The Principles of Mathematics, by Bertrand Russell, was first published in 1903. This online edition is based on the public domain text as it appears in the 1996 Norton paperback reprint of the 1938 Second Edition (ISBN 0-393-31404-9). (We have been forced to omit Russell’s Introduction to the Second Edition from this online edition, as it is still held under copyright.)

The transcription is currently incomplete. We have placed the parts that have been completed online in the hope that they will be useful.

- Preface
### Part I. The Indefinables of Mathematics

#### Chapter I. 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.

#### Chapter II. Symbolic Logic

#### Chapter III. 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
- § 41. The variable in formal implication has an unrestricted field
- § 42. A formal implication is a single propositional function, not a relation of two
- § 43. Assertions
- § 44. Conditions that a term in an implication may be varied
- § 45. Formal implication involved in rules of inference

#### Chapter IV. 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

#### 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*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

#### Chapter VI. Classes

- § 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

#### Chapter VII. 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

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

#### Chapter IX. Relations

#### Chapter X. The Contradiction

- § 100. Consequences of the contradiction
- § 101. Various statements of the contradiction
- § 102. An analogous generalized argument
- § 103. Various statements of the contradiction
- § 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

### Part II. Number

#### Chapter XI. Definition of Cardinal Numbers

#### Chapter XII. Addition and Multiplication

#### Chapter XIII. Finite and Infinite

#### Chapter XIV. Theory of Finite Numbers

#### Chapter XV. 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

#### 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 conjunctions of its parts
- § 138. How far analysis is falsification
- § 139. A class as one is an aggregate

#### Chapter XVII. Infinite Wholes

#### Chapter XVIII. Ratios and Fractions

### Part III. Quantity

#### Chapter XIX. 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 unanalyzable relation of quantities
- § 155. Equality is sameness of magnitude
- § 156. Every particular magnitude is simple
- § 157. The principle of abstraction
- § 158. Summary
- Note to Chapter XIX.

#### Chapter XX. The Range of Quantity

#### Chapter XXI. 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

#### Chapter XXII. Zero

#### Chapter XXIII. 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

### Part IV. Order

#### Chapter XXIV. The Genesis of Series

#### Chapter XXV. 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
- § 207. The second way of generating series is alone fundamental, and gives the meaning of order

#### Chapter XXVI. 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 the theory
- § 215. Monistic theory and the reasons for rejecting it
- § 216. Order requires that relations should be ultimate

#### Chapter XXVII. Difference of Sense and Difference of Sign

#### Chapter XXVIII. On the Difference Between Open and Closed Series

#### Chapter XXIX. Progressions and Ordinal Numbers

#### Chapter XXX. 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

#### Chapter XXXI. Distance

### Part V. Infinity and Continuity

#### Chapter XXXII. 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

#### Chapter XXXIII. Real Numbers

#### Chapter XXXIV. 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

#### Chapter XXXV. Cantor's First Definition of Continuity

#### Chapter XXXVI. 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

#### Chapter XXXVII. 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
`a`_{0} - § 288. Other transfinite cardinals
- § 289. Finite and transfinite cardinals form a single series by relation to greater and less

#### Chapter XXXVIII. 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

#### Chapter XXXIX. 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

#### Chapter XL. The Infinitesimal and the Improper Infinite

#### Chapter XLI. Philosophical Arguments Concerning the Infinitesimal

- § 315. Current philosophical opinions illustrated by Cohen
- § 316. Who bases the calculus upon infinitesimals
- § 317. Space and motion are here irrelevant
- § 318. Cohen regards the doctrine of limits as insufficient for the calculus
- § 319. And supposes limits to be essentially quantitative
- § 320. To involve infinitesimal differences
- § 321. And to introduce a new meaning of equality
- § 322. He identifies the inextensive with the intensive
- § 323. Consecutive numbers are supposed to be required for continuous change
- § 324. Cohen's views are to be rejected

#### Chapter XLII. The Philosophy of the Continuum

- § 325. Philosophical sense of continuity not here in question
- § 326. The continuum is composed of mutually external units
- § 327. Zeno and Weierstrass
- § 328. The argument of dichotomy
- § 329. The objectionable and the innocent kind of endless regress
- § 330. Extensional and intensional definition of a whole
- § 331. Achilles and the tortoise
- § 332. The arrow
- § 333. Change does not involve a state of change
- § 334. The argument of the measure
- § 335. Summary of Cantor's doctrine of continuity
- § 336. The continuum consists of elements

#### Chapter XLIII. The Philosophy of the Infinite

- § 337. Historical retrospect
- § 338. Positive doctrine of the infinite
- § 339. Proof that there are infinite classes
- § 340. The paradox of Tristram Shandy
- § 341. A whole and a part may be similar
- § 342. Whole and part and formal implication
- § 343. No immediate predecessor of
`ω`or`a`_{0} - § 344. Difficulty as regards the number of all terms, objects, or propositions
- § 345. Cantor's first proof that there is no greatest number
- § 346. His second proof
- § 347. Every class has more sub-classes than terms
- § 348. But this is impossible in certain cases
- § 349. Resulting contradictions
- § 350. Summary of Part V

### Part VI. Space

#### Chapter XLIV. Dimensions and Complex Numbers

- § 351. Retrospect
- § 352. Geometry is the science of series of two or more dimensions
- § 353. Non-Euclidean geometry
- § 354. Definition of dimensions
- § 355. Remarks on the definition
- § 356. The definition of dimensions is purely logical
- § 357. Complex numbers and universal algebra
- § 358. Algebraical generalization of number
- § 359. Definition of complex numbers
- § 360. Remarks on the definition

#### Chapter XLV. Projective Geometry

- § 361. Recent threefold scrutiny of geometrical principles
- § 362. Projective, descriptive, and metrical geometry
- § 363. Projective points and straight lines
- § 364. Definition of the plane
- § 365. Harmonic ranges
- § 366. Involutions
- § 367. Projective generation of order
- § 368. Möbius nets
- § 369. Projective order presupposed in assigning irrational coordinates
- § 370. Anharmonic ratio
- § 371. Assignment of coordinates to any point in space
- § 372. Comparison of projective and Euclidean geometry
- § 373. The principle of duality

#### Chapter XLVI. Descriptive Geometry

- § 374. Distinction between projective and descriptive geometry
- § 375. Method of Pasch and Peano
- § 376. Method employing serial relations
- § 377. Mutual independence of axioms
- § 378. Logical definition of the class of descriptive spaces
- § 379. Parts of straight lines
- § 380. Definition of the plane
- § 381. Solid geometry
- § 382. Descriptive geometry applies to Euclidean and hyperbolic, but not elliptic space
- § 383. Ideal elements
- § 384. Ideal points
- § 385. Ideal lines
- § 386. Ideal planes
- § 387. The removal of a suitable selection of points renders a projective space descriptive

#### Chapter XLVII. Metrical Geometry

- § 388. Metrical geometry presupposes projective or descriptive geometry
- § 389. Errors in Euclid
- § 390. Superposition is not a valid method
- § 391. Errors in Euclid (continued)
- § 392. Axioms of distance
- § 393. Stretches
- § 394. Order as resulting from distance alone
- § 395. Geometries which derive the straight line from distance
- § 396. In most spaces, magnitude of divisibility can be used instead of distance
- § 397. Meaning of magnitude of divisibility
- § 398. Difficulty of making distance independent of stretch
- § 399. Theoretical meaning of measurement
- § 400. Definition of angle
- § 401. Axioms concerning angles
- § 402. An angle is a stretch of rays, not a class of points
- § 403. Areas and volumes
- § 404. Right and left

#### Chapter XLVIII. Relation of Metrical to Projective and Descriptive Geometry

- § 405. Non-quantitative geometry has no metrical presuppositions
- § 406. Historical development of non-quantitative geometry
- § 407. Non-quantitative theory of distance
- § 408. In descriptive geometry
- § 409. And in projective geometry
- § 410. Geometrical theory of imaginary point-pairs
- § 411. New projective theory of distance

#### Chapter XLIX. Definitions of Various Spaces

#### Chapter L. The Continuity of Space

- § 416. The continuity of a projective space
- § 417. The continuity of metrical space
- § 418. An axiom of continuity enables us to dispense with the postulate of the circle
- § 419. Is space prior to points?
- § 420. Empirical premisses and induction
- § 421. There is no reason to desire our premisses to be self-evident
- § 422. Space is an aggregate of points, not a unity

#### Chapter LI. Logical Arguments Against Points

- § 423. Absolute and relative position
- § 424. Lotze's arguments against absolute position
- § 425. Lotze's theory of relations
- § 426. The subject-predicate theory of propositions
- § 427. Lotze's three kinds of Being
- § 428. Argument from the identity of indiscernibles
- § 429. Points are not active
- § 430. Argument from the necessary truths of geometry
- § 431. Points do not imply one another

#### Chapter LII. Kant's Theory of Space

### Part VII. Matter and Motion

### Part Appendices.

#### Appendix A. The Logical and Arithmetical Doctrines of Frege

- § 475. Principal points in Frege's doctrines
- § 476. Meaning and indication
- § 477. Truth-values and judgment
- § 478. Criticism
- § 479. Are assumptions proper names for the true or the false?
- § 480. Functions
- § 481. Begriff and Gegenstand
- § 482. Recapitulation of theory of propositional functions
- § 483. Can concepts be made logical subjects?
- § 484. Ranges
- § 485. Definition of
*∈*and of*relation* - § 486. Reasons for an extensional view of classes
- § 487. A class which has only one member is distinct from its only member
- § 488. Possible theories to account for this fact
- § 489. Recapitulation of theories already discussed
- § 490. The subject of a proposition may be plural
- § 491. Classes having only one member
- § 492. Theory of types
- § 493. Implication and symbolic logic
- § 494. Definition of cardinal numbers
- § 495. Frege's theory of series
- § 496. Kerry's criticisms of Frege

#### Appendix B. The Doctrine of Types

The Principles of Mathematics was written by Bertrand Russell, and published in 1903. It is now available in the Public Domain.