## 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 |

529 | |

534 | |

### Common terms and phrases

adjectives aggregate analysis argument Arithmetic assertion asymmetrical relation axiom axiom of Archimedes belongs Calculus called Cantor cardinal number Chapter class of classes class of terms class-concept collection compact series complex concept concerning considered constituents contained continuity contradiction correlation defined definition denoted discussion distance distinction distinguished divisibility entities equal equivalent existence fact false finite integers finite number follows formal implication Frege Geometry Hence holds identical implies q indefinable infinite classes infinite wholes infinitesimal infinity involved Leibniz limit logical constants logical product material implication mathematical induction means method notion null-class number of terms object obtained occur one-one relation ordinal Peano philosophical philosophy of space points possible predicate premisses present presupposed principle progression projective Geometry properties propositional function prove quantities question rational numbers real numbers regard require seems segments sense similar Socrates space stretches supposed Symbolic Logic theory transfinite transitive relation true values variable zero