The Principles of Mathematics (1903)

§ 137

It is important to realize that a whole is a new single term, distinct from each of its parts and from all of them: it is one, not many[96], and is related to the parts, but has a being distinct from theirs. The reader may perhaps be inclined to doubt whether there is any need of wholes other than unities; but the following reasons seem to make aggregates logically unavoidable. (1) We speak of one collection, one manifold, etc., and it would seem that in all these cases there really is something that is a single term. (2) The theory of fractions, as we shall shortly see, appears to depend partly upon aggregates. (3) We shall find it necessary, in the theory of extensive quantity, to assume that aggregates, even when they are infinite, have what may be called magnitude of divisibility, and that two infinite aggregates may have the same number of terms without having the same magnitude of divisibility: this theory, we shall find, is indispensable in metrical geometry. For these reasons, it would seem, the aggregate must be admitted as an entity distinct from all its constituents, and having to each of them a certain ultimate and indefinable relation.(§ 137 ¶ 1)

§ 137 n. 1. I.e. it is of the same logical type as its simple parts.