Is There Any Room for Spatial Intuition in Riemann's Philosophy of Geometry?

Abstract

In his famous Habilitationsvortrag Riemann underlines important points on the very nature of space with respect to philosophy, mathematics, and physics. Although Riemann's greatness in mathematics has been well acknowledged, and the importance and implications of his geometry studied widely by philosophers, the same does not seem to be true of his philosophy of geometry. In part, this paper is motivated by this very fact. In his Habilitationsvortrag Riemann sets aside the usual approaches that had been taken until then, and instead tries out new ideas and approaches. Riemann thought that while Euclidean geometry made an interesting proposal for the construction of a theory of space, there was in fact no a priori connection between the concept of space and the axioms of Euclidean geometry. He argued, then, that the fundamental concepts central to Euclidean geometry do not have to be part of every system of geometry imaginable. That is, the fundamental concepts of Euclidean geometry should not be thought of as necessary for the construction of all possible systems of geometry. Riemann wanted to depict nature from the perspective of its inner structures and one aspect of this endeavor entailed questioning the nature of space and geometry from heterogeneous points.