The Principles of Mathematics (1903)

§ 162

All the magnitudes dealt with hitherto have been, strictly speaking, indivisible. Thus the question arises: Are there any divisible magnitudes? Here I think a distinction must be made. A magnitude is essentially one, not many. Thus no magnitude is correctly expressed as a number of terms. But may not the quantity which has a magnitude be a sum of parts, and the magnitude a magnitude of divisibility? If so, every whole consisting of parts will be a single term possessed of the property of divisibility. The more parts it consists of, the greater is its divisibility. On this supposition, divisibility is a magnitude, of which we may have a greater or less degree; and the degree of divisibility corresponds exactly, in finite wholes, to the number of parts. But though the whole which has divisibility is of course divisible, yet its divisibility, which alone is strictly a magnitude, is not properly speaking divisible. The divisibility does not itself consist of parts, but only of the property of having parts. It is necessary, in order to obtain divsibility, to take the whole strictly as one, and to regard divisibility as its adjective. Thus although, in this case, we have numerical measurement, and all the mathematical consequences of division, yet, philosophically speaking, our magnitude is still indivisible.(§ 162 ¶ 1)

There are difficulties, however, in the way of admitting divisibility as a kind of magnitude. It seems to be not a property of the whole, but merely a relation to the parts. It is difficult to decide this point, but a good deal may be said, I think, in support of divisibility as a simple quality. The whole has a certain relation, which for convenience we may call that of inclusion, to all its parts. This relation is the same whether there by many parts or few; what distinguishes a whole of many parts is that it has many such relations of inclusion. But it seems reasonable to suppose that a whole of many parts differs from a whole of few parts in some intrinsic respect. In fact, wholes may be arranged in a series according as they have more or fewer parts, and the serial arrangement implies, as we have already seen, some series of properties differing more or less from each other, and agreeing when two wholes have the same finite number of parts, but distinct from number of parts in finite wholes. These properties can be none other than greater or less degrees of divisibility. Thus magnitude of divisibility would appear to be a simple property of a whole, distinct from the number of parts included in the whole, but correlated with it, provided this number be finite. If this view can be maintained, divisibility may be allowed to remain as a numerically measurable, but not divisible, class of magnitudes. In this class we should have to place lengths, areas, and volumes, but not distances. At a later stage, however, we shall find that the divisibility of infinite wholes, in the sense in which this is not measured by cardinal numbers, must be derived through relations in a way analogous to that in which distance is derived, and must be really a property of relations[113].(§ 162 ¶ 2)

Thus it would appear, in any case, that all magnitudes are indivisible. This is one common mark which they all possess, and so far as I know, it is the only one to be added to those enumerated in Chapter XIX. Concerning the range of quantity, there seems to be no further general proposition. Very many simple non-relational terms have magnitude, the principal exceptions being colours, points, instants and numbers.(§ 162 ¶ 3)