Repository | Book | Chapter

The development of non-Euclidean geometry in the nineteenth century led mathematicians, scientists, and philosophers to reconsider the foundations of geometry. One of the issues at stake was to redefine the notion of geometrical axiom and to establish criteria of choice among different axiomatic systems in case of equivalent geometries. The possibility of considering a variety of hypotheses concerning physical space appeared to contradict Kant's conception of geometrical axioms as a priori synthetic judgments. Therefore, Riemann called geometrical axioms hypotheses, and maintained that the geometry of physical space is a matter for empirical investigation. In order to support this view, Helmholtz pointed out the empirical origin of geometrical axioms. At the same time, he foreshadowed a conventionalist conception of geometrical axioms as definitions that can be abstracted from our experiences with solid bodies and their free mobility.This chapter is devoted to Helmholtz's objections to Kant and to two different strategies for defending the aprioricity of geometrical axioms from a neo-Kantian perspective. Cohen was able to readapt his notion of the a priori to the case of geometrical axioms because he fundamentally relativized Kant's notion. For the same reason, Cohen agreed with Helmholtz that the principles of geometry should be determined in connection with those of mechanics. Another line of argument goes back to Alois Riehl. He did not deny the empirical origin of three-dimensionality. However, he maintained that the remaining, formal properties of space suffice to assume that the geometry of space must be Euclidean. My suggestion is that Cohen developed a more plausible view, because he was not committed to any spatial structure independently of empirical science.

Full citation: