The Principles of Mathematics (1903)

§ 132

Thus the only point which remains is this: Does the notion of a term presuppose the notion of 1? For we have seen that all numbers except 0 involve in their definitions the notion of a term, and if this in turn involves 1, the definition of 1 becomes circular, and 1 will have to be allowed to be indefinable. This objection to our procedure is answered by the doctrine of § 128, that a term is not one in the sense which is relevant to Arithmetic, or in the sense which is opposed to many. The notion of any term is a logical indefinable, presupposed in formal truth and in the whole theory of the variable; but this notion is that of the variable conjunction of terms, which in no way involves the number 1. There is therefore nothing circular in defining the number 1 by means of the notion of a term or of any term.(§ 132 ¶ 1)

To sum up: Numbers are classes are classes, namely of all classes similar to a given class. Here classes have to be understood in the sense of numerical conjunctions in the case of classes having many terms; but a class may have no terms, and a class of one term is distinct from that term, so that a class is not simply the sum of its terms. Only classes have numbers; of what is commonly called one object, it is not true, at least in the sense required, to say that it is one, as appears from the fact that the object may be a class of many terms. One object seems to mean merely a logical subject in some proposition. Finite numbers are not to be regarded as generated by counting, which on the contrary presupposes them; and addition is primarily logical addition, first of propositions, then of classes, from which latter arithmetic addition is derivative. The assertion of numbers depends upon the fact that a class of many terms can be a logical subject without being arithmetically one. Thus it appeared that no philosophical argument could overthrow the mathematical theory of cardinal numbers set forth in Chapters XI to XIV.(§ 132 ¶ 2)