The Principles of Mathematics (1903)

§ 142

I come now to our second question: Must an infinite whole which contains parts be an aggregate of terms? It is often held, for example, that spaces have parts, and can be divided ad lib., but that they have no simple parts, i.e. they are not aggregates of points. The same view is put forward as regards periods of time. Now it is plain that, if our definition of a part by means of terms (i.e. of the second sense of part by means of the first) was correct, the present problem can never arise, since parts only belong to aggregates. But it may be urged that the notion of part ought to be taken as an indefinable, and that therefore it may apply to other wholes than aggregates. This will require that we should add to aggregates and unities a new kind of whole, corresponding to the second sense of part. This will be a whole which has parts in the second sense, but is not an aggregate or a unity. Such a whole seems to be what many philosophers are fond of calling a continuum, and space and time are often held to afford instances of such a whole.(§ 142 ¶ 1)

Now it may be admitted that, among infinite wholes, we find a distinction which seems relevant, but which, I believe, is in reality merely psychological. In some cases, we feel no doubt as to the terms, but great doubt as to the whole, while in others, the whole seems obvious, but the terms seem a precarious inference. The ratios between 0 and 1, for instance, are certainly indivisible entities; but the whole aggregate of ratios between 0 and 1 seems to be of the nature of a construction or inference. On the other hand, sensible spaces and times seem to be obvious wholes; but the inference to indivisible points and instants is so obscure as to be often regarded as illegitimate. This distinction seems, however, to have no logical basis, but to be wholly dependent on the nature of our senses. A slight familiarity with co-ordinate geometry suffices to make a finite length seem strictly analogous to the stretch of fractions between 0 and 1. It must be admitted nevertheless, that in cases where, as with the fractions, the indivisible parts are evident on inspection, the problem with which we are concerned does not arise. But to infer that all infinite wholes have indivisible parts merely because this is known to be the case with some of them, would certainly be rash. The general problem remains, therefore, namely: Given an infinite whole, is there a universal reason for supposing that it contains indivisible parts?(§ 142 ¶ 2)