The Principles of Mathematics (1903)

§ 140

In the present chapter the special difficulties of infinity are not to be considered: all these are postponed to Part V. My object now is to consider two questions: (1) Are there any infinite wholes? (2) If so, must an infinite whole which contains parts in the second of our three senses be an aggregate of parts in the first sense? In order to avoid the reference to the first, second, and third senses, I propose henceforward to use the following phraseology: A part in the first sense is to be called a term of the whole[99]; a part in the second sense will be called a part simply; and a part in the third sense will be called a constituent of the whole. Thus terms and parts belong to aggregates, while constituents belong to unities. The consideration of aggregates and unities, where infinity is concerned, must be separately conducted. I shall begin with aggregates.(§ 140 ¶ 1)

An infinite aggregate is an aggregate corresponding to an infinite class, aggregate which has an infinite number of terms. Such aggregates are defined by the fact that they contain parts which have as many terms as themselves. Our first question is: Are there any such aggregates?(§ 140 ¶ 2)

Infinite aggregates are often denied. Even Leibniz, favourable as he was to the actual infinite, maintained that, where infinite classes are concerned, it is possible to make valid statements about any term of the class, but not about all the terms, nor yet about the whole which (as he would say) they do not compose[100]. Kant, again, has been much criticised for maintaining that space is an infinite given whole. Many maintain that every aggregate must have a finite number of terms, and that where this condition is not fulfilled there is no true whole. But I do not believe that this view can be successfully defended. Among those who deny that space is a given whole, not a few would admit that what they are pleased to call a finite space may be a given whole, for instance, the space in a room, a box, a bag, or a book-case. But such a space is only finite in a psychological sense, i.e. in the sense that we can take it in at a glance: it is not finite in the sense that it is an aggregate of a finite number of terms, nor yet a unity of a finite number of constituents. Thus to admit that such a space can be a whole is to admit that there are wholes which are not finite. (This does not follow, it should be observed, from the admission of material objects apparently occupying finite spaces, for it is always possible to hold that such objects, though apparently continuous, consist really of a large but finite number of material points.) With respect to time, the same argument holds: to say, for example, that a certain length of time elapses between sunrise and sunset, is to admit an infinite whole, or at least a whole which is not finite. It is customary with philosophers to deny the reality of space and time, and to deny also that, if they were real, they would be aggregates. I shall endeavour to show, in Part VI, that these denials are supported by a faulty logic, and by the now resolved difficulties of infinity. Since science and common sense join in the opposite view, it will therefore be accepted; and thus, since no argument à priori can now be adduced against infinite aggregates, we derive from space and time an argument in their favour.(§ 140 ¶ 3)

Again, the natural numbers, or the fractions between 0 and 1, or the sum-total of all colours, are infinite, and seem to be true aggregates: the position that, although true propositions can be made about any number, yet there are no true propositions about all numbers, could be supported formerly, as Leibniz supported it, by the supposed contradictions of infinity, but has become, since Cantor's solution of these contradictions, a wholly unnecessary paradox. And where a collection can be defined by a non-quadratic propositional function, this must be held, I think, to imply that there is a genuine aggregate composed of the terms of the collection. It may be observed also that, if there were no infinite wholes, the word Universe would be wholly destitute of meaning.(§ 140 ¶ 4)

§ 140 n. 1. A part in this sense will also be sometimes called a simple or indivisible part.

§ 140 n. 2. Cf. Phil. Werke, ed. Gerhardt, II, p. 315; also I, p. 338, V, pp. 144–5.