The Forgotten Tradition. How the Logical Empiricists missed the Philosophical Significance of the work of Riemann, Christoffel and Ricci

Abstract. The paper attempts to show how the Logical Empiricists’ interpretation of the relation between geometry and reality emerges from a “collision” of mathematical traditions. Considering Riemann’s work as the initiator of a 19th century geometrical tradition, whose main protagonists were Helmholtz and Poincaré, the Logical Empiricists neglected the fact that Riemann’s revolutionary insight flourished rather in a non-geometrical tradition dominated by the works of Christoffel and Ricci-Curbastro roughly in the same years. I will argue that in the attempt to draw the line Riemann-Helmholtz-Poincaré-Einstein Logical Empiricists were led to argue that General Relativity raised mainly a problem of mathematical under-determination, i.e. the discovery that there are physical differences that cannot be expressed in the relevant mathematical structure of the theory. However, a historical reconstruction of the alternative line of development Riemann-Chritoffel-Ricci-Einstein shows on the contrary that the main philosophical issue raised by Einstein’s theory was rather that of mathematical over-determination, i.e. the recognition of the presence of redundant mathematical differences that do not have any correspondence in physical reality