The Principles of Mathematics (1903)

§ 31

So much of the above brief outline of Symbolic Logic is inspired by Peano, that it seems desirable to discuss his work explicitly, justfying by criticism the points in which I have departed from him.(§ 31 ¶ 1)

The question as to which of the notions of symbolic logic are to be taken as indefinable, and which of the propositions as indemonstrable is, as Professor Peano has insisted[25], to some extent arbitrary. But it is important to establish all the mutual relations of the simpler notions of logic, and to examine the consequence of taking various notions as indefinable. It is necessary to realize that definition, in mathematics, does not mean, as in philosophy, an analysis of the idea to be defined into constituent ideas. This notion, in any case, is only applicable to concepts, whereas in mathematics it is possible to define terms which are not concepts[26]. Thus also many notions are defined by symbolic logic which are not capable of philosophical definition, since they are simple and unanalyzable. Mathematical definition consists in pointing out a fixed relation to a fixed term, of which one term only is capable: this term is then defined by means of the fixed relation and the fixed term. The point in which this differs from philosophical definition may be elucidated by the remark that the mathematical definition does not point out the term in question, and that only what may be called philosophical insight reveals which it is among all the terms there are. This is due to the fact that the term is defined by a concept which denotes it unambiguously, not by actually mentioning the term denoted. What is meant by denoting, as well as the different ways of denoting, must be accepted as primitive ideas in any symbolic logic[27]: in this respect, the order adopted seems not in any degree arbitrary.(§ 31 ¶ 2)

§ 31 n. 1. E.g. F. 1901, p. 6; F. 1897, Part I, pp. 62–63.