Action Principles, Uniqueness, and Stability

illusory unity at the price of artificial hypotheses. Moreover, functional descriptions will ultimately exhaust the fact. Given conventionalism and the ideal of an ultimately stable description – though not in a absolute sense –, one accordingly wonders how Mach’s epistemology by conceptual adaptation could delimitate and keep together its single ‘grand facts’ – and so avoid the pernicious alternatives, that there is either just one grand fact, to wit, our entire world, or myriads of small facts the boundaries between which are drawn by practical interests only.

counterpart of the later view of Vienna Indeterminists that reversible processes are just idealizations or macroscopic limits of random processes. (Cf. Chapter 4)

Boltzmann emphasized that the Principle of Least Action yields all equations of motion while energeticists had to add further propositions, such as independent energy conservation for each direction in space. Although Boltzmann did not claim that the

“appearance of equations analogous to the mechanical ones in the theory of heat, electricity, and so on” (Boltzmann, 1904, p. 136/257) justified a reduction of these fields to hidden mechanical motions, he nevertheless contended:

Only it would certainly be clearer if we could explain not only all phenomena of motion in solid, liquid and gaseous bodies but also heat, light, electricity, magnetism, and gravitation by means of the idea of motions of material points in space; that is by means of a single unitary principle, instead of requiring for each of these agencies a whole inventory of entirely alien concepts like temperature, electric charge, potential and so on … (Boltzmann, 1904, p. 137/258)

Mach, instead, did not see any ontological advantage in such a unifying principle, although he

admit[s] that it is possible to discover analogies for the Principle of Least Action in the various departments without reaching them through the circuitous course of mechanics. I look upon mechanics not as the ultimate explanatory foundation of all the other provinces, but rather, owing to its superior formal development, as an admirable prototype of such an explanation. (Mach, 1988, p. 406/471)

To Mach, Boltzmann’s 1866 linkage between the Principle of Least Action and the second law was initially a surprising result.

Yet there is no reason for being surprised. When once it has been found that quantity of heat behaves like vis viva, and thus an analogue of the theorem of vis viva is applicable to it, it is not to be wondered at that the remaining mechanical principles (which are not essentially different from this principle) may also be applied in the theory of heat. (Mach, 1919, p. 364/334)

Hence, Boltzmann did not discover “a new proof of the mechanical nature of heat.”

(Ibid.) In view of Boltzmann’s subsequent piecemeal elaboration of the theory against the criticisms of Loschmidt and Zermelo, it is surprising that Mach regarded the plan of the 1866 paper as already accomplished and merely rejected its interpretation.

There are, to my mind, two sources of Mach’s misapprehension. First, similarly as the energeticists, Mach considered the Principle of Least Action and energy conservation as at bottom equivalently complete descriptions of a mechanical system, and he hardly distinguished between variations and differentials. Euler’s precaution in

‘perfecting’ this analogy is deemed as “singularly timid” (Mach, 1988, p. 457/532). In

‘A Word from Mathematics to Energetics’, Boltzmann (1905, p. 106) spotted the energeticist Helm’s principal fault precisely in this erroneous identification. Second, what for Boltzmann represented a unifying theoretical principle of mechanics, for Mach and even more for his Berlin ally Joseph Petzoldt was intimately linked to the ontological principle of uniqueness.

To Mach, the core of the Principle of Least Action lies in variation within a system of determining conditions. The feature of minimality present in it only stems from its historical origin in the world-view of a materially economical (or parsimonious) nature.

Notice that the Principle of Least Action, like all other minimum principles in mechanics, do not express other than that in the instances in question precisely so much happens as possibly can happen under the conditions, or as is determined, viz., uniquely determined by them…[T]he principle of unique determination has been better and more perspicuously elucidated than in my case by J. Petzoldt in a work entitled Maxima, Minima and Economy…: “In the case of all motions, the paths actually traversed can be interpreted as distinguished [German: ausgezeichnete] instances chosen from an infinite number of conceivable instances…”…I am in entire accord with Petzoldt when he says: “The theorems of Euler and Hamilton, and not less that of Gauss, are thus nothing more than analytic expressions for the fact of experience that the phenomena of nature are uniquely determined.” The uniqueness of the minimum is decisive. (Mach, 1988, p. 404f./470f.)

In the cited article Petzoldt argued that those curves for which the variation does not vanish occur at least pairwise. As this was at bottom a Leibnizian idea, so is its philosophical interpretation.⁵³ “Thus one can conceive the Principle of Least Action and the related theorems within their domain of validity as analytical expressions for the principle of sufficient reason.” (Petzoldt, 1890, p. 216) Pondering that a reversion of particle motion would violate the principle of uniqueness, Petzoldt in 1895 even concludes that the unidirectionality of physical and physiological processes is closely connected to this principle which he declares “the supreme law of nature” (Petzoldt, 1895, p. 203) although it was not based on positive experience, but represented a regulative condition of the possibility of knowledge. Analogously, Ostwald had proposed his ‘principle of the distinguished case’ as a generalization of all minimum principles. “If there is present an infinite number of possibilities for a process, then what actually happens is distinguished among the possible cases.” (Ostwald, 1893, p.

600) Of course, a “certain difficulty in applying the principle is to find in each case the characteristic quantity the variation of which is to vanish.” (Ibid., p. 602) However it is philosophically framed, the requirement of uniqueness does not in general suffice for a derivation of the equations of motion.

In his 1890 paper, Petzoldt set up an even higher barrier against assessing Boltzmann’s statistical interpretation of the second law than Mach had ever done when approving Boltzmann’s 1866 idea. He argues that Mach’s merely subjective principle of economy cannot provide a measure to treat objectively purposive biological structures. Instead, Fechner’s global tendency toward stability permits a reconciliation of teleology and causality on a general level.⁵⁴

Purposiveness [Zweckmäßigkeit] as large as possible is the goal of all development. Thus the teleological principle coincides with the principle of tendency toward stability, and the latter mediates between the former and the law of causality. This conception, however, signifies a generalization of the notion of purpose, since one has to declare all stable states as purposive. (Petzoldt, 1890, p. 226)

In my view, this central role of the principle of stability is a consequence of Petzoldt’s and Ostwald’s assumption that the possible worlds really existed. This hypothesis was not shared by Mach who had insisted on the individual existence of nature. (See Sect.

3.1)

In the Theory of Heat, Mach defends his principle of economy against Petzoldt by insisting, firstly, that in physics there “is no choice between the actual happening and another. For this reason I have not used the notion of economy in any way in this

53 See (Stöltzner, 2000c).

54 See (Heidelberger, 1993) for a general account of Mach’s and Petzoldt’s reception of Fechner.

domain.” (Mach, 1919, p. 393/360) But Mach does not further insist on this principal difference between economy and stability and considers it mainly as a matter of terminology. At another place, Mach detects in Petzoldt’s account the danger of

“falling into a kind of Aristotelian physics by ascribing to organisms a tendency toward stability.” (Ibid., p. 382/351)

Boltzmann had, in passing, expressed similar misgivings about possible world arguments. “Nor do I examine whether, as Herr Ostwald holds, the actual world is a special case among all possible ones, or whether the latter are just fancy combinations of the actual in a slightly modified arrangement.” (Boltzmann, 1905, p. 131) And in the lecture notes, one can find a passage in which Boltzmann rejects any strong form of modality. “All that is, is necessary; there is nothing which could be otherwise as well.” (Fasol-Boltzmann, 1990, p. 161)

Petzoldt, on his part, even extended the principle of stability to the realm of mental phenomena, to aesthetics and ethics. In this way, Mach’s principle of economy of thought is changed into an objective cognitive stability.

The economical order of a system of conceptual reactions [Begriffsreactionen] means nothing but such an arrangement for which there exists no longer a condition for further change in the relations among the single concepts and between them and the complexes of sensations and ideas eliciting them.

(Petzoldt, 1890, p. 429)

We shall see in Section 3.8. that this daring extension of Machian economy comes rather close to Planck’s convergent realism which was based on the ideal of a stable physical world view, that is, a stable system of relations between basic physical concepts expressed in a few principles.

Despite his valid criticisms against Petzoldt, Mach still cherished stability as the core of the second law.

It may be mentioned that Boltzmann, presumably without being acquainted with the views just mentioned [among them Fechner’s and Petzoldt’s], demonstrated that a physical system when left to itself, gradually goes over into “more probable states” and finally into the “most probable state”.

Closer consideration shows that this “most probable state” is at the same time the most stable. (Mach, 1919, p. 381/351)

Boltzmann (1905, p. 154/53 n. 9) frankly admitted his ignorance and retorted:

The assertion that a system of very many bodies in motion tends, bar unobservably few exceptions, to a state for which a specifiable mathematical expression denoting its probability becomes a maximum does seem to me more than the almost tautological statement that the system tends towards the most stable state. (Boltzmann, 1905, p. 154/53)

Boltzmann also observed “that all concepts of [thermodynamic] phenomenology are derived from quasi-stationary processes and no longer hold good for turbulent motion.” (Ibid., p. 148/45) And he kept repeating that far from equilibrium only the mechanical approach yielded definitive results.

To summarize the above discussions, if general ontological principles, such as uniqueness and stability, are employed in the constitution and explanation of facts, they are in danger of relapsing into Kantian a priori conditions of the possibility of knowledge or at least relativized versions thereof. Petzoldt’s move to reclassify the

an unprecedented strategy in those days; thus several neo-Kantians moved from the first to the third Critique. Clearly distinguishing minima, maxima, and economy from their objective counterparts, uniqueness and stability, Petzoldt still revealed dualistic tendencies which Mach, in his need for a reality criterion, partially endorsed. For both reasons, Mach’s redefinition of causality in terms of functional dependences could not fully thrive. By separating more clearly the facts and the theories – which do not face facts instantaneously and one by one – Boltzmann could better avail himself of the Machian conception of functional dependences as an ontological basis for physical theory. To him, uniqueness was the major requirement imposed upon theory and it rendered atomism – on the theoretical level – inevitable. Yet, Boltzmann’s setting apart facts and theory created the need for a new reality criterion because, like Mach, he rejected any a priori knowledge. In their correspondence about causality, Boltzmann puts his finger on Mach’s ontological problems and appeared – in sheer inversion to the received view – as the more determined empiricist. This shall concern us next.