The Forgotten Tradition: How the Logical Empiricists Missed the Philosophical Significance of the Work of Riemann, Christoffel and Ricci

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Erkenntnis, ISSN: 0165-0106, Vol: 78, Issue: 6, Page: 1219-1257

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Giovanelli, Marco
Springer Nature; Springer (Springer Science+Business Media B.V.)
Arts and Humanities; Mathematics
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This 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 instead 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 interpret general relativity as the last link of the chain Riemann-Helmholtz-Poincaré-Einstein, logical empiricists were led to argue that Einstein's theory of gravitation mainly raised 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 Riemann-Christoffel-Ricci-Einstein line of evolution shows on the contrary that the main philosophical issue raised by Einstein's theory was instead 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. © 2012 Springer Science+Business Media Dordrecht.