Boltzmann on Causality and Probability

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.

successfully: “With the concepts of cause and effect one cannot operate a tramway.”

(Fasol-Boltzmann, 1990, p. 280) Likewise Boltzmann treated the notion of continuum (See Sect. 3.5).

There are, however, two important differences. First, beyond restating Mach’s linkage of functional causality to psychology and sense physiology, Boltzmann connects the issue of causality much closer to classical philosophy, in particular to Kant’s theory of rationality. All laws of thought are inherited habits of thought that have become a priori conditions of knowledge, “but it seems to be no more than a logical howler of Kant’s to infer their infallibility in all cases.” (Boltzmann, 1905, p.

398/195) In the Kantian antinomies, adaptation overshoots the mark.

Indeed people racked their brains over the question whether cause and effect represent a necessary link or merely an accidental sequence, whereas one can sensibly ask only whether a specific phenomenon is always linked with a definite group of others, being their necessary consequence, or whether this group may at times be absent. (Ibid., p. 354/166)

Second, in a fragment for the lectures, titled “Cause and Effect”, Boltzmann links causality to probability.⁵⁶

Before any experience takes place, both [an accidental sequence or a causal link between phenomena]

is equally probable. But my repeated experiences render it infinitely improbable that all observed regularity would be accidental, and infinitely probable that actual actually takes place. (Fasol-Boltzmann, 1990, p. 282)

Still at the time of his philosophy lectures, Boltzmann considered probability as degree of certainty and he seems to have favored the logical interpretation of Johannes von Kries. In his 1886 academy address on the probabilistic character of the second law, Boltzmann approvingly quoted von Kries’s seminal book (1886), yet without further discussing his approach or mentioning the Spielraum (range) concept (Cf. Boltzmann, 1905, p. 37/22). As the relevant chapter of Kries’s book titled “On the Application of Probability Calculus in Theoretical Physics” was almost exclusively based on Boltzmann’s writings, one obtains the impression that Boltzmann was just glad that somebody had accomplished the task of providing a philosophical basis to his statistical mechanics. Yet Boltzmann was aware that applying logical probability requires “that the mechanical conditions of the system are known.” (Boltzmann, 1905, p. 37/22) As Martin Klein has shown, prompted by his critics Boltzmann through the years made several major changes in his use of the concept of probability, most of which were not fully noticed by his readers. The first of them in 1877 led from “a theory emphasizing kinetics and based on the special assumptions about collisions underlying the Boltzmann equation, to a theory emphasizing combinatorial statistics and independent of collision analysis.” (Klein, 1973, p. 84) In a second reinterpretation, Boltzmann “took the probability of a distribution to be the fraction of any sufficiently long time interval during which one could expect to find the gas described by this distribution.” (Ibid., p. 88)⁵⁷ And he, finally, insisted against Zermelo on “the typical or representative character of the Maxwell distribution” (Ibid., p 91) within the set of initial conditions.

56 For a more extensive discussion of ‘A5’, see (Blackmore, 1995b), ch. 7.

But despite these modifications, developing statistical mechanics as a science of its own right that studies “the properties of a complex of very many mechanical systems starting from the most varied initial conditions” (Boltzmann, 1905, p.

360/171) was aggravated by obtaining a proper concept of equiprobability. “However, this being the fundamental concept, it cannot in turn be derived and must be regarded as given.” (Ibid., p. 361/171) On January, 31^st, 1906, Boltzmann’s notes for the philosophy class read:

Knowledge by the law of causality not in the same way from experience. Source of experience. We stand⁵⁸ under its influence. One seeks probability from a priori probability. [This] only [makes] sense, if equally possible cases. Necessarily subjective from our classifications or after known causal law.

(Fasol-Boltzmann, 1990, p.145)

To my mind, Boltzmann here argues that in the same way as we necessarily order experiences by (functional) causality, we pose equiprobabilities in order to base probabilistic laws. Both are achieved either by classifications, e.g. the symmetry of a die, or according to already empirically known laws, such as: “We can infer from experience that in lotto every move is equally probable.” (Boltzmann, 1905, p. 163/75) To be sure, the Kriesian account combined aprioristic and empiricist elements because given the range (Spielraum) of possible outcomes their relative weight was determined empirically. Already at the time when he endorsed von Kries’s theory of probability, Boltzmann had accepted Mach’s empiricist notion of natural law. The Boltzmann of the philosophy notebooks is still further from Kriesian territory when he contemplates that there could be “deviations from the principle of energy [conservation], perhaps only of the second law, also from the area law, or from the center of mass law.”

(Fasol-Boltzmann, 1990, p. 106)

But Boltzmann never made the final step to base probability entirely on experience, although in a letter to Felix Klein in 1899 he expressed his misgivings about Emanuel Czuber’s abstract definition of the object of probability calculus (cf.

Höflechner, 1994, p. II 318) and paralleled them with his earlier criticisms of “the boring and uninformative definitions of number, addition, etc.” (Ibid., p. II 270).

While in the latter case he calls – citing Mach – number ‘a purely empirical concept’, he apparently was not acquainted with Fechner’s relative frequency interpretation of probability posthumously published in 1897. Recall that as Fechner and Mach, Boltzmann considered the law of causality not as an a priori precondition of experience but as a very general empirical fact. Moreover, the relative frequency interpretation would fit so neatly to his own definition of statistical mechanics as an autonomous science. Thus in effect many piers of the bridge toward this interpretation had already been set. The only problem would have been to find the appropriate collective objects (Kollektivgegenstände).

Why did Boltzmann never read Fechner; not even after Mach had publicly criticized his ignorance? Perhaps it was the context of the tendency to stability emphasized by Mach and Petzoldt which made such reading unpalatable to Boltzmann. And as Boltzmann’s rebuttal of Zermelo’s criticism depended upon the existence of a measure on the set of possible initial conditions, the Kriesian notion of range (Spielraum) provided an appropriate framework. But one could also argue the

58 Here I cannot make sense from the transcription from Boltzmann’s shorthand other that changing ‘entstehen’

(originate, emerge) into ‘stehen’. See (Blackmore, 1905a, p. 169) for a translation of the entire note.

other way round. According to Fechner’s, indeterminism arose from the novelty of initial conditions, so that even laws of nature may change on a cosmological scale.

This could have given some philosophical justification for the strange events admitted by the second law. As did Mach and all Vienna Indeterminists, Fechner held that it is impossible to ultimately decide the conflict between determinism and indeterminism.

(See Heidelberger, 1993, § 7.1)

Boltzmann’s ignorance is even more surprising if one takes into account that after 1897 the Kollektivmaßlehre was immediately discussed in the literature and that it was clearly recognized as an alternative to the Laplacian definition and to von Kries.

Shortly after Boltzmann’s death, “about 1908 Fechner’s theory of collectives apparently was standard knowledge for everyone working on probability theory and statistics in the German-speaking area.” (Ibid., p. 376) Thus matters remain puzzling. I think that Michael Heidelberger is quite right that Fechner’s thoughts about probability were too much embedded into his general and often hermetic outlook to be quickly accessible for someone who was – in stark contrast to Mach – unfamiliar with their philosophical context.⁵⁹ Major support for this conclusion derives from the fact that Franz Exner, who was familiar with Fechner’s writings, quickly accomplished the missing step towards the relative frequency interpretation and subsequently turned Boltzmann’s second law it into a rather comprehensive world view. (See Sect. 4.1.)

Boltzmann himself, however, both having rejected Mach’s principle of unique determination and ignoring Fechner’s collectives had to find an ontology for statistical mechanics in another way.