The Principles of Mathematics (1903)

§ 107

We have now briefly reviewed the apparatus of general logical notions with which Mathematics operates. In the present Part, it is to be shown how this apparatus suffices, without new indefinables or new postulates, to establish the whole theory of cardinal integers as a special branch of Logic[72]. No mathematical subject has made, in recent years, greater advances than the theory of Arithmetic. The movement in favour of correctness in deduction, inaugurated by Weierstrass, has been brilliantly continued by Dedekind, Cantor, Frege, and Peano, and attains what seems its final goal by means of the logic of relations. As the modern mathematical theory is but imperfectly known even by most mathematicians, I shall begin this Part by four chapters setting forth its outlines in a non-symbolic form. I shall then examine the process of deduction from a philosophical standpoint, in order to discover, if possible, whether any unperceived assumptions have covertly intruded themselves in the course of the argument.(§ 107 ¶ 1)

§ 107 n. 1. Cantor has shown that it is necessary to separate the study of Cardinal and Ordinal numbers, which are distinct entities, of which the former are simpler, but of which both are essential to ordinary Mathematics. On Ordinal numbers, cf. Chaps. XXIX, XXXVIII, infra.