Hilbert’s “World Equations” and His Vision of a Unified Science

Abstract

In summer 1923, a year after his lectures on the 'New Foundation of Mathematics' and half a year before the republication of his two notes on the 'Foundations of Physics,' Hilbert delivered a trilogy of lectures in Hamburg. In these lectures, Hilbert expounds in an unusually explicit manner his epistemological perspective on science as a subdiscipline of an all embracing science of mathematics. The starting point of Hilbert's considerations is the claim that the class of gravitational and electromagnetic field equations implied by his original variational formulation of 1915 provides valid candidate 'world equations,' even in view of attempts at unified field theoriesá la Weyl and Eddington based on the concept of the affine connection. We give a discussion of Hilbert's lectures and, in particular, examine his claim that Einstein in his 1923 papers on affine unified field theory only arrived at Hilbert's original 1915 theory. We also briefly comment on Hilbert's philosophical viewpoints expressed in these lectures. IntroductionIn the history of unified field theory, many contributors may be identified [Goldstein and Ritter 2003, Goenner 2004], among them certainly, and perhaps foremost, Einstein. Hilbert's place in the history of the unified field * To appear in a forthcoming volume of Einstein Studies, eds.

Einstein’s Unified Field Theory Program

“…But in the course of trying to touch base with familiar general relativistic gravitation theory and Maxwellian electromagnetism, he also performed some transformation of variables and ended up with a variational formulation that, in fact, was almost equivalent to the variational formulation of the general theory of relativity with a Maxwellian field provided by Hilbert in 1915. This eqivalence was highlighted by Hilbert in lectures held in Hamburg and Zurich in the summer and fall of 1923, and prompted his republication of a merged version of his two communications on the "Foundations of Physics" in the Mathematische Annalen in 1924 (Hilbert 1924, Majer andSauer 2005).…”

On the History of Unified Field Theories

“…In the third paper as well, Einstein’s desire to create a unified field theory satisfying all his criteria still was not fulfilled: His equations, again, did not give a singularity-free electron. In a paper on Hilbert’s vision of a unified science, Sauer and Majer recently have found out from lectures of Hilbert given in Hamburg and Zürich in 1923, that Hilbert considered Einstein’s work in affine theory a return to his own results of 1915 by “[…] a colossal detour via Levi-Civita, Weyl, Schouten, Eddington […]” [215]. It seems that, in this evaluation, Hilbert was influenced by Einstein’s proportionality between the 4-potential and the electrical current which Hilbert had assumed as early as in 1915 [161].…”

Tuning up mind's pattern to nature's own idea: Eddington's early twenties case for variational derivatives

“…The way the interactions and exchanges that occurred between them in this short period of time may have affected their respective approaches and somewhat deflected their ultimate contributions has been extensively studied in the recent years (cf. [36], [33], [7], [37], [27]). The timeline should be kept in mind here if a clear understanding of both positions is to be gained.…”