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- Bertrand Russell

BERTRAND RUSSELL

Toward the ''Principles of Mathematics'' 1900-02

Edited by Gregory H. Moore

Bertrand Russell, taken in the late 1890s. (Courtesy of Trinity College,

Cambridge)

London and New York

11

First published 1993 by Routledge New Fetter Lane, London EC4P 4EE

THE COLLECTED PAPERS OF BERTRA ND RUSSELL

Simultaneously published in the USA and Canada by Routledge 29 West 35th Street, New York, NY IOOOI Bertrand Russell's unpublished letters, Papers 1-4, 9, 11, 14, 15, 20, 23, and Appendices I-III, V.I, VI, VII.I , IX, XI © McMaster University 1993. Papers 6, 7, 19, 21, 22 © The Bertrand Russell Estate I900, I90I, I902. Paper 8 ©George Allen & Unwin 1956. Paper 10 © George Allen & Unwin 19I8. Paper 13 ©American Journal of Mathematics I902. Paper 17 © Mathematical Association 1902. Paper 18 © Encyclopaedia Britannica 1902. Appendix IV © Librairie Armand Colin 1901. Appendices v.2 and vn.2 © University of Turin 1901, 1902. Appendix VIII © Universite Catholique de Louvain 1902. Appendix x ©University of London 1902. Editorial matter and the translations printed as Papers 5, 12, 16 © Gregory H. Moore I993· Funds to edit this volume were provided by a major editorial grant from the Social Sciences and Humanities Research Council of Canada and by McMaster University. All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data

Russell, Bertrand Toward the " Principles of Mathematics", 1900-02. - (Collected Papers of Ber trand Russell; Vol. 3) I. Title II. M oore, Gregory H . III. Series 500.1

GENERAL EDITOR

John Passmore (Australian National University)

ADVISORY EDITORI AL BOARD

Sir Alfred Ayert I. Grattan-Guinness (Middlesex Polytechnic) Jock Gunn (Queen's University) Francess G. Halpenny (University of Toronto) Royden Harrison (University of Warwick) Leonard Linsky (University of Chicago) H.C.G. Matthew (St. Hugh's College, Oxford) John Passmore (Australian National University) D .F. Pears (Christ Church, Oxford) John M . Robson (University of Toronto) Alan Ryan (Princeton University) Katharine Tait

Library of Congress Cataloging-in-Publication Data

Russell, Bertrand, 1872-1970. Toward the "Principles of mathematics" 1900-02/ Rertrand Russell; edited by Gregory H. Moore. p. cm. - (The Collected papers of Bertrand Russell; vol. 3). Includes 16 previously unpublished articles by Russell and the draft manuscript of The Principles of mathematics. Includes bibliographical references and indexes. Mathematks- Philosophy. 2. Russell's paradox. I. Moore, Gregory H . Title III. Series: Russell, Bertrand, 1872-1970. Selections. 1983; v. 3. B1649.R9I 1983 vol. 3 [QA8.6] 192s- dc20 93-3505 CIP [5 ro' .1] ISBN 0-415-09405-4 Typeset in ro/I2 pt. Plantin by The Bertrand Russell Editorial Project, McMaster University Printed in Great Britain at the University Press, Cambridge

t Deceased 1989

Contents

x xii

Illustrations Abbreviations Introduction Acknowledgements Chronology

Xlll

xlix liii

PART I. DRAFTS OF THE PRINCIPLES OF MATHEMATICS

General Headnote 1 The Principles of Mathematics, Draft of 1899-1900

Part I. Number Part II. Whole and Part Part III. Quantity Part IV. Order Part v. Continuity and Infinity Part VI. Space and Time Part VII. Matter and Motion 2 Part I of the Principles, Draft of 1901 3 Plan for Book I: The Variable [1902]

3 9 15

35 54 75 106

142 160 181 209

PART II. ABSOLUTE SPACE AND TIME

General Headnote 4 Is Position in Time Absolute or Relative? [1900] 5 The Notion of Order and Absolute Position in Space and Time [1901] 6 Is Position in Time and Space Absolute or Relative? [1901]

215 219

234 259

PART III. AFTER PEANO: FOUNDATIONS OF MATHEMATICS

General Headnote 7 On the Notion of Order [1901] 8 The Logic of Relations with Some Applications to the Theory of Series [1901] Vll

310

CONTENTS

Vlll

9 Recent Italian Work on the Foundations of Mathematics [I90I] 10 Recent Work on the Principles of Mathematics [I90I] 11 Lecture II. Logic of Propositions [I90I] 12 General Theory of Well-Ordered Series [I902] 13 On Finite and Infinite Cardinal Numbers [I902] 14 Continuous Series [I902] 15 On Likeness [I 902]

CONTENTS

384 422 43I 437

PART IV. GEOMETRY

General Headnote 16 Note [I902] 17 The Teaching of Euclid [I902] 18 Geometry, Non-Euclidean [I902]

455 46I 465 470

PART V. GENERAL PHILOSOPHY

General Headnote 19 Review of Schultz, Psychologie der Axiome [I900] 20 Leibniz's Doctrine of Substance as Deduced from His Logic [I 900] 21 Review of Boutroux, L' Imagination et les mathematiques selon Descartes [I 90 I] 22 Review of Hastie, Kant's Cosmogony [I90I] 23 Do Psychical States Have Position in Space? [I902]

Draft and French Text of Paper 8 On the Logic of Relations with Applications V.I to Arithmetic and the Theory of Series Sur la logique des relations avec des V.2 applications a la theorie des series Outline of Paper 9 VI Draft and French Text of Paper 12 VII vu.I On the General Theory of Well-Ordered Series VII.2 Theorie generate des series bien ordonnees VIII French Text of Paper 16 IX Geometry IX.I On Geometry and Dimensions IX.2 Geometry in the I90I-02 Lectures x Logic and Methodology as a Subject for the B.Sc. Degree XI General Theory of Functions

v 350 363 380

507 508

ix 589 590 6I3 628 630 63I 66I 674 676 677 677 680 686

MISSING AND UNPRINTED TEXTS

69I

ANNOTATION

695

TEXTUAL NOTES

755

BIBLIOGRAPHICAL INDEX

82I

SYMBOLS INDEX

85I

GENERAL INDEX

865

511 535 538 542

APPENDICES

Identity and Diversity Do Differences Differ? I. I On Identity 1.2 Logic Founded on Diversity 1.3 On a Logic Founded on Diversity 1.4 Logic Founded on Diversity 1.5 II An Assault on Russell's Paradox III Notes on Implication and Classes III. I Note on all and Formal Implication III.2 The Variable III.3 Note on Class III.4 Analytic Theory of a C b III.5 Classes, Implication, and Formal Implication IV French Text of Paper 5

553 555 557 557 558 559 560 566 567 567 568 568 568 570

ILL UST RA TIO NS

Illustrations

VII

xi

Paper 15, "On Likeness", showing his first efforts to develop a general relation-arithmetic. Note the use of v and A for union and intersection of classes, while u is retained for "or" and for union of relations. See 440: 1-29.

VIII Two pages of Russell's notes, from mid-I902, on Frege's Grundgesetze of I893. The first does not object to Frege's dubious Postulate v. The second shows Russell noting that his definition of cardinal number was also in Frege.

frontispiece Bertrand Russell, taken in the late 1890s. (Courtesy of Trinity College, Cambridge)

between pages 38 and 39 Paper 1, 1899-1900 draft of the Principles, Parts I and III, with marginal notes on the date of composition. See 15: 1-23 and 54: I-19. II

Paper 1, Part 11, including marginal notes that reveal Peano's influence in October 1900 on the intended revision of Parts I and 11. See 35: 1-22.

III

Paper 2, table of contents and first page, of I90I version of Part I of the Principles. Russell's first statement of the logicist thesis and his first definition, revised in the Principles, of the notion of logical constant by means of the universal class. See I84: I-I85: 2I.

IV

Paper 2, giving the first extant version of Russell's Paradox. Uses predicates rather than classes. See 194: 35-195: 34.

v

Two pages of notes from September 1900 on Peano's Formulaire. The first shows Russell's initial reaction to Peano's use of mathematical induction. The second gives Russell's favorable reaction to Cantor's 1895a treatment of the uncountability of the set of all real numbers.

VI

The left page, from notes on Peano (September I900), shows the fundamental place that logical diversity then occupied among Russell's logical notions; see Appendix 1.3-5. The right page, from Appendix v. I, is an attempt to define other logical notions from that of diversity; see 593: 23-594: 9.

x

All plates are photographs of documents in the Bertrand Russell Archives at McMaster University. Plates I-VIII are shown reduced from their original size, which is given at the head of each set of textual notes.

Abbreviations

Introduction

To GIVE THE reader an uncluttered text, we have kept abbreviations to a minimum. The few necessary to the referencing system are as follows. The papers printed in the volume are given a boldface number for easy reference. For example, "On Likeness" is Paper 15. Angle brackets in the text distinguish editorial insertions from Russell's more common square brackets. Bibliographical references are usually in the form of author, date and page, e.g. "Russell 1967, 145". Consulting the Bibliographical Index shows that this reference is to The Autobiography ofBertrand Russell, l 8721914, Vol. 1 (London: Allen & Unwin, 1967), p. 145. The location of archival documents cited in the edition is the Bertrand Russell Archives at McMaster University ("RA"), unless a different location is given. In particular, letters quoted without a reference can be found in the Russell Archives. File numbers of documents in the Russell Archives are provided only when manuscripts of papers printed here are cited or when files are difficult to identify. Cross-references to annotations are preceded by "A" and followed by page and line numbers (as in "A131: 25"). Cross-references to textual notes are preceded by "T". Further abbreviations are used in the Textual Notes, but they are identified at the beginning of each set of notes.

"THE MOST IMPORTANT year in my intellectual life", Russell observed in 1944, "was the year 1900, and the most important event in this year was my visit to the International Congress of Philosophy in Paris." There, in August, Russell was profoundly influenced by Peano's mathematical logic, which became the basis for Principia Mathematica (1910-13), the master work that Russell began to write in 1902 with Alfred North Whitehead. But Russell often emphasized, in retrospect, the dramatic shifts in his intellectual development more than the continuing themes. He was affected so deeply by Peano largely because of a task that he had previously set himself. For several years he had been struggling to write a book, eventually called The Principles of Mathematics (1903), on the foundations of mathematics. Peano's logic promised Russell a way out of the formidable difficulties that beset him in writing this book, a new method for resolving old problems. These problems concerned relations and their logic. Nevertheless, his belief in the importance of relations predated his conversion to Peano's logic by several years. Volume 3 of Russell's Collected Papers deals with a highly transitional period in his contributions to philosophy and mathematics. In addition to his conversion to Peano, his philosophical views had recently undergone one fundamental shift. Later, in My Philosophical Development, he conflated these two shifts into one: There is one major division in my philosophical work: in the years 1899-1900 I adopted the philosophy of logical atomism and the technique of Peano in mathematical logic .... The change in these years was a revolution; subsequent changes have been of the nature of an evolution. 1 By 1899 he no longer considered himself to be a Kantian or Hegelian, but viewed himself rather as a pluralist and a Platonic realist. He had been l

Xll

Russell 1959, l r. See below for the sense in which Russell had adopted certain elements of logical atomism during this period, although he did not give it that name, or lecture on it, until 1918. Xlll

INTRODUCTION

INTRODUCTION

led to abandon the idealism that he had embraced for half a decade by his younger Cambridge colleague, G. E. Moore:

We have, therefore, first to arrange the postulates of the science so as to leave the minimum of contradictions; then to supply, to these postulates or ideas, such supplement as will abolish the special contradictions of the science in question, and thus pass outside to a new science, which may then be similarly treated. Thus e.g. Number, the fundamental notion of Arithmetic, involves something numerable. Hence Geometry, since space is the only directly measurable element in sensation. Geometry, again, involves something which can be located, and something which can move-for a position, by definition, cannot move. Hence matter and Physics. (1896h, 5)

XIV

He took the lead in rebellion, and I followed, with a sense of emancipation .... We ... thought that everything is real that common sense, uninfluenced by philosophy or theology, supposes real. ... The world, which had been thin and logical, suddenly became rich and varied and solid. Mathematics could be quite true, and not merely a stage in dialectic. Something of this point of view appeared in my Philosophy of Leibniz. (Russell 1944, I2)

In his book on Leibniz, Russell also rejected traditional subject-predicate logic and emphasized the importance of relations in logic. Volume 3 can be seen as Russell's journey from his book on Leibniz, finished early in I900, to the Principles of Mathematics, finished in I902 and published the following year. I. BEFORE LEIBNIZ

At the beginning of his career, while at Berlin in I895, Russell had a sudden insight about how he wanted his work to develop: I remember a cold, bright day in early spring when I walked by myself in the Tiergarten, and made projects of future work. I thought that I would write one series of books on the philosophy of the sciences from pure mathematics to physiology, and another series of books on social questions. I hoped that the two series might ultimately meet in a synthesis at once scientific and practical. My scheme was inspired by Hegelian ideas .... The moment was an important and formative one as regards my purposes. (1967, 125) Intimately connected with the first series of books was his attempt to construct a single dialectical system of all the sciences. In late I896 or early I897 he wrote a "Note on the Logic of the Sciences", which summed up his views on constructing such a system: Every science works with a certain limited number of fundamental ideas .... What we have to do, therefore, in a logic of the sciences, is to construct, with the appropriate set of ideas, a world containing no contradictions but those which unavoidably result from the incompleteness of these ideas. Within any science, all contradictions not thus unavoidable are logically condemnable ....

xv

This passage reflects two important aspects of his thought at the time. First, no science (and no part of mathematics) would be consistent. Each would generate certain contradictions, which could only be eliminated by passing to the next higher science. Second, this process would move through the sciences in a specific order. Thus arithmetic would come first, then geometry, and then physics. In practice, he began with geometry. After completing his undergraduate degree at Cambridge in I894, he began to write a dissertation to be submitted for a Trinity prize fellowship. This dissertation, on the metaphysical bearings of non-Euclidean geometry, was devoted to a defence of Kant's work in the face of the challenge posed by Bolyai and Lobachevskii. His first two published papers were on the same subject, as was his first philosophical book, An Essay on the Foundations of Geometry (I897). This work remained neo-Kantian and neo-Hegelian, in that it discussed at length the contradictions that he regarded as being inevitable in geometry (see the general headnote to Part IV, below). Before the book appeared, Russell had already begun to work on the foundations of physics. Unlike his views on geometry, his ideas on the philosophy of physics were not published at the time. He later stated that his early geometrical ideas were mainly Kantian, while those on physics were Hegelian (1959, 40). At first, he adopted a Boscovichian theory of matter, in which unextended point-atoms acted at a distance in a force field. Then in I 897 he was converted to an ether theory in which space is a plenum (1897a, 22). One of the problems that carried over from this period to his later work (first in his book on Leibniz and then in the Principles) was how to resolve the conflict between the composition of forces and the causation of particulars by particulars (cf. Paper 1, I69-7I). In his book on Leibniz, he regarded this as the "principal difficulty" in Dynamics, one which "no existing theory of Dynamics can avoid". He described it as follows:

xvi

INTRODUCTION

INTRODUCTION

When a particle is subjected to several forces, they are compounded by the parallelogram law, and the resultant is regarded as their sum .... If we are to admit particular causes, each of which, independently of all others, produces its effect, we must regard the resultant motion as compounded of its components .... But it has not been generally perceived that a sum of motions, ... or vectors generally, is a sum in a quite peculiar sense-its constituents are not parts of it .... Thus no one of the constituent causes ever really produces its effect.... (1900, 98)

ematicians. But it seemed worth while to collect and define ... some contradictions in the relation of continuous quantity to number, and also to show, what mathematicians are in danger of forgetting, that philosophical antinomies, in this sphere, find their counterpart in mathematical fallacies. These fallacies seem, to me at least, to pervade the Calculus, and even the more elaborate machinery of Cantor's collections (Mengen). (1896d, 46)

Russell intended to make a Hegelian dialectical transition from geometry to kinematics, where matter and motion are fundamental, then a transition from kinematics to dynamics, where the notion of force is introduced. Later, there would be a transition from dynamics to psychology (with the notion of conation), and finally to metaphysics, where no more contradictions would remain. At this time, he regarded the choice between monism and monadism as "the most fundamental question in metaphysics"( 1897c, 97). His concern with this question reflected the influence on him of the Oxford philosopher F. H. Bradley, who was a monist, and his teacher at Cambridge, James Ward, who was a monadist. Despite Bradley's influence, Russell retained a pluralism of logical subjects even while still committed to a neo-Hegelian dialectic (1898b, 168), a combination of views that he shared at the time with his good friend, the Cambridge philosopher J.M.E. McTaggart. Russell only sketchily developed the dialectical transitions involving kinematics and dynamics, and his work on them remained unpublished at the time. Those involving psychology and metaphysics he hardly developed at all. By 1899, before he had worked out these transitions in detail, he abandoned Hegel. During his Hegelian period, while he was formulating a philosophy of physics, Russell was led back to the earliest dialectical level mentioned above in his "Note on the Logic of the Sciences": number. In February of 1896 he began to read a book on atomism by the French neo-Kantian philosopher Arthur Hannequin, and was thereby led to read Cantor. Hannequin construed atomism broadly, including under this heading not only atoms in physics but also Cantor's attempt to treat the continuum by means of numbers. Russell was impressed by Hannequin's book and gave it a lengthy and positive review (1896d). He adopted from it a mistrust of Cantor's work and was encouraged by it to reject Cantor's infinite ordinal numbers. He expressed his mistrust in an article, "On Some Difficulties of Continuous Quantity", inspired by Hannequin: From Zeno onwards, the difficulties of continua have been felt by philosophers, and evaded, with ever subtler analysis, by math-

XVll

The concern that he expressed here about the relation between number and quantity continued until the Principles was published, though by that time his opinion of Cantor had changed from negative to positive. During his idealist period, Russell's views on logic were very different from his later views. "In Logic", he wrote in his Essay on the Foundations of Geometry, "I have learnt most from Mr. Bradley, and next to him, from Sigwart and Bosanquet" (1897b, i). There, influenced by Bradley, he considered identity and diversity to be presupposed by all relations but not themselves to be relations (1897b, 198). Soon after he abandoned idealism, toward the end of 1898, 2 he began to argue strenuously against Bradley's views on relations. Bradley had claimed that relations are not real and that they are all reducible to identity and diversity of content. In a paper given in January 1899, "The Classification of Relations", Russell contended, on the contrary, that relations are real and that diversity is a relation and is not analyzable into a pair of predicates of the related terms .... I wish now ... to show that no relation is analyzable into a pair of predicates of the related terms. Mr. Bradley has argued much and hotly against the view that relations are ever purely "external" .... I should be retaining his terminology if I described my view as the view that all relations are external. (I 899c, 143) He later called Bradley's view the doctrine of internal relations, and his own, the doctrine of external relations. In the fall of 1900, he regarded the relation of diversity as so important that he made repeated attempts, which were ultimately unsuccessful, to base his logic on it (see Appendix 1.3-5). Although in 1899 he viewed diversity as a relation, at the time he denied that identity is a relation (1899c, 140). Early in 1900, he was still uncertain whether identity is a relation, since he required that a relation have two terms (see Appendix 1.2). But by October, when he drafted his first paper on the logic of relations, he had accepted identity as a relation (see v. I, §1, *7), a view that he continued to hold thereafter.

2

Russell 1959, 54.

INTRODUCTION

INTRODUCTION

The classification of relations that Russell proposed in his I 899 paper was a formal one, with four divisions. He began by stating two axioms that a relation r might satisfy: (I) if ArB, then BrA; (2) if ArB and BrC, then ArC. If a relation r satisfied axioms (I) and (2), he called it symmetrical (and we would now call it an equivalence relation). If r satisfied (I) but not (2), he called it reciprocal. If it satisfied (2) but not (I), he called it transitive. And if it satisfied neither (I) nor (2), he called it one-sided (1899c, I38-9). This classification, which he retained at some places in Paper 1, is potentially confusing because at other places in Paper 1 and in other papers in this volume he uses "symmetrical" and "transitive" with their modern meaning, namely as satisfying axioms (I) and (2) respectively. A related source of confusion is that, not long before, he had called a relation r "asymmetrical" if it did not satisfy both (I) and (2), i.e. if r was not an equivalence relation (1898b, I91). By May I900, when he wrote Paper 4, he had given "symmetrical" its modern meaning, but he then appeared to use "asymmetrical" to mean "not satisfying axiom (I)" (4, 224). A certain vagueness continued to surround his use of "asymmetrical" in Papers 5, 6 and 7. But in them he was groping toward a distinction that he made clearly in November 1900, when he wrote his final draft of Part IV of the Principles: a relation R is asymmetrical if, for all a and b, aRb implies not bRa, and is not-symmetrical if, for some a and b, aRb does not imply bRa. More simply, "not-symmetrical" means not satisfying axiom (1), while asymmetrical is a stronger condition. It was this stronger sense of being asymmetrical that he needed in order to develop his ideas on series and order. 3 Russell's rejection of Bradley's doctrine of internal relations was one of the most fundamental changes in his philosophy. It was intimately connected with his rejection of the traditional view that every proposition is reducible to subject-predicate form. "I first realized the importance of the question of relations", he wrote later,

Leibniz's views on substance and on propositions are the subject of Paper 20 in this volume. Russell's logic, during his idealist period, was very much involved with the notion of whole and part. He ascribed to the idealist philosophers Bradley and Bosanquet the view, which he accepted in his Essay on the Foundations of Geometry, that "every judgment ... is both synthetic and analytic; it combines parts into a whole, and analyzes a whole into parts." 4 This category of whole and part, he believed, "pervades all Mathematics, except perhaps projective Geometry" (1898b, 194). But he differed from their approach to whole and part in a fundamental way. For him, but not for them, the category of whole and part was intimately connected with what he called the "Logical Calculus", which stemmed from Boole but which he used in the form found in Whitehead's Treatise on Universal Algebra (1898). Also taken from Whitehead's book was the notion of manifold, which Russell defined as "a collection of terms having that kind of unity and relation which is found associated with a common predicate" (1898b, 179)-perhaps his earliest use of the Principle of Comprehension. In 1897, while still an idealist, Russell began the book that eventually became the Principles of Mathematics (1903). 5 His first substantial attempt to draft this book, completed in July 1898, was entitled "An Analysis of Mathematical Reasoning". He summed up this draft, in its introduction, by observing that

XVlll

when I was working on Leibniz. I found-what books on Leibniz failed to make clear-that his metaphysic was explicitly based upon the doctrine that every proposition attributes a predicate to a subject and (what seemed to him almost the same thing) that every fact consists of a substance having a property. I found that this same doctrine underlies the systems of Spinoza, Hegel, and Bradley.... (1959, 61)

XlX

the main ideas of Mathematics are the following: The Manifold and (logical) addition, which are used throughout; Number, which is first introduced in Arithmetic; Order, which is introduced with the ordinal numbers; the relations of equal, greater, and less, which constitute quantity; the extensive continuum, which distinguishes extensive from intensive quantity; the idea of dimensions; and the idea of thing, ... not explicitly introduced till we come to Dynamics. These ideas will be found for the most part indefinable .... It will be found that one pervading contradiction occurs almost, if not quite, universally. This is the contradiction of a difference between two terms, without a difference in the conceptions applicable to them. I shall call it the contradiction of relativity. This, with addition and the manifold, appear to define the realm of Mathematics. (1898b, 166)

In his "Analysis of Mathematical Reasoning", Russell associated the contradiction of relativity particularly with signed quantities, the notion of

3 The stronger sense of asymmetrical first appears unambiguously in this volume in Paper 2, 194, although he is quite close to it with his use of "unsymmetrical" and "asymmetrical" in Paper 1, 35, 38.

4 Russell 1897b, 58. For Bosanquet's views, see his 1888, 1: 102:3. 5 Letter from Russell to Couturat, 25 June 1902.

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INTRODUCTION

order, and relations that ascribe different predicates to their terms-what he would later call asymmetrical transitive relations. He regarded this contradiction as unavoidable in mathematics, since, he then held with Bradley, every relation is reducible to "adjectives" (predicates) of its terms. By I900, however, he had turned this argument on its head; now, the fact that asymmetrical transitive relations lead to the contradiction of relativity, if one assumes the doctrine of internal relations, was Russell's reason for rejecting this doctrine (Paper 1, 89-90). Asymmetrical transitive relations already played a role in "The Fundamental Ideas and Axioms of Mathematics" (1899b, 266), Russell's second substantial attempt to draft the book that became the Principles. He had ceased to be a Hegelian, and the contradiction of relativity was no longer of concern. But two other antinomies, one on causality and one on infinite number, troubled him. He devoted a chapter of "Fundamental Ideas" to the second of them, and summarized it as follows:

II. FROM LEIBNIZ TO PEANO

Chap. vn. Antinomy of Infinite Number. This arises most simply from applying the idea of a totality to numbers. There is, and is not, a number of numbers. This and Causality are the only antinomies known to me. This one is more all-pervading .... No existing metaphysic avoids this antinomy. (1899b, 267) He spelled out this antinomy in more detail in the outline to a different chapter: "There are many numbers, therefore there is a number of numbers. If this be N, N + 1 is also a number, therefore there is no number of numbers" (1899b, 265). The antinomy of infinite number continued to be troublesome in his third substantial attempt, the I899-I900 draft of the Principles (Paper 1). This antinomy was very similar in form to the Paradox of the Largest Cardinal, which he formulated in January I90I and which led him in May I90I to Russell's Paradox. There were two differences in content between the antinomy of infinite number and the Paradox of the Largest Cardinal. First, in the interim he had come to accept the existence of infinite cardinal numbers, and so the phrase "number of numbers" was replaced by "cardinal number of the class of all cardinal numbers", where some of them were infinite. Second, he replaced the step in the antinomy from a number N to a larger number N + 1 by Cantor's argument from a cardinal number X"' to a larger cardinal number 2l'{"'. It is not at all clear that he consciously saw this formal similarity between the antinomy of infinite number and the Paradox of the Largest Cardinal.

XX1

We have seen that Russell's book on Leibniz, which he completed early in 1900 , reflected his changed views on relations, on subject-predicate logic, and on substance. On 24 March he wrote to the.French p~ilosopher Louis Couturat, with whom he had been correspondmg on logic and the hilosophy of mathematics for two years, that "I have been very busy with ; book on Leibniz that I have just finished." 6 He learned in April that Couturat too was engaged in a book on Leibniz's logic. On 5 May, Russell responded: It is rather curious that, without knowing it, we have encountered each other on the subject of Leibniz. My book is rather a sketch of his entire philosophy than a monograph on a part of it. Nevertheless, it is Logic that has interested me the most, and I wrote the book because I saw that the rest of the system was founded on logic. 7 During this period, Russell's letters to Moore were mainly about the proper translation of Leibniz's Latin. Russell was increasingly concerned with another question that he had mentioned in "Fundamental Ideas", one which was connected with antinomies and with his rejection of the doctrine of internal relations: absolute position (1899b, 270). Earlier, while he was still an idealist, he had accepted the relativity of motion and of position. But at that time he had argued that the "relativity of motion leads to an infinite regress in space, which is a precise counterpart of the equally fatal infinite regress in time caused by causality" (1896g, 19). The first infinite regress was parallel to what he called "the antinomy of absolute motion", and together they reinforced him in his Hegelian belief that physics would require a dialectical transition to a still higher science (1896/, 15). Even at that time, however, he had felt some attraction to absolute motion. 8 When Russell rejected idealism and the doctrine of internal relations, he also rejected the Leibnizian view that motion is purely relative, and 6 Like all of the letters between Russell and Couturat, this one was in French. Since these letters remain unpublished, we shall give the original French, which reads in this case: ''J'ai ete tres occupe par un livre sur Leibniz que je viens de finir." 7 The original reads: "II est bien curieux que sans le savoir nous nous soyons rencontres au sujet de Leibniz. Pour moi, mon livre est plutot une esquisse de sa philosophie entiere qu'un monographe sur une partie. Cependant c'est la Logique qui m'a le plus interesse, et c'est parce que j'ai vu que le reste du systeme etait fonde sur la logique que j'ai ecrit le livre. 8 For a more detailed discussion, see the general headnote to Part II (pp. 215-18).

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INTRODUCTION

embraced absolute position. He discussed this reversal in a letter of I9 May I899 to Couturat: "I believe I possess logical reasons, as well as geometric reasons, which force me to adopt absolute position in space and time". 9 These logical reasons were connected with the realist position that he shared with Moore. Russell saw his view on absolute position as depending on his other views on logic and as requiring "an entire book to justify it .... Hence I do not wish to speak prematurely of my new opinions, until I have a solid base." 10 An entire year passed before Russell made his first public pronouncement in favour of absolute position. Then, in May I900, he gave a paper to the Oxford Philosophical Society on the question "Is Position in Time Absolute or Relative?" (Paper 4). Even when speaking to that mathematically unsophisticated audience, Russell found it necessary to discuss the mathematical technicalities of asymmetrical transitive relations. He saw the question of absolute position as one which, in effect, concerns all series and not just time and space. Russell spoke again on absolute position in August I900, when his audience was the International Congress of Philosophy, then being held at Paris. He chose this topic under Couturat's influence. In June I899, Couturat had invited Russell to speak at the Congress, in the section on the logic of the sciences. In August, Russell asked for Couturat's advice on a subject for his lecture, and thereby revealed the direction of his interests at the time:

dy been discussed enough" . 12 Russell accepted the suggestion, and 1 ahrea ult was Paper 5, "The Notion of Order and Absolute Position in t e res . . h p 6 . d . S and Time", published m Frenc . In 190I aper , a revise vers10n f~~eappeared, for the benefit of Engli~h philosopher.s, in Mind. Meanwhile, in the months before leavmg fo~ t~e Pans Congress, Russell te Paper 1, his 1899-1900 draft of the Principles. It reflected many of interests that he had expressed in his August letter, and included chapt e on every one of them. But, as we have seen, this period after he acters dp , · h pted Moore's views and before he adopte eano s 1og1c was very muc cetransitional one. This was particularly true of his view of infinite num~er. In I899, in "Fundamental Ideas", he had been very concerned with the antinomy of infinite number, writing that totality for classes "seems necessary; but if we make it so, infinite number with its contradictions becomes inevitable, being the number of concepts or of numbers" (1899b, 2 66). Here again, he stood at the brink of the Paradox of th~ L~rgest Cardinal. By the time he wrote the 1899-I900 draft, he was begmmng to take seriously Cantor's infinite ordinal and cardinal numbers, although he continued to hold that Cantor's theory does not render "the antinomy of infinite number one whit less formidable" (1, I I9). In contrast, Russell had altered his earlier position by accepting Cantor's treatment of the continuum. The logic that Russell developed in the 1899-1900 draft continued to reflect in good measure his views in "Fundamental Ideas", and even several of those from his Hegelian period. In particular, his emphasis on the category of whole and part and on its connection with Boole's logical system carried over from that period. By the time Russell wrote "Fundamental Ideas", he treated whole and part as a relation, indeed an indefinable one (1899b, 266), and it remained a relation in his later work. In the 1899-1900 draft, however, the matter becomes somewhat murkier. The relation of whole to part is "so important that almost all our philosophy depends upon the theory we adopt in regard to it" (1, 38), but the relationship between this relation, logical priority, and implication is now rather complicated. Implication is taken to be a "fundamental and indefinable" relation, from which he defines logical priority as follows: The proposition p is logically prior to q if q implies p but p does not imply q. After considering whether logical priority is identical with the relation of part to whole, he concludes that they are distinct (since logical priority is not indefinable), but that they are closely connected. The relation of whole to part must, he insists, be taken as indefinable and as distinct from

XXll

I could read on the infinite, the antinomies, and arithmetic-a subject on which I have, I believe, several ideas that are new but rather close to yours; or else, on the notion of order and series, including continuity, and the works of Cantor. Or else, on the analysis of the notion of quantity. Or again, on the necessity of absolute position in space and time. 11 Couturat replied quickly, suggesting that Russell speak on "a subject as new and as personal as possible, for example absolute position, or the notion of order, rather than questions relative to sets or quantity, which have 9 The original reads: "je crois avoir des raisons logiques, ainsi que des raisons geometriques, qui me forcent a adopter la position absolue dans l'espace et le temps." IO The original reads: "un livre entier pour le justifier.... Je ne desire done pas parler prematurement de mes nouvelles opinions, jusqu'a ce que j'ai une base solide." I I The original reads: "Je pourrais lire sur l'infini, Jes antinmnies, et l'arithmetique, un sujet sur lequel j'ai plusieures idees nouvelles, je crois, mais assez rapprochees des votres; ou bien, sur la notion de l'ordre et des series, ce qui renferme la continuite, et Jes travaux de Cantor. Our bien, sur !'analyse de la notion de quantite. Ou encore, sur la necessite de la position absolue dans l'espace et le temps."

0

w:o

12

The original reads: "un sujet aussi nouveau et aussi personnel que possible, par ex., la position absolue, ou l'idee d'ordre, plutot que Jes questions relatives aux ensembles ou a la grandeur, qui ont ete deja assez discutees."

INTRODUCTION

INTRODUCTION

the relation of implication (1, 35-8). Then the relation of whole and part can be treated formally, and without arithmetic, in Boole's logic as the notion of one class being included in another. He regarded every complex term as being composed of simple parts (1, 35-6), and it is here that the roots of logical atomism are to be found. Russell's treatment of arithmetic in Paper 1 also harked back to his idealist period, and did not differ fundamentally from that in "Analysis of Mathematical Reasoning" (1898). In both, addition of integers was based on ratio, which was taken as ultimate. His treatment of space was, in large part, merely an argument in favour of absolute space, as hinted at in "Fundamental Ideas" (1899). It was his treatment of logic, number, and space that would be revolutionized by Peano's work. By contrast, Russell broke free from idealism in Paper 1 when he treated asymmetrical transitive relations. He emphasized their importance in refuting the traditional subject-predicate logic and in showing that some relations are not reducible to properties of their terms. He regarded the notion of order as philosophically important because it was bound up with these matters. These views would not change in any essential way in the

per "does have wide bearings, since the theory of relations set forth in hais destructive (if true) of every form of idealism in the d__d modern nse". To Bradley, Russell conceded on 22 June that "Moore does now s~e many objections to his theory of judgement, and that is the reason he ~oes not expand it systematically. I think both he and I feel a kind of despair on fundamental questions of logic, but I hope he may find solutions where I see none. "

XXIV

Principles (1903).

The philosophical importance of order was the central theme of Paper 7, which Russell wrote shortly before meeting Peano but which was only published afterwards, in January 1901. This paper, Russell's first published attempt to make precise his new ideas on order, clarified a change in how he viewed the notion of the "sense" of a relation. There he argued that it is "a question of considerable importance in Logic, and particularly in the theory of inference, whether ... we are to speak of one relation with two senses, or two distinct relations with the relation of difference of sense" (7, 300). In 1899 he had held the first view, that a relation R is distinct from its two senses R 1 and R 2 , in the same way that the distance between two points A and B is different from the two directed distances (that from A to B, and that from B to A). This view is put forward in "Fundamental Ideas" (1899b, 269) and most clearly in "Note on Order" (1898, 353-5). He appears, though with less clarity, to adopt the same view in the 1899-1900 draft of the Principles (1, 76). By May 1900, he adopted a more sophisticated version of the first position, treating R as an equivalence relation and R 1 and R 2 as a correlated pair of asymmetrical, transitive relations which differ in sense (4, 224). It is only in Paper 7, when both views are discussed explicitly, that he accepts the second view. He retains this view in his later work. Paper 7 led Russell to exchange letters in 1901 on the notion of order and related questions with Bradley and other philosophers, and with his friend Gilbert Murray. 13 To Murray, Russell insisted on 14 April that this I

3 See the headnote to Paper 7 for a discussion of these letters.

xxv

III. FROM PEANO TO FREGE

The radical effect that Peano had on Russell, beginning at the Paris Congress of Philosophy, was first visible in a letter written to Moore on 16 August 1900. It reveals how Russell was now, for the first time, involved with the notion of variable, which would be a central theme of Papers 2 and 3: We got back from abroad last night, after a most successful time. The Congress was admirable, and there was much first-rate discussion of mathematical philosophy. I am persuaded that Peano and his school are the best people of the present time in that line. Have you ever considered the meaning of any? I find it to be the fundamental problem of mathematical philosophy. E.g. "Any number is less by one than another number." Here any number cannot be a new concept, distinct from the particular numbers, for only these fulfil the above proposition. But can any number be an infinite disjunction? And if so, what is the ground for the proposition? The problem is the general one as to what is meant by any member of a defined class. I have tried many theories without success. Five days later, he informed Moore that he was "learning Peano's system, which is splendid-the best thing that has been done for a very long time." Russell described the period glowingly in the Autobiography, saying of Peano that as soon as the Congress was over I retired to Fernhurst to study quietly every word written by him and his disciples. It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years .... By the end of August I had become completely familiar with all the work of his school. I spent September in extending his methods to the logic of relations. It seems to me in retrospect that, through that

XXVl

INTRODUCTION

month, every day was warm and sunny. The Whiteheads stayed with us at Fernhurst, and I explained my new ideas to him. Every evening the discussion ended with some difficulty, and every morning I found that the difficulty of the previous evening had solved itself while I slept. The time was one of intellectual intoxication .... For years I had been endeavouring to analyze the fundamental notions of mathematics, such as order and cardinal numbers. Suddenly, in the space of a few weeks, I discovered what appeared to be definitive answers to the problems which had baffled me for years. And in the course of discovering these answers, I was introducing a new mathematical technique, by which regions formerly abandoned to the vaguenesses of philosophers were conquered for the precision of exact formulae. Intellectually, the month of September I900 was the highest point of my life. (1967, I44-5) Early in October, he wrote a draft of Paper 8, his article for Peano's journal on the logic of relations. This version, printed as Appendix V.I, was revised heavily before he sent the final version to Peano early the following year. The draft showed a concern with developing a logic based on diversity and with treating group theory,1 4 an important part of algebra, in terms of relations; these concerns vanished in the final version. Later in October, he composed a paper on Peano that was intended for Mind. Already on 25 August, he had written to its editor, G.F. Stout, who had been a teacher and undergraduate tutor of Russell at Cambridge and with whom he occasionally corresponded on philosophical questions. Russell urged Stout to consider publishing long critical articles in Mind, and offered to write one on Peano's school. The result was Paper 9, which for unknown reasons remained unpublished. In Paper 8 he insisted that "the logic of relations ... must serve as a foundation for mathematics, since it is always types of relations which are considered in symbolic reasoning" (8, 3I4). This paper used his logic of relations, and Peano's logic, to deal with a problem that had troubled him for years: Are numbers ultimate and indefinable? In "Fundamental Ideas" he had regarded every individual number as indefinable, rejecting in particular the view that addition can be used to define the numbers beyond 1 (1899b, 265). He retained this perspective in the I899-I900 draft of the Principles (1, I8). When he first drafted his paper for Peano, in October I900, his treatment of numbers had shifted substantially, and he thought 14 Some of the treatment of groups was more sophisticated than what appeared in the published paper. Such was the case with his generalized notion of "distance", which was simultaneously a dense series and a permutation group.

INTRODUCTION

xxvii

h t cardinal numbers, and thus 1, 2, 3, ... in particular, are definable.

~a obtain a definition of cardinal numbers, he used what Peano called "~efinition by abstraction". That is, given an equivalence relation R, there is a function x such that xRy if and only if x = y; thus, for xample, the relation of one-one correspondence between two classes x :ndy give rises to the ~unction "cardi~al number of x'.' (Peano l894a, 45). But Russell regarded 1t as necessary, m order to obtam such a function (or as he preferred to put it, a many-one relation S), to introduce a primi~ive proposition stating that any equivalence relation R can be written as the relative product of a many-one relation Sand its converse (v.I, §I, *6·2). He applied this primitive proposition, which in the Principles he called the Principle of Abstraction (1903, I66), to the relation of similarity between classes, thereby obtaining such a relation S; then he defined the class Ne of all cardinal numbers as the codomain of S. But in a marginal comment by this definition, he recognized the problem that S is not uniquely determined: "This won't do: there may be many such relations as S. Ne must be indefinable" (V.I, §3, *1·4). While in his first draft (Appendix v. I) he freely referred to the cardinal number of a class without any mention of S, in the published version (Paper 8) he took a particular S as given and only defined individual cardinal numbers in terms of S. Nevertheless, sometime between February and July I90I, he added a sentence to the effect that, for any equivalence relation R, we can always take the equivalence class of a term u as "the individual indicated by the definition by abstraction; thus for example the cardinal number of a class u would be the class of classes similar to u" (8, 320). This was the famous Frege-Russell definition of cardinal number. Russell applied it not only to the relation of one-one correspondence, in order to obtain cardinal numbers, but to any equivalence relation whatever. It is likely that Russell was led to consider this definition of the cardinal number of a class through reading an article by Peano that appeared in March I90I. Some might think, Peano remarked, that num a, the cardinal number of any class a, can be defined as the class of all classes similar to a. But he explicitly rejected this as a definition of num a, which he regarded as having properties different from the class of all classes similar to a (1901, 70). 15 In a part of the Principles written in June I90I, Russell discussed this article and argued, against Peano, that this definition of cardinal number is both adequate and appropriate (1903, II5-I6). Late in 1900, Russell began to draft a new version of Parts III-VI of the Principles, essentially the final version. "Every day throughout October, 1

5 Peano, in turn, was probably led to consider this definition of num a (and to reject it) by reading Frege's Grundgesetze (1893), which he reviewed in 1895a and which contains the same definition.

xxviii

INTRODUCTION

INTRODUCTION

November and December," he later remarked, "I wrote my ten pages, and finished the MS on the last day of the century" (1967, 145). In a boastful mood, he wrote to his friend Helen Thomas, at Bryn Mawr, about this draft: "In October I invented a new subject, which turned out to be all (of) mathematics, for the first time treated in its essence. Since then I have written 200,000 words, and I think they are all better than any I had written before." Influenced by Peano, Russell radically revised the logic found in the I899-1900 draft of the Principles. In that draft, as we saw above, implication was intimately connected to whole and part, and was treated in terms of class inclusion, following Boole and his successors. In October, as Russell began to revise that draft, he wrote an important comment in the margin:

now treated implication as having the form x ::> x 1./fx. "The enlightenment that I derived from Peano", Russell wrote later,

Note. I have been wrong in regarding the Logical Calculus as having specially to do with whole and part. Whole is distinct from Class, and occurs nowhere in the Logical Calculus, which depends on these notions: (I) implication, (2) and, (3) negation. Whole and part require the Teoria della grandezze (Bettazzi 1890),

i.e. a special form of addition, not that of the Logical Calculus.

XXIX

came mainly from two purely technical advances of which it is very difficult to appreciate the importance unless one had (as I had) spent years in trying to understand arithmetic .... The first advance consisted in separating propositions of the form "Socrates is mortal" from propositions of the form "All Greeks are mortal" .... Neither logic nor arithmetic can get far until the two forms are seen to be completely different. "Socrates is mortal" attributes predicate to a subject which it names. "All Greeks are mortal" expresses a relation of two predicates-viz. "Greek" and "mortal". The full statement of "All Greeks are mortal" is "For all possible values of x, if x is Greek, xis mortal" (i.e. x =>x 1./fx.) ..•• The statement ... says nothing about Greeks in particular, but is a statement about everything in the universe. (1959, 66) This shift in his view of logic was first apparent in Paper 2, his May draft of Part I of the Principles. The last aspect mentioned, that variables must range over everything in the universe, was apparent there when he defined logic: "And logic may be defined as (1) the study of what can be said of everything, i.e. of the propositions which hold of all entities, together with (2) the study of the constants which occur in true propositions concerning everything" (2, I87). The first aspect, whereby propositions of the form "Socrates is mortal" are separated from those of the form "All Greeks are mortal", is closely connected with his new view of what pure mathematics is: 1901

He added a second comment: "I must preface Arithmetic, as Peano does, by the true Logical Calculus, to be called Book I, the Individual. [Oct. 1900]". 16 In fact, he appears not to have written a new draft of Part I (or Book I) of the Principles until May I90l. This new draft, previously unpublished, is printed as Paper 2. Before examining Paper 2, let us consider the significance of these marginal comments. The second of them shows that, instead of beginning with number (as he had done in "Fundamental Ideas" and Paper 1), he now began Paper 2 and the Principles with logic, and only then developed the notion of number. The first marginal comment is more complicated. From early I899 until July I900, as was mentioned above, Russell's logic was based on class inclusion, along Boolean lines. From his later perspective in the Principles, this Boolean "Logical Calculus" was not the whole of logic but merely the propositional calculus, where propositions vary rather than individuals, and are only built up by using propositional connectives: implication, and, or, not. The propositional calculus only studies material implication, and so cannot quantify over variables other than propositions. The "true Logical Calculus", as he called it, was Peano's symbolic logic supplemented by Russell's new calculus of relations. Following Peano, he

~his .was his first precise statement of logicism. It differed from the version m the Principles in the way that he defined logical constants. There he .was to state that logical constants are all the notions definable in terms of 1 ~Pl'ication · (with a universal quantifier), the membership relation, the notion of "such that" (class abstraction), and the notion of relation. Here,

16 See Plate

17 Paper 2, 185. See Plate m.

II.

Pure Mathematics is the class of all propositions of the form "a implies b", where a and b are propositions each containing at least one variable, and containing no constants except logical constants or such as can be defined in terms of logical constants. And logical constants are classes or relations whose extension either includes everything or has as many terms as if it included everything. 17

xxx

INTRODUCTION

in Paper 2, he insisted that logical constants are universal in the same way as variables, i.e. each logical constant is in one-one correspondence with the universe of all terms. This universality, at least with respect to variables, was a feature that his logic shared with Frege. Peano's attitude toward such universality was more ambiguous. He did not insist that his variables had to vary over the whole universe, typically restricting them to some universe of discourse (such as all propositions or all numbers). But at times he required that his variables vary over all objects (1900, §1). Peano was instrumental in producing an important change, albeit temporary, in how Russell viewed the problem of finding the indefinable notions and the indemonstrable propositions at the base of logic. Previously Russell had believed that there was only one true, philosophically correct way of treating the foundations of mathematics. He had expressed this dogmatically in Paper 1: Every concept is necessarily either simple or complex, and it is not in our power to alter its nature in this respect. If it is complex, it should be analyzed and defined; if simple, it should be used in defining other terms, without itself receiving a definition.... It does not lie with us to choose what terms are to be indefinable; on the contrary, it is the business of philosophy to discover these terms. (1, 57)

INTRODUCTION

though there are indefinables and indemonstrables in every branch of applied mathematics, there are none in pure mathematics except such as belong to general logic. Logic, broadly speaking, is distinguished by the fact that its propositions can be put into a form in which they apply to anything whatever. All pure mathematics-Arithmetic, Analysis, and Geometry-is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic.... (10, 366-7) He noted that this reduction had already been accomplished "over the greater and more difficult part of the domain of mathematics ... ; in the few remaining cases, there is no special difficulty, and it is now being rapidly achieved" (10, 367). No doubt he had in mind the work of Cantor Dedekind, and Weierstrass, as well as that of Peano and his school. ' Paper 10 still displayed a facile self-assurance that was soon to be severely shaken from two quite different directions, when he "began to be assailed simultaneously by intellectual and emotional problems which plunged me into the darkest despair that I have ever known". The first direction was emotional, even spiritual. It arose in the month after the paper was written, on the evening when Gilbert Murray gave a public reading of his translation of Euripides' Hippolytus. After the reading Russell and his wife Alys returned home to find '

Now in Paper 2, under the influence of Peano,1 8 he wrote: The question as to which of the notions of symbolic logic are to be taken as indefinable, and which of the propositions as indemonstrable, is ... to some extent arbitrary. But it is important to establish all the mutual relations of the simpler notions of logic, and to examine the consequences of taking various notions as indefinable. (2, 203) Russell retained that passage in the Principles. By the time he finished Principia Mathematica, however, he was convinced that only the theory of types was adequate as a foundation for logic and mathematics. Russell's first statement of logicism appeared in a popular article, "Recent Work on the Principles of Mathematics", written in January 1901 (Paper 10). There he argued that I8

In the various editions of the Formulaire de mathematiques, Peano used quite different sets of primitive notions.

xxxi

Mrs. Whitehead undergoing an unusually severe bout of pain. She seemed cut off from everyone and everything by walls of agony.... Ever since my marriage, my emotional life had been calm and superficial. I had forgotten all the deeper issues, and had been content with flippant cleverness. Suddenly the ground seemed to give way beneath me, and I found myself in quite another region. Within five minutes I went through some such reflexions as the following: the loneliness of the human soul is unendurable; nothing can penetrate it except the highest intensity of the sort of love that religious teachers have preached .... . At the end of those five minutes, I had become a completely different person. For a time, a sort of mystic illumination possessed_ me .... Having been an imperialist, I became during those five mmutes a pro-Boer and a pacifist. Having for years cared only ~or exa~tness and analysis, I found myself filled with semi-mystical feelings about beauty ... and with a desire almost as profound as that of the Buddha to find some philosophy which should make human life endurable .... The mystic insight which I then imag-

XXXll

INTRODUCTION

ined myself to possess has largely faded, and the habit of analysis has reasserted itself. But something of what I thought I saw in that moment has remained always with me.... (1967, 146) The second direction was intellectual: the emergence of Russell's Paradox in May 1901. The paradox had its roots in his discovery of the Paradox of the Largest Cardinal (1959, 75). On 8 December 1900, he wrote to Couturat: I have discovered a mistake in Cantor, who maintains that there is no largest cardinal number. But the number of classes is the largest number. The best of Cantor's proofs to the contrary can be found in (Cantor 1891) .... In effect it amounts to showing that, if u is a class whose number is a, the number of classes included in u (which is 2"') is larger than a. The proof presupposes that there are classes included in u which are not individuals (members) of u; but if u = Class, that is false: every class of classes is a class. 19 This passage contained all the ingredients for the Paradox of the Largest Cardinal, but at the time he did not regard it as a paradox or contradiction. Couturat replied by doubting that "one can consider the class of all possible classes without some sort of contradiction." On 17 January 1901, Russell vigorously defended Class, the class of all classes: "If you grant that there is a contradiction in this concept, then the infinite always remains contradictory, and your work as well as that of Cantor has not solved the philosophical problem." Russell insisted that "no contradiction results" from Class since Cantor's 1891 proof does not apply to it. Russell analyzed his argument of December 1900, in effect the Paradox of the Largest Cardinal, and was thereby led to Russell's Paradox: the class of all classes that are not members of themselves is, and is not, a member of itself (1959, 75). But, strange to say, he appears not to have discussed his paradox with anyone for an entire year. In June 1902 he wrote about it to Peano and Frege, then in September to Couturat-a matter we defer to the following section. The earliest extant document containing Russell's Paradox is Paper 2. There he formulated his paradox in a way that involved no classes, but only predicates. He argued that no predicate applies to exactly those predicates that cannot be predicated of themselves. This argument led him to reject the Principle of Comprehension, which he had adopted from Peano, that "every definable collection of terms forms a class defined by a com19 This and the following letters are printed in Moore and Garciadiego 1981, 325-7.

INTRODUCTION

xxxiii

predicate". He began a search, which lasted for almost a decade, "to a class"

%~;over what properties a collection must have in order to form

(2,Despite i9s). discovermg · h'1s parad ox, R usse11 plunge d ah ead wit . h h'1s atmpt to work out his logicism in complete technical detail by developing ~ea Peanoesque symbolism all of pure mathematics, including geometry, :ithin his logic of relations. In January 1901, he had persuaded Whitehead of the importance of his logic of relations, 20 but the first evidence of their intention to collaborate on Principia Mathematica came in a letter of October from Russell to Couturat: "I intend to write with Whitehead 2 a book On the Logic of Relations, with Applications to Arithmetic, to Group Theory, and to Functions and Equations of the Logical Calculus. But we will need at least two years to complete it." In actuality, it took nine years rather than two. By June of 1901, Russell had completed his part of their first joint publication (Paper 13). 21 It was a paper on cardinal numbers treated in Peanoesque notation within Russell's logic of relations. His part of the paper dealt with finite and infinite cardinals, and contained his first symbolic statement of the Frege-Russell definition of cardinal number. It also included an attempt to find appropriate axioms for the arithmetic of transfinite cardinals. One of those axioms was a special case of what he later called the Multiplicative Axiom, more commonly known now as the Axiom of Choice. The special case (*4·3) stated that every infinite class is the union of some family of denumerable classes. The Multiplicative Axiom is the proposition that if u is a class of mutually exclusive classes other than the empty class, then there is a class v having exactly one member in common with each x in u; the axiom's name comes from the fact that it states that the "multiplicative class" of u is non-empty, and was used implicitly by Whitehead (1902) to define the multiplication of infinitely many cardinal numbers. Russell had first encountered a proposition that depends on the Multiplicative Axiom, but without knowing that he did so, in October 1900 when he took Cantor's Trichotomy of Cardinals (i.e., if a and f3 are cardinals, then a = f3 or a< f3 or f3 which, by the definition of arithmetical addition, is ~he number of the collection (Cm and Cn). Thus the number (m+n)+p is the number of the collection (Cm and Cn) and Cp, while m + (n + p) is the number of the collection Cm and (Cn and Cp). Here we have no longer any need of reference to numbers: we have only addition of terms, in the 1 Peano gives a proper proof, Formulaire 1898, §2, 012·4.

30

24

IO

20

30

1 PART I, CHAP. IV: MULTIPLICATION

sense of Chapter I, and we have to show that (Cm and Cn) and GP is the same collection as Cm and (Cn and Gp). Now what is the meaning of the brackets in these expressions? The meaning is as follows. (Cm and Cn) and GP means: To Cm, a collection Cn is added; to the resulting collection, a collection Gp is added. Cm and (Cn and Gp) means: To Cm the sum of two collections Cn and Gp is added. The associative law then asserts that the two total collections thus defined are identical. Now it may be observed that the two consist of precisely the same terms, occurring in precisely the same order. The only difference arises in the grouping. But what does the grouping amount to? One group of terms is not, for present purposes, one group, but many terms; for considered as one group, its number is one, not morn or p. Thus the fact that Cm is one collection is not relevant; what is relevant, is the fact that Cm ism different terms. Hence it is evident that the grouping of the terms is irrelevant, since in any case the collection is not a sum of groups, but a sum of separate terms. The commutative law proves that the number of a collection is the same whatever order we give to its terms. For suppose we have a collection (Cm and Cn) or Cm+n- Then m + n = n + m, and therefore Cm+n = Cn+m = (Cn and Cm)· Thus if a collection be divided into any two groups, these two groups may be permuted. By repetitions of this process, any order of the terms can be obtained from any other order, provided the number of terms is finite. A few subtleties, however, are required at this point. Of a collection A and B and C and etc., it is not strictly correct to say that its number is the same whatever order we give to the terms. For the correct statement is, that the terms of the collection as such have no order. Terms which have an order form not merely a collection, but a series. A confusion naturally arises in this matter from the fact that we cannot mention the terms without giving them an order. But this is a mere exigency of speech and writing, due to the fact that these occupy time and space respectively: it has nothing to do with the nature of the collection itself. The terms of the collection should be considered as being wholly destitute of order, or as all simultaneously proclaimed by a well-trained choir: the difference between A and B and C and C and B and A is like the difference between French and English, a mere change of symbol without any change in the thing symbolized. In spite of this, however, the proposition

h umber of the new collection is therefore m + n or n + m indifferand t ~~e possibility of combining collections does not result from the entlY· ~ mal arguments, for it is presupposed in the meaning of (m) + (n). ab~ve or bol means simply the number of the collection resulting from the Thisb.symtion of two collect10ns · Cm and Cw Thus fi na11y, on1y t h e propco~ tnfa the addition of terms are required for the commutative and aserues o . .h . 1 dd. . sociative laws as apphe? to ~nt rr.iet1 ca a. iuon.h . dd .. Th axiom last obtamed is og1ca11y pnor to t at connectmg a iuon et · Thus the foundations of the so-called addition of integers are and ra 10 . . . tained in the followmg axioms: co( ) Any two collections can be combined to form a third collection. (~) If the first two collections have a number, so has the third. In symbols (m) + (n) = (p ). c ) Numbers have an intrinsic order obtained from mutual relations 3 which may be called ratios. In virtue of this order, two numbers may be consecutive. ( ) The collection formed by adding one term to a collection of n terms 4 has the number which comes next after n. In symbols, (n) + (1) = (n + l); where (n + 1) means the number next after n. From these axioms, all the particular propositions concerned with addition of particular integers can be deduced, it being assumed that all the integers are known in their order.

m+n=n+m

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has a very important meaning, which is this: Any two collections can be combined to form a third collection. This fact alone, when we remember that a collection is wholly defined by its terms, suffices to prove the commutative law. For if two collections Cm and Cn be combined to form a new collection, the terms of the new collection have again no order inter se,

1

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IV

Multiplication r IS EVIDENT that the theory of ratio propounded in Chapter II renders multiplication and the rule of three far more fundamental than they usually are supposed to be. In favour of my view, I might urge that modern work on universal Algebra, and on the principles of a calculus in general, have more and more tended to exhibit addition and multiplication as two radically distinct operations. Either may be modified in various ways, either may be retained in a Calculus which dispenses with the other, and generally the two principles of synthesis follow very divergent courses. Thus a view which places arithmetical multiplication on a level with arithmetical addition cannot be accused of being opposed to the spirit of modern work on the general principles of symbolic reasoning. The rule of three follows at once from the relation of ratio, without our requiring any allusion to addition. If a and b be any two numbers, they have a definite ratio a:b. Given another number c, it is required to find a fourth number x such that the ratio c:x is the same as the ratio a:b. The-

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oretically, this problem, where it has a solution (i.e. where x is integral for it would be premature to introduce fractions at this stage), can be' solved by immediate inspection. I do not mean that x can be found by inspection, but that, any x being proposed, it should be obvious whether this x is or is not the required number. For the problem involves no relation except ratio, and ratios are immediate relations between integers, not logically deduced from any other properties of the numbers. But practically, it must be admitted, most people have not a sufficiently keen arithmetical vision to be able to perceive ratios immediately, except where the numbers concerned are very small. And if no indirect method of estimating ratios existed, we should be left unable to determine whether the ratio a:b is the same as or different from the ratio c:x. Cases of such inability will meet us frequently when we come to series in general. For example, the colours of the rainbow form a series, ordered by relations of greater and less immediate similarity or difference. I shall prove in Part IV that such relations are intensive quantities, of which any two are equal or unequal. Thus theoretically, given two shades of red and one shade of blue, there is a definite answer to the question what shade of blue differs from the given shade as much as the two shades of red differ; but practically this question cannot be answered. We have to examine, therefore, how it is that the corresponding question as regards numbers always can be answered. It will be well, as a preliminary, to define multiplication, which is a particular case of proportion. When, in the equation a:b = c:x, a is 1, x is said to be b multiplied by c. This is shown to be the same number as c multiplied by b, i.e. x is the same number in the two equations: l:b

=

c:x and l:c

=

b:x.

xis then spoken of as the product of band c. We shall find the numerical measurement of ratios greatly simplified by a consideration of multipli30 cation and its axioms. How are we to discover, when b and c are given, what number is b multiplied by c? It will be seen that this problem is essentially the same as that concerning proportion, but simplified by the fact that a is replaced by 1. Some axiom, we shall see, is required to connect multiplication with addition. In the previous chapter, we established the axioms necessary to addition, so that we now know how to discover what number is represented by m + n, when m and n are known. But we do not yet know how to discover a number which is twice or thrice any other number, or is some other given multiple of some given number. In order to effect this 40 we require the following axiom: "If m and n be integers, then m multiplied by n is the number of a collection formed by adding n collections each

. .

27

of m terms." This is another axiom connecting addition and

~sis~ing mbols it is m x n = m + m + ... to n terms. It follows from the

~0 · :~~

from the properties of addition already dev.eloped that x m. The axiom enables us to find a number mn havmg the same ~n as n has to l. We shall see when we come to fractions that it :auol to compare the ratio of any two numbers with that of any other nab es us om_

:0 :

w~he proof that mn = nm may be made by induction, as follows: >< = m, 1 x m = 1 + 1 + ... to m terms = m. Hence m x 1 = 1 x m. As1 me now mxn = nxm. Then

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(m+ l)n = (m+ l)+(m+ 1)+ ... ton terms = m + m + ... to n terms+ 1 + 1 + ... to n terms =mn+n =nm+n = n+n+ ... tom terms+n = n+n+ ... to (m+ 1) terms = n(m+ 1).

Hence, since m·l = l·m, it follows that mn = nm.always. 1 •• The proposition mn = nm is, in terms of ratio, the proposltlon that l:m = n:x implies 1:n = m:x. It may be instructive to observe that, when ratios are considered as the distances of terms in a series, the above proposition is closely analogous to the following: If A, B, C, D be four points on a straight line, or four moments of time, such that AB = CD, then AC= BD. This proposition is one of those that characterize series in which distances are capable of numerical measurement and, by a · suitable convention, of addition. In the spatial illustration, the proposition would commonly be held to be proved by adding (or subtracting) the distance BC to each of the distances AB, CD. But such a proof, as we shall find hereafter, presupposes a host of previous propositions, namely those by which the addition and measurement of distances are defined. To apply such a proof to ratios demands, evidently, that they should be measured by their logarithms. But this is a point to which I shall return later. It is necessary first to define fractions, and to show how to discover generally .. the equality or inequality of two ratios. I

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It should be observed that all the above proofs depend upon mathematical induction. Since the essence of Cantor's transfinite numbers is that they cannot be obtained by successive additions of 1, mathematical induction cannot reach them. It is therefore not surprising that the above properties should fail for transfinite numbers. 40

1 PART I, CHAP. V: RATIONAL FRACTIONS

Chapter v Rational Fractions N THE PRECEDING chapter, a general problem was propounded, namely, given three numbers a, b, c, to find a new number x such that a:b = c:x. This problem in its general form was not further pursued, but we contented ourselves with a discussion of the special case in which a = 1. It is now time to return to the general case. Here the problern no longer has a solution (in general), so long as we confine ourselves to integers. The general solution demands the admission of fractions. How fractions are to be specified, and how the rules of operating with fractions are to be established, it is now time to inquire. It is important first of all to distinguish fractions clearly from ratios. We have already had the ratio half, which is simply the ratio 1:2. But the number 1/ 2 is not to be a ratio; it is to be a number which can be added to other numbers, and is by no means to have the nature of an intensive magnitude. Again we wish to have 1/ 2 + 1/ 2 = 1. But the addition of ratios yields no such result. If they are added as terms, we get simply two ratios. If they are added by the convention which is usually convenient in series, namely distance AB+distance BC= distance AC, then 1/ 2 + 1/ 2 = 1/ 4 • Thus if we are to have any special arithmetical addition of ratios, it is analogous rather to the multiplication than to the addition of numbers, which again suggests logarithms as the best measures of ratios. These remarks illustrate the fact that fractions are something wholly distinct from ratios. It is common to define fractions by means of addition, somewhat in the following manner. If two collections have the same number of terms, and if their sum, taken as one whole, be one, then the number applicable to each of the original collections is 1/ 2 . There are however many objections to such an account of the matter. The sum of the two collections is many if taken as it results from addition, and is necessarily one if taken as a whole. Similarly each of the original collections is many if taken as a collection, and is one if taken as one whole. In no case is either collection one-half. No collection can have any other than an integral number: its number must be one or many. Thus addition alone, with the notion of collections, does not enable us to obtain fractions. Fractions are sometimes obtained by a reference to extensive magnitude. If A be such a magnitude, and B and C be two equal parts which together compose A, then each is said to be half of A. The objection to this definition is, that fractions are far simpler than extensive magnitude, and should be defined in some purely arithmetical manner. The notion of extensive magnitude, as we shall see in Part III, is highly complex-far

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magnitude-and really presupposes, for its t hat of intensive · · · wou ld b e most unore so than ment the not10ns of fractions. Hence it uate treat ' . eq d fine these by ns means. wise to e 'fons asserting fractions show an important difference from 1 b T he propos1 · integers. We can say: A is one, A and Bare two, etc.; ut .h e asserting .d Th . 1 t . os . A is one-half, A and Bare two-thir s, etc. ere is a ways ·.w . . . e cannot say. econd entity to which our first has a fractional relation. d of some s ' pee . A is one-half of B, A and B are two-thirds of C, etc. Thus must sayh. re they occur in other than arithmetical propositions, always .·.t'ffle ;f: ctions w e . . f ·• .. r. a ' 1 ·ons Thus we seem brought back to the mterpretation o ress re au · . . . d exP . tios which n was our object to av01d. Let us en eavour to f: actions as ra ' " h' r. · ly what is meant by "A is one-half of B . T is seems to exstate precise A dB b ;. 1 t'on whose terms are A and B. Hence an must not e ress a re a 1 . b 1 ;P · s such or at least, if they are collect10ns, they must e co .:collect10ns a ' . . f e term each. For each of them has to be one term m a re;:lecuons o on . . 1 1 . . d' d :. . . . B t · order that they may have the anthmetica re ation m icate , Iauon. u m · b : is . necessar y that B at least should be somehow connected with a num .+er it . This is effected by regardmg B as the whole composed 01 a other t han 1· .. ·•.·.co11ecuon · of terms · Regarding A in like manner, the proposition asserts . 'th h number of parts in A is one-half of the number of parts m B; . at t e ( h r . while if B has only two parts, A consists of a single term. T e ap~ 1cat10n .. of this account, in cases where A and B are not composed of a fimte number of indivisible parts, is a problem which I leave for Part II.) Thus we . are definitely brought back to ratio, and 1/ 2 appears as fundamentally a ; relation between numbers, which applies to A and B only by correlation with their respective numbers of parts. · If the above account be correct, then the interpretation of 1/ 2 + 1/ 2 = 1 ·••·is as follows: If A and B be two collections each of which has half as many terms as C, then A and B together have the same number of terms as C. .This is a particular case of our general proposition: mn = m + m + ... to ;•n terms. The interpretation of fractions whose numerator is other than •:·unity is similar. If A be a collection of m terms taken as a whole, and B •a collection of n terms taken as a whole, then A is m/nths of B. The •;•·fraction may, for convenience, be still distinguished from the ratio m:n, by taking the fraction as the complex notion applicable to the relations of . wholes, while the ratio is the simple notion applicable to the relations of numbers. A similar distinction may be made with regard to the integers. If A is one and B is one, we shall say that the whole composed of A and B is double of A and double of B, where double is a complex notion applicable to terms, and corresponding to the simple notion twice. Double, treble, etc. may be regarded as fractions whose denominator is 1: they are correlated not only with a ratio n:l, but also with an integer n. For certain formal purposes, such fractions may be identified with the corresponding

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l PART I, CHAP. VI: INFINITE COLLECTIONS

integers; but their real meaning is by no means to be so identified. We can now return to proportion, and discuss the general solution of the problem a:b = e:x. In all cases where this problem has an integral solution, the preceding discussion, in which we put a= 1, will be found adequate. But in cases where there is no integral solution, the matter is otherwise. Let us suppose a, b, e, x to be numbers applicable to collections A, B, C, X, and let us choose these collections so that every term in each of them is a whole composed of m other terms. Then each of them may be replaced by collections having respectively ma, mb, me, mx terms. We still have ma:mb = me:mx.

·on A and B: thus two is not a relation of A and B to A and to B, ec t1 . d . .0 11 such a relat10n woul reqmre that A and B should be taken as one nc~ not as two. This is precisely the point in which fractions differ h~e~umbers. It ~s this that renders fractions definable and relational, like integers, mdefinable and absolute. Ofhus fractions, in so far as they are distinguished from ratios, are not undamental, but are definable relations between collections as wholes. h.e important an? in~efinable conc~i:ts are ratios. ~hatever ~an be as. ted of a collect10n, m any proposltlon not expressmg a relation, must an integer; and hence also arithmetical addition is, at bottom, always ~dition of integers. This is the reason why fractions have to be reduced a common denominator before addition; for then the numerators may 0 e added as integers, while the common denominator merely expresses a elation of two units. But this demands an explanation of the word unita word which we have not hitherto found it necessary to employ. This topic, however, belongs to the philosophy of whole and part, and will be iscussed in Part II, where we shall find a new development of the notion f fractions.

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If we make m =a, we shall have a:b

=

ae:ax

and hence a:ae = b:ax

20

(assuming the axiom that, if m:n = m':n' and n:p m':p') or l:e = b:ax. Hence we have ax= be.

=

n':p', then m:p =

Chapter

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VI

Infinite Collections

20

assumed in the previous discussion that all the collec•·· . tions with which we had to deal had some number of terms. This • assumption, however, places considerable restrictions upon the kind ,of collections concerned. Points, instants, colours, the numbers themselves, are collections of which no definite number can be asserted. Thus ;when a collection is given, it must always remain a question whether or :not it has a number. It is indeed common to assume that all collections have numbers, and to say of such collections as the above that they have an infinite number. It is a question, with which we shall continue to be occupied throughout the greater part of this work, whether such collec- 3o tions have no number or an infinite number. But first of all, we must ~endeavour to discover marks by which we may know whether a given ; collection is finite or infinite, and even what these words mean. In Chapter III, we set up the axiom that, if each of two collections has a number, so has their sum. In arithmetical language, this amounts merely to the assertion that the sum of two numbers is a number: formally, /(m)+(n) = p. It follows from this axiom that, if any given collection has . no .number, it cannot be completely specified by successive enumeration •of Its terms, one by one. For the enumeration of one more term adds 1 to the number of the collection, and thus yields a new number. Now there 40 1.c.1· T HAS BEEN

Thus ax is always an integer. Hence x is that fraction which expresses the relation of a collection of be terms to a collection of a terms; and this fraction corresponds to the ratio be:a. The problem of division now has also a general solution. Division may be defined as the inverse of multiplication. Multiplication was defined by l:b = e:x. Division will be defined by l:x = b:e. Applying this to four collections, each of which is a whole composed of b terms, we shall find bx= e, and thus by the definition of fractions, x = e/b. It might be thought that fractions might be defined simply by ratios, i.e. e/b might be defined by l:x = b:e. This would hold good if fractions were properly numbers, as we at first hoped. But since fractions were found essentially to involve collections, they cannot be thus defined. Thus b/e must be defined as the relation between a whole composed of b terms and a whole composed of e terms, a relation which is such that b times the second collection has the same number as e times the first. Thus fractions are only formally, not philosophically, to be included among numbers. Unlike numbers, they are ideas of relation. It may be worth while to point out why numbers are not also relations. If A and B are two, this is not true of the whole composed of A and B, but of the

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are three ways in which a collection may fail to be capable of complet enumeration. Either its terms may be incapable of order, so that it is irn~ possible, by any device, either to say that some specified term A come after or before some other specified term B, or to say that A is the nt~ term of the collection; or the first may be possible, but not the second· or, though both these be possible, yet it may happen that, whatever ter~ A we take, there is always, however we may order our terms, a term 8 which comes after A. It is to be observed that any collection whose num. ber is n is capable of an order. For if its terms do not themselves form a series, they may, by speech or writing, be correlated with times or spaces· or again they may be correlated, as in counting, with the first n numbers' It follows that terms which have no order, either intrinsically or by cor~ relation, form a collection which does not have any number n that can be specified. And generally, even where terms have an order, if there is no method of arrangement by which, when the method has been specified . any assigned term can be correlated with some number n, then the terms' in question have no number that can be assigned. For, if N be the number of terms, any possible arrangement will enable us to correlate with the various terms the numbers from 1 to N. But even where the terms have an order, and where any specified term is correlated with some number n, so that it may be said to be the nth, yet, if there is always a term after any given term, then no assignable number applies to the collection. For if N were its number, we could correlate the numbers from 1 to N with its terms, and the term correlated with N would be the last. Thus in all the cases mentioned, any number whatever that may be specified is seen to be not the number of the collection. The case last specified is that of the numbers themselves. For, by the axiom in Chapter III, the sum of two numbers is a number, and therefore we may always add 1 to any given number. Strictly speaking, a new axiom is required to assert that n + 1 is not any of the numbers 1, 2, ... n - 1, n. 1 This being also assumed, it follows that there is a number after any given number, and therefore no number N that may be specified is the number of numbers. It will be observed that all the methods of specifying infinite collections hitherto mentioned involve the notion of order. This is also involved in the definition introduced by Bolzano, 2 that an infinite collection is one . whose terms may be correlated term for term with the terms of a part of · the collection. (Thus all integers may be correlated with the even integers, though these are only part of the collection of integers.) The question 1 This is best done by Peano's Axiom, Formulaire 1898, §2, 002·4, acN 0 .::) .a+,..,= 0. 2 Paradoxien des Unendlichen, §21.

l PART I, CHAP. VI: INFINITE COLLECTIONS

33

. h arises is: Can infinite collections be specified otherwise than by ) hic . ns of order. the definition of a~ infinite collection is certainly independent of order. ollection is infimte when no number N that can be specified is the ,, ~ber of its terms. 3 But we want, if possible, to find some criterion of u nite collections which shall be independent of order. Certainly some llections are known to be infinite which, as wholes, have no order. Such 0 the collection of all concepts or terms. It is impossible to say that the ints of space come before or after the numbers or the instants. But it .true that selections from among all terms do form infinite series, and it IO ay be that this fact is esse~tial to th~ recognition of the infinity of the lection. For we may take 1t as an axiom that, when part of a collection infinite, so is the whole collection. This is indeed the converse of our xiom m+n = p. If two collections be added, and one has no number, en the whole has no number. It may be said, I think, that a criterion of infinity always involves, if not order itself, yet such conditions as are also conditions of order. If there be an asymmetrical relation with two senses R1' R 2 , and a collection of terms each of which (with the possible e~ception of a single one) has each of these relations R 1' R 2 to one or more other terms of the collection, then the collection is infinite. But these con- 20 ditions also insure that the collection is a series (see Part IV). The reason of this apparent connection of infinity with order appears, however, to be mainly psychological. It is, I think, that we cannot apprehend an infinite collection as such, and can only realize its infinity when we know some .law by which new terms can be discovered ad libitum. But such a law will .give an order to our series. For if from a portion of the collection a new .portion can be inferred, and from these two portions a third, and so on, then the portions obtained by these successive steps form a series. And i.. any law which enables us to infer new terms ad libitum from a finite num;'.:ber of terms must be of this form. 30 •. It may be not without interest to point out, as an example of such a law, that the assumption "there is a number l" implies the indefinite series ,of numbers, and shows that no assignable number is the number of numbers. ~or i~ there is a number 1, then 1 is, or has Being. Consequently there is Bemg. But 1 and Being are two. Therefore there is 2. But 1 and Being and 2 are three, and so on. To this process there is evidently no end. Thus though the numbers taken simply do not imply other numbers, yet taken as entities they do imply one another. On the question whether infinite collections have no number or an infinite number, we cannot at present say much. We have seen that infinite 4o 3 Cf. Peano in Formulaire 1895, v, §1, 4 (p. 58).

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collections have no number that can be assigned, and it might be thought that any number can be assigned. Such an assumption would be safe enough as regards finite collections, but since numbers are an infinite collection, we cannot assign all numbers, and therefore there is no proof that there are no numbers which cannot be assigned. Indeed any number N which may be assigned is finite, and therefore, ifthere be infinite number, it cannot be assigned. Therefore we must leave it doubtful for the present whether or not there be infinite number. But infinite collections are absolutely undeniable, and it will be one of our main problems to free them from the contradictions which cling to them.

Part n Whole and Part Chapter

I

Meaning of Whole and Part been compelled, particularly in connection with fractions, to mention the notion of whole and part. In the discussion of integers, this notion is not required; for collections f which integers other than 1 can be asserted are essentially not wholes, ~ince every whole is o_ne, while collections are many. Indeed, as we shall see collections are pnor to wholes, and must be defined and understood before wholes can be discussed. The theory of arithmetical addition and ratio, however, is not prior to the elementary theory of whole and part, but coordinate with it. Before entering on the meaning of whole and part, I may as well point out that I use whole as strictly correlative to part, so that nothing is a whole unless it has parts. Hence simple concepts, such as numbers, points, or instants, are not wholes in the sense in which I use the word. With this proviso, we may proceed to the definition of whole and part. Terms which are one may be, as we saw in Part I, Chapter I, of two kinds. The first kind are simple, and have no logical presuppositions whatever. The second kind, on the other hand, are complex, that is, they presuppose the terms of a certain collection. Whatever is one is called a unit; and thus units are either simple or complex. A complex unit I define as a whole; all units, simple or complex, which it presupposes, I call its parts. Thus whole and part are defined in terms of logical priority. On this conit seems necessary to say a few words, though we shall find it necessary ultimately to abandon the above definition. Between two propositions A and B there may, or may not, be a certain fundamental and indefinable relation called implication. This relation may be symmetrical or unsymmetrical, but is always transitive. That is, when A implies B, it sometimes happens that B also implies A, while at other times this does not happen; but if A implies B and B implies C, then A always implies C. We say that A is prior to B when B implies A but A does not imply B; when the implication is mutual, the two are said to be coordinate. Thus Euclid's axioms, except the axiom of parallels, are prior ~o the thirty-second proposition; for it implies them, while they do not imply it. But the axiom of parallels is coordinate with the thirty-second pro~osition, for in this case the implication is mutual. Implication is the basis of all reasoning; for in order to infer B from A, we must know that

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A implies B, as well as that A is true; indeed "A implies B" is a type of proposition containing the logical essence of what is called inference. Whenever A implies B, we have also the following propositions: A's truth implies B's truth, and B's falsehood implies A's falsehood. But the implication tells us nothing as to whether A and B are true or false. When the proposition implied is of the simple form "X is", we may say that the implying proposition implies the term X; for if any term X can be implied, then that term is an entity. Thus simple terms, which are not propositions, may, by this extension, be implied by propositions. The extension is essential if logical priority is to be identified with the relation of part to whole, since it is evident that a whole may have simple parts. Wherever we have a one-sided implication, what is implied is simpler than what implies it. Thus in the above geometrical instance, the axioms before the twelfth are simpler than the thirty-second proposition. Again, to take the stock instance, "Socrates is a man implies Socrates is mortal", it is evident that the latter proposition is simpler than the former, for man is a concept of which mortal forms part. Or if we take any proposition asserting some relation between two entities A and B, it is evident that the proposition presupposes that there are such entities as A and B, and that there is such a relation as that asserted to hold between them. That is, the proposition presupposes A and B and the relation, all of which are simpler than the proposition. There will only be equal complexity where the implication is mutual: instances are "A is greater than B" and "B is less than A", or "2 is twice l" and "l is half of 2". Returning now to the definition of a whole, it is plain to begin with that every complex term presupposes the being of the simple terms which compose it. Any one of these simple terms might be, without the complex term being; but if the complex term is, then the simple terms also are. Thus all these simple terms are parts of the complex term. But further, if the complex term presupposes one or more other complex terms, then these also are parts of it. Thus if the complex term be "A implies B" where A and B are propositions, then A and B are both presupposed in the complex proposition (not their truth, but their Being), and are thus prior to it; they are also parts of it. Between a whole and any one of its parts, there is a specific simple relation, that of logical priority or of whole and part. This relation demands that the whole should be one, not many: thus the collections dealt with in the preceding Part are not wholes. Two points are important to notice in connection with wholes. The first is a division of wholes into two classes, namely (I) such as can be completely specified by enumeration of their parts, (2) such as cannot be so specified-e.g. propositions. The second point is, that no whole is simply equivalent to all its parts, but always contains something different from

l PART II, CHAP. I: WHOLE AND PART

37

····he constituents revealed by analysis. The development of these two points t st now be undertaken. •· rn~he difference between the two kinds of wholes is important, and il· l trates a fundamental point in Logic. I shall therefore repeat it in other :;rds. Any collection whatever, though as such it_ is many, yet composes '• whole, whose parts are the terms of the collect10n or any whole comaosed of some of the terms of the collection. It is highly important to p alize the difference between a whole and all its parts, even in this case ;,~here the difference is a minimum. The word collection, being singular, '•applies m~re strictly to the whole than to all the parts; but convenience IO •of expression has led me to neglect grammar, and speak of all the terms as the collection. The whole formed of the terms of the collection I call an aggregate. Such a whole is completely specified when all its simple constituents are specified: its parts have no direct connection inter se, but only the indirect connection involved in being parts of one and the same whole. But other wholes occur, which contain relations or what may be called ·predicates, not occurring simply as terms in a collection, but as relating or qualifying. Such wholes are always propositions. These are not com)pletely specified when their parts are all known. Take, as a simple in. stance, the proposition "A differs from B", where A and B are simple 20 terms. The simple parts of this whole are A and B and difference; but the enumeration of these three does not specify the whole, since there are two other wholes composed of the same parts, namely the aggregate formed of A and Band difference, and the proposition "B differs from A". In the former case, although the whole was different from all its parts, yet • it was completely specified by specifying its parts; but in the present case, .· not only is the whole different, but it is not even specified by specifying its parts. We cannot explain this fact by saying that the parts stand in . certain relations which are omitted in the analysis; for in the above case • of "A differs from B", the relation was included in the analysis. The fact 3o seems to be that a relation is one thing when it relates, and another when it is merely enumerated as a term in a collection. There are certain fundamental difficulties in this view, which however I leave aside as irrelevant to our present purpose. Similar remarks apply to A is, which is a whole composed of A and Being, but is different from the whole formed of the collection A and Being. A is one raises the same point, and so does A and Bare two. Indeed all propositions raise this point, and we may distinguish them among complex terms by the fact that they raise it. Thus we see that there are two very different classes of wholes, of which 40 the ~rst will be called aggregates, while the second will be called unities. (Unu is a word having a much wider application, since whatever is one is

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VOL. 3 TOWARD THE "PRINCIPLES", 1900-02

a unit.) Each class of wholes consists of terms not simply equivalent to all their parts; but in the case of unities, the whole is not even specified by its parts. For example, the parts A, greater than, B, may compose simply an aggregate, or the propositions "A is greater than B", "Bis greater than A". Unities thus involve problems from which aggregates are free. As aggregates are more relevant to mathematics than unities, I shall in future usually confine myself to the former. Returning now to the connection of whole and part with logical priority, we shall see reason to hold that the two are not identical in meaning, though there is a close connection between them. In the first place, logical priority is not a simple relation: implication is simple, but logical priority involves not only the proposition "B implies A", but also the proposition "A does not imply B". This state of things is perfectly realized when A is part of B: but it is necessary to regard the relation of whole to part as something simple, and different from any possible relation of one whole to another which is not part of it, which would not be the case if the relation were that of logical implication. We require that the relation of whole and part should be always asymmetrical, i.e. if A is part of B, then B is never part of A. And in this way we avoid the necessity for saying that a proposition may imply a term which is not a proposition. Again it may be doubted whether mere aggregates are propositions. "The whole composed of A and B" seems to be not a proposition, but a single term. But here some caution is necessary. The actual whole which A and B compose is a single term, different from A and from B and from the two A and B. But language has no name for such a whole, being content to specify it by means of its parts. Now when the mention of the parts is included, the whole has become a proposition, namely "a certain whole has the parts A and B". The mention of the parts, therefore, in speaking of the whole which A and B compose, must be held to be only linguistically necessary: the whole itself is to be a single term related to the parts, but having a Being distinct from that of the parts. It is, in short, one term, while they are two other terms. And the whole taken as one term is not a proposition, and is not therefore capable of forming either side in a relation of implication. Thus aggregates, as opposed to unities, are not propositions, and their parts are not logically prior to them; but the parts are prior to the assertion that the whole is composed of these parts. The reader may perhaps be inclined to doubt whether there is any need to assume wholes other than unities. The following reasons, however, seem to make such wholes unavoidable. In the first place, we speak of . one collection, one manifold, etc., and it would seem that in all these cases there really is something which is one. In the second place, the theory of fractions, as we have already partly seen, and as we shall see more fully hereafter, demands the notion of a whole. In the third place, the idea of

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