Helmholtz's Naturalized Conception of Geometry and his Spatial Theory of Signs

Abstract

I analyze the two main theses of Helmholtz's “The Applicability of the Axioms to the Physical World,” in which he argued that the axioms of Euclidean geometry are not, as his neo-Kantian opponents had argued, binding on any experience of the external world. This required two argumentative steps: 1) a new account of the structure of our representations which was consistent both with the experience of our (for him) Euclidean world and with experience of a non-Euclidean one, and 2) a demonstration of why geometric propositions are essentially connected to material and temporal aspects of experience. The effect of Helmholtz's discussion is to throw into relief an intermediate category of metrological objects—objects which are required for the properly theoretical activity of doing physical science (in this sense, a priori requirements for doing science), all while being recognizably contingent aspects of experience.