# Principles of mathematics

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### Contents

 CHAPTER I 3 constants 4 Asserts formal implications 5 Which may have any value without exception 6 Mathematics deals with types of relations 7 Applied mathematics is defined by the occurrence of constants which are not logical 8 SYMBOLIC LOGIC 9 Definition and scope of symbolic logic 10
 CHAPTER XXXIV 276 Elementary properties of limits 277 An arithmetical theory of irrationals is indispensable 278 Defects in Dedekinds axiom of continuity 279 Objections to his theory of irrationals 280 Weierstrasss theory 282 Cantors theory 283 Real numbers are segments of rationals 285

 Symbolic logic consists of three parts 11 Definition 13 Distinction between implication and formal implication 14 17 Two indefinables and ten primitive propositions in this calculus 15 The ten primitive propositions 16 Disjunction and negation defined 17 B The Calculus of Classes PAGE 20 Three new indefinables 18 The relation of an individual to its class 19 The notion of such that 20 Relation to propositional calculus 21 Identity 23 New primitive propositions 25 Mathematical and philosophical definitions 26 Peanos indefinables 27 Elementary definitions 28 Peanos primitive propositions 29 Negation and disjunction 31 Existence and the nullclass 32 IMPLICATION AND FORMAL IMPLICATION 37 Meaning of implication 33 Asserted and unasserted propositions 34 Inference does not require two premisses 35 Formal implication is to be interpreted extensionally 36 A formal implication is a single propositional function not a relation of two 38 Assertions 39 Formal implication involved in rules of inference 40 CHAPTER IV 42 Terms 43 Things and concepts 44 Concepts as such and as terms 45 Conceptual diversity 47 All verlw except perhaps is express relations 49 Relations are not particularized by their terms 50 Definition of denoting 53 Connection with subjectpredicate propositions 54 Denoting concepts obtained from predicates 55 Extensional account of all every any a and some 56 Intensional account of the same 58 Illustrations 59 The difference between all every etc lies in the objects denoted not in the way of denoting them 61 The notion of the and definition 62 The notion of the and identity 63 Summary 64 CHAPTER VI 66 Meaning of elaxx 67 Distinctions overlooked by Peano 68 The notion of and 70 All men is not analyzable into all and men 72 There are null classconcepts but there is no null class 73 The class as one except when it has one term is distinct from 74 class as many 76 Every any a and some each denote one object but an ambiguous one 77 The relation of a term to its class 78 The contradiction 79 Summary 80 CHAPTER VII 82 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 But this analysis is impossible in other cases 84 Variation of the concept in a proposition 86 Relation of propositional functions to classes 88 CHAPTER VIII 89 Formal and restricted variables 91 Duality of any and xome 92 Other classes can be defined by means of such that 93 CHAPTER IX 95 Relations of terms to themselves 96 Mi 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 THE CONTRADICTION 100 Consequences of the contradiction 102 Variable propositional functions are in general inadmissible 103 The contradiction arises from treating as one a class which is only many 104 Other prima fade possible solutions appear inadequate 105 Summary of Part I 106 CHAPTER XI 111 CHAPTER XII 117 Definition of finite numbers by mathematical induction 123 CHAPTER XV 129 Is there a more fundamental sense of number than that denned above? 130 Numbers must beclasses 131 Numbers apply to classes as many 132 Counting not fundamental in arithmetic 133 Addition of terms generates classes primarily not numbers 135 CHAPTER XVI 137 Three kinds of relation of whole and part distinguished 138 Two kinds of wholes distinguished 140 A whole is distinct from the numerical conjunction of its parts 141 CHAPTER XVII 143 Infinite unities if there are any are unknown to us 144 Are all infinite wholes aggregates of terms? 146 CHAPTER XVIII 149 Ratios are oneone relations 150 Fractions depend not upon number but upon magnitude of divisibility 151 QUANTITY 153 CHAPTER XIX 157 Quantity not fundamental in mathematics 158 Meaning of magnitude and quantity 159 Equality is not identity of number of parts 160 Equality is not an unanalysable relation of quantities 162 Equality is sameness of magnitude 164 The principle of abstraction 166 Summary 167 Note 168 CHAPTER XX 170 Distance 171 Hil Differential coefficients 173 Every magnitude is unanalysable 175 CHAPTER XXI 176 Intrinsic measurability 177 Of divisibilities 178 And of distances 179 Measure of distance and measure of stretch 181 Extensive and intensive magnitudes 182 CHAPTER XXII 184 CHAPTER XXIII 188 Three antinomies 189 Of which the antitheses depend upon an axiom of finitude 190 And the use of mathematical induction 192 Provisional sense of continuity 193 Summary of Part III 194 PART IV 197 CHAPTER XXIV 199 Generation of order by oneone relations 200 By transitive asymmetrical relations 203 By distances 204 By relations between asymmetrical relations 20 5 205 CHAPTER XXV 207 First theory 208 A relation is not between its terms 210 Second theory of between 211 Reasons for rejecting the second theory 213 Meaning of separation of couples 214 Reduction to transitive asymmetrical relations 215 This reduction is formal 216 The second way of generating series is alone fundamental and gives the meaning of order 217 Classification of relations as regards symmetry and transitiveness 218 Symmetrical transitive relations 219 Relative position 220 Are relations reducible to predications? 221 Mouadistic theory of relations 222 Monistic theory and the reasons for rejecting it 224 216 Order requires that relations should be ultimate 226 CHAPTER XXVII 227 Meaning of difference of sense 228 Difference of sign 229 And of magnitudes 230 Right and left 231 Difference of sign arises from difference of sense among transitive asymmetrical relations 232 CHAPTER XXVIII 234 Series generated by triangular relations 236 Fourterm relations 237 Closed series are such as have an arbitrary first term 238 PROGRESSIONS AND ORDINAL NUMBERS 229 Definition of progressions 239 All finite arithmetic applies to every progression 240 Definition of ordinal numbers 243 Positive and negative ordinals 244 CHAPTER XXX 245 Generalized form of mathematical induction 246 Definition of a singly infinite system 247 Dedekinds proof of mathematical induction 248 And of cardinals 249 CHAPTER XXXI 252 Definition of distance 253 Measurement of distances 254 Summary of Part IV 255 INFINITY AND CONTINUITY 256 CHAPTER XXXII 259 The supposed contradictions of infinity have been resolved 260 Correlation of series 261 Independent series and series by correlation 262 Functions 263 Functions of a variable whose values form a series 264 Functions which arc defined by formulae 267 Complete series 269 CHAPTER XXXIII 270 Segments of rationale 272 Coherent classes in a series 274
 CHAPTER XXXV 287 Cohesion 288 Perfection 291 The existence of limits must not be assumed without special grounds 293 ORDINAL CONTINUITY 276 Continuity is a purely ordinal notion 296 Only ordinal notions occur in this definition 298 Segments of general compact series 299 Segments defined by fundamental series 300 Two compact series may be combined to form a series which is not compact 303 CHAPTER XXXVII 304 Properties of cardinals 307 The smallest transfinite cardinal 310 Finite and transfinite cardinals form a single series by relation to greater and less 311 CHAPTER XXXVIII 312 Definition of w 314 An infinite class can be arranged in many types of series 315 Addition and subtraction of ordinals 317 295 Multiplication and division 318 Wellordered series 319 Series which are not wellordered 320 Ordinal numbers are types of wellordered series 321 Proofs of existencetheorems 322 There is no maximum ordinal number 323 CHAPTER XXXIX 325 Definition of a continuous function 326 Definition of the derivative of a function 328 The infinitesimal is not implied in this definition 329 Neither the infinite nor the infinitesimal is involved in this definition 330 CHAPTER XL 331 Instances of the infinitesimal 332 No infinitesimal segments in compact series 334 Orders of infinity and infinitesimality 335 Summary 337 CHAPTER XLI 338 317 Space and motion are here irrelevant 339 And supposes limits to be essentially quantitative 340 To involve infinitesimal differences 341 He identifies the inextensive with the intensive 342 Consecutive numbers are supposed to be required for continuous change 344 THE PHILOSOPHY OF THE CONTINUUM 325 Philosophical sense of continuity not here in question 346 Zeno and YVeierstrass 347 The argument of dichotomy 348 Extensional and intensional definition of a whole 349 Achilles and the tortoise 350 Change does not involve a state of change 351 The argument of the measure 352 Summary of Cantors doctrine of continuity 353 CHAPTER XLIII 355 Historical retrospect 35 5 356 Proof that there are infinite classes 357 The paradox of Tristram Shandy 358 A whole and a part may be similar 359 Whole and part and formal implication 360 No immediate predecessor of u or a0 361 Difficulty as regards the number of all terms objects or propositions 362 Cantors first proof that there is no greatest number 363 His second proof 364 Every class has more subclasses than terms 366 Resulting contradictions 367 Summary of Part V 368 CHAPTER XLIV 371 Geometry is the science of series of two or more dimensions 372 NonEuclidean geometry 374 Remarks on the definition 375 355 The definition of dimensions is purely logical 376 Algebraical generalization of number 377 Definition of complex numbers 379 CHAPTER XLV 381 Projective points and straight lines 382 Definition of the plane 384 Involutions 385 Projective generation of order 386 Mbbius nets 388 Projective order presupposed in assigning irrational coordinates 389 Anharmonic ratio 390 Comparison of projective and Euclidean geometry 301 392 CHAPTER XLVI 393 Method of Pasch and Peano 394 Method employing serial relations 395 Mutual independence of axioms 396 Logical definition of the class of descriptive spaces 397 Solid geometry 309 399 Ideal elements 400 Ideal lines 401 Ideal planes 402 The removal of a suitable selection of points renders a projective space descriptive 403 CHAPTER XLVII 404 Superposition is not a valid method 405 Errors in Euclid continued 406 Axioms of distance 407 Stretches 408 Order as resulting from distance alone 409 Geometries which derive the straight line from distance 410 In most spaces magnitude of divisibility can be used instead of distance 411 Difficulty of making distance independent of stretch 413 Theoretical meaning of measurement 414 Axioms concerning angles 415 An angle is a stretch of rays not a class of points 416 Areas and volumes 417 CHAPTER XLVIII 418 RELATION OF METRICAL TO PROJECTIVE AND DESCRIPTIVE GEOMETRY 405 Nonquantitative geometry has no metrical presuppositions 419 Historical development of nonquantitative geometry 420 Nonquantitative theory of distance 421 In descriptive geometry 423 And in projective geometry 425 Geometrical theory of imaginary pointpairs 426 New projective theory of distance 427 DEFINITIONS OF VARIOUS SPACES PAGE 412 All kinds of spaces are definable in purely logical terms 429 Definition of projective spaces of three dimensions 430 Definition of Euclidean spaces of three dimensions 432 Definition of Cliffords spaces of two dimensions 434 CHAPTER L 437 The continuity of a metrical space 438 An axiom of continuity enables us to dispense with the postulate of the circle 440 Empirical premisses and induction 441 Space is an aggregate of points not a unity 442 CHAPTER LI 445 Lotzes arguments against absolute position 446 The subjectpredicate theory of propositions 448 Lotzes three kinds of Being 449 Argument from the identity of indiscernibles 451 Points are not active 453 Argument from the necessary truths of geometry 454 CHAPTER LII 456 Mathematical reasoning requires no extralogical element 457 Kants mathematical antinomies 458 MATTER AND MOTION 461 CHAPTER LIII 465 Matter as substance 466 Relations of matter to space and time 467 Definition of matter in terms of logical constants 468 Definition of change 469 There is no such thing as a state of change 471 Occupation of a place at a time 473 CHAPTER LV 474 Causation of particulars by particulars 475 Cause and effect are not temporally contiguous 476 Is there any causation of particulars by particulars? 477 Generalized form of causality 478 CHAPTER LVI 480 CHAPTER LVII 482 The second law of motion 483 Summary of Newtonian principles 485 Causality in dynamics 486 Accelerations as caused by particulars 487 No part of the laws of motion is an a priori truth 488 CHAPTER LVIII 489 Grounds for absolute motion 490 Neumanns theory 491 Absolute rotation is still a change of relation 492 HERTZS DYNAMICS 470 Summary of Hertzs system 494 Hertzs innovations are not fundamental from the point of view of pure mathematics 495 Principles common to Hertz and Newton 496 Summary of Part VI 461 497 APPENDIX A THE LOGIC AL AND ARITHMETICAL DOCTRINES OF FREGE 475 Principal points in Freges doctrines 501 Meaning and indication 502 Criticism 503 Are assumptions proper names for the true or the false? 504 Functions 505 Recapitulation of theory of prepositional functions 508 Can concepts be made logical subjects? 510 Definition of t and of relation 512 485 Reasons for an extensional view of classes 513 Possible theories to account for this fact 514 Recapitulation of theories already discussed 515 The subject of a proposition may be plural 517 Theory of types 518 Definition of cardinal numbers 519 Freges theory of series 520 APPENDIX B THE DOCTRINE OF TYPES 497 Statement of the doctrine 523 Numbers and propositions as types 525 Are prepositional concepts individuals ? 526 Contradiction arising from the question whether there are more classes of propositions than propositions 527 Index 529 Meinongs theory 184 534