The Principles of Mathematics (1903)

§ 129

The commonly received view, as regards finite numbers, is that they result from counting, or, as some philosophers would prefer to say, from synthesizing. Unfortunately, those who hold this view have not analyzed the notion of counting: if they had done so, they would have seen that it is very complex, and presupposes the very numbers which it is supposed to generate.(§ 129 ¶ 1)

The process of counting has, of course, a psychological aspect, but this is quite irrelevant to the theory of Arithmetic. What I wish now to point out is the logical process involved in the act of counting, which is as follows. When we say one, two, three, etc., we are necessarily considering some one-one relation which holds between the numbers used in counting and the objects counted. What is meant by the one, two, three is that the objects indicated by these numbers are their correlates with respect to the relation which we have in mind. (This relation, by the way, is usually extremely complex, and is apt to involve a reference to our state of mind at the moment.) Thus we correlate a class of objects with a class of numbers; and the class of numbers consists of all the numbers from 1 up to some number n. A further process is required to show that this number of numbers is n, which is only true, as a matter of fact, when n is finite, or, in a certain wider sense, when n is a0 (the smallest of infinite numbers). Moreover the process of counting gives us no indication as to what the numbers are, as to why they form a series, or as to how it is to be proved (in the cases where it is true) that there are n numbers from 1 up to n. Hence counting is irrelevant in the foundations of Arithmetic; and with this conclusion, it may be dismissed until we come to order and ordinal numbers.(§ 129 ¶ 2)