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


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4. Schlick and the Criterion of Simplicity

In Schlick’s 1925 textbook entry on the “Philosophy of Nature” one find a passage that initially rehearses Planck’s praise for the PLA in a slightly conventionalist fashion.

Physics often finds it convenient … to state the laws of nature in such a way that it assumes the beginning and end of a natural process to be given, and derives from thence the intervening course of the process; thus it treats this course as if it were something dependent on both past and future at once. A law of this form is Hamilton’s principle, or ‘the principle of least action’, and it is of great significance for the formal construction of physics that this very principle is capable of the most universal application. In all the advances of physics it has turned out that, in contrast to many another law, the action principle preserves its validity unshaken; all newly discovered laws of nature, including those of relativity theory, can be regarded as consequences of a principle of least action, which thereby appears to assume the highest rank of formal generality. It is obviously capable of this, because its formulation involves the fewest assumptions about the particular type of reciprocal dependency among natural processes. In regard to these dependencies there is actually a considerable arbitrariness in our choice of views: the one is as legitimate as the other, so long as it does but conform to the idea of a thoroughgoing perfect determinacy of the whole. (1925, p. 433f./II35)
In his 1920 “Reflections of the Causal Principle”, Schlick had continued a similar argument as such:

This [arbitrariness of description] should be borne in mind, above all, when examining the difference, and the legitimacy, of causal and finalistic or teleological viewpoints; many erroneous questions in this area have arisen from lack of clarity in regard to the simple relationships we have discussed. (1920, p. 462/I298)


This marked a significant departure from Planck’s position. Although in his early days Planck was loathe to any teleological smack of the PLA, he stressed that it stood on a higher level than the differential equations.

“Thoroughgoing determinacy” of the processes by natural law constitutes, to Schlick’s mind, the essence of the principle of causality. But he does not understand unique determinacy in a Machian sense and rejects Mach’s famous pithy phrase “nature happens only once (Ibid., p. 462/299) as an argument against the possibility of a recurrence of identical events.21 On Schlick’s account, this uniformity – at least in a approximative sense – is a precondition for the principle of causality because otherwise there could be individual laws of nature which explicitly depend upon position and time. Such laws, in which space and time attain an absolute meaning, however, cannot be empirically distinguished from a completely lawless universe. Thus any natural law must be sufficiently general and homogeneous in space and time.22 This relative character of space and time receives further support from relativity theory.

Schlick distinguishes differential micro-laws and integral macro-laws which result as their integrals, but without mentioning the PLA.

Only the latter fall within experience, for the infinitely small is not observable. The differential laws prevailing in nature can therefore be conjectured only from the integral laws, and these inferences are never, strictly speaking, univocal, since one can always account for the observed macro-laws by various hypotheses about the underlying micro-laws. Among the various possibilities we naturally choose that marked by the greatest simplicity. It is the final aim of exact science to reduce all events to the fewest and simplest possible differential laws. (Ibid., p. 462/297)


Thus simplicity is a constitutive trait of the principle of causality. The last sentence of the quoted passage could well have been written by Planck. Unifications such as the PLA, accordingly, are not a matter of mere economy. Thus together with his teacher Planck the early Schlick could be found guilty of “a traditional misapprehension to construe an accord between the postulate of simplicity and ‘least’ principles in nature.” (Yourgrau/Mandelstam, 1968, p. 173) But such an allegation misses the point because neither Schlick nor Planck attributed any fundamental significance to the physical quantity “action”. Simplicity was a formal criterion of the laws of nature. Siding more and more with conventionalism, Schlick (1932) emphatically rejected Planck’s metaphysical belief that simplicity is an indication of our approaching absolute knowledge about the real world. It always remains possible to choose another formulation, as long as unique determinacy holds true.

For – in contrast to Frank’s account discussed below – the principle of causality was not empirical, “but rather a general expression that all events in nature are subject without exception to natural laws.” (1920, p. 461/295)

Schlick’s first theory of causality was modeled after relativity theory; there was almost no mention of statistical mechanics or atomic physics. Because of this reliance on determinism, the new quantum mechanics forced upon Schlick a major chance of position. Although not driven by the notoriously vague nature of the concept of simplicity, it a fortiori eliminated its privileged status, which had been a major point in favor of the PLA. A decade later, he revoked his earlier attempts to characterize lawlikenes by explicit conditions on natural laws.

Our mistake hitherto has been a failure to adhere with sufficient exactness to the actual procedure whereby we actually test, in science, whether … a law, a causal sequence, is or is not present.…It is quite generally the case that the meaning of a proposition is always revealed to us only through the manner of its verification. (1931, p. 149/II185)


The one and only criterion for causality was thus the fulfillment of predictions. On this line, of course, no law of nature can ever be finally verified. But, “at bottom a law of nature does not even have the logical status of an ‘assertion’, but represents, rather, a ‘prescription for the making of assertions’” (Ibid., p. 151/188) – adopting Wittgensteinian terms. Quantum mechanics teaches us “that a limit of principle is set to the exactness of prediction by the laws of nature themselves” (Ibid., p. 153/191) which puts a limit to the usefulness of this prescription. Schlick’s new concept of causality was less restrictive than the former one, and accordingly it was able to encompass all empirically founded explanations in biology. This brought his theory closer to Frank’s, however, still without sharing the aim that the principle of causality be empirical as well.

It is surprising that Schlick, the leading philosopher of relativity theory in the 1920s, never mentioned Hilbert’s respective contributions; even more so because Hilbert had considered his work as an implementation of the axiomatization program launched in the Foundations of Geometry. And Schlick did make prominent reference to them in his theory of implicit definitions according to which the basic concepts of a consistent formal system are defined just by the fact that they satisfy the axioms.


David Hilbert undertook to construct geometry on a foundation whose absolute certainty would not be placed in jeopardy at any point by an appeal to [a Kantian a priori] intuition. (1974, p. 33)

It is therefore all the more important that in implicit definition we have found an instrument that enables us to determine concepts completely…To this end, however, we have had to effect a radical separation between concept and intuition, between thought and reality. (Ibid., p. 38)


According to Majer (2002), this separation was quite far from Hilbert who took geometry as the simplest and most perfect science. When Schlick writes that “the construction of a strict deductive science has only the significance of a game with symbols”(1974, p. 37), he was even further from Hilbert’s understanding of the axiomatic method as conceptual criticism.
5. Hans Hahn’s Response to a Former Teacher
In Hahn’s case, the silence about the PLA is most surprising. When Hahn came to Göttingen for the winter semester of 1903/4, he brought with him a new approach to the problem of the Second Variation, that is, about sufficient conditions for a variational problem, developed by his teacher Gustav von Escherich. (Cf. W. Frank 1993) In 1904, Hahn wrote an entry on the recent developments in variational calculus for the Enzyklopädie der Mathematischen Wissenschaften coordinated by Felix Klein. His co-author was Planck’s former student and assistant Ernst Zermelo. During his Göttingen semester, Hahn also attended the classes of Hilbert and Minkowski,23 such that one can safely assume that he was familiar with the Göttingen approach to mathematical physics. In later years, Hahn would become an eminent figure in variational calculus whose publications “often constituted important steps in the development and the simplification of the methods of the calculus of variations.” (W. Frank, 1996, p. 1)

In 1933, a small booklet entitled Logik, Mathematik und Naturerkennen authored by Hahn appeared as the second volume of the series Einheitswissenschaft (Unified Science) edited by the Vienna Circle. Although Hilbert’s name does not appear in the booklet, its title openly alludes to the widely-read Königsberg lecture “Naturerkennen und Logik” (1930). And Hahn directly addresses Hilbert’s main topic, the relationship between thought and reality.

The usual view can then be described like this: from experience we gather certain facts and formulate them as “laws of nature”; but since by thought we apprehend the most general lawlike connections in reality (of a logical and mathematical nature), our mastery over nature on the basis of facts we have gathered by observation extends much further than our actual observations; for we also know that everything that can be inferred from what we have observed by applying logic and mathematics must be real. … This view seems to find a powerful support in the numerous discoveries made in a theoretical manner. …

But we are nevertheless of the opinion that this view is completely untenable. For upon closer reflection it appears that the role of thought is incomparably more modest than the role ascribed to it on this view. … Why should what is compelling to our thought also be compelling to the course of the world? Our only recourse would be to believe in a miraculous pre-established harmony between the course of our thought and the course of the world, an idea which is deeply mystical and ultimately theological. (1933, p. 64-66/27-28.)


This was the main charge against Hilbert: pre-established harmony however mitigated by mathematics amounts to mysticism. To be sure, Hilbert had emphasized that our knowledge of natural laws is of empirical origin, but this could have been easily assented to by a Kantian. Hahn required a firmer stand.
There seems to be no other way out of this situation than a return to a pure empiricist position, a return to the view that observation is the only source of our knowledge of facts: there is no factual knowledge a priori, no “material” a priori. Only we must avoid the mistake of earlier empiricists who would see nothing but empirical facts in the propositions of logic and mathematics; we must look around for a different view of logic and mathematics. (Ibid., p. 66/28)
To Hahn’s mind, the only way to reconcile a consistent empiricism in Mach’s tradition with modern mathematics was to deny any reality at all to the concepts of logic and mathematics, and regard them – as did Wittgenstein’s Tractatus – as mere conventions about the use of the symbols of a formal language. Mach (1988, Ch. I,1) had criticized Archimedes’ derivation of the law of the lever by means of Euclidean geometry because it implicitly pre­sup­posed factual knowledge that could only be attained by previous experiences. Such illegitimate border crossings had to be banned from the application of the new axiomatic methods in the sciences. And, on Logical Empiricist’s account, Hilbert’s concept of ‘deepening’, the heir of Archimedes’ faulty proof, did not fully sever the bond with the antique prototype. Similarly as Schlick, Hahn believed that after a rigorous separation between tautologous mathematics and empirical facts the axiomatization of the sciences could fully thrive because all theorems become tautological implications of freely chosen assumptions.
Some chapters of physics have already been axiomatized in the same sense as geometry and turned thereby into special chapters of the theory of relations. Yet they remain chapters of physics and hence of an empirical and factual science because the basic concepts that occur in them are constituted out of the given.

In doing this we may have the following goal in mind: to set up an axiomatic system by which the whole of physics is logicized and incorporated into the theory of relations. If we do this, it may well turn out that, as the axiomatic systems become more comprehensive, as they encompass more of the whole field of physics, their basic concepts become increasingly remote from reality and are connected with the given by increasingly longer, increasingly more complicated constitutive chains. All we can do is state this fact, as a peculiarity of the given; but there is no bridge that leads from here to the assertion that behind the sensible world there lies a second, ‘real’ world enjoying an independent being and differing in kind from the world of our senses, a world which we can never directly perceive. (1930, p. 44-45/27-28)


Here another anathema of Logical Empiricism rises suddenly within the axiomatic method: metaphysical realism. Thus at bottom, Hahn identifies the different philosophical contexts of the PLA in Planck’s and Hilbert’s thinking. This shows that it was Logical Empiricists’ strict containment strategy against metaphysics which both prevented a due appreciation of Hilbert’s axiomatic method and rendered the PLA their Shibboleth.

In a Hilbertian perspective, Hahn’s worries originate, firstly, from falsely equating the establishment of a single axiom system and its subsequent critical analysis with ontological reduction and unification, as Planck did. But Hilbert had rather insisted that ‘deepening the foundations’ was a methodological reduction to mathematically more basic entities, such as the PLA. This methodological aspect which included phenomenological theories must be clearly distinguished from claims which Hilbert deliberately made at places that we had actually reached a unified theory and some traits of reality in Planck’s sense. Admittedly, some ‘deepenings’ of Hilbert crossed the border set by the empiricist criterion of meaning. (Cf. Stöltzner, 2002)

Secondly, Hahn’s ideas about axiomatization treat all basic concepts on a par within a single network of logical relations. What Hilbert considered as ‘deepening the foundations’, on Hahn’s account, was either metaphysical or just an economical convention. This yielded a highly static picture of the axiomatization of science that was oriented at justification rather than providing, as did Hilbert’s axiomatic method, a critical instance for theory dynamics. Thirdly, by considering any axiom system exclusively as a system of logical relations plus constitutive definitions of the basic concepts, Hahn made the axiomatic method much more dependent on the success of a foundational program for mathematics than Hilbert ever did.

But also from a Planckian perspective, Hahn’s identification of any physical excess content of the PLA with metaphysics was not quite persuasive, because Planck regarded fundamental constants of nature – not the PLA – as the best candidate for “absolutely real” entities.


6. Frank – Mathematics and Antimetaphysics
Apart from Schlick, Philipp Frank was the key person on physical matters within the Vienna Circle. He was one of the first to work on the theory of relativity which would earn him Einstein’s former Prague professorship in 1912, but his earlier activities were devoted to the PLA and variational calculus. Most interestingly, Frank’s 1906 dissertation and the related papers were written mainly from a mathematician’s perspective and they concern a topic that is – as the author repeatedly stressed – typically absent from the treatises of mechanics including the one by his late teacher Boltzmann: sufficient conditions for a minimum the action functional or the theory of the second variation. Both were research fields of Hahn and von Escherich. Frank constantly attended their courses24, in particular Hahn’s maiden lectures on variational calculus in the summer term of 1905. Thus he was well prepared to broaden his respective mathematical knowledge in Göttingen during the summer semester of 1906 where he studied among others with Hilbert, Klein and Zermelo.25 From the university catalogue26 one can infer that Frank went into Hilbert’s lecture on continuum mechanics, and thus was quite familiar with Hilbert’s use of the axiomatic method in physics including its provisional application to phenomenological theories.

Although after Boltzmann’s death the experimentalists Franz Serafin Exner and Viktor von Lang signed the opinion for Frank’s thesis, its topic evidently resulted from his interactions with the Göttingen and Vienna mathematicians, above all with Hahn who – so Frank reports in his recollections (1961) – would join him and Otto Neurath from 1907-1912 in weekly coffee house discussions on problems of science and philosophy which are sometimes bestowed upon the name ‘First Vienna Circle’. (Cf. Uebel, 2000 & 2002)

In contrast to the publication resulting from it (1909), Frank’s handwritten dissertation leaves little doubt about the philosophical background of the project. It begins as such:
In mechanics it is proven that in virtue of the equations of motion of a material point which moves in the plane with energy h and the force function V [today: potential energy] the first variation of the integral between two points of the orbit vanishes; i.e. the orbital curves satisfy Lagrange’s necessary conditions for J becoming minimal by them. In former times this theorem which was precipitately designated as the Principle of Least Action was directly stated as such: The orbital curves minimize the integral J. Already Jakobi [sic!] observed that the orbital curves only minimize J between points a and b sufficiently close to one another; incidentally he believes that the question to what extent the orbital curves yield a factual minimum is “of no importance for mechanics in the narrower sense” (Jakobi, Vorlesungen über Dynamik, p. 48) [1866] This sentence contains a hit at [eine Pointe gegen] the view then not yet generally overcome that precisely the property of the orbital curves to be curves of minimal action were their characteristic property and that in this the wisdom of the Creator of the lex parsimoniae naturae manifest itself. That this tendency prevailed in Jakobi can be seen at a phrase on p. 45 of the same work. After Jakobi has stated the principle in a form more precise than ever done before him, he says:

“It is difficult to find a metaphysical cause for the Principle of Least Action, if it is expressed in this true form, as is necessary. There exist minima of an entirely different type, from which one can also derive the differential equations of the motion, which in this respect are much more appealing.”

One can give the theorem an even more ametaphysical form than Jakobi’s by saying: A material point moves according to the Lagrange equations appertaining to the variational problem . This casts off the last remnant of minimum-romance.

And for the time being, in this way nothing seems to be lost for mechanics. For by this formulation of the theorem one reaches the advantage of higher precision because it also embraces the following two circumstances:



  1. The orbital curves stay orbital curves even where they cease to be minimal curves of J.

  2. As I have proven in connection with a remark of Routh (Dynamik, vol.II, § 455) in the Mathematische Annalen [1906b], there exists a curve minimizing J without being an orbital curve…

Thus for the time being it seems indeed the best for mechanics to give up any reference to a maximum or a minimum as precipitately metaphysical… And in fact, in the textbooks of mechanics one nowhere finds a proof that the orbital curves yield a minimum though only between sufficiently close points. But this proof can be conducted very easily by the means of modern variational calculus. (1906a, pp.1-3)
The remainder of the thesis contains a physical interpretation of what “sufficiently close” means. But let me first explain Frank’s result 2) at the example of the ballistic throw from A to B (Fig. 1) given in (1906b). One can prove that J attains a smaller value for the piecewise continuous curve AA’B’B than for the parabola AB which is the actual orbital curve. The main reason is that the horizontal line A’B’ is the limit curve along which the projectile has zero velocity, such that h=V and the integral vanishes. As Hilbert’s 20th Problem had stated, the important point is to find the appropriate class of solutions for the PLA. This is also pivotal for the axiomatic method in physics because unintended models such as the trajectory AA’B’B have to be excluded by a reasonable physicality condition, as Hilbert attempted to do in 1915. Emphatic advocates of a universal application of the PLA, such as Planck, must assume a priori that such a criterion can always be found.

Fig. 1: An absolute minimum which is no orbital curve (from Frank, 1932, p. 83/91)
Frank now searches for a precise physical characterization of the sufficient conditions which is, however, not found in the minimality itself because for sufficiently close points a and b this can always be achieved. Instead he obtains a geometric condition for the orbital curve as a whole which expresses a global property of the set of possible dynamics and physically signifies stability against perturbation. If there are Jacobian conjugate points, i.e., two points through which all varied curves pass like through foci, within the interval [a,b] then the Lagrange equations yield no absolute minimum of J and the curve is oscillatory stable, i.e. small perturbations will repeatedly intersect the orbital curve and oscillate around it. If conjugate points exist only at infinity, J attains its absolute minimum and there is no oscillatory stability. Frank emphasizes that this classification can be achieved without integrating the equations of motion and he shows which force laws permit stable rotations around a central mass.

In Frank’s philosophical works, however, the PLA is almost absent. In his seminal book The Law of Causality and Its Limits one finds only two short sections in the chapter “Currents of thought hostile to causality”. Frank takes the PLA as example of “another widely spread manner of treating natural phenomena by analogy to human emotional life” (1932, p. 82/90). As in the dissertation he criticizes the German (and a fortiori also the English) translation of Maupertuis’ ‘principe de la moindre action’ but without mentioning that also Jacobi had done so because he, as would Mach, conceived the true meaning of the PLA in the least expenditure of work (Cf. Jacobi, 1866, p. 1 and 44). Subsequently Frank repeats the example of the ballistic throw from his dissertation and (1906b) and concludes:


It is not at all characteristic for the orbit a point-mass follows that along that orbit any magnitude assumes its smallest value. If the orbital curves satisfied another law … there would always be a magnitude that depends on the velocity (or acceleration) and which is smaller for the orbital curves than for any other curve. Just this magnitude would then be regarded as a measure of the action of nature. We should therefore be able to prove why a definite magnitude signifies the action of nature… This would mean a return to pure anthropomorphism, to the animistic world-conception of the pre-scientific age. (1932, p. 84/91f.)
In contrast to the dissertation, Frank subsequently does not enter into the area of sufficient conditions. Nor does he comment upon the successful applications the PLA has found in other domains of physics since. Instead he turns to a conclusion that even in comparison to Mach minimizes the import of the PLA.
Only a certain mathematical simplification is hidden in the minimal principles of mechanics. With its help the laws of the orbital curves can be expressed in fewer variables … From the one concept ‘length’, the whole law of the formation of straight lines can be deduced [when setting up a PLA for geodetic motion]. Something similar is the case with all orbital curves of mechanics. Complicated equations are replaced by the somewhat less complicated concept of ‘action’ or ‘effect’. This has, however, nothing to do with economical measures of nature, since such an expression exists for any group of curves, if only they obey differential equations.

For mechanics and physics, this is all actually obvious and will hardly be disputed by anybody. I have discussed it in so much detail only in order to show that in biology matters are in no way different. (Ibid., p. 84f./92)


Frank’s metaphysical worries are more specific than just the specter of realism and the material a priori which beleaguered Hahn. First of all, by granting independent significance to the PLA, Frank fears to open the door for a return of anthropomorphic design arguments within biology and beyond. Throughout the book he is at pains to show that any teleological argument is either theological by positing a higher intelligence or tautological because it contains nothing that cannot be phrased in causal terms.27 A major target of his criticism is Driesch’s entelechy which erroneously purports to provide an objective measure of life. By Frank’s exclusive focus on the physical quantity of action and his near-to complete silence about those formal and mathematical virtues, which stood behind Planck’s and Hilbert’s high esteem for the PLA, eventually puts the principle on a par with the tautologous notion of entelechy. Second, where Planck saw a smack of teleology, Frank just states that “the difference is in fact not between a specification of the initial state alone or of the initial and final states, but there is always a specification through several points, and the question is only whether the points A and B are close to each other or distant” (Ibid., p. 96/101) because, for instance, any determination of an initial velocity requires two distant points. Such a view, of course, requires a notion of causality at least as wide as Mach’s functional dependencies – even one that comes close to tautology, as the author frankly admits.

But Frank’s worries are not limited to biology. In another booklet of the series Einheitswissenschaft, Frank pursued a containment strategy similar to his anti-vitalism against the slogan that “the new physics is not mechanistic but mathematical.” (1935, p. 169/111) His targets are General Smuts and James Jeans who held that the fall of the mechanical world view and the rise of abstract mathematical entities led to a return of spiritual elements within modern science, so that “the universe is now more like a great [organic] idea than a great machine.” (Ibid., p. 171/112) But, so Frank contends, geodesics in space-time and quantum mechanical probabilities are by no means different from Newtonian gravitational forces. Whoever desires to find spiritual analogies will succeed both in classical and in modern physics. Hence, the “assertion that the new physics is not ‘mechanical’ but ‘mathemati­cal’ only means that the formulae of relativity and quantum mechanics contra­dict those of the old mechanics or to put it more precisely, agree with them only for small velocities and large masses.”(Ibid., p.172/113) Moreover,

[t]he laws of physics consist of mathematical relations between quantities, as well as of directions on how these quantities can be related to actual observations, and in this respect nothing has changed even in the twentieth century. The equations have changed, the quantities are different, and the directions, too, are therefore no longer the same; but the general scheme according to which a physical theory is constructed still has the same fundamental character today as it had in Newton’s time. (Ibid., p. 198/128f.)
This was, of course, hardly a basis to assess the unifying force attributed to the PLA and Hilbert’s program of the axiomatization of the sciences. In this perspective, the only passage of Frank’s biography of Einstein that refers to Hilbert is no longer astonishing, though with a little wink as to the motivations of the mathematician’s work.
Hilbert once said: “Every boy in the streets of our mathematical Göttingen understands more about four-dimensional geometry than Einstein. Yet, despite that, Einstein did the work and not the mathematicians.” And he once asked a gathering of mathematicians: “Do you know why Einstein said the most original and profound things about space and time that have been said in our generation? Because he had learnt nothing about all the philosophy and mathematics of time and space.” (1947, p. 249f.)

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