Carnap and Weyl on the Foundations of Geometry and Relativity Theory

Abstract

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 correspondingly viewed 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.

An earlier version of this paper was presented at a workshop on Semantical Aspects of Space-Time Theories at the Zentrum für interdisziplinäre Forschung at the Universität Bielefeld in May 1994. I am indebted to the participants for helpful comments. I am also, and especially, indebted to comments from and discussions with Thomas Ryckman. All translations from the German are my own.