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THE PRINCIPLES

OF

MATHEMATICS

BY

BERTRAND RUSSELL M.A.,

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

VOL L ^^' â€¢^

Cambridge :

at the University Press

1903

I^^^ii

CamhritigE :

PRINTED BY J. AND C. F. CLAY,

AT THE UNIVERSITY PRESS.

1 .sn37n

PREFACE.

THE present work has two main objects. One of these, the proof

that all pure mathematics deals exclusively with concepts definable

in terms of a very small number of fundamental logical concepts, and

that all its propositions are deducible from a very small number of

fundamental logical principles, is undertaken in Parts II. â€” VII. of this

Volume, and will be established by strict symbolic reasoning in Volume ii. -

The demonstration of this thesis has, if I am not mistaken, all the

certainty and precision of which mathematical demonstrations are capable.

As the thesis is very recent among mathematicians, and is almost

universally denied by philosophers, I have undertaken, in this volume,

to defend its various parts, as occasion arose, against such adverse

theories as appeared most widely held or most difficult to disprove.

I have also endeavoured to present, in language as untechnical as

possible, the more important stages in the deductions by which the

thesis is established.

The other object of this work, which occupies Part I., is the

explanation of the fundamental concepts which mathematics accepts

as indefinable. This is a purely philosophical task, and I cannot flatter

myself that I have done more than indicate a vast field of inquiry, and

give a sample of the methods by which the inquiry may be conducted.

The discussion of indefinables â€” which forms the chief part of philosophical

logic^is the endeavour to see clearly, and to make others see clearly,

the entities concerned, in order that the mind may have that kind of

acquaintance with them which it has with redness or the taste of a

pineapple. Where, as in the present case, the indefinables are obtained

primarily as the necessary residue in a process of analysis, it is often

easier to know that there must be such entities than actually to perceive

them ; there is a process analogous to that which resulted in the discovery

of Neptune, with the difference that the final stage â€” the search with a

mental telescope for the entity which has been inferred â€” is often the

most difficult part of the undertaking. In the case of classes, I must

confess, I have failed to perceive any concept fulfilling the conditions

vi Preface

I requisite for the notion of class. And the contradiction discussed in

Chapter x. proves that something is amiss, but what this is I have

hitherto failed to discover.

The second volume, in which I have had the great good fortune

to secure the collaboration of Mr A. N. Whitehead, will be addressed

exclusively to mathematicians ; it will contain chains of deductions,

from the premisses of symbolic logic through Arithmetic, finite and

infinite, to Geometry, in an order similar to that adopted in the present

volume ; it will also contain various original developments, in which the

method of Professor Peano, as supplemented by the Logic of Relations,

has shown itself a powerful instrument of mathematical investigation.

1 The present volume, which may be regarded either as a commentary

j upon, or as an introduction to, the second volume, is addressed in equal

measure to the philosopher and to the mathematician ; but some parts

will be more interesting to the one, others to the other. I should advise

mathematicians, unless they are specially interested in Symbolic Logic,

to begin with Part IV., and only refer to earlier parts as occasion arises.

The following portions are more specially philosophical : Part I.

(omitting Chapter ii.); Part II., Chapters xi., xv., xvi., xvii. ; Part III.;

Part IV., Â§207, Chapters xxvi., xxvii., xxxi.; Part V., Chapters xli.,

XLii., xLiii. ; Part VI., Chapters l., li., lii. ; Part VII., Chapters liii.,

Liv., Lv., Lvii., Lviii.; and the two Appendices, which belong to Part I.,

and should be read in connection with it. Professor Frege's work, which

largely anticipates my own, was for the most part unknown to me when

.the printing of the present work began ; I had seen his Grundgesetze

der Arithmetik, but, owing to the great difficulty of his symbolism, I had

failed to grasp its importance or to understand its contents. The only

method, at so late a stage, of doing justice to his work, was to devote

an Appendix to it ; and in some points the views contained in the

Appendix differ from those in Chapter vi., especially in Â§Â§71, 73, 74.

On questions discussed in these sections, I discovered errors after passing

the sheets for the press ; these errors, of which the chief are the denial

of the null-class, and the identification of a term with the class whose

only member it is, are rectified in the Appendices. The subjects

treated are so difficult that I feel little confidence in my present

opinions, and regard any conclusions which may be advocated as

essentially hypotheses.

A few words as to the origin of the present work may serve to

show the importance of the questions discussed. About six years ago,

I began an investigation into the philosophy of Dynamics. I was

met by the difficulty that, when a particle is subject to several forces,

Preface vii

no one of the component accelerations actually occurs, but only

the resultant acceleration, of which they are not parts ; this fact

rendered illusory such causation of particulars by particulars as is

affirmed, at first sight, by the law of gravitation. It appeared also that

the difficulty in regard to absolute motion is insoluble on a relational

theory of space. From these two questions I was led to a re-examination

of the principles of Geometry, thence to the philosophy of continuity

and infinity, and thence, with a view to discovering the meaning of the

word any, to Symbolic Logic. The final outcome, as regards the

philosophy of Dynamics, is perhaps rather slender ; the reason of this

is, that almost all the problems of Dynamics appear to me empirical,

and therefore outside the scope of such a work as the present. Many

very interesting questions have had to be omitted, especially in Parts

VI. and â– VII., as not relevant to my purpose, which, for fear of

misunderstandings, it may be well to explain at this stage.

When actual objects are counted, or when Geometry and Dynamics

are applied to actual space or actual matter, or when, in any other way,

mathematical reasoning is applied to what exists, the reasoning employed

has a form not dependent upon the objects to which it is applied being

just those objects that they are, but only upon their having certain

general properties. In pure mathematics, actual objects in the world

of existence will never be in question, but only hypothetical objects

having those general properties upon which depends whatever deduction

is being considered ; and these general properties will always be

expressible in terms of the fundamental concepts which I have called

logical constants. Thus when space or motion is spoken of in pure

mathematics, it is not actual space or actual motion, as we know them

in experience, that are spoken of, but any entity possessing those abstract

general properties of space or motion that are employed in the reasonings

of geometry or dynamics. The question whether these properties belong,

as a matter of fact, to actual space or actual motion, is irrelevant to pure

mathematics, and therefore to the present work, being, in my opinion,

a purely empirical question, to be investigated in the laboratory or the

observatory. Indirectly, it is true, the discussions connected with pure

mathematics have a very important bearing upon such empirical questions,

since mathematical space and motion are held by many, perhaps most,

philosophers to be self-contradictory, and therefore necessarily difi^erent

from actual space and motion, whereas, if the views advocated in the

following pages be valid, no such self-contradictions are to be found in

mathematical space and motion. But extra-mathematical considerations

of this kind have been almost wholly excluded from the present work.

viii Pi^eface

/ On fundamental questions of philosophy, my position, in all its chief

features, is derived from Mr G. E. Moore. I have accepted from him

the non-existential nature of propositions (except such as happen to

assert existence) and their independence of any knowing mind ; also

the pluralism which regards the world, both that of existents and

that of entities, as composed of an infinite number of mutually

independent entities, with relations which are ultimate, and not

reducible to adjectives of their terms or of the whole which these

compose7\ Before learning these views from him, I found myself

completely unable to construct any philosophy of arithmetic, whereas

their acceptance brought about an immediate liberation from a large

number of difficulties which I believe to be otherwise insuperable.

The doctrines just mentioned are, in my opinion, quite indispensable

to any even tolerably satisfactory philosophy of mathematics, as I hope

the following pages will show. But I must leave it to my readers to

judge how far the reasoning assumes these doctrines, and how far it

supports them. Formally, my premisses are simply assumed ; but the

fact that they allow mathematics to be true, which most cuiTent

philosophies do not, is surely a powerful argument in their favour.

In Mathematics, my chief obligations, as is indeed evident, are to

Georg Cantor and Professor Peano. If I had become acquainted

sooner with the work of Professor Frege, I should have owed a

great deal to him, but as it is I arrived independently at many

results which he had already established. At every stage of my work,

I have been assisted more than I can express by the suggestions, the

criticisms, and the generous encouragement of Mr A. N. Whitehead;

he also has kindly read my proofs, and greatly improved the final

expression of a very large number of passages. Many useful hints

I owe also to Mr W. E. Johnson ; and in the more philosophical parts

of the book I owe much to Mr G. E. Moore besides the general position

which underlies the whole.

In the endeavour to cover so wide a field, it has been impossible to

acquire an exhaustive knowledge of the literature. There are doubtless

many important works with which I am unacquainted ; but where the

labour of thinking and writing necessarily absorbs so much time, such

ignorance, however regrettable, seems not wholly avoidable.

Many words will be found, in the course of discussion, to be defined

in senses apparently departing widely from common usage. Such

departures, I must ask the reader to believe, are never wanton, but have

been made with great reluctance. In philosophical matters, they have

been necessitated mainly by two causes. First, it often happens that

I'

i

Preface ix

two cognate notions are both to be considered, and that language has

two names for the one, but none for the other. It is then highly

convenient to distinguish between the two names commonly used as

synonyms, keeping one for the usual, the other for the hitherto nameless

sense. The other cause arises from philosophical disagreement with

received views. Where two qualities are commonly supposed inseparably

conjoined, but are here regarded as separable, the name which has

applied to their combination will usually have to be restricted to one

or other. For example, propositions are commonly regarded as (1) true

or 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 false as such, and

this name can scarcely be other than proposition. In such a case, the

departure from usage is in no degree arbitrary. As regards mathematical

terms, the necessity for establishing the existence-theorem in each case â€”

i.e. the proof that there are entities of the kind in question â€” has led to

many definitions which appear widely different from the notions usually

attached to the terms in question. Instances of this are the definitions

of cardinal, ordinal and complex numbers. In the two former of these,

and in many other cases, the definition as a class, derived from the

principle of abstraction, is mainly recommended by the fact that it

leaves no doubt as to the existence-theorem. But in many instances of

such apparent departure from usage, it may be doubted whether more

has been done than to give precision to a notion which had hitherto

been more or less vague.

For publishing a work containing so many unsolved difficulties, my

apology is, that investigation revealed no near prospect of adequately

resolving the contradiction discussed in Chapter x., or of acquiring a

better insight into the nature of classes. The repeated discovery of errors

in solutions which for a time had satisfied me caused these problems to

appear such as would have been only concealed by any seemingly satis-

factory theories which a slightly longer reflection might have produced ;

it seemed better, therefore, merely to state the difficulties, than to wait

until I had become persuaded of the truth of some ahiiost 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 and courtesy

in regard to the present volume.

London J

December, 1902.

TABLE OF CONTENTS

Preface ............. v

PA.RT I.

THE INDEFINABLES OF MATHEMATICS.

CHAPTER I.

DEFINITION OF PURE MATHEMATICS.

1. Definition of pure mathematics ........ 3

2. The principles of mathematics are no longer controversial ... 3

3. Pure mathematics uses only a few notions, and these are logical

constants ........... 4

4. All pure mathematics follows formally from twenty premisses . . 4

5. Asserts formal implications ......... 5

6. And employs variables ......... 5

7. Which may have any value without exception ..... 6

8. Mathematics deals with types of relations ...... 7

9. Applied mathematics is defined by the occurrence of constants which

are not logical .......... 8

10. Relation of mathematics to logic ........ 8

CHAPTER II.

SYMBOLIC LOGIC.

11. Definition and scope of symbolic logic ....... 10

12. The indefinables of symbolic logic ....... 10

13. Symbolic logic consists of three parts . . . . . . . 11

A. The Propositional Calculus.

14. Definition ..........

15. Distinction between implication and formal implication

16. Implication indefinable .......

17. Two indefinables and ten primitive propositions in this calculus

18. The ten primitive propositions ......

19. Disjunction and negation defined .....

13

14

14

15

16

17

Xll

Table of Contents

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

B. The Calculus of Classes.

PAGK

Three new indefinables ......... 18

The relation of an individual to its class ...... 19

Prepositional functions ......... 19

The notion of such that . . . . . . . . . 20

Two new primitive propositions . . . . . . . . 20

Relation to prepositional calculus ....... 21

Identity .23

C. The Calculus of Relations.

The logic of relations essential to mathematics ..... 23

New primitive propositions ......... 24

Relative products .......... 25

Relations with assigned domains ........ 26

D. Peano's Symbolic Logic.

Mathematical and philosophical definitions ...... 26

Peano's indefinables . . . . . ... . . . 27

Elementary definitions . . . . . . . ... 28

Peano's primitive propositions . . . ... . . 29

Negation and disjunction . . . . . . . . . 31

Existence and the null-class . . . . . . . . 32

CHAPTER III.

IMPLICATION AND FORMAL IMPLICATION.

37- Meaning of implication . . . . . . . . , 33

38. Asserted and unasserted propositions . . . . . . . 34

39. Inference does not require two premisses ...... 35

40. Formal implication is to be interpreted extensionally .... 36

41. The variable in a formal implication has an unrestricted field . . 36

42. A formal implication is a single propositional function^ not a relation

of two 38

43. Assertions ............ 39

44. Conditions that a term in an implication may be varied ... 39

45. Formal implication involved in rules of inference .... 40

CHAPTER IV.

PROPER NAMES, ADJECTIVES AND VERBS.

46. Proper names, adjectives and verbs distinguished

47. Terms ......

48. Tilings and concepts ....

49. Concepts as such and as terms

50. Conceptual diversity ....

51. Meaning and the subject-predicate logic

52. Verbs and truth .....

53. All verbs, except perhaps is, express relatio

54. Relations pr.r se and relating relations

65. Relations are not particularized by their terms

42

43

44

45

46

47

47

49

49

50

Table of Contents

xiu

CHAPTER V.

DENOTING.

56. Definition of denoting .........

57. Connection with subject-predicate propositions ....

58. Denoting concepts obtained from predicates . .

59. Extensional account of all, every, any, a and so7)ie ...

60. Intensional account of the same .......

61. Illustrations ..........

62. The difference between all, every, etc. lies in the objects denoted^ not

in the way of denoting them ......

68. l^he notion of the and definition .......

64. The notion of the and identity . . â– .

Q6. Summary . . . . . . . . ' .

PAGE

53

54

55

56

58

59

61

62

63

64

CHAPTER VI.

CLASSES. N

66. Combination of intensional and extensional standpoints required

67. Meaning of class .....

68. Intensional and extensional genesis of classes

69. Distinctions overlooked by Peano

70. The class as one and as many

71. The notion of and .....

72. All men is not analyzable into all and men .

73. There are null class-concepts, but there is no null class

74. The class as one, except when it has one term, is distinct from the

class as many .........

75. Every, any, a and some each denote one object, but an ambiguous one

76. The relation of a term to its class ......

77. The relation of inclusion between classes .....

78. The contradiction .........

79. Summary ...........

66

67

67

68

68

69

72

73

76

77

77

78

79

80

CHAPTER VH.

PROPOSITIONAL FUNCTIONS.

80. Indefinability of such that ......... 82

81. Where a fixed relation to a fixed term is asserted, a propositional

function can be analyzed into a variable subject and a constant

assertion ........... 83

82. But this analysis is impossible in other cases ..... 84

83. Variation of the concept in a proposition ...... 86

84. Relation of propositional functions to classes ..... 88

85. A propositional function is in general not analyzable into a constant

and a variable element ........ 88

XIV

Table of Contents

CHAPTER VIII.

THE VARIABLE.

86. Nature of the variable

87. Relation of the variable to any

88. Formal and restricted variables .

89. Formal implication presupposes any

90. Duality of any and some

91. The c\ass-conce]it propositioual fimction is indefinable

92. Other classes can be defined by means of such that

93. Analysis of the variable .....

PAGE

89

89

91

91

92

92

93

93

\ CHAPTER IX.

RELATIONS.

94. Characteristics of relations .......

95. Relations of terms to themselves ......

96. The domain and the converse domain of a relation

97. Logical sum, logical product and relative product of relations

98. A relation is not a class of couples .....

99. Relations of a relation to its terms .....

95

96

97

98

99

99

CHAPTER X.

THE CONTRADICTION.

100. Consequences of the contradiction ....... 101

101. Various statements of the contradiction ...... 102

102. An analogous generalized argument ....... 102

103. Variable propositional functions are in general inadmissible . . 103

104. The contradiction arises from treating as one a class which is only

many ............ 104

105. Other prima facie possible solutions appear inadequate .... 105

106. Summary of Part I 106

Table of Contents xv

PART II.

NUMBER.

CHAPTER XI.

DEFINITION OF CARDINAL NUMBERS.

PAGE

107. Plan of Part II Ill

108. Mathematical meaning of definition ....... Ill

109. Definition of numbers by abstraction . . . . , . .112

110. Objections to this definition ........ 114

111. Nominal definition of numbers . . . . . . . . 115

CHAPTER XII.

ADDITION AND MULTIPLICATION.

112. Only integers to be considered at present

113. Definition of arithmetical addition

114. Dependence upon the logical addition of classes .

115. Definition of multiplication .....

116. Connection of addition, multiplication and exponentiation

iir

117

118

119

119^

CHAPTER Xni.

FINITE AND INFINITE.

117. Definition of finite and infinite ........ 121

118. Definition of Qq 121

119. Definition of finite numbers by mathematical induction . . . 123

CHAPTER XIV.

THEORY OF FINITE NUMBERS.

120. Peano's indefinables and primitive propositions 124

121. Mutual independence of the latter ....... 125

122. Peano really defines progressions^ not finite numbers . . . .125

123. Proof of Peano's primitive propositions ...... 127

R.

XVI

Table of Contents

CHAPTER XV.

ADDITION OF TERMS AND ADDITION OF CLASSES.

124. Philosopliy and mathematics distinguished .....

125. Is there a more fundamental sense of number than that defined above ?

126. Numbers must be classes ......

127. Numbers apply to classes as many

128. One is to be asserted, not of terms, but of unit classes

129. Counting- not fundamental in arithmetic

130. Numerical conjunction and plurality ....

131. Addition of terms generates classes primarily^ not numbers

132. A term is indefinable, but not the number 1

PAGE

129

130

131

132

132

133

133

135

135

CHAPTER XVI.

WHOLE AND PART.

133. Single terms may be either simple or complex

134. Whole and part cannot be defined by logical priority .

135. Three kinds of relation of whole and part distinguished

136. Two kinds of wholes distinguished . .

137. A whole is distinct from the numerical conjunction of its parts

138. How far analysis is falsification ......

139. A class as one is an aggregate ......

137

137

138

140

141

141

141

CHAPTER XVII.

INFINITE WHOLES.

140. Infinite aggregates must be admitted .

141. Infinite unities, if there are any, are unknown to us

142. Are all infinite wholes aggregates of terms ?

143. Grounds in favour of this view ....

143

144

146

146

CHAPTER XVIII.

RATIOS AND FRACTIONS.

144. Definition of ratio .......... 149

145. Ratios are one-one relations ........ 150

146. Fractions are concerned with relations of whole and part . . . 1.50

147- Fractions depend, not upon number, but upon magnitude of divisibility 151

148. Summary of Part II 152

Table of Contents

xvii

PAUT III.

QUANTITY.

CHAPTER XIX.

THE MEANING OF MAGNITUDE.

149. Previous views on tlie relation of number and quantity

150. Quantity not fundamental in mathematics .

151. Meaning of magnitude and quantity

152. Three possible theories of equality to be examined

153. Equality is not identity of number of parts

154. Equality is not an unanalyzable relation of quantities

155. Equality is sameness of magnitude

156. Every particular magnitude is simple

157. The principle of abstraction

158. Summary .....

Note

PAGE

157

1.58

1.59

159

160

162

164

164

166

167

168

CHAPTER XX.

THE RANGE OF QUANTITY.

159. Divisibility does not belong to all quantities ....

160. Distance ...........

161. Differential coefficients ........

162. A magnitude is never divisible, but may be a magnitude of divisibility

163. Every magnitude is unanalyzable .......

170

171

173

173

174

^

164.

165.

166.

167.

168.

169.

170.

171.

172.

173.

174.

175.

176.

177.

178.

CHAPTER XXI.

NUMBERS AS EXPRESSING MAGNITUDES: MEASUREMENT.

Definition of measurement ......... 176

Possible grounds for holding all magnitudes to be measurable . . 176

Intrinsic measurability ......... 177

Of divisibilities 178

And of distances . . . . . . . . . â€¢ 179

Measure of distance and measure of stretch ...... 181

Distance-theories and stretch-theories of geometry . . . . 181

Extensive and intensive magnitudes ....... 182

CHAPTER XXII.

SCIENCE LIBRARY

â€¢I

Digitized by the Internet Archive

in 2010 with funding from

Boston Library Consortium IVIember Libraries

http://www.archive.org/details/principlesofmath01russ

THE PRINCIPLES

OF

MATHEMATICS

BY

BERTRAND RUSSELL M.A.,

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE

VOL L ^^' â€¢^

Cambridge :

at the University Press

1903

I^^^ii

CamhritigE :

PRINTED BY J. AND C. F. CLAY,

AT THE UNIVERSITY PRESS.

1 .sn37n

PREFACE.

THE present work has two main objects. One of these, the proof

that all pure mathematics deals exclusively with concepts definable

in terms of a very small number of fundamental logical concepts, and

that all its propositions are deducible from a very small number of

fundamental logical principles, is undertaken in Parts II. â€” VII. of this

Volume, and will be established by strict symbolic reasoning in Volume ii. -

The demonstration of this thesis has, if I am not mistaken, all the

certainty and precision of which mathematical demonstrations are capable.

As the thesis is very recent among mathematicians, and is almost

universally denied by philosophers, I have undertaken, in this volume,

to defend its various parts, as occasion arose, against such adverse

theories as appeared most widely held or most difficult to disprove.

I have also endeavoured to present, in language as untechnical as

possible, the more important stages in the deductions by which the

thesis is established.

The other object of this work, which occupies Part I., is the

explanation of the fundamental concepts which mathematics accepts

as indefinable. This is a purely philosophical task, and I cannot flatter

myself that I have done more than indicate a vast field of inquiry, and

give a sample of the methods by which the inquiry may be conducted.

The discussion of indefinables â€” which forms the chief part of philosophical

logic^is the endeavour to see clearly, and to make others see clearly,

the entities concerned, in order that the mind may have that kind of

acquaintance with them which it has with redness or the taste of a

pineapple. Where, as in the present case, the indefinables are obtained

primarily as the necessary residue in a process of analysis, it is often

easier to know that there must be such entities than actually to perceive

them ; there is a process analogous to that which resulted in the discovery

of Neptune, with the difference that the final stage â€” the search with a

mental telescope for the entity which has been inferred â€” is often the

most difficult part of the undertaking. In the case of classes, I must

confess, I have failed to perceive any concept fulfilling the conditions

vi Preface

I requisite for the notion of class. And the contradiction discussed in

Chapter x. proves that something is amiss, but what this is I have

hitherto failed to discover.

The second volume, in which I have had the great good fortune

to secure the collaboration of Mr A. N. Whitehead, will be addressed

exclusively to mathematicians ; it will contain chains of deductions,

from the premisses of symbolic logic through Arithmetic, finite and

infinite, to Geometry, in an order similar to that adopted in the present

volume ; it will also contain various original developments, in which the

method of Professor Peano, as supplemented by the Logic of Relations,

has shown itself a powerful instrument of mathematical investigation.

1 The present volume, which may be regarded either as a commentary

j upon, or as an introduction to, the second volume, is addressed in equal

measure to the philosopher and to the mathematician ; but some parts

will be more interesting to the one, others to the other. I should advise

mathematicians, unless they are specially interested in Symbolic Logic,

to begin with Part IV., and only refer to earlier parts as occasion arises.

The following portions are more specially philosophical : Part I.

(omitting Chapter ii.); Part II., Chapters xi., xv., xvi., xvii. ; Part III.;

Part IV., Â§207, Chapters xxvi., xxvii., xxxi.; Part V., Chapters xli.,

XLii., xLiii. ; Part VI., Chapters l., li., lii. ; Part VII., Chapters liii.,

Liv., Lv., Lvii., Lviii.; and the two Appendices, which belong to Part I.,

and should be read in connection with it. Professor Frege's work, which

largely anticipates my own, was for the most part unknown to me when

.the printing of the present work began ; I had seen his Grundgesetze

der Arithmetik, but, owing to the great difficulty of his symbolism, I had

failed to grasp its importance or to understand its contents. The only

method, at so late a stage, of doing justice to his work, was to devote

an Appendix to it ; and in some points the views contained in the

Appendix differ from those in Chapter vi., especially in Â§Â§71, 73, 74.

On questions discussed in these sections, I discovered errors after passing

the sheets for the press ; these errors, of which the chief are the denial

of the null-class, and the identification of a term with the class whose

only member it is, are rectified in the Appendices. The subjects

treated are so difficult that I feel little confidence in my present

opinions, and regard any conclusions which may be advocated as

essentially hypotheses.

A few words as to the origin of the present work may serve to

show the importance of the questions discussed. About six years ago,

I began an investigation into the philosophy of Dynamics. I was

met by the difficulty that, when a particle is subject to several forces,

Preface vii

no one of the component accelerations actually occurs, but only

the resultant acceleration, of which they are not parts ; this fact

rendered illusory such causation of particulars by particulars as is

affirmed, at first sight, by the law of gravitation. It appeared also that

the difficulty in regard to absolute motion is insoluble on a relational

theory of space. From these two questions I was led to a re-examination

of the principles of Geometry, thence to the philosophy of continuity

and infinity, and thence, with a view to discovering the meaning of the

word any, to Symbolic Logic. The final outcome, as regards the

philosophy of Dynamics, is perhaps rather slender ; the reason of this

is, that almost all the problems of Dynamics appear to me empirical,

and therefore outside the scope of such a work as the present. Many

very interesting questions have had to be omitted, especially in Parts

VI. and â– VII., as not relevant to my purpose, which, for fear of

misunderstandings, it may be well to explain at this stage.

When actual objects are counted, or when Geometry and Dynamics

are applied to actual space or actual matter, or when, in any other way,

mathematical reasoning is applied to what exists, the reasoning employed

has a form not dependent upon the objects to which it is applied being

just those objects that they are, but only upon their having certain

general properties. In pure mathematics, actual objects in the world

of existence will never be in question, but only hypothetical objects

having those general properties upon which depends whatever deduction

is being considered ; and these general properties will always be

expressible in terms of the fundamental concepts which I have called

logical constants. Thus when space or motion is spoken of in pure

mathematics, it is not actual space or actual motion, as we know them

in experience, that are spoken of, but any entity possessing those abstract

general properties of space or motion that are employed in the reasonings

of geometry or dynamics. The question whether these properties belong,

as a matter of fact, to actual space or actual motion, is irrelevant to pure

mathematics, and therefore to the present work, being, in my opinion,

a purely empirical question, to be investigated in the laboratory or the

observatory. Indirectly, it is true, the discussions connected with pure

mathematics have a very important bearing upon such empirical questions,

since mathematical space and motion are held by many, perhaps most,

philosophers to be self-contradictory, and therefore necessarily difi^erent

from actual space and motion, whereas, if the views advocated in the

following pages be valid, no such self-contradictions are to be found in

mathematical space and motion. But extra-mathematical considerations

of this kind have been almost wholly excluded from the present work.

viii Pi^eface

/ On fundamental questions of philosophy, my position, in all its chief

features, is derived from Mr G. E. Moore. I have accepted from him

the non-existential nature of propositions (except such as happen to

assert existence) and their independence of any knowing mind ; also

the pluralism which regards the world, both that of existents and

that of entities, as composed of an infinite number of mutually

independent entities, with relations which are ultimate, and not

reducible to adjectives of their terms or of the whole which these

compose7\ Before learning these views from him, I found myself

completely unable to construct any philosophy of arithmetic, whereas

their acceptance brought about an immediate liberation from a large

number of difficulties which I believe to be otherwise insuperable.

The doctrines just mentioned are, in my opinion, quite indispensable

to any even tolerably satisfactory philosophy of mathematics, as I hope

the following pages will show. But I must leave it to my readers to

judge how far the reasoning assumes these doctrines, and how far it

supports them. Formally, my premisses are simply assumed ; but the

fact that they allow mathematics to be true, which most cuiTent

philosophies do not, is surely a powerful argument in their favour.

In Mathematics, my chief obligations, as is indeed evident, are to

Georg Cantor and Professor Peano. If I had become acquainted

sooner with the work of Professor Frege, I should have owed a

great deal to him, but as it is I arrived independently at many

results which he had already established. At every stage of my work,

I have been assisted more than I can express by the suggestions, the

criticisms, and the generous encouragement of Mr A. N. Whitehead;

he also has kindly read my proofs, and greatly improved the final

expression of a very large number of passages. Many useful hints

I owe also to Mr W. E. Johnson ; and in the more philosophical parts

of the book I owe much to Mr G. E. Moore besides the general position

which underlies the whole.

In the endeavour to cover so wide a field, it has been impossible to

acquire an exhaustive knowledge of the literature. There are doubtless

many important works with which I am unacquainted ; but where the

labour of thinking and writing necessarily absorbs so much time, such

ignorance, however regrettable, seems not wholly avoidable.

Many words will be found, in the course of discussion, to be defined

in senses apparently departing widely from common usage. Such

departures, I must ask the reader to believe, are never wanton, but have

been made with great reluctance. In philosophical matters, they have

been necessitated mainly by two causes. First, it often happens that

I'

i

Preface ix

two cognate notions are both to be considered, and that language has

two names for the one, but none for the other. It is then highly

convenient to distinguish between the two names commonly used as

synonyms, keeping one for the usual, the other for the hitherto nameless

sense. The other cause arises from philosophical disagreement with

received views. Where two qualities are commonly supposed inseparably

conjoined, but are here regarded as separable, the name which has

applied to their combination will usually have to be restricted to one

or other. For example, propositions are commonly regarded as (1) true

or 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 false as such, and

this name can scarcely be other than proposition. In such a case, the

departure from usage is in no degree arbitrary. As regards mathematical

terms, the necessity for establishing the existence-theorem in each case â€”

i.e. the proof that there are entities of the kind in question â€” has led to

many definitions which appear widely different from the notions usually

attached to the terms in question. Instances of this are the definitions

of cardinal, ordinal and complex numbers. In the two former of these,

and in many other cases, the definition as a class, derived from the

principle of abstraction, is mainly recommended by the fact that it

leaves no doubt as to the existence-theorem. But in many instances of

such apparent departure from usage, it may be doubted whether more

has been done than to give precision to a notion which had hitherto

been more or less vague.

For publishing a work containing so many unsolved difficulties, my

apology is, that investigation revealed no near prospect of adequately

resolving the contradiction discussed in Chapter x., or of acquiring a

better insight into the nature of classes. The repeated discovery of errors

in solutions which for a time had satisfied me caused these problems to

appear such as would have been only concealed by any seemingly satis-

factory theories which a slightly longer reflection might have produced ;

it seemed better, therefore, merely to state the difficulties, than to wait

until I had become persuaded of the truth of some ahiiost 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 and courtesy

in regard to the present volume.

London J

December, 1902.

TABLE OF CONTENTS

Preface ............. v

PA.RT I.

THE INDEFINABLES OF MATHEMATICS.

CHAPTER I.

DEFINITION OF PURE MATHEMATICS.

1. Definition of pure mathematics ........ 3

2. The principles of mathematics are no longer controversial ... 3

3. Pure mathematics uses only a few notions, and these are logical

constants ........... 4

4. All pure mathematics follows formally from twenty premisses . . 4

5. Asserts formal implications ......... 5

6. And employs variables ......... 5

7. Which may have any value without exception ..... 6

8. Mathematics deals with types of relations ...... 7

9. Applied mathematics is defined by the occurrence of constants which

are not logical .......... 8

10. Relation of mathematics to logic ........ 8

CHAPTER II.

SYMBOLIC LOGIC.

11. Definition and scope of symbolic logic ....... 10

12. The indefinables of symbolic logic ....... 10

13. Symbolic logic consists of three parts . . . . . . . 11

A. The Propositional Calculus.

14. Definition ..........

15. Distinction between implication and formal implication

16. Implication indefinable .......

17. Two indefinables and ten primitive propositions in this calculus

18. The ten primitive propositions ......

19. Disjunction and negation defined .....

13

14

14

15

16

17

Xll

Table of Contents

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

B. The Calculus of Classes.

PAGK

Three new indefinables ......... 18

The relation of an individual to its class ...... 19

Prepositional functions ......... 19

The notion of such that . . . . . . . . . 20

Two new primitive propositions . . . . . . . . 20

Relation to prepositional calculus ....... 21

Identity .23

C. The Calculus of Relations.

The logic of relations essential to mathematics ..... 23

New primitive propositions ......... 24

Relative products .......... 25

Relations with assigned domains ........ 26

D. Peano's Symbolic Logic.

Mathematical and philosophical definitions ...... 26

Peano's indefinables . . . . . ... . . . 27

Elementary definitions . . . . . . . ... 28

Peano's primitive propositions . . . ... . . 29

Negation and disjunction . . . . . . . . . 31

Existence and the null-class . . . . . . . . 32

CHAPTER III.

IMPLICATION AND FORMAL IMPLICATION.

37- Meaning of implication . . . . . . . . , 33

38. Asserted and unasserted propositions . . . . . . . 34

39. Inference does not require two premisses ...... 35

40. Formal implication is to be interpreted extensionally .... 36

41. The variable in a formal implication has an unrestricted field . . 36

42. A formal implication is a single propositional function^ not a relation

of two 38

43. Assertions ............ 39

44. Conditions that a term in an implication may be varied ... 39

45. Formal implication involved in rules of inference .... 40

CHAPTER IV.

PROPER NAMES, ADJECTIVES AND VERBS.

46. Proper names, adjectives and verbs distinguished

47. Terms ......

48. Tilings and concepts ....

49. Concepts as such and as terms

50. Conceptual diversity ....

51. Meaning and the subject-predicate logic

52. Verbs and truth .....

53. All verbs, except perhaps is, express relatio

54. Relations pr.r se and relating relations

65. Relations are not particularized by their terms

42

43

44

45

46

47

47

49

49

50

Table of Contents

xiu

CHAPTER V.

DENOTING.

56. Definition of denoting .........

57. Connection with subject-predicate propositions ....

58. Denoting concepts obtained from predicates . .

59. Extensional account of all, every, any, a and so7)ie ...

60. Intensional account of the same .......

61. Illustrations ..........

62. The difference between all, every, etc. lies in the objects denoted^ not

in the way of denoting them ......

68. l^he notion of the and definition .......

64. The notion of the and identity . . â– .

Q6. Summary . . . . . . . . ' .

PAGE

53

54

55

56

58

59

61

62

63

64

CHAPTER VI.

CLASSES. N

66. Combination of intensional and extensional standpoints required

67. Meaning of class .....

68. Intensional and extensional genesis of classes

69. Distinctions overlooked by Peano

70. The class as one and as many

71. The notion of and .....

72. All men is not analyzable into all and men .

73. There are null class-concepts, but there is no null class

74. The class as one, except when it has one term, is distinct from the

class as many .........

75. Every, any, a and some each denote one object, but an ambiguous one

76. The relation of a term to its class ......

77. The relation of inclusion between classes .....

78. The contradiction .........

79. Summary ...........

66

67

67

68

68

69

72

73

76

77

77

78

79

80

CHAPTER VH.

PROPOSITIONAL FUNCTIONS.

80. Indefinability of such that ......... 82

81. Where a fixed relation to a fixed term is asserted, a propositional

function can be analyzed into a variable subject and a constant

assertion ........... 83

82. But this analysis is impossible in other cases ..... 84

83. Variation of the concept in a proposition ...... 86

84. Relation of propositional functions to classes ..... 88

85. A propositional function is in general not analyzable into a constant

and a variable element ........ 88

XIV

Table of Contents

CHAPTER VIII.

THE VARIABLE.

86. Nature of the variable

87. Relation of the variable to any

88. Formal and restricted variables .

89. Formal implication presupposes any

90. Duality of any and some

91. The c\ass-conce]it propositioual fimction is indefinable

92. Other classes can be defined by means of such that

93. Analysis of the variable .....

PAGE

89

89

91

91

92

92

93

93

\ CHAPTER IX.

RELATIONS.

94. Characteristics of relations .......

95. Relations of terms to themselves ......

96. The domain and the converse domain of a relation

97. Logical sum, logical product and relative product of relations

98. A relation is not a class of couples .....

99. Relations of a relation to its terms .....

95

96

97

98

99

99

CHAPTER X.

THE CONTRADICTION.

100. Consequences of the contradiction ....... 101

101. Various statements of the contradiction ...... 102

102. An analogous generalized argument ....... 102

103. Variable propositional functions are in general inadmissible . . 103

104. The contradiction arises from treating as one a class which is only

many ............ 104

105. Other prima facie possible solutions appear inadequate .... 105

106. Summary of Part I 106

Table of Contents xv

PART II.

NUMBER.

CHAPTER XI.

DEFINITION OF CARDINAL NUMBERS.

PAGE

107. Plan of Part II Ill

108. Mathematical meaning of definition ....... Ill

109. Definition of numbers by abstraction . . . . , . .112

110. Objections to this definition ........ 114

111. Nominal definition of numbers . . . . . . . . 115

CHAPTER XII.

ADDITION AND MULTIPLICATION.

112. Only integers to be considered at present

113. Definition of arithmetical addition

114. Dependence upon the logical addition of classes .

115. Definition of multiplication .....

116. Connection of addition, multiplication and exponentiation

iir

117

118

119

119^

CHAPTER Xni.

FINITE AND INFINITE.

117. Definition of finite and infinite ........ 121

118. Definition of Qq 121

119. Definition of finite numbers by mathematical induction . . . 123

CHAPTER XIV.

THEORY OF FINITE NUMBERS.

120. Peano's indefinables and primitive propositions 124

121. Mutual independence of the latter ....... 125

122. Peano really defines progressions^ not finite numbers . . . .125

123. Proof of Peano's primitive propositions ...... 127

R.

XVI

Table of Contents

CHAPTER XV.

ADDITION OF TERMS AND ADDITION OF CLASSES.

124. Philosopliy and mathematics distinguished .....

125. Is there a more fundamental sense of number than that defined above ?

126. Numbers must be classes ......

127. Numbers apply to classes as many

128. One is to be asserted, not of terms, but of unit classes

129. Counting- not fundamental in arithmetic

130. Numerical conjunction and plurality ....

131. Addition of terms generates classes primarily^ not numbers

132. A term is indefinable, but not the number 1

PAGE

129

130

131

132

132

133

133

135

135

CHAPTER XVI.

WHOLE AND PART.

133. Single terms may be either simple or complex

134. Whole and part cannot be defined by logical priority .

135. Three kinds of relation of whole and part distinguished

136. Two kinds of wholes distinguished . .

137. A whole is distinct from the numerical conjunction of its parts

138. How far analysis is falsification ......

139. A class as one is an aggregate ......

137

137

138

140

141

141

141

CHAPTER XVII.

INFINITE WHOLES.

140. Infinite aggregates must be admitted .

141. Infinite unities, if there are any, are unknown to us

142. Are all infinite wholes aggregates of terms ?

143. Grounds in favour of this view ....

143

144

146

146

CHAPTER XVIII.

RATIOS AND FRACTIONS.

144. Definition of ratio .......... 149

145. Ratios are one-one relations ........ 150

146. Fractions are concerned with relations of whole and part . . . 1.50

147- Fractions depend, not upon number, but upon magnitude of divisibility 151

148. Summary of Part II 152

Table of Contents

xvii

PAUT III.

QUANTITY.

CHAPTER XIX.

THE MEANING OF MAGNITUDE.

149. Previous views on tlie relation of number and quantity

150. Quantity not fundamental in mathematics .

151. Meaning of magnitude and quantity

152. Three possible theories of equality to be examined

153. Equality is not identity of number of parts

154. Equality is not an unanalyzable relation of quantities

155. Equality is sameness of magnitude

156. Every particular magnitude is simple

157. The principle of abstraction

158. Summary .....

Note

PAGE

157

1.58

1.59

159

160

162

164

164

166

167

168

CHAPTER XX.

THE RANGE OF QUANTITY.

159. Divisibility does not belong to all quantities ....

160. Distance ...........

161. Differential coefficients ........

162. A magnitude is never divisible, but may be a magnitude of divisibility

163. Every magnitude is unanalyzable .......

170

171

173

173

174

^

164.

165.

166.

167.

168.

169.

170.

171.

172.

173.

174.

175.

176.

177.

178.

CHAPTER XXI.

NUMBERS AS EXPRESSING MAGNITUDES: MEASUREMENT.

Definition of measurement ......... 176

Possible grounds for holding all magnitudes to be measurable . . 176

Intrinsic measurability ......... 177

Of divisibilities 178

And of distances . . . . . . . . . â€¢ 179

Measure of distance and measure of stretch ...... 181

Distance-theories and stretch-theories of geometry . . . . 181

Extensive and intensive magnitudes ....... 182

CHAPTER XXII.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63

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