The Principles of Mathematics (1903)

§ 141

We must, then, admit infinite aggregates. It remains to ask a more difficult question, namely: Are we to admit infinite unities? This question may also be stated in the form: Are there any infinitely complex propositions? This question is one of great logical importance, and we shall require much care both in stating and in discussing it.(§ 141 ¶ 1)

The first point is to be clear on the meaning of an infinite unity. A unity will be infinite when the aggregate of all its constituents is infinite, but this scarcely constitutes the meaning of an infinite unity. In order to obtain the meaning, we must introduce the notion of a simple constituent. We may observe, to begin with, that a constituent of a constituent is a constituent of the unity, i.e. this form of the relation of part to whole, like the second, but unlike the first form, is transitive. A simple constituent may now be defined as a constituent which itself has no constituents. We may assume, in order to eliminate the question concerning aggregates, that no constituent of our unity is to be an aggregate, or, if there be a constituent which is an aggregate, then this constituent is to be taken as simple. (This view of an aggregate is rendered legitimate by the fact that an aggregate is a single term, and does not have that kind of complexity which belongs to propositions.) With this the definition of a simple constituent is completed.(§ 141 ¶ 2)

We may now define an infinite unity as follows: A unity is finite when, and only when, the aggregate of its simple constituents is finite. In all other cases a unity is said to be infinite. We have to inquire whether there are any such unities[101].(§ 141 ¶ 3)

If a unity is infinite, it is possible to find a constituent unity, which again contains a constituent unity, and so on without end. If there be any unities of this nature, two cases are primâ facie possible. (1) There may be simple constituents of our unity, but these must be infinite in number. (2) There may be no simple constituents at all, but all constituents, without exception, may be complex; or, to take a slightly more complicated case, it may happen that, although there are some simple constituents, yet these and the unities composed of them do not constitute all the constituents of the original unity. A unity of either of these two kinds will be called infinite. The two kinds, though distinct, may be considered together.(§ 141 ¶ 4)

An infinite unity will be an infinitely complex proposition: it will not be analyzable in any way into a finite number of constituents. It thus differs radically from assertions about infinite aggregates. For example, the proposition any number has a successor is composed of a finite number of constituents: the number of concepts entering into it can be enumerated, and in addition to these there is an infinite aggregate of terms denoted in the way indicated by any, which counts as one constituent. Indeed it may be said that the logical purpose which is served by the theory of denoting is, to enable propositions of finite complexity to deal with infinite classes of terms: this object is effected by all, any, and every, and if it were not effected, every general proposition about an infinite class would have to be infinitely complex. Now, for my part, I see no possible way of deciding whether propositions of infinite complexity are possible or not; but this at least is clear, that all the propositions known to us (and, it would seem, all propositions that we can know) are of finite complexity. It is only by obtaining such propositions about infinite classes that we are enabled to deal with infinity; and it is a remarkable and fortunate fact that this method is successful. Thus the question whether or not there are infinite unities must be left unresolved; the only thing we can say, on this subject, is that no such unities occur in any department of human knowledge, and therefore none such are relevant to the foundations of mathematics.(§ 141 ¶ 5)

§ 141 n. 1. In Leibniz's philosophy, all contingent things are infinite unities.