## The Principles of Mathematics |

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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 numbers concept concerning considered contained continuity contradiction correlation defined definition denoted descriptive Geometry discussion distance distinction distinguish divisibility entities equal equivalent Euclidean space existence fact false finite number follows formal implication Frege given greater Hence hold ideal points identical implies indefinable infinite classes infinitesimal infinity integers kind Leibniz less limit logical constants logical product magnitude material implication mathematical induction means metrical notion null-class number of terms object obtained one-one relation ordinal Peano philosophical plane possible predicate premisses presupposed principle projective Geometry projective space properties propositional function prove quantities question rational numbers real numbers regard seems segments sense Socrates straight line stretch supposed Symbolic Logic theory transfinite transitive transitive relation true values variable zero

### 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.