THEPRINCIPLES

OFMATHEMATICS

by

Bertrand Russell, M.A.,

Late Fellow of Trinity College, Cambridge

Vol. I

First published in 1903 by Cambridge University Press. Thisonline edition (version 0.16: 16 Sep 2019) is based on that edi-tion, with various typographical corrections. Missing here isthe Introduction to the 1937 second edition, which is not yetin the public domain. Rather than publishing a second vol-ume, Russell and his co-author A.N. Whitehead publishedthe three volumes of Principia Mathematica in 1910–1913.Original pagenumbers aremarked in themargins. Thepage ci-tations in the Table of Contents and Index refer to these num-bers.

PREFACE

The present work has two main objects. One of these, the vproof that all pure mathematics deals exclusively with con-cepts definable in terms of a very small number of fundamen-tal logical concepts, and that all its propositions are deduciblefrom a very small number of fundamental logical principles, isundertaken in Parts II–VII of this Volume, and will be estab-lished by strict symbolic reasoning in Volume II. The demon-stration of this thesis has, if I am not mistaken, all the cer-tainty and precision of which mathematical demonstrationsare capable. As the thesis is very recent among mathemati-cians, and is almost universally denied by philosophers, I haveundertaken, in this volume, to defend its various parts, as oc-casion arose, against such adverse theories as appeared mostwidely held or most difficult to disprove. I have also endeav-oured to present, in language as untechnical as possible, themore important stages in the deductions by which the thesisis established.

The other object of this work, which occupies Part I, is theexplanation of the fundamental concepts which mathemat-ics accepts as indefinable. This is a purely philosophical task,and I cannot flatter myself that I have done more than indi-cate a vast field of inquiry, and give a sample of the meth-ods by which the inquiry may be conducted. The discussionof indefinables—which forms the chief part of philosophical

ii Bertrand Russell The Principles of Mathematics iii

logic—is the endeavour to see clearly, and to make others seeclearly, the entities concerned, in order that the mind mayhave that kind of acquaintance with them which it has withredness or the taste of a pineapple. Where, as in the presentcase, the indefinables are obtained primarily as the necessaryresidue in a process of analysis, it is often easier to know thatthere must be such entities than actually to perceive them;there is a process analogous to that which resulted in the dis-covery of Neptune, with the difference that the final stage—the search with a mental telescope for the entity which hasbeen inferred—is often the most difficult part of the under-taking. In the case of classes, I must confess, I have failed toperceive any concept fulfilling the conditions requisite for thevinotion of class. And the contradiction discussed in Chapter xproves that something is amiss, butwhat this is I have hithertofailed to discover.

The second volume, in which I have had the great good for-tune to secure the collaboration of Mr A.N. Whitehead, willbe addressed exclusively to mathematicians; it will containchains of deductions, from the premisses of symbolic logicthrough Arithmetic, finite and infinite, to Geometry, in an or-der similar to that adopted in the present volume; it will alsocontain various original developments, in which the methodof Professor Peano, as supplemented by the Logic ofRelations,has shown itself a powerful instrument of mathematical inves-tigation.

Thepresent volume,whichmaybe regarded either as a com-mentary upon, or as an introduction to, the second volume,is addressed in equal measure to the philosopher and to themathematician; but some parts will be more interesting to theone, others to the other. I should advise mathematicians, un-less they are specially interested in Symbolic Logic, to beginwith Part IV, and only refer to earlier parts as occasion arises.

The following portions are more specially philosophical: PartI (omitting Chapter ii); Part II, Chapters xi, xv, xvi, xvii; PartIII; Part IV, §207, Chapters xxvi, xxvii, xxxi; Part V, Chaptersxli, xlii, xliii; Part VI, Chapters l, li, lii; Part VII, Chapters liii,liv, lv, lvii, lviii; and the twoAppendices, which belong to PartI, and should be read in connection with it. Professor Frege’swork, which largely anticipates my own, was for the most partunknown to me when the printing of the present work began;I had seen his Grundgesetze der Arithmetik, but, owing to thegreat difficulty of his symbolism, I had failed to grasp its im-portance or to understand its contents. The only method, atso late a stage, of doing justice to his work, was to devote anAppendix to it; and in some points the views contained in theAppendix differ from those in Chapter vi, especially in §§71,73, 74. On questions discussed in these sections, I discoverederrors after passing the sheets for the press; these errors, ofwhich the chief are the denial of the null-class, and the identi-fication of a term with the class whose only member it is, arerectified in the Appendices. The subjects treated are so dif-ficult that I feel little confidence in my present opinions, andregard any conclusions which may be advocated as essentiallyhypotheses.

A few words as to the origin of the present work may serveto show the importance of the questions discussed. About sixyears ago, I began an investigation into the philosophy of Dy-namics. I was met by the difficulty that, when a particle is sub-ject to several forces, no one of the component accelerations viiactually occurs, but only the resultant acceleration, of whichthey are not parts; this fact rendered illusory such causationof particulars by particulars as is affirmed, at first sight, by thelaw of gravitation. It appeared also that the difficulty in re-gard to absolute motion is insoluble on a relational theory ofspace. From these two questions I was led to a re-examination

iv Bertrand Russell The Principles of Mathematics v

of the principles of Geometry, thence to the philosophy ofcontinuity and infinity, and thence, with a view to discover-ing the meaning of the word any, to Symbolic Logic. The finaloutcome, as regards the philosophy of Dynamics, is perhapsrather slender; the reason of this is, that almost all the prob-lems of Dynamics appear to me empirical, and therefore out-side the scope of such a work as the present. Many very in-teresting questions have had to be omitted, especially in PartsVI and VII, as not relevant to my purpose, which, for fear ofmisunderstandings, it may be well to explain at this stage.

When actual objects are counted, or when Geometry andDynamics are applied to actual spaceor actualmatter, orwhen,in any other way, mathematical reasoning is applied to whatexists, the reasoning employed has a form not dependentupon the objects to which it is applied being just those objectsthat they are, but only upon their having certain general prop-erties. In puremathematics, actual objects in theworld of exis-tence will never be in question, but only hypothetical objectshaving those general properties upon which depends what-ever deduction is being considered; and these general proper-tieswill alwaysbe expressible in termsof the fundamental con-cepts which I have called logical constants. Thus when spaceor motion is spoken of in pure mathematics, it is not actualspace or actual motion, as we know them in experience, thatare spoken of, but any entity possessing those abstract generalproperties of space or motion that are employed in the reason-ings of geometry or dynamics. The question whether theseproperties belong, as a matter of fact, to actual space or ac-tual motion, is irrelevant to pure mathematics, and thereforeto the present work, being, in my opinion, a purely empiricalquestion, to be investigated in the laboratory or the observa-tory. Indirectly, it is true, the discussions connectedwith puremathematics have a very important bearing upon such empir-

ical questions, since mathematical space and motion are heldby many, perhaps most, philosophers to be self-contradictory,and therefore necessarily different from actual space and mo-tion, whereas, if the views advocated in the following pagesbe valid, no such self-contradictions are to be found in mathe-matical space and motion. But extra-mathematical considera-tions of this kind have been almost wholly excluded from thepresent work.

On fundamental questions of philosophy, my position, in viiiall its chief features, is derived from Mr G. E. Moore. I haveaccepted from him the non-existential nature of propositions(except such as happen to assert existence) and their indepen-dence of any knowing mind; also the pluralism which regardsthe world, both that of existents and that of entities, as com-posed of an infinite number of mutually independent enti-ties, with relations which are ultimate, and not reducible toadjectives of their terms or of the whole which these com-pose. Before learning these views from him, I found myselfcompletely unable to construct any philosophy of arithmetic,whereas their acceptance brought about an immediate liber-ation from a large number of difficulties which I believe tobe otherwise insuperable. The doctrines just mentioned are,in my opinion, quite indispensable to any even tolerably sat-isfactory philosophy of mathematics, as I hope the followingpages will show. But I must leave it to my readers to judge howfar the reasoning assumes these doctrines, and how far it sup-ports them. Formally, my premisses are simply assumed; butthe fact that they allow mathematics to be true, which mostcurrent philosophies do not, is surely a powerful argument intheir favour.

In Mathematics, my chief obligations, as is indeed evident,are to Georg Cantor and Professor Peano. If I had become ac-quainted sooner with the work of Professor Frege, I should

vi Bertrand Russell The Principles of Mathematics vii

have owed a great deal to him, but as it is I arrived indepen-dently at many results which he had already established. Atevery stage of my work, I have been assisted more than I canexpress by the suggestions, the criticisms, and the generousencouragement of Mr A.N. Whitehead; he also has kindlyread my proofs, and greatly improved the final expression ofa very large number of passages. Many useful hints I owe alsoto Mr W.E. Johnson; and in the more philosophical parts ofthe book I owe much to Mr G. E. Moore besides the generalposition which underlies the whole.

In the endeavour to cover so wide a field, it has been im-possible to acquire an exhaustive knowledge of the literature.There are doubtless many important works with which I amunacquainted; but where the labour of thinking and writingnecessarily absorbs so much time, such ignorance, however re-grettable, seems not wholly avoidable.

Manywordswill be found, in the course of discussion, to bedefined in senses apparently departing widely from commonusage. Such departures, I must ask the reader to believe, arenever wanton, but have been made with great reluctance. Inphilosophical matters, they have been necessitated mainly bytwo causes. First, it often happens that two cognate notionsixare both to be considered, and that language has two namesfor the one, but none for the other. It is then highly conve-nient to distinguish between the two names commonly usedas synonyms, keeping one for the usual, the other for the hith-erto nameless sense. The other cause arises from philosoph-ical disagreement with received views. Where two qualitiesare commonly supposed inseparably conjoined, but are hereregarded as separable, the name which has applied to theircombination will usually have to be restricted to one or other.For example, propositions are commonly regarded as (1) trueor false, (2) mental. Holding, as I do, that what is true or false

is not in general mental, I require a name for the true or falseas such, and this name can scarcely be other than proposition.In such a case, the departure from usage is in no degree arbi-trary. As regards mathematical terms, the necessity for estab-lishing the existence-theorem in each case—i.e. the proof thatthere are entities of the kind in question—has led tomany def-initions which appear widely different from the notions usu-ally attached to the terms in question. Instances of this arethe definitions of cardinal, ordinal and complex numbers. Inthe two former of these, and in many other cases, the defini-tion as a class, derived from the principle of abstraction, ismainly recommended by the fact that it leaves no doubt as tothe existence-theorem. But in many instances of such appar-entdeparture fromusage, itmaybedoubtedwhethermorehasbeen done than to give precision to a notion which had hith-erto been more or less vague.

For publishing a work containing so many unsolved dif-ficulties, my apology is, that investigation revealed no nearprospect of adequately resolving the contradiction discussedinChapter x, or of acquiring a better insight into the nature ofclasses. The repeated discovery of errors in solutions whichfor a time had satisfied me caused these problems to appearsuch as would have been only concealed by any seemingly sat-isfactory theorieswhich a slightly longer reflectionmight haveproduced; it seemed better, therefore, merely to state the diffi-culties, than to wait until I had become persuaded of the truthof some almost certainly erroneous doctrine.

My thanks are due to the Syndics of the University Press,and to their Secretary, Mr R.T. Wright, for their kindness andcourtesy in regard to the present volume.

LONDONDecember, 1902.

viii Bertrand Russell The Principles of Mathematics ix

TABLE OF CONTENTS

PAGEPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

PART ITHE INDEFINABLES OF MATHEMATICS

CHAPTER IDEFINITION OF PURE MATHEMATICS

1. Definition of pure mathematics . . . . . . . . . . . . . . . . . . . . . . 32. The principles of mathematics are no longer

controversial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Pure mathematics uses only a few notions,

and these are logical constants . . . . . . . . . . . . . . . . . . . . . . 44. All pure mathematics follows formally from twenty

premisses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. Asserts formal implications . . . . . . . . . . . . . . . . . . . . . . . . . . 56. And employs variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. Which may have any value without exception . . . . . . . 68. Mathematics deals with types of relations . . . . . . . . . . . . 79. Applied mathematics is defined by the occurrence of

constants which are not logical . . . . . . . . . . . . . . . . . . . . . . 810. Relation of mathematics to logic . . . . . . . . . . . . . . . . . . . . . 8

CHAPTER IISYMBOLIC LOGIC

11. Definition and scope of symbolic logic . . . . . . . . . . . . . . 1012. The indefinables of symbolic logic . . . . . . . . . . . . . . . . . . 1013. Symbolic logic consists of three parts . . . . . . . . . . . . . . . . 11

A. The Propositional Calculus.14. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315. Distinction between implication and

formal implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1416. Implication indefinable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417. Two indefinables and ten primitive propositions

in this calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518. The ten primitive propositions . . . . . . . . . . . . . . . . . . . . . 1619. Disjunction and negation defined . . . . . . . . . . . . . . . . . . 17

B. The Calculus of Classes.20. Three new indefinables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821. The relation of an individual to its class . . . . . . . . . . . . . 1922. Propositional functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923. The notion of such that . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024. Two new primitive propositions . . . . . . . . . . . . . . . . . . . . 2025. Relation to propositional calculus . . . . . . . . . . . . . . . . . . 2126. Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

C. The Calculus of Relations.27. The logic of relations essential to mathematics . . . . . . 2328. New primitive propositions . . . . . . . . . . . . . . . . . . . . . . . . 2429. Relative products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2530. Relations with assigned domains . . . . . . . . . . . . . . . . . . . 26

D. Peano’s Symbolic Logic.31. Mathematical and philosophical definitions . . . . . . . . . 2632. Peano’s indefinables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2733. Elementary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2834. Peano’s primitive propositions . . . . . . . . . . . . . . . . . . . . . 29

x Bertrand Russell The Principles of Mathematics xi

35. Negation and disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . 3136. Existence and the null-class . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER IIIIMPLICATION AND FORMAL IMPLICATION

37. Meaning of implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3338. Asserted and unasserted propositions . . . . . . . . . . . . . . 3439. Inference does not require two premisses . . . . . . . . . . . . 3540. Formal implication is to be interpreted extensionally 3641. The variable in a formal implication has

an unrestricted field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3642. A formal implication is a single propositional

function, not a relation of two . . . . . . . . . . . . . . . . . . . . . . 3843. Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3944. Conditions that a term in an implication may

be varied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3945. Formal implication involved in rules of inference . . . 40

CHAPTER IVPROPER NAMES, ADJECTIVES AND VERBS

46. Proper names, adjectives and verbs distinguished . . . 4247. Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4348. Things and concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4449. Concepts as such and as terms . . . . . . . . . . . . . . . . . . . . . . 4550. Conceptual diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4651. Meaning and the subject-predicate logic . . . . . . . . . . . . 4752. Verbs and truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4753. All verbs, except perhaps is, express relations . . . . . . . 4954. Relations per se and relating relations . . . . . . . . . . . . . . . 4955. Relations are not particularized by their terms . . . . . . . 50

CHAPTER VDENOTING

56. Definition of denoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5357. Connection with subject-predicate propositions . . . . 5458. Denoting concepts obtained from predicates . . . . . . . . 5559. Extensional account of all, every , any, a and some . . . . 5660. Intensional account of the same . . . . . . . . . . . . . . . . . . . . 5861. Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5962. The difference between all, every, etc. lies in the

objects denoted, not in the way of denoting them . . . . 6163. The notion of the and definition . . . . . . . . . . . . . . . . . . . . 6264. The notion of the and identity . . . . . . . . . . . . . . . . . . . . . . 6365. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

CHAPTER VICLASSES

66. Combination of intensional and extensionalstandpoints required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

67. Meaning of class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6768. Intensional and extensional genesis of classes . . . . . . . 6769. Distinctions overlooked by Peano . . . . . . . . . . . . . . . . . . 6870. The class as one and as many . . . . . . . . . . . . . . . . . . . . . . . 6871. The notion of and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6972. All men is not analyzable into all and men . . . . . . . . . . . 7273. There are null class-concepts, but there is no

null class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7374. The class as one, except when it has one term,

is distinct from the class as many . . . . . . . . . . . . . . . . . . . 7675. Every, any, a and some each denote one object,

but an ambiguous one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7776. The relation of a term to its class . . . . . . . . . . . . . . . . . . . 7777. The relation of inclusion between classes . . . . . . . . . . . 78

xii Bertrand Russell The Principles of Mathematics xiii

78. The contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7979. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

CHAPTER VIIPROPOSITIONAL FUNCTIONS

80. Indefinability of such that . . . . . . . . . . . . . . . . . . . . . . . . . . . 8281. Where a fixed relation to a fixed term is asserted, a

propositional function can be analyzed into a variablesubject and a constant assertion . . . . . . . . . . . . . . . . . . . . 83

82. But this analysis is impossible in other cases . . . . . . . . 8483. Variation of the concept in a proposition . . . . . . . . . . . 8684. Relation of propositional functions to classes . . . . . . . 8885. A propositional function is in general not analyzable

into a constant and a variable element . . . . . . . . . . . . . . 88

CHAPTER VIIITHE VARIABLE

86. Nature of the variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8987. Relation of the variable to any . . . . . . . . . . . . . . . . . . . . . . 8988. Formal and restricted variables . . . . . . . . . . . . . . . . . . . . . 9189. Formal implication presupposes any . . . . . . . . . . . . . . . . 9190. Duality of any and some . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9291. The class-concept propositional function is indefinable 9292. Other classes can be defined by means of such that . . . 9393. Analysis of the variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

CHAPTER IXRELATIONS

94. Characteristics of relations . . . . . . . . . . . . . . . . . . . . . . . . . 9595. Relations of terms to themselves . . . . . . . . . . . . . . . . . . . 9696. The domain and the converse domain of a relation . . 97

97. Logical sum, logical product and relative product ofrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

98. A relation is not a class of couples . . . . . . . . . . . . . . . . . . 9999. Relations of a relation to its terms . . . . . . . . . . . . . . . . . . 99

CHAPTER XTHE CONTRADICTION

100. Consequences of the contradiction . . . . . . . . . . . . . . . . 101101. Various statements of the contradiction . . . . . . . . . . . 102102. An analogous generalized argument . . . . . . . . . . . . . . . 102103. Variable propositional functions are in general

inadmissible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103104. The contradiction arises from treating as one a class

which is only many . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104105. Other primâ facie possible solutions appear

inadequate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105106. Summary of Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

PART IINUMBER

CHAPTER XIDEFINITION OF CARDINAL NUMBERS

107. Plan of Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111108. Mathematical meaning of definition . . . . . . . . . . . . . . . . 111109. Definition of numbers by abstraction . . . . . . . . . . . . . . . 112110. Objections to this definition . . . . . . . . . . . . . . . . . . . . . . . 114111. Nominal definition of numbers . . . . . . . . . . . . . . . . . . . . 115

xiv Bertrand Russell The Principles of Mathematics xv

CHAPTER XIIADDITION AND MULTIPLICATION

112. Only integers to be considered at present . . . . . . . . . . . 117113. Definition of arithmetical addition . . . . . . . . . . . . . . . . . 117114. Dependence upon the logical addition of classes . . . . 118115. Definition of multiplication . . . . . . . . . . . . . . . . . . . . . . . 119116. Connection of addition, multiplication and

exponentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

CHAPTER XIIIFINITE AND INFINITE

117. Definition of finite and infinite . . . . . . . . . . . . . . . . . . . . 121118. Definition of α0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121119. Definition of finite numbers by mathematical

induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

CHAPTER XIVTHEORY OF FINITE NUMBERS

120. Peano’s indefinables and primitive propositions . . . . 124121. Mutual independence of the latter . . . . . . . . . . . . . . . . . 125122. Peano really defines progressions, not finite numbers 125123. Proof of Peano’s primitive propositions . . . . . . . . . . . . 127

CHAPTER XVADDITION OF TERMS AND ADDITION OF CLASSES

124. Philosophy and mathematics distinguished . . . . . . . . 129125. Is there a more fundamental sense of number

than that defined above? . . . . . . . . . . . . . . . . . . . . . . . . . . 130126. Numbers must be classes . . . . . . . . . . . . . . . . . . . . . . . . . . 131127. Numbers apply to classes as many . . . . . . . . . . . . . . . . . . 132

128. One is to be asserted, not of terms, but of unit classes 132129. Counting not fundamental in arithmetic . . . . . . . . . . . 133130. Numerical conjunction and plurality . . . . . . . . . . . . . . . 133131. Addition of terms generates classes primarily, not

numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135132. A term is indefinable, but not the number 1 . . . . . . . . . . 135

CHAPTER XVIWHOLE AND PART

133. Single terms may be either simple or complex . . . . . . 137134. Whole and part cannot be defined by logical priority 137135. Three kinds of relation of whole and part

distinguished . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138136. Two kinds of wholes distinguished . . . . . . . . . . . . . . . . 140137. A whole is distinct from the numerical conjunction

of its parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141138. How far analysis is falsification . . . . . . . . . . . . . . . . . . . . 141139. A class as one is an aggregate . . . . . . . . . . . . . . . . . . . . . . 141

CHAPTER XVIIINFINITE WHOLES

140. Infinite aggregates must be admitted . . . . . . . . . . . . . . . 143141. Infinite unities, if there are any, are unknown to us . 144142. Are all infinite wholes aggregates of terms? . . . . . . . . . 146143. Grounds in favour of this view . . . . . . . . . . . . . . . . . . . . 146

CHAPTER XVIIIRATIOS AND FRACTIONS

144. Definition of ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149145. Ratios are one-one relations . . . . . . . . . . . . . . . . . . . . . . . 150146. Fractions are concerned with relations of whole

and part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

xvi Bertrand Russell The Principles of Mathematics xvii

147. Fractions depend, not upon number, but uponmagnitude of divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

148. Summary of Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

PART IIIQUANTITY

CHAPTER XIXTHE MEANING OF MAGNITUDE

149. Previous views on the relation of numberand quantity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

150. Quantity not fundamental in mathematics . . . . . . . . . 158151. Meaning of magnitude and quantity . . . . . . . . . . . . . . . 159152. Three possible theories of equality to be examined . . 159153. Equality is not identity of number of parts . . . . . . . . . 160154. Equality is not an unanalyzable relation of quantities 162155. Equality is sameness of magnitude . . . . . . . . . . . . . . . . . 164156. Every particular magnitude is simple . . . . . . . . . . . . . . . 164157. The principle of abstraction . . . . . . . . . . . . . . . . . . . . . . . 166158. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

CHAPTER XXTHE RANGE OF QUANTITY

159. Divisibility does not belong to all quantities . . . . . . . . 170160. Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171161. Differential coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 173162. A magnitude is never divisible, but may be a

magnitude of divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . 173163. Every magnitude is unanalyzable . . . . . . . . . . . . . . . . . . 174

CHAPTER XXINUMBERS AS EXPRESSING MAGNITUDES:

MEASUREMENT

164. Definition of measurement . . . . . . . . . . . . . . . . . . . . . . . . 176165. Possible grounds for holding all magnitudes to be

measurable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176166. Intrinsic measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177167. Of divisibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178168. And of distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179169. Measure of distance and measure of stretch . . . . . . . . 181170. Distance-theories and stretch-theories of geometry . 181171. Extensive and intensive magnitudes . . . . . . . . . . . . . . . 182

CHAPTER XXIIZERO

172. Difficulties as to zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184173. Meinong’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184174. Zero as minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185175. Zero distance as identity . . . . . . . . . . . . . . . . . . . . . . . . . . 186176. Zero as a null segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186177. Zero and negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186178. Every kind of zero magnitude is in a sense

indefinable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

CHAPTER XXIIIINFINITY, THE INFINITESIMAL, AND CONTINUITY

179. Problems of infinity not specially quantitative . . . . . . 188180. Statement of the problem in regard to quantity . . . . . 188181. Three antinomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189182. Of which the antitheses depend upon an axiom of

finitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

xviii Bertrand Russell The Principles of Mathematics xix

183. And the use of mathematical induction . . . . . . . . . . . . 192184. Which are both to be rejected . . . . . . . . . . . . . . . . . . . . . 192185. Provisional sense of continuity . . . . . . . . . . . . . . . . . . . . 193186. Summary of Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

PART IVORDER

CHAPTER XXIVTHE GENESIS OF SERIES

187. Importance of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199188. Between and separation of couples . . . . . . . . . . . . . . . . . 199189. Generation of order by one-one relations . . . . . . . . . . 200190. By transitive asymmetrical relations . . . . . . . . . . . . . . . 203191. By distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204192. By triangular relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204193. By relations between asymmetrical relations . . . . . . . 205194. And by separation of couples . . . . . . . . . . . . . . . . . . . . . . 205

CHAPTER XXVTHE MEANING OF ORDER

195. What is order? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207196. Three theories of between . . . . . . . . . . . . . . . . . . . . . . . . . 207197. First theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208198. A relation is not between its terms . . . . . . . . . . . . . . . . . . 210199. Second theory of between . . . . . . . . . . . . . . . . . . . . . . . . . . 211200. There appear to be ultimate triangular relations . . . . . 211201. Reasons for rejecting the second theory . . . . . . . . . . . . 213202. Third theory of between to be rejected . . . . . . . . . . . . . . 213203. Meaning of separation of couples . . . . . . . . . . . . . . . . . . 214204. Reduction to transitive asymmetrical relations . . . . . . 215

205. This reduction is formal . . . . . . . . . . . . . . . . . . . . . . . . . . . 216206. But is the reason why separation leads to order . . . . . 216207. The second way of generating series is alone

fundamental, and gives the meaning of order . . . . . . . 216

CHAPTER XXVIASYMMETRICAL RELATIONS

208. Classification of relations as regards symmetry andtransitiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

209. Symmetrical transitive relations . . . . . . . . . . . . . . . . . . . 219210. Reflexiveness and the principle of abstraction . . . . . . 219211. Relative position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220212. Are relations reducible to predications? . . . . . . . . . . . . 221213. Monadistic theory of relations . . . . . . . . . . . . . . . . . . . . 222214. Reasons for rejecting this theory . . . . . . . . . . . . . . . . . . 222215. Monistic theory and the reasons for rejecting it . . . . 224216. Order requires that relations should be ultimate . . . 226

CHAPTER XXVIIDIFFERENCE OF SENSE AND DIFFERENCE OF SIGN

217. Kant on difference of sense . . . . . . . . . . . . . . . . . . . . . . . 227218. Meaning of difference of sense . . . . . . . . . . . . . . . . . . . . 228219. Difference of sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228220. In the cases of finite numbers . . . . . . . . . . . . . . . . . . . . . 229221. And of magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229222. Right and left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231223. Difference of sign arises from difference of sense

among transitive asymmetrical relations . . . . . . . . . . . 232

xx Bertrand Russell The Principles of Mathematics xxi

CHAPTER XXVIIION THE DIFFERENCE BETWEEN OPEN AND CLOSED

SERIES

224. What is the difference between open and closedseries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

225. Finite closed series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234226. Series generated by triangular relations . . . . . . . . . . . . 236227. Four-term relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237228. Closed series are such as have an arbitrary first term 238

CHAPTER XXIXPROGRESSIONS AND ORDINAL NUMBERS

229. Definition of progressions . . . . . . . . . . . . . . . . . . . . . . . . 239230. All finite arithmetic applies to every progression . . . 240231. Definition of ordinal numbers . . . . . . . . . . . . . . . . . . . . 242232. Definition of “nth” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243233. Positive and negative ordinals . . . . . . . . . . . . . . . . . . . . . 244

CHAPTER XXXDEDEKIND’S THEORY OF NUMBER

234. Dedekind’s principal ideas . . . . . . . . . . . . . . . . . . . . . . . . 245235. Representation of a system . . . . . . . . . . . . . . . . . . . . . . . . 245236. The notion of a chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246237. The chain of an element . . . . . . . . . . . . . . . . . . . . . . . . . . 246238. Generalized form of mathematical induction . . . . . . . 246239. Definition of a singly infinite system . . . . . . . . . . . . . . . 247240. Definition of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247241. Dedekind’s proof of mathematical induction . . . . . . 248242. Objections to his definition of ordinals . . . . . . . . . . . . 248243. And of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

CHAPTER XXXIDISTANCE

244. Distance not essential to order . . . . . . . . . . . . . . . . . . . . 252245. Definition of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253246. Measurement of distances . . . . . . . . . . . . . . . . . . . . . . . . . 254247. In most series, the existence of distances is doubtful 254248. Summary of Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

PART VINFINITY AND CONTINUITY

CHAPTER XXXIITHE CORRELATION OF SERIES

249. The infinitesimal and space are no longer requiredin a statement of principles . . . . . . . . . . . . . . . . . . . . . . . . 259

250. The supposed contradictions of infinity have beenresolved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

251. Correlation of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260252. Independent series and series by correlation . . . . . . . 262253. Likeness of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262254. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263255. Functions of a variable whose values form a series . . 264256. Functions which are defined by formulae . . . . . . . . . . 267257. Complete series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

CHAPTER XXXIIIREAL NUMBERS

258. Real numbers are not limits of series of rationals . . . 270259. Segments of rationals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271260. Properties of segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272261. Coherent classes in a series . . . . . . . . . . . . . . . . . . . . . . . 274

Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

xxii Bertrand Russell The Principles of Mathematics xxiii

CHAPTER XXXIVLIMITS AND IRRATIONAL NUMBERS

262. Definition of a limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276263. Elementary properties of limits . . . . . . . . . . . . . . . . . . . . 277264. An arithmetical theory of irrationals is

indispensable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277265. Dedekind’s theory of irrationals . . . . . . . . . . . . . . . . . . . 278266. Defects in Dedekind’s axiom of continuity . . . . . . . . . 279267. Objections to his theory of irrationals . . . . . . . . . . . . . 280268. Weierstrass’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282269. Cantor’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283270. Real numbers are segments of rationals . . . . . . . . . . . . 285

CHAPTER XXXVCANTOR’S FIRST DEFINITION OF CONTINUITY

271. The arithmetical theory of continuity is due toCantor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

272. Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288273. Perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290274. Defect in Cantor’s definition of perfection . . . . . . . . . 291275. The existence of limits must not be assumed

without special grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

CHAPTER XXXVIORDINAL CONTINUITY

276. Continuity is a purely ordinal notion . . . . . . . . . . . . . . 296277. Cantor’s ordinal definition of continuity . . . . . . . . . . 296278. Only ordinal notions occur in this definition . . . . . . 298279. Infinite classes of integers can be arranged in a

continuous series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298280. Segments of general compact series . . . . . . . . . . . . . . . . 299

281. Segments defined by fundamental series . . . . . . . . . . . 300282. Two compact series may be combined to form a

series which is not compact . . . . . . . . . . . . . . . . . . . . . . . 303

CHAPTER XXXVIITRANSFINITE CARDINALS

283. Transfinite cardinals differ widely from transfiniteordinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

284. Definition of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304285. Properties of cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306286. Addition, multiplication and exponentiation . . . . . . 307287. The smallest transfinite cardinal α0 . . . . . . . . . . . . . . . . 309288. Other transfinite cardinals . . . . . . . . . . . . . . . . . . . . . . . . 310289. Finite and transfinite cardinals form a single series

by relation to greater and less . . . . . . . . . . . . . . . . . . . . . . 311

CHAPTER XXXVIIITRANSFINITE ORDINALS

290. Ordinals are classes of serial relations . . . . . . . . . . . . . . 312291. Cantor’s definition of the second class of ordinals . . 312292. Definition of ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314293. An infinite class can be arranged in many types

of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315294. Addition and subtraction of ordinals . . . . . . . . . . . . . . 317295. Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . 318296. Well-ordered series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319297. Series which are not well-ordered . . . . . . . . . . . . . . . . . 320298. Ordinal numbers are types of well-ordered series . . . 321299. Relation-arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321300. Proofs of existence-theorems . . . . . . . . . . . . . . . . . . . . . . 322301. There is no maximum ordinal number . . . . . . . . . . . . . 323302. Successive derivatives of a series . . . . . . . . . . . . . . . . . . . 323

xxiv Bertrand Russell The Principles of Mathematics xxv

CHAPTER XXXIXTHE INFINITESIMAL CALCULUS

303. The infinitesimal has been usually supposedessential to the calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

304. Definition of a continuous function . . . . . . . . . . . . . . . 326305. Definition of the derivative of a function . . . . . . . . . . 328306. The infinitesimal is not implied in this definition . . . 329307. Definition of the definite integral . . . . . . . . . . . . . . . . . . 329308. Neither the infinite nor the infinitesimal is involved

in this definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

CHAPTER XLTHE INFINITESIMAL AND THE IMPROPER INFINITE

309. A precise definition of the infinitesimal is seldomgiven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

310. Definition of the infinitesimal and the improperinfinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

311. Instances of the infinitesimal . . . . . . . . . . . . . . . . . . . . . . 332312. No infinitesimal segments in compact series . . . . . . . . 334313. Orders of infinity and infinitesimality . . . . . . . . . . . . . . 335314. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

CHAPTER XLIPHILOSOPHICAL ARGUMENTS CONCERNING THE

INFINITESIMAL

315. Current philosophical opinions illustrated byCohen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

316. Who bases the calculus upon infinitesimals . . . . . . . . 338317. Space and motion are here irrelevant . . . . . . . . . . . . . . 339318. Cohen regards the doctrine of limits as insufficient

for the calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

319. And supposes limits to be essentially quantitative . . 340320. To involve infinitesimal differences . . . . . . . . . . . . . . . . 341321. And to introduce a new meaning of equality . . . . . . . 341322. He identifies the inextensive with the intensive . . . . 342323. Consecutive numbers are supposed to be required

for continuous change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344324. Cohen’s views are to be rejected . . . . . . . . . . . . . . . . . . . 344

CHAPTER XLIITHE PHILOSOPHY OF THE CONTINUUM

325. Philosophical sense of continuity not here inquestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

326. The continuum is composed of mutually externalunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

327. Zeno and Weierstrass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347328. The argument of dichotomy . . . . . . . . . . . . . . . . . . . . . . 348329. The objectionable and the innocent kind of endless

regress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348330. Extensional and intensional definition of a whole . . 349331. Achilles and the tortoise . . . . . . . . . . . . . . . . . . . . . . . . . . 350332. The arrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350333. Change does not involve a state of change . . . . . . . . . . 351334. The argument of the measure . . . . . . . . . . . . . . . . . . . . . . 352335. Summary of Cantor’s doctrine of continuity . . . . . . . . 353336. The continuum consists of elements . . . . . . . . . . . . . . . 353

CHAPTER XLIIITHE PHILOSOPHY OF THE INFINITE

337. Historical retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355338. Positive doctrine of the infinite . . . . . . . . . . . . . . . . . . . . 356339. Proof that there are infinite classes . . . . . . . . . . . . . . . . . 357340. The paradox of Tristram Shandy . . . . . . . . . . . . . . . . . . 358

xxvi Bertrand Russell The Principles of Mathematics xxvii

341. A whole and a part may be similar . . . . . . . . . . . . . . . . . 359342. Whole and part and formal implication . . . . . . . . . . . . 360343. No immediate predecessor of ω or α0 . . . . . . . . . . . . . . . 361344. Difficulty as regards the number of all terms, objects,

or propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362345. Cantor’s first proof that there is no greatest number 363346. His second proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364347. Every class has more sub-classes than terms . . . . . . . . 366348. But this is impossible in certain cases . . . . . . . . . . . . . . 366349. Resulting contradictions . . . . . . . . . . . . . . . . . . . . . . . . . . 367350. Summary of Part V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

PART VISPACE

CHAPTER XLIVDIMENSIONS AND COMPLEX NUMBERS

351. Retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371352. Geometry is the science of series of two or more

dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372353. Non-Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . 372354. Definition of dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 374355. Remarks on the definition . . . . . . . . . . . . . . . . . . . . . . . . . 375356. The definition of dimensions is purely logical . . . . . . 376357. Complex numbers and universal algebra . . . . . . . . . . . 376358. Algebraical generalization of number . . . . . . . . . . . . . . 377359. Definition of complex numbers . . . . . . . . . . . . . . . . . . . 378360. Remarks on the definition . . . . . . . . . . . . . . . . . . . . . . . . . 379

CHAPTER XLVPROJECTIVE GEOMETRY

361. Recent threefold scrutiny of geometrical principles . 381362. Projective, descriptive and metrical geometry . . . . . . . 381363. Projective points and straight lines . . . . . . . . . . . . . . . . 382364. Definition of the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384365. Harmonic ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384366. Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385367. Projective generation of order . . . . . . . . . . . . . . . . . . . . . 386368. Möbius nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388369. Projective order presupposed in assigning irrational

coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389370. Anharmonic ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390371. Assignment of coordinates to any point in space . . . 390372. Comparison of projective and Euclidean geometry . 391373. The principle of duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

CHAPTER XLVIDESCRIPTIVE GEOMETRY

374. Distinction between projective and descriptivegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

375. Method of Pasch and Peano . . . . . . . . . . . . . . . . . . . . . . . 394376. Method employing serial relations . . . . . . . . . . . . . . . . . 395377. Mutual independence of axioms . . . . . . . . . . . . . . . . . . . 396378. Logical definition of the class of descriptive spaces . 397379. Parts of straight lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397380. Definition of the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398381. Solid geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399382. Descriptive geometry applies to Euclidean and

hyperbolic, but not elliptic space . . . . . . . . . . . . . . . . . . 399383. Ideal elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400384. Ideal points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

xxviii Bertrand Russell The Principles of Mathematics xxix

385. Ideal lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401386. Ideal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402387. The removal of a suitable selection of points renders

a projective space descriptive . . . . . . . . . . . . . . . . . . . . . . 403

CHAPTER XLVIIMETRICAL GEOMETRY

388. Metrical geometry presupposes projective ordescriptive geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

389. Errors in Euclid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404390. Superposition is not a valid method . . . . . . . . . . . . . . . 405391. Errors in Euclid (continued) . . . . . . . . . . . . . . . . . . . . . . 406392. Axioms of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407393. Stretches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408394. Order as resulting from distance alone . . . . . . . . . . . . 409395. Geometries which derive the straight line from

distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410396. In most spaces, magnitude of divisibility can be used

instead of distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411397. Meaning of magnitude of divisibility . . . . . . . . . . . . . . . 411398. Difficulty of making distance independent of stretch 413399. Theoretical meaning of measurement . . . . . . . . . . . . . . 414400. Definition of angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414401. Axioms concerning angles . . . . . . . . . . . . . . . . . . . . . . . . 415402. An angle is a stretch of rays, not a class of points . . . 416403. Areas and volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417404. Right and left . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

CHAPTER XLVIIIRELATION OF METRICAL TO PROJECTIVE AND

DESCRIPTIVE GEOMETRY

405. Non-quantitative geometry has no metricalpresuppositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

406. Historical development of non-quantitativegeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

407. Non-quantitative theory of distance . . . . . . . . . . . . . . . 421408. In descriptive geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 423409. And in projective geometry . . . . . . . . . . . . . . . . . . . . . . . 425410. Geometrical theory of imaginary point-pairs . . . . . . . 426411. New projective theory of distance . . . . . . . . . . . . . . . . . 427

CHAPTER XLIXDEFINITIONS OF VARIOUS SPACES

412. All kinds of spaces are definable in purely logicalterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

413. Definition of projective spaces of three dimensions . 430414. Definition of Euclidean spaces of three dimensions . 432415. Definition of Clifford’s spaces of two dimensions . . 434

CHAPTER LTHE CONTINUITY OF SPACE

416. The continuity of a projective space . . . . . . . . . . . . . . . 437417. The continuity of a metrical space . . . . . . . . . . . . . . . . . 438418. An axiom of continuity enables us to dispense

with the postulate of the circle . . . . . . . . . . . . . . . . . . . . 440419. Is space prior to points? . . . . . . . . . . . . . . . . . . . . . . . . . . . 440420. Empirical premisses and induction . . . . . . . . . . . . . . . . 441421. There is no reason to desire our premisses to be

self-evident . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441422. Space is an aggregate of points, not a unity . . . . . . . . . 442

CHAPTER LILOGICAL ARGUMENTS AGAINST POINTS

423. Absolute and relative position . . . . . . . . . . . . . . . . . . . . . 445

xxx Bertrand Russell The Principles of Mathematics xxxi

424. Lotze’s arguments against absolute position . . . . . . . . 446425. Lotze’s theory of relations . . . . . . . . . . . . . . . . . . . . . . . . 446426. The subject-predicate theory of propositions . . . . . . 448427. Lotze’s three kinds of Being . . . . . . . . . . . . . . . . . . . . . . . 449428. Argument from the identity of indiscernibles . . . . . . . 451429. Points are not active . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452430. Argument from the necessary truths of geometry . . . 454431. Points do not imply one another . . . . . . . . . . . . . . . . . . . 454

CHAPTER LIIKANT’S THEORY OF SPACE

432. The present work is diametrically opposed to Kant . 456433. Summary of Kant’s theory . . . . . . . . . . . . . . . . . . . . . . . . 456434. Mathematical reasoning requires no extra-logical

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457435. Kant’s mathematical antinomies . . . . . . . . . . . . . . . . . . . 458436. Summary of Part VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

PART VIIMATTER AND MOTION

CHAPTER LIIIMATTER

437. Dynamics is here considered as a branch of puremathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

438. Matter is not implied by space . . . . . . . . . . . . . . . . . . . . . 465439. Matter as substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466440. Relations of matter to space and time . . . . . . . . . . . . . . 467441. Definition of matter in terms of logical constants . . 468

CHAPTER LIVMOTION

442. Definition of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469443. There is no such thing as a state of change . . . . . . . . . 471444. Change involves existence . . . . . . . . . . . . . . . . . . . . . . . . 471445. Occupation of a place at a time . . . . . . . . . . . . . . . . . . . . 472446. Definition of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472447. There is no state of motion . . . . . . . . . . . . . . . . . . . . . . . . 473

CHAPTER LVCAUSALITY

448. The descriptive theory of dynamics . . . . . . . . . . . . . . . 474449. Causation of particulars by particulars . . . . . . . . . . . . . 475450. Cause and effect are not temporally contiguous . . . . 476451. Is there any causation of particulars by particulars? . 477452. Generalized form of causality . . . . . . . . . . . . . . . . . . . . . 478

CHAPTER LVIDEFINITION OF A DYNAMICAL WORLD

453. Kinematical motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480454. Kinetic motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

CHAPTER LVIINEWTON’S LAWS OF MOTION

455. Force and acceleration are fictions . . . . . . . . . . . . . . . . . 482456. The law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482457. The second law of motion . . . . . . . . . . . . . . . . . . . . . . . . 483458. The third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483459. Summary of Newtonian principles . . . . . . . . . . . . . . . . 485460. Causality in dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486461. Accelerations as caused by particulars . . . . . . . . . . . . . 487462. No part of the laws of motion is an à priori truth . . . . 488

xxxii Bertrand Russell The Principles of Mathematics xxxiii

CHAPTER LVIIIABSOLUTE AND RELATIVE MOTION

463. Newton and his critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489464. Grounds for absolute motion . . . . . . . . . . . . . . . . . . . . . 490465. Neumann’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490466. Streintz’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491467. Mr Macaulay’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491468. Absolute rotation is still a change of relation . . . . . . . 492469. Mach’s reply to Newton . . . . . . . . . . . . . . . . . . . . . . . . . . 492

CHAPTER LIXHERTZ’S DYNAMICS

470. Summary of Hertz’s system . . . . . . . . . . . . . . . . . . . . . . . 494471. Hertz’s innovations are not fundamental from the

point of view of pure mathematics . . . . . . . . . . . . . . . . . 495472. Principles common to Hertz and Newton . . . . . . . . . . 496473. Principle of the equality of cause and effect . . . . . . . . 496474. Summary of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497

APPENDIX ATHE LOGICAL AND ARITHMETICAL DOCTRINES

OF FREGE

475. Principal points in Frege’s doctrines . . . . . . . . . . . . . . . . 501476. Meaning and indication . . . . . . . . . . . . . . . . . . . . . . . . . . . 502477. Truth-values and judgment . . . . . . . . . . . . . . . . . . . . . . . 502478. Criticism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503479. Are assumptions proper names for the true or

the false? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504480. Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505481. Begriff and Gegenstand . . . . . . . . . . . . . . . . . . . . . . . . . . . 507482. Recapitulation of theory of propositional functions 508

483. Can concepts be made logical subjects? . . . . . . . . . . . . 510484. Ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510485. Definition of ϵ and of relation . . . . . . . . . . . . . . . . . . . . . . 512486. Reasons for an extensional view of classes . . . . . . . . . . 513487. A class which has only one member is distinct from

its only member . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513488. Possible theories to account for this fact . . . . . . . . . . . . 514489. Recapitulation of theories already discussed . . . . . . . . 516490. The subject of a proposition may be plural . . . . . . . . . 516491. Classes having only one member . . . . . . . . . . . . . . . . . . 517492. Theory of types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518493. Implication and symbolic logic . . . . . . . . . . . . . . . . . . . . . 518494. Definition of cardinal numbers . . . . . . . . . . . . . . . . . . . . 519495. Frege’s theory of series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520496. Kerry’s criticisms of Frege . . . . . . . . . . . . . . . . . . . . . . . . . 520

APPENDIX BTHE DOCTRINE OF TYPES

497. Statement of the doctrine . . . . . . . . . . . . . . . . . . . . . . . . . 523498. Numbers and propositions as types . . . . . . . . . . . . . . . . 526499. Are propositional concepts individuals? . . . . . . . . . . . 526500. Contradiction arising from the question whether

there are more classes of propositions thanpropositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

1 Bertrand Russell The Principles of Mathematics 2

PART I

THE INDEFINABLES OFMATHEMATICS

CHAPTER I

DEFINITION OF PUREMATHEMATICS

1. Pure Mathematics is the class of all propositions of the 3form “p implies q,” where p and q are propositions contain-ing one or more variables, the same in the two propositions,and neither p nor q contains any constants except logical con-stants. And logical constants are all notions definable in termsof the following: Implication, the relation of a term to a classof which it is a member, the notion of such that, the notion ofrelation, and such further notions as may be involved in thegeneral notion of propositions of the above form. In additionto these, mathematics uses a notion which is not a constituentof the propositions which it considers, namely the notion oftruth.

2. The above definition of pure mathematics is, no doubt,somewhat unusual. Its various parts, nevertheless, appear tobe capable of exact justification—a justification which it willbe the object of the present work to provide. It will be shownthat whatever has, in the past, been regarded as pure mathe-matics, is included in our definition, and that whatever else isincluded possesses those marks by which mathematics is com-monly though vaguely distinguished from other studies. Thedefinition professes to be, not an arbitrary decision to use acommonword in anuncommon signification, but rather a pre-cise analysis of the ideas which, more or less unconsciously,are implied in the ordinary employment of the term. Our

3 Bertrand Russell The Principles of Mathematics 4

methodwill therefore beoneof analysis, andourproblemmaybe called philosophical—in the sense, that is to say, that weseek to pass from the complex to the simple, from the demon-strable to its indemonstrable premisses. But in one respect nota few of our discussions will differ from those that are usuallycalledphilosophical. We shall be able, thanks to the labours ofthemathematicians themselves, to arrive at certainty in regardto most of the questions with which we shall be concerned;and among those capable of an exact solution we shall findmany of the problems which, in the past, have been involvedin all the traditional uncertainty of philosophical strife. Thenature of number, of infinity, of space, time and motion, andof mathematical inference itself, are all questions to which,4in the present work, an answer professing itself demonstrablewith mathematical certainty will be given—an answer which,however, consists in reducing the above problems to problemsin pure logic, which last will not be found satisfactorily solvedin what follows.

3. The Philosophy ofMathematics has been hitherto as con-troversial, obscure and unprogressive as the other branchesof philosophy. Although it was generally agreed that mathe-matics is in some sense true, philosophers disputed as to whatmathematical propositions really meant: although somethingwas true, no two people were agreed as to what it was that wastrue, and if something was known, no one knew what it wasthat was known. So long, however, as this was doubtful, itcould hardly be said that any certain and exact knowledgewasto be obtained in mathematics. We find, accordingly, that ide-alists have tended more and more to regard all mathematicsas dealing with mere appearance, while empiricists have heldeverything mathematical to be approximation to some exacttruth about which they had nothing to tell us. This state ofthings, it must be confessed, was thoroughly unsatisfactory.

Philosophy asks of Mathematics: What does it mean? Math-ematics in the past was unable to answer, and Philosophy an-swered by introducing the totally irrelevant notion of mind.But now Mathematics is able to answer, so far at least as to re-duce the whole of its propositions to certain fundamental no-tions of logic. At this point, the discussion must be resumedby Philosophy. I shall endeavour to indicate what are the fun-damental notions involved, to prove at length that no othersoccur in mathematics, and to point out briefly the philosoph-ical difficulties involved in the analysis of these notions. Acomplete treatment of these difficulties would involve a trea-tise on Logic, which will not be found in the following pages.

4. Therewas, until very lately, a special difficulty in the prin-ciples of mathematics. It seemed plain that mathematics con-sists of deductions, and yet the orthodox accounts of deduc-tion were largely or wholly inapplicable to existing mathemat-ics. Not only the Aristotelian syllogistic theory, but also themodern doctrines of Symbolic Logic, were either theoreticallyinadequate to mathematical reasoning, or at any rate requiredsuch artificial forms of statement that they could not be practi-cally applied. In this fact lay the strength of the Kantian view,which asserted that mathematical reasoning is not strictly for-mal, but always uses intuitions, i.e. the à priori knowledge ofspace and time. Thanks to the progress of Symbolic Logic, es-pecially as treated by Professor Peano, this part of the Kantianphilosophy is nowcapable of a final and irrevocable refutation.By the help of ten principles of deduction and ten other pre-misses of a general logical nature (e.g. “implication is a rela-tion”), all mathematics can be strictly and formally deduced;and all the entities that occur in mathematics can be definedin terms of those that occur in the above twenty premisses. In 5this statement,Mathematics includesnot onlyArithmetic andAnalysis, but also Geometry, Euclidean and non-Euclidean,

5 Bertrand Russell The Principles of Mathematics 6

rational Dynamics, and an indefinite number of other studiesstill unborn or in their infancy. The fact that all Mathematicsis Symbolic Logic is one of the greatest discoveries of our age;and when this fact has been established, the remainder of theprinciples of mathematics consists in the analysis of SymbolicLogic itself.

5. The general doctrine that all mathematics is deductionby logical principles from logical principles was strongly ad-vocated by Leibniz, who urged constantly that axioms oughtto be proved and that all except a few fundamental notionsought to be defined. But owing partly to a faulty logic, partlyto belief in the logical necessity of Euclidean Geometry, hewas led into hopeless errors in the endeavour to carry out indetail a view which, in its general outline, is now known tobe correct*. The actual propositions of Euclid, for example,do not follow from the principles of logic alone; and the per-ception of this fact led Kant to his innovations in the theoryof knowledge. But since the growth of non-Euclidean Geom-etry, it has appeared that pure mathematics has no concernwith the question whether the axioms and propositions of Eu-clid hold of actual space or not: this is a question for appliedmathematics, to be decided, so far as any decision is possible,by experiment and observation. What pure mathematics as-serts is merely that the Euclidean propositions follow fromthe Euclidean axioms—i.e. it asserts an implication: any spacewhich has such and such properties has also such and suchother properties. Thus, as dealt with in pure mathematics,theEuclidean andnon-EuclideanGeometries are equally true:in each nothing is affirmed except implications. All proposi-tions as to what actually exists, like the space we live in, be-long to experimental or empirical science, not tomathematics;when they belong to applied mathematics, they arise from giv-

*On this subject, cf. Couturat, La Logique de Leibniz, Paris, 1901.

ing to one or more of the variables in a proposition of puremathematics some constant value satisfying the hypothesis,and thus enabling us, for that value of the variable, actuallyto assert both hypothesis and consequent instead of assertingmerely the implication. We assert always in mathematics thatif a certain assertion p is true of any entity x, or of any set ofentities x, y, z, . . . , then some other assertion q is true of thoseentities; but we do not assert either p or q separately of ourentities. We assert a relation between the assertions p and q,which I shall call formal implication.

6. Mathematical propositions are not only characterized bythe fact that they assert implications, but also by the fact thatthey contain variables. The notion of the variable is one ofthe most difficult with which Logic has to deal, and in thepresent work a satisfactory theory as to its nature, in spite of 6much discussion, will hardly be found. For the present, I onlywish to make it plain that there are variables in all mathemat-ical propositions, even where at first sight they might seem tobe absent. Elementary Arithmetic might be thought to forman exception: 1 + 1 = 2 appears neither to contain variablesnor to assert an implication. But as a matter of fact, as will beshown in Part II, the true meaning of this proposition is: “If xis one and y is one, and x differs from y, then x and y are two.”And this proposition both contains variables and asserts animplication. We shall find always, in all mathematical propo-sitions, that the words any or some occur; and these words arethe marks of a variable and a formal implication. Thus theabove proposition may be expressed in the form: “Any unitand any other unit are two units.” The typical proposition ofmathematics is of the form “φ(x, y, z, . . .) implies ψ(x, y, z, . . .),whatever values x, y, z, . . . may have”; where φ(x, y, z, . . .) andψ(x, y, z, . . .), for every set of values of x, y, z, . . ., are proposi-tions. It is not asserted that φ is always true, nor yet that ψ

7 Bertrand Russell The Principles of Mathematics 8

is always true, but merely that, in all cases, when φ is false asmuch as when φ is true, ψ follows from it.

The distinction between a variable and a constant is some-what obscured by mathematical usage. It is customary, for ex-ample, to speak of parameters as in some sense constants, butthis is a usagewhichwe shall have to reject. A constant is to besomething absolutely definite, concerning which there is noambiguity whatever. Thus 1, 2, 3, e, π, Socrates, are constants;and so are man, and the human race, past, present and future,considered collectively. Proposition, implication, class, etc.are constants; but a proposition, any proposition, somepropo-sition, are not constants, for these phrases do not denote onedefinite object. And thus what are called parameters are sim-ply variables. Take, for example, the equation ax+ by+ c = 0,considered as the equation to a straight line in a plane. Herewe say that x and y are variables, while a, b, c are constants. Butunless we are dealing with one absolutely particular line, saythe line fromaparticular point in London to a particular pointin Cambridge, our a, b, c are not definite numbers, but standfor any numbers, and are thus also variables. And in Geom-etry nobody does deal with actual particular lines; we alwaysdiscuss any line. The point is that we collect the various cou-ples x, y into classes of classes, each class beingdefined as thosecouples that have a certain fixed relation to one triad (a, b, c).But from class to class, a, b, c also vary, and are therefore prop-erly variables.

7. It is customary in mathematics to regard our variables asrestricted to certain classes: in Arithmetic, for instance, theyare supposed to stand for numbers. But this only means thatif they stand for numbers, they satisfy some formula, i.e. thehypothesis that they are numbers implies the formula. This,then, is what is really asserted, and in this proposition it is no7longer necessary that our variables should benumbers: the im-

plication holds equally when they are not so. Thus, for exam-ple, the proposition “x and y are numbers implies (x + y)2 =x2+2xy+y2” holds equally if for x and ywe substitute Socratesand Plato*: both hypothesis and consequent, in this case, willbe false, but the implication will still be true. Thus in everyproposition of pure mathematics, when fully stated, the vari-ables have an absolutelyunrestrictedfield: any conceivable en-tity may be substituted for any one of our variables withoutimpairing the truth of our proposition.

8. We can now understand why the constants in mathemat-ics are to be restricted to logical constants in the sense definedabove. The process of transforming constants in a proposi-tion into variables leads to what is called generalization, andgives us, as it were, the formal essence of a proposition. Math-ematics is interested exclusively in types of propositions; if aproposition p containing only constants be proposed, and fora certain one of its terms we imagine others to be successivelysubstituted, the result will in general be sometimes true andsometimes false. Thus, for example, we have “Socrates is aman”; here we turn Socrates into a variable, and consider “x isa man.” Some hypotheses as to x, for example, “x is a Greek,”insure the truth of “x is a man”; thus “x is a Greek” implies “xis aman,” and this holds for all values of x. But the statement isnot oneof puremathematics, because it dependsupon thepar-ticular nature of Greek and man. We may, however, vary thesetoo, and obtain: If a and b are classes, and a is contained in b,then “x is an a” implies “x is a b.” Here at last we have a propo-sition of puremathematics, containing three variables and theconstants class, contained in, and those involved in the notionof formal implications with variables. So long as any term in

*It is necessary to suppose arithmetical addition and multiplication de-fined (as may be easily done) so that the above formula remains significantwhen x and y are not numbers.

9 Bertrand Russell The Principles of Mathematics 10

our proposition can be turned into a variable, our propositioncan be generalized; and so long as this is possible, it is the busi-ness of mathematics to do it. If there are several chains of de-duction which differ only as to the meaning of the symbols,so that propositions symbolically identical become capable ofseveral interpretations, the proper course, mathematically, isto form the class of meanings which may attach to the sym-bols, and to assert that the formula in question follows fromthe hypothesis that the symbols belong to the class in ques-tion. In this way, symbols which stood for constants becometransformed into variables, andnew constants are substituted,consisting of classes to which the old constants belong. Casesof such generalization are so frequent that many will occur atonce to every mathematician, and innumerable instances willbe given in the present work. Whenever two sets of termshave mutual relations of the same type, the same form of de-8duction will apply to both. For example, the mutual relationsof points in a Euclidean plane are of the same type as those ofthe complex numbers; hence plane geometry, considered as abranch of pure mathematics, ought not to decide whether itsvariables are points or complex numbers or some other set ofentities having the same type of mutual relations. Speakinggenerally, we ought to deal, in every branch of mathematics,with any class of entities whose mutual relations are of a spec-ified type; thus the class, as well as the particular term consid-ered, becomes a variable, and the only true constants are thetypes of relations andwhat they involve. Nowa typeof relationis to mean, in this discussion, a class of relations characterizedby the above formal identity of the deductions possible in re-gard to the various members of the class; and hence a type ofrelations, as will appear more fully hereafter, if not already ev-ident, is always a class definable in terms of logical constants*.

*One-one, many-one, transitive, symmetrical, are instances of types of

We may therefore define a type of relations as a class of rela-tions defined by some property definable in terms of logicalconstants alone.

9. Thus pure mathematics must contain no indefinables ex-cept logical constants, and consequently no premisses, or in-demonstrable propositions, but such as are concerned exclu-sively with logical constants and with variables. It is preciselythis that distinguishes pure from applied mathematics. In ap-plied mathematics, results which have been shown by puremathematics to follow from somehypothesis as to the variableare actually asserted of some constant satisfying the hypothe-sis in question. Thus terms which were variables become con-stant, and a new premiss is always required, namely: this par-ticular entity satisfies the hypothesis in question. Thus for ex-ample Euclidean Geometry, as a branch of pure mathematics,consists wholly of propositions having the hypothesis “S is aEuclidean space.” If we go on to: “The space that exists is Eu-clidean,” this enables us to assert of the space that exists theconsequents of all the hypotheticals constituting EuclideanGeometry, where now the variable S is replaced by the con-stantactual space. But by this stepwepass frompure to appliedmathematics.

10. The connection of mathematics with logic, accordingto the above account, is exceedingly close. The fact that allmathematical constants are logical constants, and that all thepremisses ofmathematics are concernedwith these, gives, I be-lieve, the precise statement of what philosophers have meantin asserting that mathematics is à priori. The fact is that, whenonce the apparatus of logic has been accepted, all mathemat-ics necessarily follows. The logical constants themselves areto be defined only by enumeration, for they are so fundamen-tal that all the properties by which the class of them might be

relations with which we shall be often concerned.

11 Bertrand Russell The Principles of Mathematics 12

defined presuppose some terms of the class. But practically,9the method of discovering the logical constants is the analysisof symbolic logic, which will be the business of the followingchapters. The distinction of mathematics from logic is very ar-bitrary, but if a distinction is desired, itmaybemade as follows.Logic consists of the premisses of mathematics, together withall other propositions which are concerned exclusively withlogical constants and with variables but do not fulfil the abovedefinition ofmathematics (§1). Mathematics consists of all theconsequences of the above premisses which assert formal im-plications containing variables, together with such of the pre-misses themselves as have these marks. Thus some of the pre-misses of mathematics, e.g. the principle of the syllogism, “ifp implies q and q implies r, then p implies r,” will belong tomathematics, while others, such as “implication is a relation,”will belong to logic but not to mathematics. But for the desireto adhere to usage, we might identify mathematics and logic,and define either as the class of propositions containing onlyvariables and logical constants; but respect for tradition leadsme rather to adhere to the above distinction, while recogniz-ing that certain propositions belong to both sciences.

From what has now been said, the reader will perceive thatthe present work has to fulfil two objects, first, to show thatall mathematics follows from symbolic logic, and secondly todiscover, as far as possible, what are the principles of symboliclogic itself. The first of these objects will be pursued in the fol-lowingParts, while the secondbelongs toPart I.Andfirst of all,as a preliminary to a critical analysis, itwill benecessary to givean outline of Symbolic Logic considered simply as a branch ofmathematics. This will occupy the following chapter.

CHAPTER II

SYMBOLIC LOGIC

11. Symbolic or Formal Logic—I shall use these terms as 10synonyms—is the study of the various general types of de-duction. The word symbolic designates the subject by an ac-cidental characteristic, for the employment of mathematicalsymbols, here as elsewhere, is merely a theoretically irrelevantconvenience. The syllogism in all its figures belongs to Sym-bolic Logic, and would be the whole subject if all deductionwere syllogistic, as the scholastic tradition supposed. It is fromthe recognition of asyllogistic inferences that modern Sym-bolic Logic, from Leibniz onward, has derived the motive toprogress. Since the publication of Boole’s Laws of Thought(1854), the subject has been pursued with a certain vigour, andhas attained to a very considerable technical development*.Nevertheless, the subject achieved almost nothing of utility ei-ther to philosophy or to other branches of mathematics, untilit was transformed by the new methods of Professor Peano†.

*By far the most complete account of the non-Peanesque methods willbe found in the three volumes of Schröder, Vorlesungen über die Algebra derLogik, Leipzig, 1890, 1891, 1895.

†See Formulaire de Mathématiques, Turin, 1895, with subsequent edi-tions in later years; also Revue de Mathématiques, Vol. vii. No. 1 (1900). Theeditions of the Formulaire will be quoted as F. 1895 and so on. The Revuede Mathématiques, which was originally the Rivista di Matematica, will bereferred to as R. d.M.

13 Bertrand Russell The Principles of Mathematics 14

Symbolic Logic has now become not only absolutely essentialto every philosophical logician, but also necessary for the com-prehension of mathematics generally, and even for the suc-cessful practice of certain branches of mathematics. How use-ful it is in practice can only be judged by those who have ex-perienced the increase of power derived from acquiring it; itstheoretical functions must be briefly set forth in the presentchapter‡.

12. Symbolic Logic is essentially concerned with inference11in general*, and is distinguished fromvarious special branchesof mathematics mainly by its generality. Neither mathemat-ics nor symbolic logic will study such special relations as (say)temporal priority, but mathematics will deal explicitly withthe class of relations possessing the formal properties of tem-poral priority—propertieswhich are summedup in thenotionof continuity†. And the formal properties of a relation maybe defined as those that can be expressed in terms of logicalconstants, or again as those which, while they are preserved,permit our relation to be varied without invalidating any in-ference in which the said relation is regarded in the light ofa variable. But symbolic logic, in the narrower sense whichis convenient, will not investigate what inferences are possi-ble in respect of continuous relations (i.e. relations generatingcontinuous series); this investigation belongs to mathematics,but is still too special for symbolic logic. What symbolic logicdoes investigate is the general rules by which inferences are

‡In what follows the main outlines are due to Professor Peano, exceptas regards relations; even in those cases where I depart from his views, theproblems considered have been suggested to me by his works.

*I may as well say at once that I, do not distinguish between inferenceand deduction. What is called induction appears to me to be either dis-guised deduction or a mere method of making plausible guesses.

†See below, Part V, Chap. xxxvi.

made, and it requires a classification of relations or proposi-tions only in so far as these general rules introduce particularnotions. The particular notions which appear in the propo-sitions of symbolic logic, and all others definable in terms ofthese notions, are the logical constants. The number of inde-finable logical constants is not great: it appears, in fact, to beeight or nine. These notions alone form the subject-matter ofthe whole of mathematics: no others, except such as are defin-able in terms of the original eight or nine, occur anywhere inArithmetic, Geometry, or rational Dynamics. For the techni-cal study of Symbolic Logic, it is convenient to take as a sin-gle indefinable the notion of a formal implication, i.e. of suchpropositions as “x is a man implies x is a mortal, for all val-ues of x”—propositions whose general type is: “φ(x) impliesψ(x) for all values of x,” where φ(x), ψ(x), for all values of x,are propositions. The analysis of this notion of formal impli-cation belongs to the principles of the subject, but is not re-quired for its formal development. In addition to this notion,we require as indefinables the following: Implication betweenpropositions not containing variables, the relation of a termto a class of which it is a member, the notion of such that, thenotion of relation, and truth. By means of these notions, allthe propositions of symbolic logic can be stated.

13. The subject ofSymbolicLogic consists of threeparts, thecalculus of propositions, the calculus of classes, and the calcu-lus of relations. Between the first two, there is, within limits, acertainparallelism,whicharises as follows: In any symbolic ex-pression, the letters may be interpreted as classes or as propo- 12sitions, and the relation of inclusion in the one case may be re-placed by that of formal implication in the other. Thus, for ex-ample, in the principle of the syllogism, if a, b, c be classes, anda is contained in b, b in c, then a is contained in c; but if a, b, cbe propositions, and a implies b, b implies c, then a implies c.

15 Bertrand Russell The Principles of Mathematics 16

A great deal has been made of this duality, and in the later edi-tions of theFormulaire, Peano appears tohave sacrificed logicalprecision to its preservation*. But, as amatter of fact, there aremany ways in which the calculus of propositions differs fromthat of classes. Consider, for example, the following: “If p, q, rare propositions, and p implies q or r, then p implies q or p im-plies r.” This proposition is true; but its correlative is false,namely: “If