The Principles of Mathematics (1903)

§ 131

Addition, it should be carefully observed, is not primarily a method of forming numbers, but of forming classes or collections. If we add B to A, we do not obtain the number 2, but we obtain A and B, which is a collection of two terms, or a couple. And a couple is defined as follows: u is a couple if u has terms, and if, if x be a term of u, there is a term different from x, but if x, y be different terms of u, and z differs from x and from y, then every class to which z belongs differs from u. In this definition, only diversity occurs, together with the notion of a class having terms. It might no doubt be objected that we have to take just two terms x, y in the above definition: but as a matter of fact any finite number can be defined by induction without introducing more than one term. For, if n has been defined, a class u has n+1 terms, if a term x be a term of u the number of terms of u which differ from x is n. And the notion of the arithmetical sum n+1 is obtained from that of the logical sum of a class of n terms and a class of one term. When we say 1+1=2, it is not possible that we should mean 1 and 1, since there is only one 1: if we take 1 as an individual, 1 and 1 is nonsense, while if we take it as a class, the rule of Symbolic Logic applies, according to which 1 and 1 is 1. Thus in the corresponding logical proposition, we have on the left-hand side terms of which 1 can be asserted, and on the right-hand side we have a couple. That is, 1+1=2 means one term and one term are two terms, or, stating the proposition in terms of variables, if u has one term and v has one term, and u differs from v, their logical sum has two terms. It is to be observed that on the left-hand side we have a numerical conjunction of propositions, while on the right-hand side we have a proposition concerning a numerical conjunction of terms. But the true premiss, in the above proposition, is not the conjunction of the three propositions, but their logical product. This point, however, has little importance in the present connection.(§ 131 ¶ 1)