The Principles of Mathematics (Google eBook)

Front Cover
1903
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Contents

I
v
II
1
III
3
IV
10
V
33
VI
42
VII
53
VIII
66
XXXV
245
XXXVI
252
XXXVII
257
XXXVIII
259
XXXIX
270
XL
276
XLI
287
XLII
296

IX
82
X
89
XI
95
XII
101
XIII
109
XIV
111
XV
117
XVI
121
XVII
124
XVIII
129
XIX
137
XX
143
XXI
149
XXII
155
XXIII
157
XXIV
170
XXV
176
XXVI
184
XXVII
188
XXVIII
197
XXIX
199
XXX
207
XXXI
218
XXXII
227
XXXIII
234
XXXIV
239
XLIII
304
XLIV
312
XLV
325
XLVI
331
XLVII
338
XLVIII
346
XLIX
355
L
369
LI
371
LII
381
LIII
393
LIV
404
LV
419
LVI
429
LVII
437
LVIII
445
LIX
456
LX
463
LXI
465
LXII
469
LXIII
474
LXIV
480
LXV
482
LXVI
489
LXVII
494

Common terms and phrases

Popular passages

Page 106 - are : Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as are involved in formal implication, which we found ( 93) to be the following : propositional function, class *, denoting, and any or every term.
Page 11 - the following : Implication between propositions not containing variables, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and truth. By means of these notions, all the propositions of symbolic logic can be .stated.
Page 338 - with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing without limit do always converge, and to which they approach nearer than by any given difference*." But when we turn to such works as Cohen's, we find the dx and
Page 252 - which goes before, and that which follows; there is distance or interval. Relative things have their quantity, as well as absolute ones. For instance, ratios or proportions in mathematics have their quantity, and are measured by logarithms; and yet they are relations. And therefore, though time and space consist in relations, yet they have their quantity
Page 27 - Class, the relation of an individual to a class of which it is a member, the notion of a term, implication where both propositions contain the same variables, ie formal implication, the simultaneous affirmation of two propositions, the notion of definition, and the negation of a proposition.
Page v - that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles, is undertaken in Parts
Page 469 - Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T", provided that the two propositions differ only by the fact that T occurs in the one where
Page 459 - as it concerns time, must be rejected as false, and the argument concerning space, since it depends upon that regarding time, falls also. Antithesis. "The world has no beginning, and no limits in space, but is infinite both in respect of time and space." The proof of this proposition assumes the infinity of pure time and space, and argues that these imply events and things
Page 8 - priori. The fact is that, when once the apparatus of logic has been accepted, all mathematics necessarily follows. The logical constants themselves are to be defined only by enumeration, for they are so fundamental that all the properties by which the class of them might be defined presuppose some terms of the
Page 33 - it is plain .that where we validly infer one proposition from another, we do so in virtue of a relation which holds between the two propositions whether we perceive it or not : the mind, in fact, is as purely receptive in inference as common sense supposes it to be in perception of sensible objects.

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