The Principles of Mathematics (1903)

§ 108

It is often held that both number and particular numbers are indefinable. Now definability is a word which, in Mathematics, has a precise sense, though one which is relative to some given set of notions[73]. Given any set of notions, a term is definable by means of these notions when, and only when, it is the only term having to certain of these notions a certain relation which itself is one of the said notions. But philosophically, the word definition has not, as a rule, been employed in this sense; it has, in fact, been restricted to the analysis of an idea into its constituents. This usage is inconvenient and, I think, useless; moreover it seems to overlook the fact that wholes are not, as a rule, determinate when their constituents are given, but are themselves new entities (which may be in some sense simple), defined, in the mathematical sense, by certain relations to their constituents. I shall, therefore, in future, ignore the philosophical sense, and speak only of mathematical definability. I shall, however, restrict this notion more than is done by Professor Peano and his disciples. They hold that the various branches of Mathematics have various indefinables, by means of which the remaining ideas of the said subjects are defined. I hold--and it is an important part of my purpose to prove--that all Pure Mathematics (including Geometry and even rational Dynamics) contains only one set of indefinables, namely the fundamental logical concepts discussed in Part I. When the various logical constants have been enumerated, it is somewhat arbitrary which of them we regard as indefinable, though there are apparently some which must be indefinable in any theory. But my contention is, that the indefinables of Pure Mathematics are all of this kind, and that the presence of any other indefinables indicates that the subject belongs to Applied Mathematics. Moreover, of the three kinds of definition admitted by Peano--the nominal definition, the definition by postulates, and the definition by abstraction[74]--I recognize only the nominal: the others, it would seem, are only necessitated by Peano's refusal to regard relations as part of the fundamental apparatus of logic, and by his somewhat undue haste in regarding as an individual what is really a class. These remarks will be best explained by considering their application to the definition of cardinal numbers.(§ 108 ¶ 1)

§ 108 n. 1. See Peano, F. 1901, p. 6 ff. and Padoa, Théorie Algébrique des Nombres Entiers, Congrès, Vol. III, p. 314 ff.

§ 108 n. 2. Cf. Burali-Forti, Sur les différentes définitions du nombre réel, Congrès, III, p. 294 ff.