Note 7 — Chapter III · RIEMANN’S DISCOVERIES AND CONGRUENCE

This difficulty of counting points might be obviated to a certain degree were space to be considered discrete or atomic; for in that case we might count the atoms of space separating points and thereby establish absolute comparisons between distances. But here again the procedure would be artificial, for it would be nullified unless we were to assume that the spatial voids separating the successive atoms were always the same in magnitude; furthermore (as Dr. Silberstein points out in his book, “The Theory of Relativity”), we should be in a quandary to know how a succession of atoms would have to be defined, since this definition would depend on a definition of order. At any rate, we need not concern ourselves with atomic or discrete manifolds, for Riemann assumed that mathematical space was a continuous manifold. In view of the quantum phenomena we may eventually be led to modify these views and to attribute a discrete nature to space, but this is a vague possibility which there is no advantage in discussing in the present state of our knowledge.