Boltzmann and Hertz on the Bild -Conception of Physical Theory

The neo-Kantian tradition in philosophy and in physics is aptly illustrated, among others, in Cassirer Ernst , The problem of knowledge: Philosophy, science, and history since Hegel (New Haven and London, 1969), ch. 9: “Darwinism as a dogma and as a principle of knowledge”, 160–75. As regards the debate on phenomenology and Darwinism and their relation to neo-Kantism: Elkana Y. , “Boltzmann's scientific program and its alternatives”, in Elkana (ed.), The interaction between science and philosophy (Atlantic City, N.J., 1974), 243–74, p. 264. According to Elkana, Boltzmann's Darwinist ideas were part of a debate which was central to the neo-Kantian epistemology of the end of the nineteenth century. The later formulations by Piaget and Chomsky, rebutting the Sapir-Whorf ideas, originated in this debate. For relevant references see also: Janik Allan and Toulmin Stephen , Wittgenstein's Vienna (New York, 1973), and other sources quoted in ref. 2.

Wallner F. , “Boltzmann, Hertz und Wittgenstein”, 143–54; Hoerz Herbert , “Helmholtz und Boltzmann”, 191–206; and Broda Engebert , “Boltzmann and Darwin”, 129–42 (espec. the sections on “The importance of Darwinism for Boltzmann” and “Boltzmann and Kant”), in Sexl Roman and Blackmore John (eds), L. Boltzmann Gesamtausgabe, viii: Ausgewaehlte Abhandlungen der Internationalen Tagung Wien 1981 (Graz, 1982).

One of the first explicit mentions of Einstein's adhesion to Kant's “Copernicanism”, is in Einstein A. , “Physics and reality”, The journal of the Franklin Institute, ccxxi (1936), 313–47. According to Einstein, one of the greatest discoveries has been Emmanuel Kant's affirmation that a real world would be meaningless if separated from its comprehensibility. In Einstein's 1933 Herbert Spencer Lecture, he stated that “the axiomatic foundations of theoretical physics cannot be deduced by experience, but, on the contrary they are a free creation of the intellect”. In his by-now-classical essay, Gerald Holton shows how Einstein's epistemology shifted from empiricism to rational realism; Holton G. , “Mach, Einstein and the search for reality”, in his Thematic origins of scientific thought: Kepler to Einstein (Cambridge, Mass., 1975), 219–59.

Boltzmann L. , Theoretical physics and philosophical problems (Dordrecht and Boston, 1974), an almost integral English transl. of Boltzmann L. , Populaere Schriften (Leipzig, 1905; Braunschweig/Wiesbaden, 1979). See index of names: Goethe, Schiller, Kant, Schopenhauer.

Broda , “Boltzmann and Darwin” (ref. 2), 138.

Kaiser Walter , “Boltzmanns mechanische Darstellung von Thermodynamik und Elektrodynamic”, in Sexl and Blackmore (eds), op. cit. (ref. 2), 207–30.

Ehrenfest P. and Ehrenfest T. , The conceptual foundations of the statistical approach in mechanics (Ithaca, N.Y., 1959), an English transl. of the essay the Ehrenfests wrote for the Encyclopedie der mathematischen Wissenschaften in 1912.

Dugas R. , Histoire de la mécanique (Neuchatel, 1950), 426–8, 430–3. Also, Jungnickel Christa and McCormmach Russell , Intellectual mastery of nature: Theoretical physics from Ohm to Einstein (Chicago, 1986), ii, 212.

“… by the experiments above sketched the propagation in time of a supposed action-at-a-distance is for the first time proved. This fact forms the philosophical result of the experiments; and, indeed, in a certain sense the most important result”, in Hertz Heinrich , Electric waves, being researches on the propagation of electric action with finite velocity through space (London, 1893; New York, 1962), 21. This is an English transl. of his Untersuchungen ueber die Ausbreitung der elektrisken Kraft, Gesammelte Werke , ii (Leipzig, 1914; first edn, 1891).

Kuhn Thomas has significantly documented Boltzmann's wavering between two conceptions, in Kuhn T. S. , Black-body theory and the quantum discontinuity 1894–1912 (Oxford and New York, 1978), ch. 2, 38–67. Klein Martin J. , “The development of Boltzmann's statistical ideas”, in Cohen E. G. D. and Thirring W. (eds), The Boltzmann equation, theory and applications (Berlin, 1973), 53–99, underlines some of Boltzmann's ambiguities.

Boltzmann L. , “On the methods of theoretical physics”, in Boltzmann , op. cit. (ref. 4), 5–12. The above essay, the opening article in Boltzmann's Populaere Schriften, was requested by the editors and first published in Katalog mathematischer und mathematisch-physikalischer Modelle … (Munich, 1982).

Boltzmann , “On the methods …” (ref. 11).

ibid., 11.

Boltzmann L. , “On the development of the methods of theoretical physics”, in op. cit. (ref. 4), 77–100; Populaere Schriften, 120–49.

Boltzmann , “On the development…”, 90 note.

I have found what I consider some non-Darwinian features of Boltzmann's Bild-conception at an earlier time in 1890, in his conference at Graz University (1890): The Bild “serves as a guiding star in all our thoughts and experiments, i.e., in completing, as it were, the thinking process …” (my italics), ( Boltzmann , “Ueber die Bedeutung von Theorien”, Populaere Schriften (ref. 4), 76–80, p. 77). In “On the question of objective existence of processes in inanimate nature” (1897) ( ibid ., 57–76, p. 58), he had already attempted a coherent presentation of an evolutionary epistemology. There he attacked some of Hertz's a priori requirements on physical theory.

Boltzmann , “On the development …” (ref. 14), 83.

Controversies do exist concerning the English translation of the German word Bild, which was originally used by Kant in his Critique of pure reason. Cohen Robert S. (op. cit. (ref. 25), Preface, p. x) objects to rendering it with the English “image”, noting that Braithwaite uses “internal pictures”. It is remarkable that Hertz, in describing Bilder as “representations”, preferred to use consistently the word “Darstellung”, rather than “Vorstellung”, to indicate the active participation of the mind ( Janik and Toulmin , op. cit. (ref. 1), 140).

Boltzmann , “On the development …” (ref. 14), 83.

ibid., 87. Boltzmann also quotes Rowland's experiment on the modification of electrostatic forces through the relative motion of charges as a verification of Weber's action-at-a-distance theory of electrodynamics.

ibid., 90–91.

ibid., 94.

ibid., 94.

ibid., 96.

Hertz H. , The principles of mechanics presented in a new form (New York, 1956), Introduction, 2–3. A translation (1899) of Hertz H. , Prinzipien der Mechanick in neuen Zusammenhange dargestellt (Leipzig, 1984).

In a modern presentation of Hertz's Prinzipien, Robert S. Cohen thus summarizes Hertz's conception of a physical theory ( Cohen R. : “Hertz's philosophy of science, an introductory essay”, in Hertz , The principles of mechanics , p. v): “… a theory [in mechanics] should be logically clear, in the sense of free from contradictions; second, the theory must be correct, agreeing with the observed material motions; third, the theory should be appropriate, utilizing notions which are neither ambiguous nor extraneous”. Concerning PM, Cohen comments: “His first requirement is frankly metaphysical, nor would he preclude any other metaphysical queries for he says: ‘A doubt which makes an impression on our mind cannot be removed by calling it metaphysical … we cannot a priori demand from nature simplicity …. But with regard to images (Bilder) of our own creation we can lay down requirements’” (ibid., p. v). Cohen's conclusion is that “Hertz is a neo-Kantian whose a-priori speculations function as the fundamental axioms of an axiom system”. Note the remarkable similarity between Hertz's conception in his Principles (Introduction, 1–2) of the relation between concepts and observables and Einstein's statement that our concepts have no other relations with things but that which passes between the check and the coat (Einstein, “Physics and reality” (ref. 3)).

Hertz , The principles, 2.

ibid., 8.

ibid., 6.

ibid., 3.

ibid., 8.

ibid., 12.

ibid., 26–41.

ibid., 11.

ibid., 26.

Hertz , Electric waves (ref. 9), Introduction.

ibid., 19.

ibid., 21.

ibid., 21.

ibid., 4.

ibid., 4. It is not far-fetched to conclude that the fundamental equations of mechanics, sharing with electromagnetic equations the property of being common to the various modes of representation, symbolize at its best the axiomatic structure of theory. In this sense I also explain Hertz's famous dictum on Maxwell's equations.

Hertz , The principles (ref. 25), 23. Arthur Miller sees the Hertzian requirement as an exigency that mechanical principles can be compatible with Kant's forms of internal intuition, i.e. the a priori transcendental forms of Euclidean space and absolute time. Miller provides evidence to support this thesis. Miller A. I. , “On the origins, methods, and legacy of L. Boltzmann's mechanics”, in Sexl and Blackmore (eds), op. cit. (ref. 2), 231–63, p. 243.

Hertz , The principles (ref. 25), 45.

Ibid . In this statement Hertz seems to support an orthodox Kantian position rather than a neo-Kantian revision of the role of the forms of intuition. However, perhaps as a tribute to his neo-Kantian master Helmholtz, Hertz adds a somewhat contrasting comment, for it is somehow at odds with his previously orthodox Kantian adherence to Euclidean space: “It is immaterial to us whether these properties [of space] are regarded as being given by the laws of our internal intuition, or as consequences of thought which necessarily follow from arbitrary definitions” (ibid., 45). Notice that Hertz explicitly refers in this passage to both the Kantian transcendental forms: Internal intuition and intellectual categories.

Hertz , The principles (ref. 25), 8.

ibid., 45.

ibid., 8.

Accordingly Cassirer E. (op. cit. (ref. 1), 103ff) illustrates how “the fundamental concepts of physics are [for Hertz] the expression of a highly intellectual process”. Broda in his essay (ref. 2) seems to support the thesis that Hertz's conception of the a priori refers to Kant's “Formen der Aushauung”, neglecting Kant's Kategorien.

Hertz , The principles (ref. 25), 33.

ibid., 270–1.

ibid., 144.

ibid., 144.

Wilson A. D. , “Boltzmann's philosophical education and its bearing on his mature scientific epistemology”, a paper presented at the Rome Congress on Boltzmann, February 1989, and kindly given to me by the author when the main lines of my paper were completed. Wilson's essay contributes valuable new information on the above problem and provides additional support for my present thesis.

Boltzmann , “On the fundamental principles and equations of mechanics” (1899) in Boltzmann , op. cit. (ref. 4), 101–28.

ibid., 105.

Boltzmann gives examples of variations in the laws of thought, comparing “the direction of thought of a naive [sic!] people like the Greeks with that of the medieval scholastics and this again with the present situation”. In Boltzmann's sense, laws of thought are regular connections between concepts whose truthfulness can be tested by experience (for instance, experience uncovered the “falseness” of the connection between parallelism in gravitational attraction and flatness of the Earth; this example is shown by Boltzmann in his 1905 papers). Laws of thought in the Hertzian sense are instead regulative principles which shape the form of theories, i.e. their axiomatic structure; as such, they cannot be tested termwise but as a relational whole (Cassirer, op. cit. (ref. 1), 106).

Boltzmann , “On the fundamental principles” (ref. 54), 105.

Boltzmann , “On statistical mechanics” (1904), in Boltzmann , op. cit. (ref. 4), 159–71.

ibid., 166.

ibid., 166.

ibid., 169.

ibid., 168.

Boltzmann , “On the fundamental principles” (ref. 54), 111.

See, for details, Miller , op. cit. (ref. 42).

Boltzmann , “On the fundamental principles” (ref. 54), 119.

Hertz , The principles (ref. 25), 26.

Boltzmann , “On the fundamental principles” (ref. 54), 117.

ibid., 118.

Boltzmann , “On a thesis of Schopenhauer's” (1905), in Boltzmann, op. cit. (ref. 4), 185–98, p. 195.

Boltzmann , “On the principles of mechanics” (1900), ibid., 129–52, p. 133.

Broda , in op. cit. (ref. 2), rightly refers to Boltzmann's “absolute Darwinism”. Besides my thesis of Boltzmann's early confrontation with the neo-Kantian tradition and his final acceptance of philosphical Darwinism, I acknowledge other interesting and detailed researches on the same topic. Among them, Blackmore J. , “Boltzmann's concessions to Mach's philosophy of science”, in Sexl and Blackmore (eds), op. cit. (ref. 2), 155–90, discusses Boltzmann's ideas within the framework of his own personal definitions of realism and idealism and co-related thesis that Boltzmann switched from epistemological realism to idealism, i.e. to “a type of phenomenalism more rigid than that of Mach” (p. 162). Blackmore considers Kant's philosophy as a crusade against causal-realism and as a fight against metaphysics, which influenced Mach's and Boltzmann's idealism. I find that in this perspective, Boltzmann's antipathy for Kant's philosophy is difficult to explain. In the present paper I stick to what I consider a more historical approach to Boltzmann's philosophical and cultural problems. My view is that Boltzmann's views clustered around a conflict of atomism and phenomenology with a neo-Kantian tradition and found a compromise in the emerging Darwinism.

Boltzmann , “On the principles” (ref. 70), 135.

Cohen R. , “Prefatory notes to Hertz's Principles”, op. cit. (ref. 25), p. x. D'Agostino S. “Pourquoi Hertz et non pas Maxwell a-t-il découvert les ondes électriques?”, Centaurus, xxxix (1989), 66–76.

Boltzmann , “On statistical mechanics” (ref. 58), 68.

Boltzmann , “Lectures on the principles of mechanics”, in Boltzmann , op. cit. (ref. 4), 223–65, Part I, “Fundamental concepts”, 225.

ibid., 253. Also, Boltzmann , “On the fundamental principles” (ref. 54), 108; “On statistical mechanics” (ref. 58), 166.

Boltzmann , “Lectures” (ref. 75), 225.

ibid., 223, 255.

For example, Kuhn remarks (op. cit. (ref. 10), 70) that for Boltzmann probability calculus was primarily “a technique for evading paradoxes” and that “the mechanicistic approach to gas theory … exemplified by the H-theorem was always … [his] fundamental tool”. Somewhat at odds with Kuhn's remarks, Martin Klein maintains that, after his 1876 answer to Loschmidt, Boltzmann changed his views, deepening his understanding as a result of a reflection on Loschmidt's objections, but that Boltzmann gave no indication of this change in his answer. Boltzmann wrote as if he were merely elaborating on what was implicitly present in his 1872 memoir ( Klein , op. cit. (ref. 10), 73).

Boltzmann , “Lectures” (ref. 75), 259.

ibid., 259.

Boltzmann , “On certain questions on the theory of gases”, in Boltzmann , op. cit. (ref. 4), 201–9, p. 201; Nature, li (1895), 413–15, p. 413.

This is Martin Klein's remark in Klein, op. cit. (ref. 10), 98.

Pouthas J. and Oms J. , “L. Boltzmann and the second law of the theory of heat“, presented at the International Conference on L. Boltzmann, Vienna, 1981. Concerning Boltzmann's polemic with Planck: Von Meyenn Karl , “Boltzmann als kritiker und Rezensent”, in Sexl and Blackmore (eds), op. cit. (ref. 2), 97–127, pp. 105–6.

Boltzmann , Vorlesungen ueber Maxwell Theorie der Elektricitaet und des Lichtes (Leipzig, 1891), Vorwort, p. IV.