¹ ‘Invariance in Physical Theory’, Proceedings of the American Philosophical Society 93, 1949; as reprinted in Wigner (1995), pp. 283–93.

² van Fraassen (1990); Brading and Castellani (2003); and very recently, Debs and Redhead (2007).

³ Gross (1995).

⁴ ‘Symmetry in Nature’ (1973), in Wigner (1995), p. 382.

⁵ ‘Violations of Symmetry in Physics’ (1965), in Wigner (1995), p. 355.

⁶ In the 1905 Summer Semester Lectures Logische Principien des mathematischen Denkens, Hilbert writes ‘The general idea of [the axiomatic method] always lies behind any theoretical and practical thinking’, as cited and translated in Michael Hallett, ‘Hilbert's Axiomatic Method and the Laws of Thought’, in George (1994), p. 162.

⁷ David Hilbert, Neubegründung der Mathematik. Erste Mitteilung, Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität 1 (1922), as reprinted in Hilbert (1935), pp. 157–77, 161. For Hilbert, the basic objects of number theory, the positive integers or rather the signs that are their symbolic counterparts, are given in a quasi-spatial, but not in spatial or temporal intuition.

⁸ David Hilbert, Winter Semester lectures 1922–3 Wissen und mathematisches Denken. Ausgearbeitet von Wilhelm Ackermann. Mathematische Institut Göttingen. Published in a limited edition, Göttingen, 1988.

⁹ Hilbert's 1905 Summer Semester Göttingen lectures, op. cit. note 6, already characterized the general idea of the axiomatic method as stressing the consistency, independence, and completeness of an axiom system.

¹⁰ Hilbert (1918) as reprinted in Hilbert (1935), pp. 146–56, 148: ‘The procedure of the axiomatic method (…) amounts to a deepening of the foundations of the individual domains of knowledge, just as becomes necessary for every edifice that one wishes to extend and build higher while preserving its stability.’

¹¹ Lecturing in 1918 on the ‘Basic Ideas of General Relativity’, Hilbert stressed that the new conceptions of space, time and motion of Einstein's theory were still compatible with ‘the traditional intuition’ of ‘everyday life, our practice and custom’: ‘Thus we have listed all the essential features of the old conception of space, time and motion. But (…) it is still absolutely necessary to bring to mind how excellent this conception of space-time has proved to be. As far as natural sciences and their applications are concerned, we find that everything is based on this conception. And in this construction everything fits together perfectly. Even the boldest speculations of physicists and astronomers are brilliantly confirmed in the minutest detail so that one can say that the experiences of everyday life, our practice and custom, the traditional intuition and the most demanding sciences were in complete agreement and most pleasant harmony with each other.’ As cited and translated in Majer (1995), p. 274.

¹² Brading and Ryckman (2008).

¹³ In our defense of this mode of proceeding, there is the assessment of Felix Klein, who noted, in 1921: ‘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)’. Klein (1917); as reprinted with additional remarks in Klein (1921), p. 566, fn 8.

¹⁴ Corry, Renn, and Stachel (1997).

¹⁵ See Gray (2000) for discussion and an English translation of Hilbert's 1900 lecture.

¹⁶ Hilbert's use of the term ‘world parameters’ in place of the standard locution ‘space-time coordinates’ is instructive. As expressly stated in his (1917), and as Mie also observed, it is intended to highlight the analogy Hilbert sought to draw between the arbitrariness of parameter representations of curves in the calculus of variations, and the arbitrariness of coordinates on a space-time manifold. Hilbert was, of course, a grand master of the calculus of variations, as this communication will demonstrate. In both cases, objective significance will accrue only to objects invariant under arbitrary transformation of the parameters, respectively, coordinates. As Hilbert used precisely the same language of ‘world parameters’ also in the Proofs, this is prima facie evidence that his views regarding the lack of physical meaningfulness accruing to space-time coordinates were already in place.

¹⁷ Hilbert (1915a). Also (1915b), p. 396.

¹⁸ The form of equations 4 and 5 is trivially algebraically different between (1915a) and (1915b). Here the published version (1915b) is followed. For ease of comparison with the text, Hilbert's non-standard designation of the electromagnetic potential as well as his practice of using roman letters as indices for that potential and for the ‘world parameters’ is followed.

¹⁹ Hilbert (1915a), pp. 2–3; (1915b), p. 397. Hilbert regards the invariant H as the additive sum of two general invariants H = K + L, where K (the Riemann scalar) represents the source-free gravitational Lagrangian and L is the source term associated with the addition of matter fields (the electromagnetic field in Hilbert's theory). As Klein (1917), p. 481, first pointed out, there are therefore eight identities available; four associated with K and four with L. According to Klein, the identities associated with L reveal that the conservation laws of the matter field equations are consequences of the gravitational field equations, and he concluded that they therefore ‘have no physical significance’. This redundancy in the field equations, a feature of the generally invariant structure of the theory, prompted Hilbert's interpretation of the electromagnetic equations as a consequence of the gravitational equations.

²⁰ Hilbert (1915a), pp. 3–4.

²¹ See e.g., Stachel, ‘The Meaning of General Covariance: The Hole Story’, in Earman et al. (eds) (1993), pp. 129–60.

²² Hilbert (1915a), 7.

²³ Renn and Stachel (1999), p. 32, note Einstein's conviction ‘(e)ven before Einstein developed the hole argument’, that energy–momentum conservation requires such a restriction.

²⁴ The topic of energy–momentum in general relativity did not go away either: it was the subject of ongoing discussions between Hilbert, Einstein and Klein, and remains a delicate issue.

²⁵ Hilbert (1917).

²⁶ In brief, general invariance (or, general covariance) allows for local diffeomorphic point-transformations among the solutions to the field equations; for causal (time-like) curves, this permits inversion of the ordering of space-time points on such curves. Obviously, this can lead to causal anomalies. Hilbert first recognized that the full invariance (or covariance) of a generally invariant theory must be broken in order to adequately establish causality (in terms of the initial value and Cauchy problems). On the other hand, Hilbert held that the fundamental equations of physics must be time-reversal invariant, and so favoring the causal ordering by adopting particular coordinate systems is but an ‘anthropomorphic’ epistemic condition.

²⁷ ‘Proper coordinate systems’ are defined by the satisfaction of 4 inequalities among the components of the metric tensor. Where x ₄ is the time coordinate, using Hilbert's numbering (31),

²⁸ Pauli termed them ‘Wirklichkeitsverhältniße’ in Pauli (1921), English translation (1958), §22.

²⁹ Hilbert (1917), p. 58. For a definition of the aforementioned inequalities (31), see footnote 27.

³⁰ Hilbert, Grundgedanken der Relativitätstheorie, Göttingen Summer Semester Lectures 1921; as cited by Majer (1995), p. 284.

³¹ Hilbert (1926), p. 190.

³² Majer (1993a).

³³ E.g., Hilbert (1992), p. 91: ‘The transition to the broader system [containing ideal elements] either can be effected in a constructive manner, in which the new elements are developed through mathematical construction from the older ones, or in an axiomatic manner, in which the new system is characterized through relational properties. In the second case, it requires a proof that the supposition of a system with the desired condition does not in itself lead to contradiction.’

³⁴ Majer (1993b), p. 191.

³⁵ Hilbert (1930); reprinted in Hilbert (1935), pp. 383, 385.