At the end of the nineteenth century and, even more, in the early years of the twentieth century, the philosophy of geometry experienced unprecedented pressures and tensions. For the revolutionary new developments in the mathematical foundations of geometry and, even more, the application of many of these new mathematical ideas to nature in Einstein's theory of relativity seemed to suggest irresistibly that all earlier attempts to comprehend philosophically the relationship between geometry on the one hand and our experience of nature on the other were radically mistaken. In particular, the Kantian understanding of this relationship – according to which, geometry functions as an a priori “transcendental condition” of the possibility of our scientific experience of nature, and space is viewed correspondingly as a “pure form of our sensible intuition” – seemed to be wholly undermined by the new mathematical-physical developments. The question then – for philosophers, mathematicians, and physicists alike – was what new understanding of the relationship between geometry on the one hand and our experience of nature on the other was to be put in its place.

The variety of mutually incompatible answers that were given to this question is remarkable, in that precisely this variety reflects the true complexity – the manifold pressures and tensions – engendered by the radically new philosophical, mathematical, and physical situation. It is especially remarkable, in particular, how seldom a straightforwardly empiricist understanding of the relationship between geometry and experience – according to which geometry is an empirical theory like any other whose validity is straightforwardly verified or falsified by experience – was represented.