Collision of Traditions. The Emergence of Logical Empiricism Between\ud the Riemannian and Helmholtzian Traditions

This paper attempts to explain the emergence of the logical empiricist\ud philosophy of space and time as a collision of mathematical traditions.\ud The historical development of the ``Riemannian'' and ``Helmholtzian''\ud traditions in 19th century mathematics is investigated. Whereas Helmholtz's\ud insistence on rigid bodies in geometry was developed group theoretically\ud by Lie and philosophically by Poincaré, Riemann's Habilitationsvotrag\ud triggered Christoffel's and Lipschitz's work on quadratic differential\ud forms, paving the way to Ricci's absolute differential calculus.\ud The transition from special to general relativity is briefly sketched\ud as a process of escaping from the Helmholtzian tradition and entering\ud the Riemannian one. Early logical empiricist conventionalism,\ud it is argued, emerges as the failed attempt to interpret Einstein's\ud reflections on rods and clocks in general relativity through the conceptual\ud resources of the Helmholtzian tradition. Einstein's epistemology\ud of geometry should, in spite of his rhetorical appeal to Helmholtz and\ud Poincaré, be understood in the wake the Riemannian tradition\ud and of its aftermath in the work of Levi-Civita, Weyl, Eddington, and\ud others.\u