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Riemann’s and Helmholtz-Lie’s problems of space from Weyl’s relativistic perspective. (English) Zbl 1382.81007

Summary: I reconstruct Riemann’s and Helmholtz-Lie’s problems of space, from some perspectives that allow for a fruitful comparison with Weyl. In Part II. of his inaugural lecture, Riemann justifies that the infinitesimal metric is the square root of a quadratic form. Thanks to Finsler geometry, I clarify both the implicit and explicit hypotheses used for this justification. I explain that Riemann-Finsler’s kind of method is also appropriate to deal with indefinite metrics. Nevertheless, Weyl shares with Helmholtz a strong commitment to the idea that the notion of group should be at the center of the foundations of geometry. Riemann missed this point, and that is why, according to Weyl, he dealt with the problem of space in a “too formal” way. As a consequence, to solve the problem of space, Weyl abandoned Riemann-Finsler’s methods for group-theoretical ones. However, from a philosophical point of view, I show that Weyl and Helmholtz are in strong opposition. The meditation on Riemann’s inaugural lecture, and its clear methodological separation between the infinitesimal and the finite parts of the problem of space, must have been crucial for Weyl, while searching for strong epistemological foundations for the group-theoretical methods, avoiding Helmholtz’s unjustified transition from the finite to the infinitesimal.

MSC:

81P05 General and philosophical questions in quantum theory
00A79 Physics
00A30 Philosophy of mathematics
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
20G45 Applications of linear algebraic groups to the sciences
22E70 Applications of Lie groups to the sciences; explicit representations
83A05 Special relativity
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)

Biographic References:

Weyl, Hermann
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