Note 147 — Chapter IV · SPACE AND TIME

It must be carefully noted that distance as thus defined is not properly a quantitative relation, and involves no notion of magnitude, but only of relative place in a series. It should also be observed that in assuming the existence of such a unique relation between every pair of points, it is tacitly taken for granted that the number of dimensions of the spatial order is finite. In a space of an infinite number of dimensions, such unique relation would be impossible. (See Russell, Foundations of Geometry, p. 161 ff.) Our justification for making this assumption, as also for taking time to be of one dimension only, seems to be that it is indispensable for all those practical purposes which depend on our ability to create a science of Geometry, and that we have no positive ground for assuming the opposite. Thus ultimately the assumption appears to be of the nature of a postulate.