Geometry and Experience: Epistemological Aspects of Spatial Intuition in Mathematics after Lobachevsky and Riemann

This work is devoted to the problem of spatial intuition in the geometry of XIX - early XX centuries, its connection with the empirical applicability of mathematics and epistemological questions about the status of mathematical concepts. The historical line of development of spatial intuition in this paper is analyzed on a concrete plot from the history of geometry and physics. This story opens with the formulation of non-Euclidean geometries by Riemann and Lobachevsky and ends with discussions about the theory of relativity, related to its geometric basis and Einstein's views on the status of geometry. One of the central figures here is Helmholtz, around whose works European mathematicians have initiated extensive discussions about the possibility of physiological or physical justification of geometry, about the relationship between geometry and perception, geometry and the fundamental concepts of solid state and motion in physics. This subplot, covered in the third chapter, has an unquestionable connection with the ideas of Lobachevsky and Riemann about the empirical nature of geometric concepts, which are outlined in the second chapter. The same themes are then raised again, inspired by the emergence of the Theory of Relativity, whose author, Einstein, was, as shown in chapter four, a diligent Helmholtz reader and developed some of his ideas (not without the help of Moritz Schlick). As a separate philosophical subtopic, the same chapter analyzed the development of Ernst Cassirer's views on the relationship between geometry and physics in the context of his program of rehabilitation of transcendental philosophy and the difficulty of reconciling its principles with the latest achievements in mathematics and physics.