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

In the same year when Planck completed his encyclopedia entry on the PLA, David Hilbert gave an independent derivation of the field equations of general relativity by means of a single action principle. Corry, Renn, and Stachel (1997) have recently uncovered that the galley proofs of Hilbert’s “Die Grundlagen der Physik” (1916) did not contain the explicit form of the field equations. The printed version, however, appeared only after Einstein’s (1916a) seminal paper that used a more pedestrian but physically more transparent derivation. For PLA advocates the question of priority might seem crucial because it could demonstrate the principle’s superior heuristic power. However, such an understanding would be rather short-sighted because; in effect, it would severely limit the import of Hilbert’s ambitious program of the axiomatization of physics to a matter of finishing order. As I shall argue, the PLA represents a core instance of this program which laid the foundations of modern mathematical physics.

Moreover, both “Hilbert and Einstein saw their achievements of November 1915 as the culmination of year-long efforts of scientific research along their respective research programs” (Sauer, 1999, p. 566). These were by no means identical and followed a different heuristics. David Rowe even diagnoses a basic difference in their physical agendas: “it was microphysics and not gravitation that Hilbert saw as the central problem area. His eyes had been set all along on the possibility of linking Einstein’s theory of gravitation with Mie’s theory of matter by exploiting the formalisms of invariant theory and variational methods.” (Rowe, 2001, p. 404). As Rowe’s papers shows in detail, Einstein pursued a zigzag course concerning the PLA. Originally he did not dissent from the Machian outlook, according to which the PLA was a mere mathematical reformulation of the respective differential equations. He used it where appropriate, in particular when discussing Planck’s reformulation of the special theory¹⁰. But already the Entwurf theory published in the following year with Marcel Grossmann (1913) and the subsequent paper on “The formal basis of general relativity” (Einstein, 1914) put the PLA in a much more prominent position, perhaps thus contributing to Hilbert’s interest in the topics. But the approach involved “some messy considerations about the use of certain variational principles” (Rowe, 2001, p. 394) which had to yield to the detailed criticism of Tullio Levi-Civita. Yet in Einstein’s definitive theory (1916a) the PLA only appeared as a calculatory device and without being generally covariant. Most interestingly, there exists a manuscript presumably intended as an appendix to this paper which contains an argument close to Hilbert’s derivation (Einstein, 1996). It found entrance only into a later paper (1916b) after the derivation had become standard. Rowe concludes that “in 1916 Einstein had rather less interest in such formal niceties: what he wanted most of all was a theory that would enable him to attack the physical phenomena he had long ago predicted on the basis of his early attempts to extend the principle of relativity to non-inertial frames.” (2001, p. 410)

Einstein’s zigzag path clearly shows that the PLA never became a core element of his methodology. This certainly had some influence on the attitudes of Moritz Schlick, who would become the leading philosopher of relativity and a close friend of Einstein, and of Philipp Frank, who by 1916 was already an active contributor in relativistic physics. But a similar attitude seems to have been widespread within the physics community of the day, and is was mainly due to Felix Klein’s insistence that Hilbert’s paper made it into Wolfgang Pauli’s (1921) report.¹¹

In Hilbert’s case, the PLA was part and parcel of the axiomatic method. Already before 1915, he had built his axiomatizations of mechanics and continuum theory (Cf. Corry, 1997) upon this simple unifying principle right in the vein which Planck demanded. But the “Grundlagen” eventually prompted Klein’s pithy remark to Pauli about Hilbert’s “fanatical belief in variational principles, the opinion that the essence of nature could be explained by mere mathematical reflection.” (Pauli, 1979, p. 31) One should, however, not extend Klein’s negative judgment about the PLA to Hilbert’s axiomatic method tout court. First, to a writer so precise as Klein the hastily written “Grundlagen” (1916) tasted after appallingly messy mathematics. Second, there is an important philosophical distinction concerning the “essence of nature”. While the physicist Planck considered the steps towards a unified world-view as ontological reductions, the mathematician Hilbert was in first place after unifications in a methodological sense that were based on the central role of mathematics among all sciences. This permitted Hilbert to simultaneously apply his axiomatic method to phenomenological or even preliminary theories and to succumb to exalted hopes in a final unified field theory at other times, in particular in his work on relativity theory. On either way the PLA could serve as a guide. But only the first strategy was viable to Logical Empiricists, and they failed to embrace this important distinction because of their philosophy of mathematics. I shall develop this argument starting from a closer look at the “Grundlagen” and then outline Hilbert’s axiomatization program in general and its relation to the calculus of variations.

In the first published version Hilbert poses two axioms. (I) Mie’s axiom of the world function H demands that the variation of vanishes for each gravitational potential g_ and each electromagnetic potential q_s, where g is the determinant of g_ and d=d₁d₂d₃d₄ is the differential of the world parameters _k uniquely fixing the world points. H contains gravitational arguments, the g_ and their first an second partial derivatives with respect to the _k, and electromagnetic arguments, the q_s and their first partial derivatives with respect to the _k. Axiom (II) states that H be invariant with respect to an arbitrary transformation of the world parameters _k. Hilbert considers this axiom as “the simplest mathematical expression of the requirement that the coordinates in themselves do not possess any physical significance, but only represent an enumeration of the world points” (1924, p. 50). In a footnote Hilbert connects this axiom to Einstein’s idea of general invariance (today: ‘covariance’) and remarks that “Hamilton’s principle, however, only plays a minor part in Einstein whose functions H are not at all general invariants.” (1916, p. 396, 1924, p. 50). As shown above, also for Planck general invariance represented a main virtue of the PLA that was on a par with the (absolute) fundamental constants and other universal characteristics, such as the black-body spectrum.

Hilbert now formulates a theorem that he calls the “leitmotif for the construction of his theory” (1916, p. 396), but he does not provide a proof. In the 1924 reprint of the “Grundlagen” a weakened version is proven as theorem II.¹² A footnote acknowledges that Emmy Noether has provided the proof of a more general result which today is called Noether’s second theorem.¹³ In 1924 he tacitly skipped an unwarranted claim which had presumably been the physical motivation for calling the theorem a ‘leitmotif’ and for initially contemplating the title “Die Grundgleichungen der Physik” (as it appears in the galley proofs), to wit, that “the electrodynamic phenomena are an effect of gravitation.” (1916, p. 397)¹⁴ Accordingly, the introduction of 1924 is rather careful as to what extent the field-theoretic ideal of unity will prove to be a definitive one in the future development of physics.

Although sufficient for a derivation of Noether’s theorem and the Bianchi identities, axioms (I) and (II) do not fix H uniquely, such that Hilbert introduces two further axioms of a more physical kind. (III) Demanding the additivity of pure gravity and electromagnetism H=R+L, with R being the usual Riemann scalar curvature and L not containing second derivatives of the g_, guarantees that no higher than second order derivatives of the g_ appear in the field equations, such that one obtains a reasonable dynamics. Axiom (IV) “further elucidates the connection of the theory with experience” (1924, p. 57) by specifying the signature of the metric in order to obtain the required 3+1 pseudo-geometry for space-time.

In addition, there are two supplementary conditions concerning causality and regularity. A restriction on g guarantees that the time coordinate respects causal order, i.e., that cause and effect are not transformed to equal times. This condition, however, is in general not compatible with the relations between pure gravity and Mie’s electrodynamics. Hilbert tries a way out by postulating a meaning criterion: “From knowing the state variables at present, all future statements about them follow necessarily and uniquely, provided they are physically meaningful.”(1924, p. 64) He believes that
only a sharper comprehension of the idea basic to the principle of general relativity is needed in order to maintain the principle of causality also within the new physics. That is, in accordance with the essence of the new principle of relativity we have to require not only the invariance of the general laws of physics, but also attribute an invariant character to each single statement in physics, in case it shall be physically meaningful – chiming with that, after all, every physical fact must be established by light clocks, i.e., by instruments of an invariant character. (1916, p. 61, 1924, p. 63)
In a footnote of 1924, Hilbert discusses a simple example of an invariant electromagnetic Lagrangian fulfilling this causality condition from which he concludes that the condition typically corresponds to a restriction on the initial conditions. Since Gödel’s rotating universe of 1949, however, we know that even for pure gravity there exist solutions which globally violate causal chronology. Still today no simple condition is in sight that would correspond to Hilbert’s intuition.

The boom of research into singularities since the 1960s has shown that also Hilbert’s insistence on the regularity of all physical solutions¹⁵ was too restrictive. Nevertheless, “solutions with non-regular points are an important mathematical means for approximating characteristic regular solutions” (Hilbert, 1916, p. 70, 1924, p. 73). Hilbert’s belief in the regularity of physical solutions also reflected a characteristic trait of variational calculus which is the core of the 19^th problem, to wit, that solutions of a variational problem are typically nicer¹⁶ than had been initially required from the candidates. In the 20^th problem Hilbert was very optimistic about the whole field: “Has not every regular variation problem a solution, provided certain assumptions regarding the given boundary conditions are satisfied…, and provided also if need be that the notion of a solution shall be suitably extended?” (1900, p.289/470) Thiele (1997) rightly stresses that for each definition of ‘solution’ one solves, in effect, a different problem. Hilbert was strongly convinced that such existence proofs for the variational problem could be achieved more easily than for the related (differential) boundary value problem. Rowe (2001, p. 415) surmises that this was also a main motive in how Hilbert approached general relativity.

In variational calculus there is thus no ignorabimus. In virtue of their key role in science variational problems involve yet another philosophical principle above and beyond Hilbert’s deep-seated mathematical optimism. In both the “Problems” (1900, p. 257/440) and his Königsberg address (1930, p. 960), Hilbert professes the faith of a non-Leibnizian pre-established harmony between mathematics and experience. Does Hilbert also join in with a teleological interpretation of the PLA? For Leibniz, “[t]here is evidently in all things a principle of determination which is derived from a maximum or a minimum.” (1697, p. 487) Moreover, Hilbert’s belief (substantially mitigated in 1924) in the ideal of unified field physics which supplanted his earlier mechanical reductionism (Cf. Corry, 1999b) suggests that he shared Planck’s belief in ontological reductionism.

But in this way, a substantial part of Hilbert’s activities in mathematical physics would be incomprehensible, for instance his axiomatic treatment of Kirchhoff’s law of radiation that was a (merely descriptive) phenomenological theory in Mach’s sense. Recalling an ensuing polemic between Hilbert and the experimentalist Ernst Pringsheim, Max Born – who had been an assistant to Hilbert – resumes that¹⁷

According to Ulrich Majer, Hilbert’s attitude in gas theory was similar. “From an axiomatic point of view … the macroscopic-phenomenological approach is as suited for an axiomatic investigation as the microscopic-molecular one.” (2001, p. 20) Here is a passage from the introduction to Hilbert’s second lecture course on the mechanics of continua in the winter of 1906/7, the class which Frank attended in the semester before.

As goal of mathematical physics we can perhaps describe, to treat also all not purely me­chanical phenomena according to the model of point mechanics; hence … on the basis of Ha­milton’s principle, perhaps after ap­propriately generalizing it. Physics has … already gained brilliant successes in this direction …

Even if the keen hypotheses, which have been made in the realm of mole­cular phys­ics, sometimes certainly come close to the truth because the pre­dictions are often con­firmed in a surprising manner, one has to characterize the achievements still as small and often as rather insecure, be­cause the hypotheses are in many cases still in need of supple­mentation and they some­times fail completely. … Such considerations recommend it as advisable to ta­ke meanwhile a completely different, yet a directly opposite path in the treat­ment of physics – as it indeed has happened. Namely, one tries from the start to pro­duce as little detailed ideas as possible of the physical process, but fixes instead only its general parameters, which determine its exter­nal development; then one can by axio­matic physical assumptions determine the form of the Lagrangian function L as function of the parameters and their differential quo­tients. If the development is given by the minimal principle _t1 ^t₂L dt = Min., then we can infer ge­neral properties of the state of motion solely from the assumptions with respect to the form of L, without any closer knowledge of the processes … The prese­ntation of physics just in­dicated, … which permits the deduction of essential state­ments from formal assumptions about L, shall be the core of my lecture. (quoted acc. to Majer, 2001, p. 18)

The ideal of geometrization looks more reductionism than it actually was because it did not embrace the full physical content of the theory. In Hilbert’s axiomatization of general relativity there are two types of axioms, the purely geometrical ones (I,II), the physical specifications (III, IV), and two supplementary conditions. This distinction between different level of axioms is a general feature throughout Hilbert’s axiomatizations. It already appeared in the Sixth Problem where Hilbert listed the further elements of his program.

Further, the mathematician has the duty to test exactly in each instance whether the new axioms are compatible with the previous ones. The physicist, as his theories develop, often finds himself forced by the results of his experiments to make new hypotheses, while he depends, with respect to the compatibility of the new hypotheses with the old axioms, solely upon these experiments or upon a certain physical intuition. (1900. p. 272f./454f.)

Hilbert’s examples for external consistency between theories make the concept not fully clear. Kinetic theory is consistent with thermodynamics, and Einsteinian gravity possesses a well-defined Newtonian limit, while quantum theory contradicts Maxwell’s equations, such that a new foundation of electrodynamics is called for. Does Hilbert, accordingly, subscribe to physical reductionism or ontological unification which would make the external consistency an internal one? This seems, first, to run against his proposal to modify each axiom in order to check whether it is really independent and whether one could formulate other consistent theories.¹⁸ Hilbert, however, does not mention the conventionalist’s contention that even two entirely different theories may describe the same factual domain. Second, such an understanding misses the mathematical nature of the notion of ‘deepening the foundations’ which starts from the analysis of the independence of the axioms. It is an heir of the ancient attempts, e.g. of Archimedes, to prove the fundamental presuppositions of science themselves. These reductions

are not in themselves proofs, but basically only make it possible to trace things back to certain deeper propositions, which in turn now to be regarded as new axioms …. The actual axioms of geometry, arithmetic, statics, mechanics, radiation theory, or thermo­dy­namics arose in this way …. The procedure of the axiomatic method, as is expressed here, amounts to deepening the foundations of the individual domains of knowledge – a deep­ening that is necessary for every edifice that one wishes to expand and to build higher while preserving its stability. (1918, p. 407/1109)
One can distinguish at least eight types of ‘deepenings’ of different strength,¹⁹ the most simple one being just to drop a dependent axiom. Of interest here are the following. (i) Hilbert lauds both Boltzmann and Hertz for having deepened the foundations of Lagrange’s mechanics containing arbitrary forces and constraints to either forces without constraints or constraints without forces.²⁰ The fact that there are two conceptually inequivalent deepenings demonstrates that Hilbert’s ‘deepenings’ do not necessarily aspire at ontological reductions where one would expect unique basic entities. (ii) Hilbert’s deepenings may also arrive at a physically non-standard formula­tion.

The axioms of classical mechanics can be deepened if, using the axiom of continuity [which is a very deep mathematical concept], one imagines continuous motions to be decomposed into small uniform rectilinear motions caused by discrete impulses and following one another in rapid succession. One then applies Bertrand’s maximum principle as the essential axiom of mechanics, according to which the motion that actually occurs after each impulse is that which always maximizes the kinetic energy of the system with respect to all motions that are compatible with the law of the conservation of energy. (1918, p. 409/1111)

To recap, in the concept of ‘deepening the foundations’ we obtain, to my mind, a more precise form of Hilbert’s belief in a pre-established harmony between mathematics and the sciences. Typically, the PLA led to such deepenings. But the deepening concept remained nebulous in Hilbert’s published writings and it united both claims of methodological reduction and reductions which deliberately crossed the border between physics and mathematics. It is also important to note the multi-layered structure of Hilbert’s deepenings which is most clearly visible in the three groups of axioms and conditions of the “Grundlagen”. Moreover, this layering was not necessarily unique as shown in case (i) above. Both features are generic for the PLA which embraces simultaneously the general fact that a domain of physical be expressed by an integral principle and the specific Lagrangian of the problem which need not be unique. Let me now turn to how Logical Empiricists positioned themselves within this setting.