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THE PRINCIPLES

OF

MATHEMATICS



BY



BERTRAND RUSSELL M.A.,

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE



VOL L ^^' •^









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. — 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 — 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 — the search with a
mental telescope for the entity which has been inferred — 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., §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 §§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 —
i.e. the proof that there are entities of the kind in question — 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 . . . . . . . . . • 179

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

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

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

CHAPTER XXII.