The Principles of Mathematics (1903)

§ 136

The difference between the kinds of wholes is important, and illustrates a fundamental point in Logic. I shall therefore repeat it in other words. Any collection whatever, if defined by a non-quadratic propositional function, though as such it is many, yet composes a whole, whose parts are the terms of the collection or any whole composed of some of the terms of the collection. It is highly important to realize the difference between a whole and all its parts, even in this case where the difference is a minimum. The word collection, being singular, applies more strictly to the whole than to all the parts; but convenience of expression has led me to neglect grammar, and speak of all the terms as the collection. The whole formed of the terms of the collection I call an aggregate. Such a whole is completely specified when all its simple constituents are specified; its parts have no direct connection inter se, but only the indirect connection involved in being parts of one and the same whole. But other wholes occur, which contain relations or what may be called predicates, not occurring simply as terms in a collection, but as relating or qualifying. Such wholes are always propositions. These are not completely specified when their parts are all known. Take, as a simple instance, the proposition A differs from B, where A and B are simple terms. The simple parts of this whole are A and B and difference; but the enumeration of these does not specify the whole, since there are two other wholes composed of the same parts, namely the aggregate formed of A and B and difference, and the proposition B differs from A. In the former case, although the whole was different from all its parts, yet it was completely specified by specifying its parts; but in the present case, not only is the whole different, but it is not even specified by specifying its parts. We cannot explain this fact by saying that the parts stand in certain relations which are omitted in the analysis; for in the above case of A differs from B, the relation was included in the analysis. The fact seems to be that a relation is one thing when it relates, and another when it is merely enumerated as a term in a collection. There are certain fundamental difficulties in this view, which however I leave aside as irrelevant to our present purpose[95].(§ 136 ¶ 1)

Similar remarks apply to A is, which is a whole composed of A and Being, but is different from the whole formed of the collection A and Being. A is one raises the same point, and so does A and B are two. Indeed all propositions raise this point, and we may distinguish them among complex terms by the fact that they raise it.(§ 136 ¶ 2)

Thus we see that there are two very different classes of wholes, of which the first will be called aggregates, while the second will be called unities. (Unit is a word having a quite different application, since whatever is a class which is not null, and is such that, if x and y be members of it, x and y are identical, is a unit.) Each class of wholes consists of terms not simply equivalent to all their parts; but in the case of unities, the whole is not even specified by its parts. For example, the parts A, greater than, B, may compose simply an aggregate, or either of the propositions A is greater than B, B is greater than A. Unities thus involve problems from which aggregates are free. As aggregates are more specially relevant to mathematics than unities, I shall in future generally confine myself to the former.(§ 136 ¶ 3)