Bertrand Russell - Principles of Mathematics PDF

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Principles of Mathemat ics “Unless we are very much mistaken, its lucid application and develop- ment of the great discoveries of Peano and Cantor mark the opening of a new epoch in both philosophical and mathematical thought” – The Spectator Bertrand Russell Principles of Mathematics London and Ne...

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Principles of Mathemat ics “Unless we are very much mistaken, its lucid application and development of the great discoveries of Peano and Cantor mark the opening of a new epoch in both philosophical and mathematical thought” – The Spectator

Bertrand

Russell Principles of Mathematics

London and New York

First published in 1903 First published in the Routledge Classics in 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business This edition published in the Taylor & Francis e-Library, 2009. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 2010 The Bertrand Russell Peace Foundation Ltd Introduction © 1992 John G. Slater All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-203-86476-X Master e-book ISBN

ISBN 10: 0-415-48741-2 ISBN 10: 0-203-86476-X (ebk) ISBN 13: 978-0-415-48741-2 ISBN 13: 978-0-203-86476-0 (ebk)

C ONTENTS

introduction to the 1992 edition introduction to the second edition preface PART I THE INDEFINABLES OF MATHEMATICS 1

2

Definition of Pure Mathematics 1. Definition of pure mathematics 2. The principles of mathematics are no longer controversial 3. Pure mathematics uses only a few notions, and these are logical constants 4. All pure mathematics follows formally from twenty premisses 5. Asserts formal implications 6. And employs variables 7. Which may have any value without exception 8. Mathematics deals with types of relations 9. Applied mathematics is defined by the occurrence of constants which are not logical 10. Relation of mathematics to logic Symbolic Logic 11. Definition and scope of symbolic logic 12. The indefinables of symbolic logic 13. Symbolic logic consists of three parts

xxv xxxi xliii 1 3 3 3 4 4 5 6 6 7 8 8 10 10 11 12

vi

contents

A. The Propositional Calculus 14. 15.

Definition 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

20. 21. 22. 23. 24. 25. 26.

27. 28. 29. 30.

31. 32. 33. 34. 35. 36. 3

13 13 14 14 15 16 17

B. The Calculus of Classes

18

Three new indefinables The relation of an individual to its class Propositional functions The notion of such that Two new primitive propositions Relation to propositional calculus Identity

18 19 19 20 20 21 23

C. The Calculus of Relations

23

The logic of relations essential to mathematics New primitive propositions Relative products Relations with assigned domains

23 24 25 26

D. Peano’s Symbolic Logic

27

Mathematical and philosophical definitions Peano’s indefinables Elementary definitions Peano’s primitive propositions Negation and disjunction Existence and the null-class

27 27 28 30 31 32

Implication and Formal Implication 37. Meaning of implication 38. Asserted and unasserted propositions 39. Inference does not require two premisses 40. Formal implication is to be interpreted extensionally

34 34 35 37 37

contents

41. 42. 43. 44. 45. 4

5

6

The variable in a formal implication has an unrestricted field A formal implication is a single propositional function, not a relation of two Assertions Conditions that a term in an implication may be varied Formal implication involved in rules of inference

Proper Names, Adjectives and Verbs 46. Proper names, adjectives and verbs distinguished 47. Terms 48. Things 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 relations 54. Relations per se and relating relations 55. Relations are not particularized by their terms 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 some 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 63. The notion of the and definition 64. The notion of the and identity 65. Summary Classes 66. Combination of intensional and extensional standpoints required 67. Meaning of class 68. Intensional and extensional genesis of class

37 39 40 40 41 43 43 44 45 46 47 48 49 50 50 51 54 54 55 56 57 59 60

63 64 65 66 67 67 68 68

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viii contents 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 7

8

9

Distinctions overlooked by Peano The class as one and as many The notion of and All men is not analysable into all and men There are null class-concepts, but there is no null-class The class as one, except when it has one term, is distinct from the class as many Every, any, a and some each denote one object, but an ambiguous one The relation of a term to its class The relation of inclusion between classes The contradiction Summary

Propositional Functions 80. Indefinability of such that 81. Where a fixed relation to a fixed term is asserted, a propositional function can be analysed into a variable subject and a constant assertion 82. But this analysis is impossible in other cases 83. Variation of the concept in a proposition 84. Relation of propositional functions to classes 85. A propositional function is in general not analysable into a constant and a variable element

69 69 70 73 74 77 77 78 79 80 81 82 82

83 84 86 88 88

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 class-concept propositional function is indefinable 92. Other classes can be defined by means of such that 93. Analysis of the variable

89 89 89 91 91 92

Relations 94. Characteristics of relations 95. Relations of terms to themselves 96. The domain and the converse domain of a relation

95 95 96

93 93 93

97

contents

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 10

The Contradiction 100. Consequences of the contradiction 101. Various statements of the contradiction 102. An analogous generalized argument 103. Variable propositional functions are in general inadmissible 104. The contradiction arises from treating as one a class which is only many 105. Other primâ facie possible solutions appear inadequate 106. Summary of Part I

98 99 99 101 101 102 102 103 104 105 106

PART II NUMBER

109

11

Definition of Cardinal Numbers 107. Plan of Part II 108. Mathematical meaning of definition 109. Definition of numbers by abstraction 110. Objections to this definition 111. Nominal definition of numbers

111 111 111 112 114 115

12

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

118 118 118 119 120

Finite and Infinite 117. Definition of finite and infinite 118. Definition of α0 119. Definition of finite numbers by mathematical induction

122 122 123

13

14

Theory of Finite Numbers 120. Peano’s indefinables and primitive propositions 121. Mutual independence of the latter

121

124 125 125 126

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contents

122. Peano really defines progressions, not finite numbers 123. Proof of Peano’s primitive propositions 15

126 128

Addition of Terms and Addition of Classes 124. Philosophy 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

130 130

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

138 138

142 142 143

17

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

144 144 145 147 147

18

Ratios and Fractions 144. Definition of ratio 145. Ratios are one-one relations 146. Fractions are concerned with relations of whole and part 147. Fractions depend, not upon number, but upon magnitude of divisibility 148. Summary of Part II

150 150 151

16

131 132 133 133 134 135 136 136

138 139 141

151 152 153

contents

PART III QUANTITY

155

19

157

20

21

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The Meaning of Magnitude 149. Previous views on the 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 unanalysable relation of quantities 155. Equality is sameness of magnitude 156. Every particular magnitude is simple 157. The principle of abstraction 158. Summary Note

157 158 159 159 160 162 164 164 166 167 168

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 unanalysable

170 170 171 173

Numbers as Expressing Magnitudes: Measurement 164. Definition of measurement 165. Possible grounds for holding all magnitudes to be measurable 166. Intrinsic measurability 167. Of divisibilities 168. And of distances 169. Measure of distance and measure of stretch 170. Distance-theories and stretch-theories of geometry 171. Extensive and intensive magnitudes

176 176 177 178 178 180 181

Zero 172. Difficulties as to zero 173. Meinong’s theory 174. Zero as minimum 175. Zero distance as identity 176. Zero as a null segment

184 184 184 185 186 186

173 174

181 182

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contents

23

177. Zero and negation 178. Every kind of zero magnitude is in a sense indefinable

187

Infinity, the Infinitesimal and Continuity 179. Problems of infinity not specially quantitative 180. Statement of the problem in regard to quantity 181. Three antinomies 182. Of which the antitheses depend upon an axiom of finitude 183. And the use of mathematical induction 184. Which are both to be rejected 185. Provisional sense of continuity 186. Summary of Part III

189 189 189 190

187

191 193 193 194 195

PART IV ORDER

199

24

The Genesis of Series 187. Importance of order 188. Between and separation of couples 189. Generation of order by one-one relations 190. By transitive asymmetrical relations 191. By distances 192. By triangular relations 193. By relations between asymmetrical relations 194. And by separation of couples

201 201 201 202 205 206 206 207 207

25

The Meaning of Order 195. What is order? 196. Three theories of between 197. First theory 198. A relation is not between its terms 199. Second theory of between 200. There appear to be ultimate triangular relations 201. Reasons for rejecting the second theory 202. Third theory of between to be rejected 203. Meaning of separation of couples 204. Reduction to transitive asymmetrical relations 205. This reduction is formal 206. But is the reason why separation leads to order

209 209 209 210 212 213 214 215 215 216 217 218 218

contents

207. The second way of generating series is alone fundamental, and gives the meaning of order 26

27

28

29

218

Asymmetrical Relations 208. Classification of relations as regards symmetry and transitiveness 209. Symmetrical transitive relations 210. Reflexiveness and the principle of abstraction 211. Relative position 212. Are relations reducible to predications? 213. Monadistic theory of relations 214. Reasons for rejecting this theory 215. Monistic theory and the reasons for rejecting it 216. Order requires that relations should be ultimate

220

Difference of Sense and Difference of Sign 217. Kant on difference of sense 218. Meaning of difference of sense 219. Difference of sign 220. In the cases of finite numbers 221. And of magnitudes 222. Right and left 223. Difference of sign arises from difference of sense among transitive asymmetrical relations

229 229 230 230 231 231 233

On the Difference Between Open and Closed Series 224. What is the difference between open and closed series? 225. Finite closed series 226. Series generated by triangular relations 227. Four-term relations 228. Closed series are such as have an arbitrary first term

236

Progressions and Ordinal Numbers 229. Definition of progressions 230. All finite arithmetic applies to every progression 231. Definition of ordinal numbers 232. Definition of “nth” 233. Positive and negative ordinals

220 221 221 222 223 224 224 226 228

234

236 236 238 239 240 241 241 242 244 245 246

xiii

xiv contents 30

Dedekind’s Theory of Number 234. Dedekind’s principal ideas 235. Representation of a system 236. The notion of a chain 237. The chain of an element 238. Generalized form of mathematical induction 239. Definition of a singly infinite system 240. Definition of cardinals 241. Dedekind’s proof of mathematical induction 242. Objections to his definition of ordinals 243. And of cardinals

247 247 247 248 248 248 249 249 250 250 251

31

Distance 244. Distance not essential to order 245. Definition of distance 246. Measurement of distances 247. In most series, the existence of distances is doubtful 248. Summary of Part IV

254 254 255 256 256 257

PART V INFINITY AND CONTINUITY

259

32

261

33

The Correlation of Series 249. The infinitesimal and space are no longer required in a statement of principles 250. The supposed contradictions of infinity have been resolved 251. Correlation of series 252. Independent series and series by correlation 253. Likeness of relations 254. Functions 255. Functions of a variable whose values form a series 256. Functions which are defined by formulae 257. Complete series Real Numbers 258. Real numbers are not limits of series of rationals 259. Segments of rationals 260. Properties of segments 261. Coherent classes in a series Note

261 262 262 264 264 265 266 269 271 272 272 273 274 276 276

contents

34

35

36

37

Limits and Irrational Numbers 262. Definition of a limit 263. Elementary properties of limits 264. An arithmetical theory of irrationals is indispensable 265. Dedekind’s theory of irrationals 266. Defects in Dedekind’s axiom of continuity 267. Objections to his theory of irrationals 268. Weierstrass’s theory 269. Cantor’s theory 270. Real numbers are segments of rationals

278 278 279 280 281 281 282 284 285 288

Cantor’s First Definition of Continuity 271. The arithmetical theory of continuity is due to Cantor 272. Cohesion 273. Perfection 274. Defect in Cantor’s definition of perfection 275. The existence of limits must not be assumed without special grounds

290

Ordinal Continuity 276. Continuity is a purely ordinal notion 277. Cantor’s ordinal definition of continuity 278. Only ordinal notions occur in this definition 279. Infinite classes of integers can be arranged in a continuous series 280. Segments of general compact series 281. Segments defined by fundamental series 282. Two compact series may be combined to form a series which is not compact

299 299 299

Transfinite Cardinals 283. Transfinite cardinals differ widely from transfinite ordinals 284. Definition of cardinals 285. Properties of cardinals 286. Addition, multiplication and exponentiation 287. The smallest transfinite cardinal α0 288. Other transfinite cardinals 289. Finite and transfinite cardinals form a single series by relation to greater and less

307

290 291 293 294 296

301 302 302 303 306

307 307 309 310 312

314 314

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xvi contents 38

39

40

41

Transfinite Ordinals 290. Ordinals are classes of serial relations 291. Cantor’s definition of the second class of ordinals 292. Definition of ω 293. An infinite class can be arranged in many types of series 294. Addition and subtraction of ordinals 295. Multiplication and division 296. Well-ordered series 297. Series which are not well-ordered 298. Ordinal numbers are types of well-ordered series 299. Relation-arithmetic 300. Proofs of existence-theorems 301. There is no maximum ordinal number 302. Successive derivatives of a series

316 316 316 318 319 321 322 323 324 325 325 326 327 327

The Infinitesimal Calculus 303. The infinitesimal has been usually supposed essential to the calculus 304. Definition of a continuous function 305. Definition of the derivative of a function 306. The infinitesimal is not implied in this definition 307. Definition of the definite integral 308. Neither the infinite nor the infinitesimal is involved in this definition

330

The Infinitesimal and the Improper Infinite 309. A precise definition of the infinitesimal is seldom given 310. Definition of the infinitesimal and the improper infinite 311. Instances of the infinitesimal 312. No infinitesimal segments in compact series 313. Orders of infinity and infinitesimality 314. Summary

336

Philosophical Arguments Concerning the Infinitesimal 315. Current philosophical opinions illustrated by Cohen 316. Who bases the calculus upon infinitesimals 317. Space and...