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

Institute for Science and Technology Studies

University of Bielefeld

P.O. Box 100131

D-33501 Bielefeld

Germany

Principle of Least Action, calculus of variations, Hilbert’s axiomatic method in physics, Mach-Planck controversy, Logical Empiricism, Moritz Schlick, Hans Hahn, Philipp Frank.

Over the centuries, no other principle of classical physics has to a larger extent nourished exalted hopes into a universal theory, has constantly been plagued by mathematical counterexamples, and has ignited metaphysical controversies about causality and teleology than did the Principle of Least Action (henceforth PLA).¹ After some decades of relative neglect, by the end of the 19^th century the PLA and its kin enjoyed a remarkable renaissance on three levels.

Since the work of Hermann von Helmholtz, the PLA had become a very successful scheme applicable not only in mechanics, but also in electrodynamics, thermodynamics and relativity theory. Did this spectacular success indicate that physicists possessed – to cite Helmholtz – “a valuable heuristic principle and leitmotif in striving for a formulation of the laws of new classes of phenomena” (Helmholtz, 1886, p. 210), or were these principles – as Ernst Mach held – just useful rules that served the economy of thought in various domains of experience?

A second important reorientation took place in variational calculus, the mathematical discipline on which the PLA was based and which had accompanied it through more than two centuries of philosophical debates. Karl Weierstraß’ critical investigations demonstrated that the precise relationship between the PLA and the differential equations resulting from it was extremely subtle, and that physicists’ customary reasoning in solving important cases only obtained under supplementary conditions. The generations of Euler and Lagrange typically had identified the PLA and the differential equations resulting from it regardless of their metaphysical attitude towards the PLA and the quantity of action. In the 19^th century, variational calculus was regarded as a very useful method in analysis the application of which however required considerable caution. Gauß, Jacobi, and many others obtained several important rigorous results, but only Weierstraß found the first sufficient condition for the variational integral to actually attain its minimum value. In three of his most influential twenty-three “Mathematical Problems”, David Hilbert (1900) filed a plea to rigorously and systematically develop variational calculus in the direction shown by Weierstraß, and in the 23^rd problem he supplied new technical means to do so.

It was, thirdly, Mach’s interpretation of the history of the PLA which permitted a fresh start on the philosophical level. All natural teleology associated with the PLA – so he argued in The Science of Mechanics (first edition 1883) – was the product of an epoch that was theologically tempered. The main obstacle for empiricists to assess the PLA, the claim that it revealed a superior harmony or material teleology [German: Zweckmäßigkeit]² inaccessible to empirical investigations, thus disappeared. Moreover, well-entrenched teleological explanations in biology could be embraced within the Machian notion of causality: functional dependencies between the determining elements – a notion that was intended to cut back the metaphysical concepts of cause and effect to their empirical basis of factual relations.

Logical Empiricists saw themselves to a large extent in Mach’s footsteps, but they rejected his empiricist philosophy of mathematics and sided with Boltzmann as to the indispensability of non-observational terms in scientific theories. Largely accepting Mach’s empiricist notion of causality, they were not biased by a Kantian approach that would a priori give preference to differential equations and Newtonian local determinism over the PLA. While in Kant’s Critique of Pure Reason (Newtonian) causality was ranked as a (synthetic a priori) category, the Critique of Judgment treated all teleological maxims only as regulative principles directing human judgment.

At first glance, these three changes could have permitted Logical Empiricists to reject the classical metaphysical claim that all natural laws boil down to a fixed set of minimality principles, but cherish the PLA as a mathematical principle that was almost universally valid in physics – and thus presumably more fundamental than differential equations and the concept of causality built upon them. They could have further argued that this general principle receives its concrete physical content by supplying the Lagrangian characteristic for the particular theories of mechanics, relativity, etc.; likewise Newton’s axioms are specified by the force acting on the material bodies.

An interpretation of this kind never came to the fore; not even in a version mitigated by conventionalism according to which the PLA represents an empirically equivalent formulation that is simpler in important respects and permits a unified approach. Browsing through the writings of Logical Empiricists one finds instead almost no mention of one of the most-discussed and most philosophy-laden physical principles of the day. It is the aim of the present paper to explain why. As a matter of fact, the silence was not a matter of ignorance. Moritz Schlick had been a student of Max Planck, one of the key advocates of the PLA. Philipp Frank wrote his dissertation on the PLA from the modern mathematical perspective, and Hans Hahn was a leading researcher in variational calculus. Moreover, both Hahn and Frank spent one semester at Göttingen where they studied under another main advocate of the PLA, David Hilbert. To be sure, there were major conceptual differences between the approaches of Planck and Hilbert. But these differences would have been a topic worthy of attention for philosophers of science, not the least because they involved two founding fathers of that modern science which Logical Empiricism wanted to be the most natural philosophy of. Why then did the PLA rather become a philosophical Shibboleth to Schlick, Hahn, and Frank?

First of all, one might be inclined to cite its teleological connotations. Yourgrau and Mandelstam, in this vein, hold that “[t]he belief in a purposive power functioning throughout the universe, antiquated and naive as this faith may appear, is the inevitable consequence of the opinion that minimum principles with their distinctive properties are signposts towards a deeper understanding of nature and not simply alternative formulations of differential equations in mechanics.” (1968, p. 174) In reality, to their mind, “variational principles evince greater propinquity to derived mathematico-physical theorems than to fundamental laws,” (Ibid., p. 178f.) such that all teleology ascribed to them “presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess.” (Ibid., p. 175) Here I disagree. There are subtle differences between the local differential equations and the integral PLA, even though the quantity of action has no fundamental physical importance. To be sure, these mathematical intricacies cannot back metaphysical claims about a general teleology in the style of Maupertuis. However, insisting that the PLA possesses particular mathematical characteristics which support a merely formal unification of physical theories per se does not require a metaphysical stand at all.

Both its staunchest advocates and those remaining silent about the PLA shared the conviction that final causation, material or organismic teleology, and analogies with human behavior had to be kept out of physics. The only exception are some passages of the late Planck written in the context of the relation of science and religion. Moreover, none of the protagonists of the debate under investigation considered the PLA as an instance of backward causation. The history of physical teleology might alternatively suggest a relationship between the PLA and the problem of determinism. This reached back to the classical criticism which Richard Bentley had leveled against the explanatory completeness of Newton’s celestial mechanics.³ Although to some protagonists of the present story, Ludwig Boltzmann’s statistical mechanics had made it a viable option that the basic processes of nature were indeterministic, neither PLA-advocates nor Logical Empiricists contemplated any relation between the PLA and the Second Law of Thermodynamics.⁴ Rather did they explicitly restrict the validity of the PLA to reversible phenomena regardless of their views on causality.

Thus the present paper has to seek an answer on a different route. (i) To Logical Empiricists, the mathematical universality claimed for the PLA represented an illegitimate border crossing between physics and mathematics because, on their account, there was no way how mathematics could contribute to the factual content of a scientific theory. In their perspective, the PLA was nothing but an equivalent mode of mathematical description. (ii) Logical Empiricists widely held that the price to be paid to reconcile Mach’s empiricism with modern mathematics was to consider mathematics as a science of tautologous transformations. This did not permit them to attribute any other advantage to the PLA than calculatory economy. (iii) The same containment strategy against metaphysics also prevented a due appreciation of Hilbert’s axiomatic method in the empirical sciences. In the end, both Hilbert and Planck were – at least implicitly – charged of realist metaphysics. This neglected the two levels present in the PLA and in Hilbert’s axiomatizations. There were general mathematical arguments – such as coordinate invariance or the non-trivial fact that a variational principle could be set up – and there were particular physical axioms or the specific Lagrangians. To Logical Empiricists, all that was just a homogeneous set of logical relations coordinated to observations.

Reconstructing debates and silence, I shall investigate Mach’s stand first. At surface value, the PLA represented merely an economical reformulation of the differential equations of motion. But Mach also adopted a principle of unique determination that had become quite popular among energeticists and his Berlin ally Joseph Petzoldt, a principle that in their hands even resounded classical Leibnizian ideas. Second, I discuss Planck’s and Hilbert’s pleas for the PLA. Although Planck was well aware that the PLA represented a universal scheme rather than a world formula, he considered it as an important step towards the ideal aim of attaining knowledge about the real world. Hilbert, as a matter of fact, repeatedly alluded to a (non-Leibnizian) pre-established harmony between nature and thought, but his mathematical reductionism expressed in the slogan ‘deepening the foundations’ (Tieferlegung) was rather methodologically oriented. It was grounded in his joint beliefs in the unity of mathematics and that all mathematical problems ultimately receive a definitive answer in a suitable sense.

If ‘deepening the foundations’ were to suggest that empirical content could be anchored in mathematics proper, Logical Empiricists had to vigorously object and deem Hilbert’s reverence of Leibniz as a sure sign of out-dated metaphysics. In the remaining sections I shall show how indeed Schlick, Hahn and Frank each argued on this line. While Hahn advocates basically the pure form of my above-stated thesis, in Schlick and Frank there exists also a link between the notion of causality and the PLA. Schlick’s early esteem for the PLA was influenced by the fact that simplicity represented a constitutive feature of causal laws, a view he was to abandon in 1931. Frank’s concept of causality was more liberal than Schlick’s and intended to embrace all allegedly teleological explanations in the life sciences. But Frank’s containment strategy did not halt at biological teleology, and he carried on against the slogan that “the new physics was mathematical”.⁵ This very general criticism of Frank is at odds with the fact that The Law of Causality and its Limits (1932, Ch. III, 22) brings up a rather intricate example in which the PLA fails to recover the equations of motion.⁶ Silence in Frank’s case thus means not to bridge this gap and ignore virtually all the sophisticated philosophical problems raised by Helmholtz, Planck and Hilbert.

Mach emphasizes that the core of the PLA is the variation of the determining circumstances. It roots in the general principle of continuity that guides scientific research. The feature of minimality present, on the other hand, only stems from the PLA’s historical origin. “The important thing, therefore, is not the maximum or minimum, but the removal of work from this state; work being the factor determinative of the alteration.” (Ibid., p. 476/555) Thus Mach concludes that “the principle of vis viva … is the real foundation of the theorem of least action.” (Ibid., p. 409/474) But the dependence of the determining circumstances contains yet an aspect more general than energeticism.

While energeticists conceived in their principle a substantialist reduction of all physical quantities to energy, Mach was at pains to insist that unification has an economical advantage, but is ontologically neutral. Although Mach approved Petzoldt’s uniqueness argument, he rejected its employment as a condition on possible worlds. There “is no choice between the actual happening and another.”(1919, p. 393/360)

In his criticism, Planck distorts Mach’s anti-substantialist ontology of neutral monism into sensualism, holding “that there are no other realities than one’s own sensations and that all natural science in the last analysis is only an economic adaptation [Anpassung] of our thoughts to our sensations by which we are driven by the struggle for existence …. The essential and only elements of the world are sensations.”(Planck, 1908, p. 20/129) In his Leading Thoughts Mach counters with his famous slogan about the task of science: “Adaptation of thoughts to facts and adaptation of facts to each other.”(Mach, 1910, p. 226/133f.)⁹ Against Planck’s belief, Machian facts are not isolated sensations, but they are constituted by relatively stable functional dependencies between these elements. “One recognizes the relations between condition and conditioned, the equations which cover greater or less domains, as the inherent permanency, substantiality, as that whose ascertainment makes possible a stable world picture.” (Mach, 1919, p. 431/390) While Mach’s relational ontology avoids any absolutist commitments, to Planck’s lights, an increased constancy of the world picture warrants stronger ontological conclusions. “This constancy which is independent of every human – especially every intellectual – individuality, is that which we now call the real [das Reale].”(Planck, 1908, p. 22/131)

In the 1930s Planck’s attitude has shifted significantly towards metaphysics. In his oft-delivered talk “Religion and Natural Science”, the PLA is presented as a most comprehensive and strictly valid law – even within the newest physics – “the proper formulation of which in every unbiased observer arouses the impression as if nature be governed by a rational and purposeful [German: zweckbewußt] will … Photons [deflected by the gravitational field of a star] act like rational beings. They select among all possible curves offered to them always that one by which they reach the goal most quickly.” (1937, p. 302) The PLA thus introduces the causa finalis into physics, but both the teleological and the causal approach represent only different mathematical forms for the same fact. Nonetheless, in a religious perspective it is important that there exists a regularity independent of man “which admits a formulation that corresponds to purposive action. This represents a rational order of the world, to which both nature and man are subjected to.”(Ibid., p. 303) Putting these passage alongside with Planck’s earlier statements on the PLA one might conclude that “his advocacy of a teleological interpretation of this law is characterized by a certain measure of contradiction” (Yourgrau/Mandelstam, 1968, p. 164) and “that Planck, despite his admonition that all anthropomorphism should be eliminated from exact science, himself succumbed to the very errors he denounced.” (Ibid., p. 180) But one must be careful how Planck relates theological or moral and scientific matters. “[B]oth ways [of stepwise perfection of knowledge] do not diverge, but run parallel to one another and they meet in the distant infinity of the respective goal.” (Planck, 1937, p. 306) Only in this unreachable limit is it possible “to identify the world order of natural science and the god of the religions” (Ibid., p. 304) Already in earlier works Planck (1923) had insisted on a clear separation of the scientific issue of determinism and the ethical problem of the freedom of the will, a position which he emphatically maintained in the view of quantum mechanics (1936). However convincing Planck’s separation might appear, it is sufficient in the present context that he found Logical Empiricists on his side (cf. Frank, 1932, Ch. X,9 and 1936, Sec. iv). Of course, both parties strongly disagreed as to whether the metaphysical questions concerning absolute reality and the freedom of the will were at all meaningful.

Planck’s reverence for the PLA has a more specific side, too. “The fundamental importance of the Principle of Least Action became generally recognized only when it proved its applicability to such systems whose mechanism is either completely unknown or too complex to think of a reduction to ordinary coordinates.” (1915, p. 76) In contrast to the differential equations of motion the PLA as an integral principle is independent of any choice of coordinates and a fortiori invariant under coordinate transformations. As had Boltzmann, Planck emphasizes that the PLA is stronger than the principle of energy conservation, but full clarity is obtained only in relativity theory where the PLA “contains all four world coordinates in fully symmetrical order” (1910b, p. 38) and is invariant under Lorentz-transformation while energy and momentum are not. Thus, the PLA unites the energeticist view of nature based on energy conservation and the mechanical view of nature based on the conservation of momentum.

In the same year Planck took a similar stance in another field. He admitted that his law of black-body radiation required a fundamental break with classical electrodynamics in favor of an elementary discontinuity in nature because classical physics unavoidably yielded Jeans’s law, in blatant contradiction even to everyday experience.

That universality and invariance are the main tenets of Planck’s ideal of physics, can also be seen at the other pillar of his world view, the constants of nature, in particular the quantum of action. Planck’s and Boltzmann’s constants characterizing thermal radiation plus the gravitational constant provide a universal system of units that does not depend on convention. In later years Planck would consider these fundamental constants as an important step towards the ideal aim of absolute knowledge.

Although Planck emphasized the value of mathematical precision regarding the PLA, within his epistemology there is surprisingly little reflection about the role of mathematics in physics, and the author is not really consistent. In 1914 he considered mathematics, at least partially, as “an empirical science about intellectual culture” (1944, p. 55). While such a formulation appears in accord with an empiricist foundation of mathematics in the style of Mach and Boltzmann, in the following year, Planck insisted on a principal difference between mathematics and physics. Unlike physical theories, mathematical theories cannot contradict one another, “such that in mathematics one cannot speak of an opposition of theories, but only of an opposition of methods.” (Ibid., p. 79) Thus the history of mathematics is not driven by the mutual modification of competing theories which are typical for the history of physics. And concerning general relativity Planck remarked in 1926 “that a theory the complete content of which can be expressed in a single mathematical formula can contradict itself as little as can two different inferences drawn from the same equation.” (Ibid., p. 172) Thus, on Planck’s account, consistency was a worthy goal of physics,, at least at a mature stage of a theory’s formal development. This sounds close to one of the cornerstones of Hilbert’s program of the axiomatization of the sciences.