The Principles of Mathematics (1903)

§ 69

The best formal treatment of classes in existence is that of Peano[55]. But in this treatment a number of distinctions of great philosophical importance are overlooked. Peano, not I think quite consciously, identifies the class with the class-concept; thus the relation of an individual to its class is, for him, expressed by is a. For him, 2 is a number is a proposition in which a term is said to belong to the class number. Nevertheless, he identifies the equality of classes, which consists in their having the smae terms, with identity--a proceeding which is quite illegitimate when the class is regarded as the class-concept. In order to perceive that man and featherless biped are not identical it is quite unnecessary to take a hen and deprive the poor bird of its feathers. Or, to take a less complex instance, it is plain that even prime is not identical with integer next after 1. Thus when we identify the class with the class-concept, we must admit that two classes may be equal without being identical. Nevertheless, it is plain that they have the same terms. Thus there is some object which is positively identical when two class-concepts are equal; and this object, it would seem, is more properly called the class. Neglecting the plucked hen, the class of featherless bipeds, every one would say, is the same as the class of men; the class of even primes is the same as the class of integers next after 1. Thus we must not identify the class with the class-concept, or regard Socrates is a man as expressing the relation of an individual t oa class of which it is a member. This has two consequences (to be established presently) which prevent the philosophical acceptance of Peano's formalism. The first consequence is, that there is no such thing as the null-class, though there are null cass-concepts. The second is, that a class having only one term is to be identified, contrary to Peano's usage, with that one term. I should not propose, however, to alter his practice or his notation in consequence of either of these points; rather I should regard them as proofs that Symbolic Logic ought to concern itself, as far as notation goes, with class-concepts rather than classes.(§ 69 ¶ 1)

§ 69 n. 1. Neglecting Frege, who is discussed in the Appendix.