The Principles of Mathematics (1903)

§ 125

In the present chapter, we shall examine the question whether the above definition of cardinal numbers in any way presupposes some more fundamental sense of number. There are several ways in which this may be supposed to be the case. In the first place, the individuals which compose classes seem to be each in some sense one, and it might be thought that a one-one relation could not be defined without introducing the number 1. In the second place, it may very well be questioned whether a class which has only one term can be distinguished from that one term. And in the third place, it may be held that the notion of class presupposes number in a sense different from that above defined: it may be maintained that classes arise from the addition of individuals, as indicated by the wrod and, and that the logical addition of classes is subsequent to this addition of individuals. These questions demand a new inquiry into the meaning of one and of class, and here, I hope, we shall find ourselves aided by the theories set forth in Part I.(§ 125 ¶ 1)

As regards the fact that any individual or term is in some sense one, this is of course undeniable. But it does not follow that the notion of one is presupposed when individuals are spoken of: it may be, on the contrary, that the notion of term or individual is the fundamental one, from which that of one is derived. This view was adopted in Part I, and there seems no reason to reject it. And as for one-one relations, they are defined by means of identity, without any mention of one, as follows: R is a one-one relation if, when x and x′ have the relation R to y, and x has the relation R to y and y′, then x and x′ are identical, and so are y and y′. It is true that here x, y, x′, y′ are each one term, but this is not (it would seem) in any way presupposed in the definition. This disposes (pending a new inquiry into the nature of classes) of the first of the above objections.(§ 125 ¶ 2)

The next question is as to the distinction between a class containing only one member, and the one member which it contains. If we could identify a class with its defining predicate or class-concept, no difficulty would arise on this point. When a certain predicate attaches to one and only one term, it is plain that that term is not identical with the predicate in question. But if two predicates attach to precisely the same terms, we should say that, although the predicates are different, the classes which they define are identical, i.e. there is only one class which both define. If, for example, all featherless bipeds are men, and all men are featherless bipeds, the classes men and featherless bipeds are identical, though man differs from featherless biped. This shows that a class cannot be identified with its class-concept or defining predicate. There might seem to be nothing left except the actual terms, so that when there is only one term, that term would have to be identical with the class. Yet for many formal reasons this view cannot give the meaning of the symbols which stand for classes in symbolic logic. For example, consider the class of numbers which, added to 3, give 5. This is a class containing no terms except the number 2. But we can say that 2 is a member of this class, i.e. it has to the class that peculiar indefinable relation which terms have to the classes they belong to. This seems to indicate that the class is different from the one term. The point is a prominent one in Peano's Symbolic Logic, and is connected with his distinction between the relation of an individual to its class and the relation of a class to another in which it is contained. Thus the class of numbers which, added to 3, give 5, is contained in the class of numbers, but is not a number; whereas 2 is a number, but is not a class contained in the class of numbers. To identify the two relations which Peano distinguishes is to cause havoc in the theory of infinity, and to destroy the formal precision of many arguments and definitions. It seems, in fact, indubitable that Peano's distinction is just, and that some way must be found of discriminating a term from a class containing that term only.(§ 125 ¶ 3)