The Principle of Least Action as the Logical Empiricist’s Shibboleth

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Let me add a short envoi how to place this paper within the emerging field of history of philosophy of science. The intense research conducted during the last two decades has taught us that Logical Empiricism was no homogeneous movement, and that in various members starkly different philosophical backgrounds came to bear. Yet undoubtedly there were important cohesive elements originating from the sciences, such as modern logic and relativity theory. Rather than insisting exclusively on a particular interpretation of these theories, the identification with scientific modernism as a “world conception” played its important historical role. This also involved positioning oneself within various traditions in the history of science and philosophy, among them the Principle of Least Action.
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* This work was supported by the SFB F012 “Theorien- und Paradigmenpluralismus in den Wissenschaften” of the Austrian Science Fund. Thanks go to Paul Weingartner and Gerhard Schurz for their constant encouragement. I am particularly indebted to Veronika Hofer, Ulrich Majer, David Rowe, Tilman Sauer, and Matthias Schramm for their suggestions and critical comments to earlier drafts of this paper. And, of course, I acknowledge the anonymous referees for their constructive criticism.

1 There is a fairly decent historical literature on the PLA and variational calculus up to 1900; among them (Goldstine, 1980), (Pulte, 1989), and (Schramm, 1985). On Helmholtz’s very influential work, see (Hecht, 1994). An overview of the physical applications if given in the classic of Lanczos (1986), and more recently (Lemons, 1997). The book of Yourgrau and Mandelstam (1968) combines a detailed study of the physical applications of the PLA with an analysis of its significance in natural philosophy (Ch. 14). Their negative conclusion rests upon the presupposition that the PLA and the differential equations are completely equivalent. As almost all the physical and philosophical literature they thus neglect the mathematical subtleties of variational calculus which played a substantial role in Planck’s and, in particular, in Hilbert’s appreciation of the PLA. As I shall show also Hahn, Frank, and, arguably, Schlick were well aware these results. Unfortunately reasons of space make it impossible to enter into further mathematical detail here. But the coherence of the philosophical argument presented here only requires to prove historically that Hahn and Frank were well-versed in variational calculus.

2 When it comes to philosophy, the German “Zweckmäßigkeit” is notoriously difficult to translate. “Teleology”, “finality”, and “purposiveness” capture only part of it and since Kant’s Critique of Judgment “Zweckmäßigkeit” also denotes the systemic structure found in organisms. For reasons of historical continuity with the philosophical tradition around the PLA, I will stick to the term “teleology” even though one of the protagonists of this paper, Moritz Schlick (1925), explicitly distinguishes biological “Zweckmäßigkeit” and metaphysical “Teleologie”.

3 See (Schramm, 1985, Ch. 2).

4 The only exception is, interestingly, Boltzmann (1866) himself who attempted to relate Clausius’ Second Law of Thermodynamics to the PLA. But he succeeded only for strictly periodic systems – not quite a generic case in thermodynamics. Although he would subsequently assign ever increasing importance to statistical concepts in understanding the Second Law, as late as in 1899 he returned to his old idea when closing his lectures at Clark University: “It turns out that the analogies with the Second Law are neither simply identical to the Principle of Least Action, nor to Hamilton’s Principle, but that they are closely related to each of them.” (1905, p. 306)

5 One might wonder why Hans Reichenbach’s name is missing here. Admittedly, his views on relativity theory are today more influential in philosophy than Schlick’s. But – or so I conjecture – his own axiomatization of relativity theory was guided by the mentioned philosophical views about the relationship between mathematics and physics which made the PLA a Shibboleth. In particular, Reichenbach’s axioms intend a simple coordination of the mathematical concepts to basic experiences rather than a simple unifying principle. Accordingly, there was not much use for the PLA within Reichenbach’s formal approach from the very beginning.

6 In Section 6 below I shall deal in more detail with this example involving sufficient conditions for the PLA to yield a minimum.

7 The authorized English translation by McCormack unfortunately uses the word ‘principle’ in all cases.

8 Leibniz’s essay remained unpublished until 1890, the same year when Petzoldt wrote his first paper; cf. (Stöltzner, 2000).

9 I am indebted to Veronika Hofer for elucidating to me the biological aspects of Mach’s thoughts. This has conducted me to translate Anpassung in a biological manner rather than by the physicist’s curve ‘fitting’ chosen by Blackmore. Notice that while Mach intends a unified theory of biology and physics, Planck limits himself to physics.

10 ‘Note added in proof’ to (Einstein, 1912). I owe thanks to Tilman Sauer for this hint.

11 See the footnote to the letter of 8 March 1921 in (Pauli, 1979, p. 27).

12 On the import and history of these early ‘folk theorems’, see (Rowe, 2001). There are important unmarked differences between both printed versions of Hilbert’s paper, a fact which Rowe judges “a blatant abuse of power” (p. 418) on Hilbert’s part in his capacity of chief editor of the Mathematische Annalen. Thus, I always give the page numbers of both versions in cases where they coincide..

13 This should not be confused with Noether’s first theorem that relates one-parameter group symmetries and conserved quantities, which today plays the more prominent role in mathematical physics.

14 For more details on this point, see (Corry, 1999a) and (Sauer, 1999) who provides a detailed analysis of the theorem.

15 John Earman stresses that Hilbert’s definition of regularity is “defective in failing to capture the distinction between genuine singularities…and mere coordinate singularities” (1995, p. 6), such as the horizon of a Schwarzschild black hole. Earman’s book also provides most competent information about causality-violating spacetimes.

16 Variational problems enjoying this property are also called ‘regular’, but in a sense different from being just non-singular. See (Gray, 2000, pp. 117-133) for a history of Hilbert’s three problems on variational calculus.

17 For the wider context and the subsequent changes of Born’s perspective, see (Schirrmacher, 2001).

18 This is, to my mind, the main reason why Hilbert lists Mach in the sixth problem because apart from Mach’s historical-critical inquiries that sketch alternative histories, there is hardly anything attractive to Hilbert on the methodological level.

19 See (Stöltzner, 2002) where the concept of elimination appearing in (Hilbert, 1930) is interpreted as a descendant of ‘deepening the foundations’.

20 In view of Boltzmann’s (1905, p. 269/ 113) negative judgment, it is quite surprising that Hilbert remains silent about the great difficulties to find those supplementary axioms which make Hertz’s theory at all applicable.

21 Schlick has also a low opinion of Mach’s role in bringing about relativity theory; cf. (1920, p. 471/313). And in the Erkenntnislehre, he largely took Planck’s side in the polemics with Mach, cf. (1974, p. 99).

22 Typically, such a homogeneity can be expressed by the invariance of the law under an appropriate class of coordinate transformations.

23 In a Curriculum Vitae written for his habilitation (Personalakt at the Archive of the University of Vienna), Hahn lists the lectures of Hilbert and Minkowski and seminars of Hilbert, Klein, and Minkowski.

24 See his Nationale (a list of the courses a student enrolled and paid for) at the Archive of the University of Vienna.

25 See his Curriculum Vitae in the Habilitationsakt at the Archive of the University of Vienna.

26 For Hilbert, the catalogue lists a “Seminar on the theory of functions” (together with Klein and Minkowski) a lecture on “Continuum Mechanics”, and an introductory class “Differential and Integral Calculus I” (together with Carathéodory) which was certainly not on the agenda of a visiting scientist. I thank Ulrich Majer for this information.

27 Most instructive here is Frank’s criticism of Ludwig von Bertalanffy’s “attempts to formulate vitalism ‘positivistically’” – so the title of the respective section IV.19. For details, see (Hofer 2002).

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