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

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Abstract

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.

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Notes

  1. For sake of historical accuracy along the paper we will try to remain faithful to the original notations. used by the various authors considered.

  2. where \(\frac{E_{rk}}{E}\) are the inverse matrix of \(\omega_{ik}\)

  3. Einstein 1915d

  4. cf. Reichenbach’s letter to Schlick on December, 6 1926 mentioned in (Schlick 2006, vol. 6, 175)

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Acknowledgments

I would like to thank Don Howard for many insightful discussions and one of the anonymous referees for providing a detailed list of helpful suggestions.

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    Marco Giovanelli

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Giovanelli, M. The Forgotten Tradition: How the Logical Empiricists Missed the Philosophical Significance of the Work of Riemann, Christoffel and Ricci. Erkenn 78, 1219–1257 (2013). https://doi.org/10.1007/s10670-012-9407-2

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