# The Principles of Mathematics

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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

### 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.