan class=\"pagenum\" id=\"Page_200\"\u003e[Pg 200]\u003c/span\u003e\r\nlogic, if it were perfected, would be such a language.\r\nWe should have symbols for variables, such as \"\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e\" and \"\u003cimg style=\"vertical-align: -0.05ex; width: 1.665ex; height: 1.595ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-72.png\" alt=\"\" data-tex=\"\\mathrm R\"\u003e\"\r\nand \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.109ex; height: 1.464ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-57.png\" alt=\"\" data-tex=\"y\"\u003e,\u003c/span\u003e\" arranged in various ways; and the way of arrangement\r\nwould indicate that something was being said to be true of\r\nall values or some values of the variables. We should not need\r\nto know any words, because they would only be needed for giving\r\nvalues to the variables, which is the business of the applied\r\nmathematician, not of the pure mathematician or logician.\r\nIt is one of the marks of a proposition of logic that, given a\r\nsuitable language, such a proposition can be asserted in such a\r\nlanguage by a person who knows the syntax without knowing\r\na single word of the vocabulary.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut, after all, there are words that express form, such as \"is\"\r\nand \"than.\" And in every symbolism hitherto invented for\r\nmathematical logic there are symbols having constant formal\r\nmeanings. We may take as an example the symbol for incompatibility\r\nwhich is employed in building up truth-functions.\r\nSuch words or symbols may occur in logic. The question is:\r\nHow are we to define them?\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nSuch words or symbols express what are called \"logical\r\nconstants.\" Logical constants may be defined exactly as\r\nwe defined forms; in fact, they are in essence the same thing.\r\nA fundamental logical constant will be that which is in common\r\namong a number of propositions, any one of which can result\r\nfrom any other by substitution of terms one for another. For\r\nexample, \"Napoleon is greater than Wellington\" results from\r\n\"Socrates is earlier than Aristotle\" by the substitution of\r\n\"Napoleon\" for \"Socrates,\" \"Wellington\" for \"Aristotle,\"\r\nand \"greater\" for \"earlier.\" Some propositions can be obtained\r\nin this way from the prototype \"Socrates is earlier than Aristotle\"\r\nand some cannot; those that can are those that are of\r\nthe form \"\u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 4.068ex; height: 2.009ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-313.png\" alt=\"\" data-tex=\"x\\mathrm Ry\"\u003e,\u003c/span\u003e\" \u003ci\u003ei.e.\u003c/i\u003e express dual relations. We cannot obtain\r\nfrom the above prototype by term-for-term substitution such\r\npropositions as \"Socrates is human\" or \"the Athenians gave\r\nthe hemlock to Socrates,\" because the first is of the subject-predicate\r\n\u003cspan class=\"pagenum\" id=\"Page_201\"\u003e[Pg 201]\u003c/span\u003e\r\nform and the second expresses a three-term relation.\r\nIf we are to have any words in our pure logical language, they\r\nmust be such as express \"logical constants,\" and \"logical\r\nconstants\" will always either be, or be derived from, what is in\r\ncommon among a group of propositions derivable from each\r\nother, in the above manner, by term-for-term substitution. And\r\nthis which is in common is what we call \"form.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIn this sense all the \"constants\" that occur in pure mathematics\r\nare logical constants. The number 1, for example, is\r\nderivative from propositions of the form: \"There is a term \u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e\r\nsuch that \u003cimg style=\"vertical-align: -0.464ex; width: 2.643ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-300.png\" alt=\"\" data-tex=\"\\phi x\"\u003e is true when, and only when, \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 0.98ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-94.png\" alt=\"\" data-tex=\"c\"\u003e.\u003c/span\u003e\" This is a\r\nfunction of \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e,\u003c/span\u003e and various different propositions result from\r\ngiving different values to \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e.\u003c/span\u003e We may (with a little omission\r\nof intermediate steps not relevant to our present purpose) take\r\nthe above function of \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e as what is meant by \"the class determined\r\nby \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is a unit class\" or \"the class determined by \u003cimg style=\"vertical-align: -0.464ex; width: 1.348ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-306.png\" alt=\"\" data-tex=\"\\phi\"\u003e is a\r\nmember of 1\" (1 being a class of classes). In this way, propositions\r\nin which 1 occurs acquire a meaning which is derived from\r\na certain constant logical form. And the same will be found\r\nto be the case with all mathematical constants: all are logical\r\nconstants, or symbolic abbreviations whose full use in a proper\r\ncontext is defined by means of logical constants.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nBut although all logical (or mathematical) propositions can\r\nbe expressed wholly in terms of logical constants together with\r\nvariables, it is not the case that, conversely, all propositions\r\nthat can be expressed in this way are logical. We have found\r\nso far a necessary but not a sufficient criterion of mathematical\r\npropositions. We have sufficiently defined the character of the\r\nprimitive \u003ci\u003eideas\u003c/i\u003e in terms of which all the ideas of mathematics\r\ncan be \u003ci\u003edefined\u003c/i\u003e, but not of the primitive \u003ci\u003epropositions\u003c/i\u003e from which\r\nall the propositions of mathematics can be \u003ci\u003ededuced\u003c/i\u003e. This is a\r\nmore difficult matter, as to which it is not yet known what the\r\nfull answer is.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nWe may take the axiom of infinity as an example of a proposition\r\nwhich, though it can be enunciated in logical terms,\r\n\u003cspan class=\"pagenum\" id=\"Page_202\"\u003e[Pg 202]\u003c/span\u003e\r\ncannot be asserted by logic to be true. All the propositions of\r\nlogic have a characteristic which used to be expressed by saying\r\nthat they were analytic, or that their contradictories were self-contradictory.\r\nThis mode of statement, however, is not satisfactory.\r\nThe law of contradiction is merely one among logical\r\npropositions; it has no special pre-eminence; and the proof\r\nthat the contradictory of some proposition is self-contradictory\r\nis likely to require other principles of deduction besides the\r\nlaw of contradiction. Nevertheless, the characteristic of logical\r\npropositions that we are in search of is the one which was felt,\r\nand intended to be defined, by those who said that it consisted\r\nin deducibility from the law of contradiction. This characteristic,\r\nwhich, for the moment, we may call \u003ci\u003etautology\u003c/i\u003e, obviously\r\ndoes not belong to the assertion that the number of individuals\r\nin the universe is \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e,\u003c/span\u003e whatever number \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e may be. But for the\r\ndiversity of types, it would be possible to prove logically that\r\nthere are classes of \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e terms, where \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e is any finite integer; or even\r\nthat there are classes of \u003cimg style=\"vertical-align: -0.375ex; width: 2.37ex; height: 1.945ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-170.png\" alt=\"\" data-tex=\"\\aleph_{0}\"\u003e terms. But, owing to types, such\r\nproofs, as we saw in Chapter XIII., are fallacious. We are left\r\nto empirical observation to determine whether there are as many\r\nas \u003cimg style=\"vertical-align: -0.025ex; width: 1.357ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-46.png\" alt=\"\" data-tex=\"n\"\u003e individuals in the world. Among \"possible\" worlds,\r\nin the Leibnizian sense, there will be worlds having one, two,\r\nthree, … individuals. There does not even seem any logical\r\nnecessity why there should be even one individual\u003ca id=\"FNanchor_43_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_43_1\" class=\"fnanchor\"\u003e[43]\u003c/a\u003e\u0026mdash;why, in\r\nfact, there should be any world at all. The ontological proof\r\nof the existence of God, if it were valid, would establish the\r\nlogical necessity of at least one individual. But it is generally\r\nrecognised as invalid, and in fact rests upon a mistaken view of\r\nexistence\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e it fails to realise that existence can only be asserted\r\nof something described, not of something named, so that it is\r\nmeaningless to argue from \"this is the so-and-so\" and \"the\r\nso-and-so exists\" to \"this exists.\" If we reject the ontological\r\n\u003cspan class=\"pagenum\" id=\"Page_203\"\u003e[Pg 203]\u003c/span\u003e\r\nargument, we seem driven to conclude that the existence of a\r\nworld is an accident\u0026mdash;\u003ci\u003ei.e.\u003c/i\u003e it is not logically necessary. If that\r\nbe so, no principle of logic can assert \"existence\" except under\r\na hypothesis, \u003ci\u003ei.e.\u003c/i\u003e none can be of the form \"the propositional\r\nfunction so-and-so is sometimes true.\" Propositions of this\r\nform, when they occur in logic, will have to occur as hypotheses\r\nor consequences of hypotheses, not as complete asserted propositions.\r\nThe complete asserted propositions of logic will all\r\nbe such as affirm that some propositional function is \u003ci\u003ealways\u003c/i\u003e true.\r\nFor example, it is always true that if \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e and \u003cimg style=\"vertical-align: -0.439ex; width: 1.041ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-280.png\" alt=\"\" data-tex=\"q\"\u003e implies \u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e\r\nthen \u003cimg style=\"vertical-align: -0.439ex; width: 1.138ex; height: 1.439ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-136.png\" alt=\"\" data-tex=\"p\"\u003e implies \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.02ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-164.png\" alt=\"\" data-tex=\"r\"\u003e,\u003c/span\u003e or that, if all \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e\u0027\u003c/span\u003es are \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e\u0027\u003c/span\u003es\r\nand \u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is an \u003cimg style=\"vertical-align: -0.025ex; width: 1.448ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-63.png\" alt=\"\" data-tex=\"\\alpha\"\u003e then\r\n\u003cimg style=\"vertical-align: -0.025ex; width: 1.294ex; height: 1.025ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-56.png\" alt=\"\" data-tex=\"x\"\u003e is a \u003cspan class=\"nowrap\"\u003e\u003cimg style=\"vertical-align: -0.439ex; width: 1.281ex; height: 2.034ex;\" src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-introduction-to-mathematical-philosophy-64.png\" alt=\"\" data-tex=\"\\beta\"\u003e.\u003c/span\u003e Such propositions may occur in logic, and their truth\r\nis independent of the existence of the universe. We may lay\r\nit down that, if there were no universe, \u003ci\u003eall\u003c/i\u003e general propositions\r\nwould be true; for the contradictory of a general proposition\r\n(as we saw in Chapter XV.) is a proposition asserting existence,\r\nand would therefore always be false if no universe existed.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_43_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_43_1\"\u003e\u003cspan class=\"label\"\u003e[43]\u003c/span\u003e\u003c/a\u003eThe primitive propositions in \u003ci\u003ePrincipia Mathematica\u003c/i\u003e are such as to\r\nallow the inference that at least one individual exists. But I now view\r\nthis as a defect in logical purity.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nLogical propositions are such as can be known \u003ci\u003ea priori\u003c/i\u003e, without\r\nstudy of the actual world. We only know from a study of\r\nempirical facts that Socrates is a man, but we know the correctness\r\nof the syllogism in its abstract form (\u003ci\u003ei.e.\u003c/i\u003e when it is stated\r\nin terms of variables) without needing any appeal to experience.\r\nThis is a characteristic, not of logical propositions in themselves,\r\nbut of the way in which we know them. It has, however, a\r\nbearing upon the question what their nature may be, since there\r\nare some kinds of propositions which it would be very difficult\r\nto suppose we could know without experience.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is clear that the definition of \"logic\" or \"mathematics\"\r\nmust be sought by trying to give a new definition of the old\r\nnotion of \"analytic\" propositions. Although we can no longer\r\nbe satisfied to define logical propositions as those that follow\r\nfrom the law of contradiction, we can and must still admit that\r\nthey are a wholly different class of propositions from those that\r\nwe come to know empirically. They all have the characteristic\r\nwhich, a moment ago, we agreed to call \"tautology.\" This,\r\n\u003cspan class=\"pagenum\" id=\"Page_204\"\u003e[Pg 204]\u003c/span\u003e\r\ncombined with the fact that they can be expressed wholly in terms\r\nof variables and logical constants (a logical constant being something\r\nwhich remains constant in a proposition even when \u003ci\u003eall\u003c/i\u003e\r\nits constituents are changed)\u0026mdash;will give the definition of logic\r\nor pure mathematics. For the moment, I do not know how to\r\ndefine \"tautology.\"\u003ca id=\"FNanchor_44_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_44_1\" class=\"fnanchor\"\u003e[44]\u003c/a\u003e\r\nIt would be easy to offer a definition\r\nwhich might seem satisfactory for a while; but I know of none\r\nthat I feel to be satisfactory, in spite of feeling thoroughly\r\nfamiliar with the characteristic of which a definition is wanted.\r\nAt this point, therefore, for the moment, we reach the frontier\r\nof knowledge on our backward journey into the logical foundations\r\nof mathematics.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp class=\"nind\"\u003e\u003ca id=\"Footnote_44_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_44_1\"\u003e\u003cspan class=\"label\"\u003e[44]\u003c/span\u003e\u003c/a\u003eThe importance of \"tautology\" for a definition of mathematics was\r\npointed out to me by my former pupil Ludwig Wittgenstein, who was\r\nworking on the problem. I do not know whether he has solved it, or even\r\nwhether he is alive or dead.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nWe have now come to an end of our somewhat summary introduction\r\nto mathematical philosophy. It is impossible to convey\r\nadequately the ideas that are concerned in this subject so long\r\nas we abstain from the use of logical symbols. Since ordinary\r\nlanguage has no words that naturally express exactly what we\r\nwish to express, it is necessary, so long as we adhere to ordinary\r\nlanguage, to strain words into unusual meanings; and the reader\r\nis sure, after a time if not at first, to lapse into attaching the usual\r\nmeanings to words, thus arriving at wrong notions as to what is\r\nintended to be said. Moreover, ordinary grammar and syntax\r\nis extraordinarily misleading. This is the case, \u003ci\u003ee.g.\u003c/i\u003e, as regards\r\nnumbers; \"ten men\" is grammatically the same form as\r\n\"white men,\" so that 10 might be thought to be an adjective\r\nqualifying \"men.\" It is the case, again, wherever propositional\r\nfunctions are involved, and in particular as regards existence and\r\ndescriptions. Because language is misleading, as well as because\r\nit is diffuse and inexact when applied to logic (for which it was\r\nnever intended), logical symbolism is absolutely necessary to\r\nany exact or thorough treatment of our subject. Those readers,\r\n\u003cspan class=\"pagenum\" id=\"Page_205\"\u003e[Pg 205]\u003c/span\u003e\r\ntherefore, who wish to acquire a mastery of the principles of\r\nmathematics, will, it is to be hoped, not shrink from the labour\r\nof mastering the symbols\u0026mdash;a labour which is, in fact, much less\r\nthan might be thought. As the above hasty survey must have\r\nmade evident, there are innumerable unsolved problems in the\r\nsubject, and much work needs to be done. If any student is\r\nled into a serious study of mathematical logic by this little\r\nbook, it will have served the chief purpose for which it has been\r\nwritten.\r\n\u003cspan class=\"pagenum\" id=\"Page_206\"\u003e[Pg 206]\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\u0027chapter\u0027\u003e\r\n\u003ch2 title=\u0027INDEX\u0027\u003e\u003ca id=\"INDEX\"\u003e\u003c/a\u003eINDEX\u003c/h2\u003e\r\n\u003cp class=\"indx\"\u003e\r\nAggregates, \u003ca href=\"#Page_12\"\u003e12\u003c/a\u003e\r\n\u003cbr\u003e\r\nAlephs, \u003ca href=\"#Page_83\"\u003e83\u003c/a\u003e, \u003ca href=\"#Page_92\"\u003e92\u003c/a\u003e, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_125\"\u003e125\u003c/a\u003e\r\n\u003cbr\u003e\r\nAliorelatives, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003ci\u003eAll\u003c/i\u003e, \u003ca href=\"#Page_158\"\u003e158\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nAnalysis, \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e\r\n\u003cbr\u003e\r\nAncestors, \u003ca href=\"#Page_25\"\u003e25\u003c/a\u003e, \u003ca href=\"#Page_33\"\u003e33\u003c/a\u003e\r\n\u003cbr\u003e\r\nArgument of a function, \u003ca href=\"#Page_47\"\u003e47\u003c/a\u003e, \u003ca href=\"#Page_108\"\u003e108\u003c/a\u003e\r\n\u003cbr\u003e\r\nArithmetising of mathematics, \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e\r\n\u003cbr\u003e\r\nAssociative law, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e\r\n\u003cbr\u003e\r\nAxioms, \u003ca href=\"#Page_1\"\u003e1\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nBetween, \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e ff., \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e\r\n\u003cbr\u003e\r\nBolzano, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e\r\n\u003cbr\u003e\r\nBoots and socks, \u003ca href=\"#Page_126\"\u003e126\u003c/a\u003e\r\n\u003cbr\u003e\r\nBoundary, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e, \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nCantor, Georg, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e, \u003ca href=\"#Page_79\"\u003e79\u003c/a\u003e, \u003ca href=\"#Page_85\"\u003e85\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_89\"\u003e89\u003c/a\u003e,\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003e\u003ca href=\"#Page_95\"\u003e95\u003c/a\u003e, \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e, \u003ca href=\"#Page_136\"\u003e136\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nClasses, \u003ca href=\"#Page_12\"\u003e12\u003c/a\u003e, \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e, \u003ca href=\"#Page_181\"\u003e181\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ereflexive, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e, \u003ca href=\"#Page_127\"\u003e127\u003c/a\u003e, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003esimilar, \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nClifford, W. K., \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e\r\n\u003cbr\u003e\r\nCollections, infinite, \u003ca href=\"#Page_13\"\u003e13\u003c/a\u003e\r\n\u003cbr\u003e\r\nCommutative law, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e\r\n\u003cbr\u003e\r\nConjunction, \u003ca href=\"#Page_147\"\u003e147\u003c/a\u003e\r\n\u003cbr\u003e\r\nConsecutiveness, \u003ca href=\"#Page_37\"\u003e37\u003c/a\u003e, \u003ca href=\"#Page_38\"\u003e38\u003c/a\u003e, \u003ca href=\"#Page_81\"\u003e81\u003c/a\u003e\r\n\u003cbr\u003e\r\nConstants, \u003ca href=\"#Page_202\"\u003e202\u003c/a\u003e\r\n\u003cbr\u003e\r\nConstruction, method of, \u003ca href=\"#Page_73\"\u003e73\u003c/a\u003e\r\n\u003cbr\u003e\r\nContinuity, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eCantorian, \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eDedekindian, \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ein philosophy, \u003ca href=\"#Page_105\"\u003e105\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eof functions, \u003ca href=\"#Page_106\"\u003e106\u003c/a\u003e ff.\u003c/span\u003e\r\n\u003cbr\u003e\r\nContradictions, \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nConvergence, \u003ca href=\"#Page_115\"\u003e115\u003c/a\u003e\r\n\u003cbr\u003e\r\nConverse, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_49\"\u003e49\u003c/a\u003e\r\n\u003cbr\u003e\r\nCorrelators, \u003ca href=\"#Page_54\"\u003e54\u003c/a\u003e\r\n\u003cbr\u003e\r\nCounterparts, objective, \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e\r\n\u003cbr\u003e\r\nCounting, \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nDedekind, \u003ca href=\"#Page_69\"\u003e69\u003c/a\u003e, \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e\r\n\u003cbr\u003e\r\nDeduction, \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nDefinition, \u003ca href=\"#Page_3\"\u003e3\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eextensional and intensional, \u003ca href=\"#Page_12\"\u003e12\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nDerivatives, \u003ca href=\"#Page_100\"\u003e100\u003c/a\u003e\r\n\u003cbr\u003e\r\nDescriptions, \u003ca href=\"#Page_139\"\u003e139\u003c/a\u003e, \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e\r\n\u003cbr\u003e\r\nDescriptions, \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e\r\n\u003cbr\u003e\r\nDimensions, \u003ca href=\"#Page_29\"\u003e29\u003c/a\u003e\r\n\u003cbr\u003e\r\nDisjunction, \u003ca href=\"#Page_147\"\u003e147\u003c/a\u003e\r\n\u003cbr\u003e\r\nDistributive law, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e\r\n\u003cbr\u003e\r\nDiversity, \u003ca href=\"#Page_87\"\u003e87\u003c/a\u003e\r\n\u003cbr\u003e\r\nDomain, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_49\"\u003e49\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nEquivalence, \u003ca href=\"#Page_183\"\u003e183\u003c/a\u003e\r\n\u003cbr\u003e\r\nEuclid, \u003ca href=\"#Page_67\"\u003e67\u003c/a\u003e\r\n\u003cbr\u003e\r\nExistence, \u003ca href=\"#Page_164\"\u003e164\u003c/a\u003e, \u003ca href=\"#Page_171\"\u003e171\u003c/a\u003e, \u003ca href=\"#Page_177\"\u003e177\u003c/a\u003e\r\n\u003cbr\u003e\r\nExponentiation, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e, \u003ca href=\"#Page_120\"\u003e120\u003c/a\u003e\r\n\u003cbr\u003e\r\nExtension of a relation, \u003ca href=\"#Page_60\"\u003e60\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nFictions, logical, \u003ca href=\"#Page_14\"\u003e14\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e, \u003ca href=\"#Page_45\"\u003e45\u003c/a\u003e, \u003ca href=\"#Page_137\"\u003e137\u003c/a\u003e\r\n\u003cbr\u003e\r\nField of a relation, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e, \u003ca href=\"#Page_53\"\u003e53\u003c/a\u003e\r\n\u003cbr\u003e\r\nFinite, \u003ca href=\"#Page_27\"\u003e27\u003c/a\u003e\r\n\u003cbr\u003e\r\nFlux, \u003ca href=\"#Page_105\"\u003e105\u003c/a\u003e\r\n\u003cbr\u003e\r\nForm, \u003ca href=\"#Page_198\"\u003e198\u003c/a\u003e\r\n\u003cbr\u003e\r\nFractions, \u003ca href=\"#Page_37\"\u003e37\u003c/a\u003e, \u003ca href=\"#Page_64\"\u003e64\u003c/a\u003e\r\n\u003cbr\u003e\r\nFrege, \u003ca href=\"#Page_7\"\u003e7\u003c/a\u003e, \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e, \u003ca href=\"#Page_25\"\u003e25\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e, \u003ca href=\"#Page_95\"\u003e95\u003c/a\u003e, \u003ca href=\"#Page_146\"\u003e146\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e\r\n\u003cbr\u003e\r\nFunctions, \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003edescriptive, \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_180\"\u003e180\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eintensional and extensional, \u003ca href=\"#Page_186\"\u003e186\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003epredicative, \u003ca href=\"#Page_189\"\u003e189\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003epropositional, \u003ca href=\"#Page_46\"\u003e46\u003c/a\u003e, \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003epropositional, \u003ca href=\"#Page_155\"\u003e155\u003c/a\u003e;\u003c/span\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nGap, Dedekindian, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e ff., \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e\r\n\u003cbr\u003e\r\nGeneralisation, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e\r\n\u003cbr\u003e\r\nGeometry, \u003ca href=\"#Page_29\"\u003e29\u003c/a\u003e, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e, \u003ca href=\"#Page_67\"\u003e67\u003c/a\u003e, \u003ca href=\"#Page_74\"\u003e74\u003c/a\u003e, \u003ca href=\"#Page_100\"\u003e100\u003c/a\u003e, \u003ca href=\"#Page_145\"\u003e145\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eanalytical, \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nGreater and less, \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e, \u003ca href=\"#Page_90\"\u003e90\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nHegel, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e\r\n\u003cbr\u003e\r\nHereditary properties, \u003ca href=\"#Page_21\"\u003e21\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nImplication, \u003ca href=\"#Page_146\"\u003e146\u003c/a\u003e, \u003ca href=\"#Page_153\"\u003e153\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eformal, \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nIncommensurables, \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e\r\n\u003cbr\u003e\r\nIncompatibility, \u003ca href=\"#Page_147\"\u003e147\u003c/a\u003e ff., \u003ca href=\"#Page_200\"\u003e200\u003c/a\u003e\r\n\u003cbr\u003e\r\nIncomplete symbols, \u003ca href=\"#Page_182\"\u003e182\u003c/a\u003e\r\n\u003cbr\u003e\r\nIndiscernibles, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e\r\n\u003cbr\u003e\r\nIndividuals, \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e, \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e, \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e\r\n\u003cbr\u003e\r\nInduction, mathematical, \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e ff., \u003ca href=\"#Page_87\"\u003e87\u003c/a\u003e, \u003ca href=\"#Page_93\"\u003e93\u003c/a\u003e,\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003e\u003ca href=\"#Page_185\"\u003e185\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nInductive properties, \u003ca href=\"#Page_21\"\u003e21\u003c/a\u003e\r\n\u003cbr\u003e\r\nInference, \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e\r\n\u003cbr\u003e\r\nInfinite, \u003ca href=\"#Page_28\"\u003e28\u003c/a\u003e; of rationals, \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eCantorian, \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eof cardinals, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eand series and ordinals, \u003ca href=\"#Page_89\"\u003e89\u003c/a\u003e ff.\u003c/span\u003e\r\n\u003cbr\u003e\r\nInfinity, axiom of, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e, \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e ff.,\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003e\u003ca href=\"#Page_202\"\u003e202\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nInstances, \u003ca href=\"#Page_156\"\u003e156\u003c/a\u003e\r\n\u003cbr\u003e\r\nIntegers, positive and negative, \u003ca href=\"#Page_64\"\u003e64\u003c/a\u003e\r\n\u003cbr\u003e\r\nIntervals, \u003ca href=\"#Page_115\"\u003e115\u003c/a\u003e\r\n\u003cbr\u003e\r\nIntuition, \u003ca href=\"#Page_145\"\u003e145\u003c/a\u003e\r\n\u003cbr\u003e\r\nIrrationals, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e, \u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e\u003cbr\u003e\r\n\u003cspan class=\"pagenum\" id=\"Page_207\"\u003e[Pg 207]\u003c/span\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nKant, \u003ca href=\"#Page_145\"\u003e145\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nLeibniz, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e, \u003ca href=\"#Page_192\"\u003e192\u003c/a\u003e\r\n\u003cbr\u003e\r\nLewis, C. I., \u003ca href=\"#Page_153\"\u003e153\u003c/a\u003e, \u003ca href=\"#Page_154\"\u003e154\u003c/a\u003e\r\n\u003cbr\u003e\r\nLikeness, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e\r\n\u003cbr\u003e\r\nLimit, \u003ca href=\"#Page_29\"\u003e29\u003c/a\u003e, \u003ca href=\"#Page_69\"\u003e69\u003c/a\u003e ff., \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eof functions, \u003ca href=\"#Page_106\"\u003e106\u003c/a\u003e ff.\u003c/span\u003e\r\n\u003cbr\u003e\r\nLimiting points, \u003ca href=\"#Page_99\"\u003e99\u003c/a\u003e\r\n\u003cbr\u003e\r\nLogic, \u003ca href=\"#Page_159\"\u003e159\u003c/a\u003e, \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e, \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003emathematical, \u003ca href=\"#Page_v\"\u003ev\u003c/a\u003e, \u003ca href=\"#Page_201\"\u003e201\u003c/a\u003e, \u003ca href=\"#Page_206\"\u003e206\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nLogicising of mathematics, \u003ca href=\"#Page_7\"\u003e7\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nMaps, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e, \u003ca href=\"#Page_60\"\u003e60\u003c/a\u003e ff., \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e\r\n\u003cbr\u003e\r\nMathematics, \u003ca href=\"#Page_194\"\u003e194\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nMaximum, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\r\n\u003cbr\u003e\r\nMedian class, \u003ca href=\"#Page_104\"\u003e104\u003c/a\u003e\r\n\u003cbr\u003e\r\nMeinong, \u003ca href=\"#Page_169\"\u003e169\u003c/a\u003e\r\n\u003cbr\u003e\r\nMethod, \u003ca href=\"#Page_vi\"\u003evi\u003c/a\u003e\r\n\u003cbr\u003e\r\nMinimum, \u003ca href=\"#Page_70\"\u003e70\u003c/a\u003e, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\r\n\u003cbr\u003e\r\nModality, \u003ca href=\"#Page_165\"\u003e165\u003c/a\u003e\r\n\u003cbr\u003e\r\nMultiplication, \u003ca href=\"#Page_118\"\u003e118\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nMultiplicative axiom, \u003ca href=\"#Page_92\"\u003e92\u003c/a\u003e, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nNames, \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e, \u003ca href=\"#Page_182\"\u003e182\u003c/a\u003e\r\n\u003cbr\u003e\r\nNecessity, \u003ca href=\"#Page_165\"\u003e165\u003c/a\u003e\r\n\u003cbr\u003e\r\nNeighbourhood, \u003ca href=\"#Page_109\"\u003e109\u003c/a\u003e\r\n\u003cbr\u003e\r\nNicod, \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e, \u003ca href=\"#Page_149\"\u003e149\u003c/a\u003e, \u003ca href=\"#Page_151\"\u003e151\u003c/a\u003e\r\n\u003cbr\u003e\r\nNull-class, \u003ca href=\"#Page_23\"\u003e23\u003c/a\u003e, \u003ca href=\"#Page_132\"\u003e132\u003c/a\u003e\r\n\u003cbr\u003e\r\nNumber, cardinal, \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e ff., \u003ca href=\"#Page_56\"\u003e56\u003c/a\u003e, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e ff., \u003ca href=\"#Page_95\"\u003e95\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ecomplex, \u003ca href=\"#Page_74\"\u003e74\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003efinite, \u003ca href=\"#Page_20\"\u003e20\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003einductive, \u003ca href=\"#Page_27\"\u003e27\u003c/a\u003e, \u003ca href=\"#Page_78\"\u003e78\u003c/a\u003e, \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003einfinite, \u003ca href=\"#Page_77\"\u003e77\u003c/a\u003e ff.;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eirrational, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e, \u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003emaximum? \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003emultipliable, \u003ca href=\"#Page_130\"\u003e130\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003enatural, \u003ca href=\"#Page_2\"\u003e2\u003c/a\u003e ff., \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003enon-inductive, \u003ca href=\"#Page_88\"\u003e88\u003c/a\u003e, \u003ca href=\"#Page_127\"\u003e127\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ereal, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e, \u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ereflexive, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e, \u003ca href=\"#Page_127\"\u003e127\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003erelation, \u003ca href=\"#Page_56\"\u003e56\u003c/a\u003e, \u003ca href=\"#Page_94\"\u003e94\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eserial, \u003ca href=\"#Page_57\"\u003e57\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nOccam, \u003ca href=\"#Page_184\"\u003e184\u003c/a\u003e\r\n\u003cbr\u003e\r\nOccurrences, primary and secondary,\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003e\u003ca href=\"#Page_179\"\u003e179\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nOntological proof, \u003ca href=\"#Page_203\"\u003e203\u003c/a\u003e\r\n\u003cbr\u003e\r\nOrder 29ff.; cyclic, \u003ca href=\"#Page_40\"\u003e40\u003c/a\u003e\r\n\u003cbr\u003e\r\nOscillation, ultimate, \u003ca href=\"#Page_111\"\u003e111\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\n\u003ci\u003eParmenides\u003c/i\u003e, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e\r\n\u003cbr\u003e\r\nParticulars, \u003ca href=\"#Page_140\"\u003e140\u003c/a\u003e ff., \u003ca href=\"#Page_173\"\u003e173\u003c/a\u003e\r\n\u003cbr\u003e\r\nPeano, \u003ca href=\"#Page_5\"\u003e5\u003c/a\u003e ff., \u003ca href=\"#Page_23\"\u003e23\u003c/a\u003e, \u003ca href=\"#Page_24\"\u003e24\u003c/a\u003e, \u003ca href=\"#Page_78\"\u003e78\u003c/a\u003e, \u003ca href=\"#Page_81\"\u003e81\u003c/a\u003e, \u003ca href=\"#Page_131\"\u003e131\u003c/a\u003e, \u003ca href=\"#Page_163\"\u003e163\u003c/a\u003e\r\n\u003cbr\u003e\r\nPeirce, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e\r\n\u003cbr\u003e\r\nPermutations, \u003ca href=\"#Page_50\"\u003e50\u003c/a\u003e\r\n\u003cbr\u003e\r\nPhilosophy, mathematical, \u003ca href=\"#Page_v\"\u003ev\u003c/a\u003e, \u003ca href=\"#Page_1\"\u003e1\u003c/a\u003e\r\n\u003cbr\u003e\r\nPlato, \u003ca href=\"#Page_138\"\u003e138\u003c/a\u003e\r\n\u003cbr\u003e\r\nPlurality, \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e\r\n\u003cbr\u003e\r\nPoincaré, \u003ca href=\"#Page_27\"\u003e27\u003c/a\u003e\r\n\u003cbr\u003e\r\nPoints, \u003ca href=\"#Page_59\"\u003e59\u003c/a\u003e\r\n\u003cbr\u003e\r\nPosterity, \u003ca href=\"#Page_22\"\u003e22\u003c/a\u003e ff., \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e; proper, \u003ca href=\"#Page_36\"\u003e36\u003c/a\u003e\r\n\u003cbr\u003e\r\nPostulates, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_73\"\u003e73\u003c/a\u003e\r\n\u003cbr\u003e\r\nPrecedent, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\r\n\u003cbr\u003e\r\nPremisses of arithmetic, \u003ca href=\"#Page_5\"\u003e5\u003c/a\u003e\r\n\u003cbr\u003e\r\nPrimitive ideas and propositions, \u003ca href=\"#Page_5\"\u003e5\u003c/a\u003e, \u003ca href=\"#Page_202\"\u003e202\u003c/a\u003e\r\n\u003cbr\u003e\r\nProgressions, \u003ca href=\"#Page_8\"\u003e8\u003c/a\u003e, \u003ca href=\"#Page_81\"\u003e81\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nPropositions, \u003ca href=\"#Page_155\"\u003e155\u003c/a\u003e; analytic, \u003ca href=\"#Page_204\"\u003e204\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eelementary, \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nPythagoras, \u003ca href=\"#Page_4\"\u003e4\u003c/a\u003e, \u003ca href=\"#Page_67\"\u003e67\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nQuantity, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_195\"\u003e195\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nRatios, \u003ca href=\"#Page_64\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e, \u003ca href=\"#Page_133\"\u003e133\u003c/a\u003e\r\n\u003cbr\u003e\r\nReducibility, axiom of, \u003ca href=\"#Page_191\"\u003e191\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003ci\u003eReferent\u003c/i\u003e, \u003ca href=\"#Page_48\"\u003e48\u003c/a\u003e\r\n\u003cbr\u003e\r\nRelation numbers, \u003ca href=\"#Page_56\"\u003e56\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nRelations, asymmetrical \u003ca href=\"#Page_31\"\u003e31\u003c/a\u003e, \u003ca href=\"#Page_42\"\u003e42\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003econnected, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003emany-one, \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eone-many, \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_45\"\u003e45\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eone-one, \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e, \u003ca href=\"#Page_47\"\u003e47\u003c/a\u003e, \u003ca href=\"#Page_79\"\u003e79\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ereflexive, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eserial, \u003ca href=\"#Page_34\"\u003e34\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003esimilar, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003esquares of, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003esymmetrical, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_44\"\u003e44\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003etransitive, \u003ca href=\"#Page_16\"\u003e16\u003c/a\u003e, \u003ca href=\"#Page_32\"\u003e32\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\n\u003ci\u003eRelatum\u003c/i\u003e, \u003ca href=\"#Page_48\"\u003e48\u003c/a\u003e\r\n\u003cbr\u003e\r\nRepresentatives, \u003ca href=\"#Page_120\"\u003e120\u003c/a\u003e\r\n\u003cbr\u003e\r\nRigour, \u003ca href=\"#Page_144\"\u003e144\u003c/a\u003e\r\n\u003cbr\u003e\r\nRoyce, \u003ca href=\"#Page_80\"\u003e80\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nSection, Dedekindian, \u003ca href=\"#Page_69\"\u003e69\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eultimate, \u003ca href=\"#Page_111\"\u003e111\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nSegments, \u003ca href=\"#Page_72\"\u003e72\u003c/a\u003e, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\r\n\u003cbr\u003e\r\nSelections, \u003ca href=\"#Page_117\"\u003e117\u003c/a\u003e\r\n\u003cbr\u003e\r\nSequent, \u003ca href=\"#Page_98\"\u003e98\u003c/a\u003e\r\n\u003cbr\u003e\r\nSeries, \u003ca href=\"#Page_29\"\u003e29\u003c/a\u003e ff.; closed, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ecompact, \u003ca href=\"#Page_66\"\u003e66\u003c/a\u003e, \u003ca href=\"#Page_93\"\u003e93\u003c/a\u003e, \u003ca href=\"#Page_100\"\u003e100\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003econdensed in itself, \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eDedekindian, \u003ca href=\"#Page_71\"\u003e71\u003c/a\u003e, \u003ca href=\"#Page_73\"\u003e73\u003c/a\u003e, \u003ca href=\"#Page_101\"\u003e101\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003egeneration of, \u003ca href=\"#Page_41\"\u003e41\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003einfinite, \u003ca href=\"#Page_89\"\u003e89\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eperfect, \u003ca href=\"#Page_102\"\u003e102\u003c/a\u003e, \u003ca href=\"#Page_103\"\u003e103\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003ewell-ordered, \u003ca href=\"#Page_92\"\u003e92\u003c/a\u003e, \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nSheffer, \u003ca href=\"#Page_148\"\u003e148\u003c/a\u003e\r\n\u003cbr\u003e\r\nSimilarity, of classes, \u003ca href=\"#Page_15\"\u003e15\u003c/a\u003e ff.;\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eof relations, \u003ca href=\"#Page_83\"\u003e83\u003c/a\u003e;\u003c/span\u003e\u003cbr\u003e\r\n\u003cspan style=\"margin-left: 1em;\"\u003eof relations, \u003ca href=\"#Page_52\"\u003e52\u003c/a\u003e\u003c/span\u003e\r\n\u003cbr\u003e\r\nSome, \u003ca href=\"#Page_158\"\u003e158\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nSpace, \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_140\"\u003e140\u003c/a\u003e\r\n\u003cbr\u003e\r\nStructure, \u003ca href=\"#Page_60\"\u003e60\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nSub-classes, \u003ca href=\"#Page_84\"\u003e84\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nSubjects, \u003ca href=\"#Page_142\"\u003e142\u003c/a\u003e\r\n\u003cbr\u003e\r\nSubtraction, \u003ca href=\"#Page_87\"\u003e87\u003c/a\u003e\r\n\u003cbr\u003e\r\nSuccessor of a number, \u003ca href=\"#Page_23\"\u003e23\u003c/a\u003e, \u003ca href=\"#Page_35\"\u003e35\u003c/a\u003e\r\n\u003cbr\u003e\r\nSyllogism, \u003ca href=\"#Page_197\"\u003e197\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nTautology, \u003ca href=\"#Page_203\"\u003e203\u003c/a\u003e, \u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003ci\u003eThe\u003c/i\u003e, \u003ca href=\"#Page_167\"\u003e167\u003c/a\u003e, \u003ca href=\"#Page_172\"\u003e172\u003c/a\u003e ff.\r\n\u003cbr\u003e\r\nTime, \u003ca href=\"#Page_61\"\u003e61\u003c/a\u003e, \u003ca href=\"#Page_86\"\u003e86\u003c/a\u003e, \u003ca href=\"#Page_140\"\u003e140\u003c/a\u003e\r\n\u003cbr\u003e\r\nTruth-function, \u003ca href=\"#Page_147\"\u003e147\u003c/a\u003e\r\n\u003cbr\u003e\r\nTruth-value, \u003ca href=\"#Page_146\"\u003e146\u003c/a\u003e\r\n\u003cbr\u003e\r\nTypes, logical, \u003ca href=\"#Page_53\"\u003e53\u003c/a\u003e, \u003ca href=\"#Page_135\"\u003e135\u003c/a\u003e ff., \u003ca href=\"#Page_185\"\u003e185\u003c/a\u003e, \u003ca href=\"#Page_188\"\u003e188\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nUnreality, \u003ca href=\"#Page_168\"\u003e168\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nValue of a function, \u003ca href=\"#Page_47\"\u003e47\u003c/a\u003e, \u003ca href=\"#Page_108\"\u003e108\u003c/a\u003e\r\n\u003cbr\u003e\r\nVariables, \u003ca href=\"#Page_10\"\u003e10\u003c/a\u003e, \u003ca href=\"#Page_161\"\u003e161\u003c/a\u003e, \u003ca href=\"#Page_199\"\u003e199\u003c/a\u003e\r\n\u003cbr\u003e\r\nVeblen, \u003ca href=\"#Page_58\"\u003e58\u003c/a\u003e\r\n\u003cbr\u003e\r\nVerbs, \u003ca href=\"#Page_141\"\u003e141\u003c/a\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nWeierstrass, \u003ca href=\"#Page_97\"\u003e97\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e\r\n\u003cbr\u003e\r\nWells, H. G., \u003ca href=\"#Page_114\"\u003e114\u003c/a\u003e\r\n\u003cbr\u003e\r\nWhitehead, \u003ca href=\"#Page_64\"\u003e64\u003c/a\u003e, \u003ca href=\"#Page_76\"\u003e76\u003c/a\u003e, \u003ca href=\"#Page_107\"\u003e107\u003c/a\u003e, \u003ca href=\"#Page_119\"\u003e119\u003c/a\u003e\r\n\u003cbr\u003e\r\nWittgenstein, \u003ca href=\"#Page_205\"\u003e205\u003c/a\u003e \u003ci\u003en.\u003c/i\u003e\r\n\u003cbr\u003e\r\n\u003cbr\u003e\r\nZermelo, \u003ca href=\"#Page_123\"\u003e123\u003c/a\u003e, \u003ca href=\"#Page_129\"\u003e129\u003c/a\u003e\r\n\u003cbr\u003e\r\nZero, \u003ca href=\"#Page_65\"\u003e65\u003c/a\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\nPRINTED IN GREAT BRITAIN BY NEILL AND CO., LTD., EDINBURGH.\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\"transnote\"\u003e\r\n\u003cp class=\"center\"\u003e\u003cb\u003eTRANSCRIBER\u0027S NOTES\u003c/b\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\r\nMinor typographical corrections and presentational changes have been\r\nmade without comment.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThis ebook was produced using OCR text generously provided by the\r\nUniversity of Toronto through the Internet Archive.\r\n\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cbr\u003e\u003cbr\u003e\u003cbr\u003e\u003c/p\u003e\r\n\u003c/article\u003e"}],"SectionSequence":["Back Link","Work Title","Deck","Author","Period","Era","Composition","Date Note","Region","Terra Avita","Terra Avita Region","Modern Country","Original Title","Language","Primary Discipline","Secondary Discipline","Tradition","Full Versions","Core Thesis","Classification","Arguments","Influence","Significance","Evidence Note","Full Text"],"Counts":{"ContextCards":3,"GeoCards":4,"DisciplineCards":2,"Links":11,"Sections":25,"Styles":3,"Scripts":1}}