In the context of 19th-century physics, geometry was quite naturally interpreted as the science of space, space itself being conceived as a self-subsisting entity, no less real than the spatial things moving across it. Paradoxically, however, the propositions of this science did not seem to be liable to empirical corroboration or refutation. Since the times of the Greeks, no geometer had ever
... [Show full abstract] thought of subjecting his conclusions to the verdict of experiment. And philosophers, from Plato to Kant, viewed geometry as the one unquestionable instance of non-trivial a priori knowledge, i.e. knowledge relevant to things that exist, yet not dependent on our experience of them. Even such an extreme empiricist as Hume regarded geometry as a non-empirical science, concerned not with matters of fact, but with relations of ideas. The discovery of non-Euclidean geometries shattered the unanimity of philosophers on this point. The existence of a variety of equally consistent systems of geometry was immediately thought to lend support to a different view of this science. The established Euclidean system could now be regarded as a physical theory, highly corroborated by experience, but liable to be eventually proved inexact. We have seen that Gauss and Lobachevsky, Riemann and Helmholtz took this empiricist view of geometry.
