Hilbert's ‘Foundations of Physics’: Gravitation and electromagnetism within the axiomatic method

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Abstract

In November and December 1915, Hilbert presented two communications to the Göttingen Academy of Sciences under the common title ‘The Foundations of Physics’. Versions of each eventually appeared in the Nachrichten of the Academy. Hilbert's first communication has received significant reconsideration in recent years, following the discovery of printer's proofs of this paper, dated 6 December 1915. The focus has been primarily on the ‘priority dispute’ over the Einstein field equations. Our contention, in contrast, is that the discovery of the December proofs makes it possible to see the thematic linkage between the material that Hilbert cut from the published version of the first communication and the content of the second, as published in 1917. The latter has been largely either disregarded or misinterpreted, and our aim is to show that (a) Hilbert's two communications should be regarded as part of a wider research program within the overarching framework of ‘the axiomatic method’ (as Hilbert expressly stated was the case), and (b) the second communication is a fine and coherent piece of work within this framework, whose principal aim is to address an apparent tension between general invariance and causality (in the precise sense of Cauchy determination), pinpointed in Theorem I of the first communication. This is not the same problem as that found in Einstein's ‘hole argument’—something that, we argue, never confused Hilbert.

Introduction

In November and December 1915, Hilbert gave two presentations to the Royal Göttingen Academy of Sciences under the common title ‘The Foundations of Physics’. Distinguished as ‘First Communication’ (Hilbert, 1915b) and ‘Second Communication’ (Hilbert, 1917), two papers (or ‘notes’, as they are widely known) eventually appeared in the Nachrichten of the Academy.1 The First Communication, which quickly entered the canon of classical general relativity, has recently become the object of renewed scholarly scrutiny since the discovery of a set of printer's proofs (Hilbert, 1915a) dated 6 December 1915 (henceforth the ‘Proofs’).2 With the exception of Renn and Stachel (1999), the Second Communication has not been given the same detailed reconsideration. However, the analysis of the Second Communication by Renn and Stachel seriously misrepresents its aims, content, and significance, and also its links to the First Communication. Our aim in this paper is to show that Hilbert's Second Communication is a natural continuation of his First Communication, that it contains important and interesting further developments of that project, and above all that it sheds needed illumination on Hilbert's assessment of the epistemological novelty posed by a generally covariant physics.

Hilbert's notes on ‘Foundations of Physics’ traditionally have been assessed solely in terms of the contributions they made to general relativity, as that theory is known in its completed form.3 From this vantage point, they present a mixed record of achievement, ranging from genuine insight (the Riemann scalar as the suitable invariant for the gravitational action) through incomprehension (Hilbert's interpretation of electromagnetism as a consequence of gravitation) to abject failure (attachment to the untenable electromagnetic theory of matter of Gustav Mie). The usual implication is that Hilbert's principal intent in November 1915 was to arrive at a theory of gravitation based on the principle of general covariance in one blinding flash, masterfully wielding an arsenal of axiomatized advanced mathematics. Thus arose ‘the legend of a royal road to general relativity’ (Renn & Stachel, 1999, p. 1) through the axiomatic method, whilst Hilbert's reputed remark that ‘physics is much too difficult for physicists’ has been widely understood to epitomize a haughty mathematical arrogance (Reid, 1970, p. 127). Correspondingly, some historians of general relativity have concluded that Hilbert ‘attached a kind of metaphysical significance to variational methods’ (Rowe, 1999, p. 201), while others have regarded Hilbert's approach as evincing an optimistic demonstration of the Göttingen-based ideology of a ‘pre-established harmony’ between mathematics and physics (Pyenson, 1985).4 Finally, some have relished pointing out where Hilbert's elaborate mathematical constructions were either inadequate to the complexities of the initial value problem in general relativity5 or simply led to hopelessly failed physics (Stachel, 1992; Renn & Stachel, 1999, pp. 77, 81–83).

Our contention is that viewing Hilbert's notes solely in terms of contributions made to general relativity as that theory is canonically understood, radically occludes internal motivations, which are largely logical and epistemological, and so casts them in a misleading light. In so doing, the explicitly stated epistemological intent of the ‘axiomatic method’ is willfully ignored, as are Hilbert's own express assertions regarding his construction as a triumph of that method. Although understandable in terms of the intellectual small change of ‘textbook’ histories, such accounts overlook or downplay fundamental philosophical and methodological differences with Einstein, in emphasis as well as in detail, concerning the significance of general covariance, arguably impeding clarification on that vexed issue for decades. But set within the logical and epistemological context of the ‘axiomatic method’, Hilbert's two notes may be seen to have the common goal of pinpointing, and then charting a path toward resolution of, the tension between causality and general covariance that, in the infamous ‘hole argument’, had stymied Einstein from 1913 to the autumn of 1915.6 Unlike Einstein's largely informal and heuristic extraction from the clutches of the ‘hole argument’, Hilbert stated the difficulty in a mathematically precise manner as an ill-posed Cauchy problem in the theory of partial differential equations, and then indicated how it can be resolved. As we will show, material cut from the proofs establishes this essential thematic linkage between the two notes and redeems Hilbert's claim that tension between causality and general covariance, precisely formulated in Theorem I of the First Communication, was the ‘point of departure’ for his axiomatic investigation.

Einstein and Hilbert were engaged in qualitatively different enterprises that only partially overlapped. In contrast to Einstein, Hilbert's goals were at least as much logical and epistemological, according to the character of the axiomatic method, as they were physical. We concur with the judgment of Felix Klein, who wrote, in 1921, that ‘there can be no talk of a question of priority, since both authors pursued entirely different trains of thought (and to be sure, to such an extent that the compatibility of the results did not at once seem assured)’.7 For Hilbert, the principal outcomes arrived at by the axiomatic method concern his revisiting the principle of causality, and his revisions of Kantian epistemology, in the light of generally covariant physics.

The structure of our paper is as follows.

In Section 2 we present what we call ‘the essential context’: Hilbert's axiomatic method and its presupposition of central tenets of Kantian epistemology.

In Section 3, we briefly review the published version of Hilbert's First Communication, proceeding in Section 4 to emphasize differences in content between the December Proofs and the published version. Since this subject has been extensively treated in Sauer (1999), in Renn and Stachel (1999), and more recently in Corry (2004), our treatment will highlight only the central features, omitting many details that can be found in these sources. We shall see that a passage cut from the proofs elucidates the problem pinpointed by Theorem I, and this, we claim, provides essential thematic linkage to Hilbert's Second Communication (see Section 6). Section 5 returns to the topic of the axiomatic method, and examines the aims and achievements of this method as it appears in the First Communication.

Section 6 concerns Hilbert's Second Communication. Following a brief introduction, we review the secondary literature, and then turn our attention to an exposition of the content of the Second Communication. We see how Hilbert now sought to resolve the challenge posed by Theorem I—the tension between general covariance and causality. We show that Hilbert's employment of the axiomatic method identified an epistemological novelty emerging in generally covariant physics regarding the constitution of physical objectivity as this is understood in a broadly transcendental idealist sense, most prominently displayed in his subsequent remarks regarding the axiom of general invariance. In further addressing the related matter of the vexing problem of causality in the new physics of general covariance, deemed a ‘pseudo-geometry’ and not, as before, as field physics set within a ‘background geometry’, Hilbert sought to remove all global ‘pseudo-Euclidean’ presuppositions, equivalent to ‘action-at-a-distance’.

Armed with our new understanding of Hilbert's ‘problem of causality’, Section 7 explains why this is not the same problem as Einstein faced in his ‘hole argument’.

In Section 8, we look beyond Hilbert's 1915 and 1917 papers to consider his further reflections on the epistemological significance of the respective principles of general covariance and causality and the ground of Hilbert's subordination of the latter to the former. We explain in detail the revisions of Kant that Hilbert believed were required in the face of the new generally covariant physics.

Section snippets

The essential context: Hilbert's axiomatic method and Kantian epistemology

There are two pieces of context that we believe are crucial to correctly understanding Hilbert's treatment of generally covariant physics: his axiomatic method, and his appeal to Kantian epistemology. These themes infuse our discussion throughout.

Hilbert's First Communication opens with a declaration that his investigation of the foundations of physics is undertaken ‘in the sense of the axiomatic method’ (‘im Sinne der axiomatischen Methode’), and it concludes with the striking claim that the

Hilbert's First Communication on ‘The Foundations of Physics’ (published version)

According to the annotation on the published version of the paper, Hilbert's First Communication was presented at the 20 November 1915 session of the Royal Göttingen Academy of Sciences. Traditionally, the date of submission was the only date appearing on publications in the Nachrichten of the Academy (Rowe, 2001, p. 418). However, with the discovery of the December Proofs in 1993 it was learned that the version submitted on 20 November differs considerably from that appearing in the published

Comparison of the Proofs with the published version of the First Communication

Felix Klein is reported to have commented on the ‘completely disordered’ character of Hilbert's First Communication, remarking that it was ‘evidently a product of great exertion and excitement’).34 We do not know whether Klein was referring to the Proofs or the published version, or perhaps to both, but there is evidence that Klein and Einstein each encountered

The First Communication and the axiomatic method

We recall that the task of the axiomatization of physics was the sixth in the famous list of 23 mathematical problems Hilbert posed at the International Congress of Mathematicians in Paris in 1900.

The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part …. If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number

Hilbert's Second Communication on ‘The Foundation of Physics’

On 4 December 1915, ‘The Foundation of Physics (Second Communication)’ was presented to the Göttingen Academy. A further presentation under that rubric was delivered on 26 February 1916. However, both of these were withdrawn before publication, and no version of either has apparently survived (Sauer, 1999, p. 557 and n. 120, p. 560 and n. 129; 2001, p. 3). From Hilbert's correspondence, it may be conjectured that one principal topic concerned a much anticipated derivation of ‘the electron’ from

Hilbert and the ‘hole argument’

The tension between general covariance and causality is just the problem encountered by Einstein in the ‘hole argument’. But in Hilbert's hands, is there any reason to think that, as with Einstein, the difficulty arose from wrongly attributing physical meaning to the spacetime coordinates? That Hilbert's four non-covariant energy equations do implicate such a misunderstanding is affirmed by Renn and Stachel (1999, pp. 77, 83):

… we will discuss Hilbert's treatment of the problem of causality in

Hilbert's revision of Kant in the light of general invariance

In the context of a discussion of Hilbert's views on foundations of mathematics, Peckhaus (1994, p. 91) has recently remarked that

Hilbert's preference for Kantianism was largely incompatible with Kant's philosophy.

However, this statement should be qualified, for its truth largely depends on what is meant by ‘Kant's philosophy’. To many of the neo-Kantians active in the first quarter of the twentieth century, Kant's philosophy was largely a work in progress, not a finished edifice, and nearly

Conclusions

Theorem I of Hilbert's First Communication, based on the axiom of general invariance, provided a purely mathematical diagnostic of the tension between causality and general covariance appearing as a problem of causal underdetermination. The material cut from the Proofs is his first attempt at resolving this tension.

The Second Communication (1917) presents Hilbert's later, and different, attempt at a resolution. For Hilbert, general covariance is a refinement, first identified as such by

Acknowledgments

We would like to thank Carlos Casanova, Sol Feferman, Brandon Fogel, Michael Friedman, Ulrich Majer, Brian Pitts, Tilman Sauer, and Joseph Zepeda, for their comments and suggestions. We are especially grateful to two anonymous referees for their detailed comments and advice. Material based on this research was presented at the Philosophy of Physics Research Seminar, Oxford University (2002); HOPOS (2004); the Logical Methods Workshop, Stanford University (2005); the Belgian Society for Logic

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