Note 8 — Chapter III · RIEMANN’S DISCOVERIES AND CONGRUENCE

Were it not for this restriction, comparisons of distance in space situated in different places could never be obtained by transporting rods from one place to another; for since the measurements yielded by our standard rods when they had reached their point of destination would depend essentially on the route they had followed, they could scarcely be called rigid. When the restriction is adhered to we obtain the most general type of Riemannian spaces or geometries exemplified by the three major types, the Euclidean type, the Riemannian type and the Lobatchewskian type. Weyl, however, dispenses with Riemann’s postulate and thereby obtains a more generalised type of space or geometry.

The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discussed a posteriori and not a priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.

So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.