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

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