Abstract
Poincaré is a pre-eminent figure: as one of the greatest of mathematicians; as a contributor of prime importance to the development of physical theory at a time when physics was undergoing a profound transformation; and as a philosopher. However, I think that Poincaré, with all this virtue, made a serious philosophical mistake. In Poincaré’s own work, this error seems to me to have kept him from several fundamental discoveries in physics. The hypothesis that Poincaré would have made these discoveries if he had not been misled by a philosophical error is not one that lends itself to conclusive assessment; but what I wish to do is to lay out the main circumstances of the case so as to make clear, at least, that the issue of philosophic principle involved, and the questions of fundamental physics under discussion, are of considerable mutual relevance.
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Notes
Henri Poincaré, “La dynamique de l’électron,” Rendiconti del Circolo matematico di Palermo 21 (1906), 129-176; reprinted in Oeuvres de Henri Poincaré, vol. 9 (Paris: Gauthier-Villars, 1954), pp. 494-550. A brief summary of the main results had previously been given in the Comptes rendus of the Académie des Sciences, Paris, 140, 1504-8 (5 June 1905); Poincaré, Oeuvres, vol. 9, pp. 489-93.
Poincaré’s name for the theory contained in this paper: he presents it as a reformulation of the theory of Lorentz, which he has “been led to modify and complete in certain points of detail”.
The phenomenon was reported first in 1898 by André Broca in the Paris Comptes rendus. The subject was taken up again in 1904 by Paul Villard, who concluded that one was dealing here with a new kind of rays–uncharged but magnetic (it was Villard who gave to the rays the name “magneto-cathodic”). It is clearly to Villard’s account that Poincaré refers, as suggesting a fundamentally new process that may put the whole theory in jeopardy. Discussion continued in the pages of the Comptes rendus until 1911; I have found no later reference to these rays, which at any rate certainly did not prove to be a fundamental and simple new process, but rather a behavior under complex conditions of ordinary cathode rays. (For a little more detail, see Howard Stein, “After the Baltimore Lectures,” in Kargon and Achinstein, eds., Kelvin’s Baltimore Lectures [etc.], p. 397, n. 29.)
In translations of Poincaré’s paper in C. W. Kilmister, ed., Special Theory of Relativity (Pergamon, 1970), and (by H. M. Schwartz) in the American Journal of Physics 39 (1971), Poincaré is represented as saying that the whole theory is put in jeopardy by the discovery of cathode rays–an astonishing idea. Arthur I. Miller, in his extensive essay “A Study of Henri Poincaré’s ‘Sur la Dynamique de l’Électron,’” Archive for History of the Exact Sciences 10 (1973), p. 320, n. 290, remarks that the qualifier “magneto” was omitted from Kilmister’s translation, and adds: “It is not clear what Poincaré meant by a “magnetocathode ray.” On the other hand, in his book Albert Einstein’s Special Theory of Relativity (Reading, Mass.: Addison Wesley, 1981), pp. 334–5, Miller says: “In the last paragraph of the Introduction to the 1906 version, Poincaré wrote, concerning Lorentz’s theory, that at ‘this moment the entire theory may well be threatened’ by Kaufmann’s new 1905 data.” He offers us not a word as to how Poincaré’s own phrase “the discovery of the magnetocathodic rays” can bear the construction he puts upon it; he simply omits that phrase, and replaces it by his own words–which thus purport, with no justification whatever, to paraphrase Poincaré. (Miller then concludes–ibid., n. 5–on the basis of this gross misreading, that the paragraph in question must be a late addition to the paper, added by Poincaré just before the work went to press.)
Such interpretations have been put forward, e.g., by Stanley Goldberg, “Henri Poincaré and Einstein’s Theory of Relativity,” American Journal of Physics 35 (1967), 934–44; Arthur Miller,in the works cited in n. 2 and in “The Physics of Einstein’s Relativity Paper of 1905 and the Electromagnetic World Picture of 1905,” American Journal of Physics 45 (1977), 1040–48; and Tetu Hirosige, “The Ether Problem, the Mechanistic Worldview, and the Origins of the Theory of Relativity,” Historical Studies in the Physical Sciences 7 (1976), 3–82.
Cf. Heinrich Hertz, Untersuchungen über die Ausbreitung der elektrischen Kraft (2nd ed., 1894), pp. 9, 18, 298.
Poincaré, Oeuvres, vol. 9 (Paris, 1954), pp. 369ff. (published originally in L’Éclairage électrique.).
They are due respectively to Hertz, Helmholtz, Helmholtz and Reif, Lorentz, and J. J. Thomson.
Poincaré (ibid., p. 395) says “conservation of electricity and magnetism” (admitting the possibility of a true magnetic charge).
Hertz, “Über die Grundgleichungen der Elektrodynamik für ruhende Körper,” in the volume cited above, n. 4; see p. 235; and cf. also “Über die Grundgleichungen der Elektrodynamik für bewegte Körper,” ibid., p. 284: “It seems moreover not to have been noticed that this system of pressures in general leaves the interior of a homogeneous body, in particular the ether, at rest only when the acting forces have a potential.”
“Á propos de la théorie de M. Larmor,” §7; Poincaré, Oeuvres, vol. 9, pp. 391–2 (emphases added).
“Grundgleichungen für bewegte Körper,” p. 284.
Ibid., pp. 294–5.
That is, the Fresnel-Fizeau results.
Poincaré, Oeuvres, vol. 9, pp. 412–13.
Ibid., pp. 381–2.
Ibid., p. 373.
Conservation of charge; Fizeau’s experiment.
Conservation of momentum for “ponderable” matter.
Science and Hypothesis, Introduction, in Henri Poincaré, The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method, trans. George Bruce Halsted (Lancaster, Pa.: The Science Press, 1946), pp. 28, 134, 136.
Ibid., p. 136.
Ibid., p. 134.
Ibid., p. 28.
Ibid.
Ibid.
Ibid., p. 29, and chs. ii-vi; cf. also pp. 123–5.
To avoid misunderstanding: the “disguised conventions” of Poincaré are not, in his characterization, synthetic a priori; they are (as “disguised definitions”) rather, in effect, analytic. But the subjects governed by them are exactly those regarded by Kant as governed by synthetic a priori principles of a mathematical nature (except for the arithmetic of the whole numbers, where Poincaré claims that a bona fide synthetic a priori principle does rule: namely, the principle of proof by mathematical induction).
Ibid., p. 135.—For completeness, it should be added that there is still another class of hypotheses mentioned by Poincaré at this point. In the Introduction, he spoke of three such classes, the disguised conventions being one; here, however, where he is discussing specifically hypotheses in physics (in distinction from mathematics), he no longer mentions the conventions—these have already been considered at length; but he does speak of another “third” kind of hypothesis, characterized as “those which are perfectly natural and from which one can scarcely escape.” These are, like the conventions, assumptions of a mathematical sort: assumptions of continuity and differentiability, or of symmetry; and Poincaré says of them: “[They] form, as it were, the common basis of all the theories of mathematical physics. They are the last that ought to be abandoned.”
Ibid., pp. 134-5, 136.
“Á propos de la théorie de M. Larmor,” Part I, §1; Poincaré, Oeuvres, vol. 9, pp. 369–373.
Science and Hypothesis, p. 141.
Ibid., p. 142.—This attitude of Poincaré’s accounts nicely for his lack of concern with the difficulties for Hertz’s electrodynamics of moving bodies that arise from atomism.
Ibid., p. 152.
I have slightly modified the translation of this last quotation, to put it into closer accord with the French.
The Value of Science, in The Foundations of Science, p. 325.
Ausbreitung der elektrischen Kraft, p. 23.
Électricité et optique, Part I, pp. xvi-xviii.
Ibid., ch. i.
Ibid., ch. ii-iii.
Ibid., p. 79.
Ibid., p. vii; emphasis in original. Cf. also Science and Hypothesis, p. 176; this chapter (xii) of the latter book consists of extracts, slightly modified, from the introductions to Poincaré’s Théorie mathématique de la lumière and Électricité et optique (Part I).
Ibid., pp. ix-xiv.
John Nash, “C1 Isometric Imbeddings,” Annals of Mathematics 60 (1954), 383-95; “The Imbedding Theorem for Riemannian Manifolds,” Annals of Mathematics 63 (1956), 20-63.
Science and Hypothesis, pp. 127–8.
See, e.g., Science and Method, in The Foundations of Science, p. 363.
An Enquiry Concerning Human Understanding, Sect. 7, Part I, next to last paragraph.
“La théorie de Lorentz et le principe de reaction,” Archives néerlandaises des Sciences exactes et naturelles, 2nd series, 5 (1900); Poincaré, Oeuvres, vol. 9, pp. 464-488.
Max Abraham, “Prinzipien der Dynamik des Elektrons,” Annalen der Physik, 4th series, 10 (1903), 105-179; see p. 110. (Abraham also cites J. J. Thomson, who proposed a notion of electromagnetic momentum in 1893 on the basis of his rather idiosyncratic—and somewhat vague—theory of “moving Faraday tubes of force.”)
Poincaré, Oeuvres, vol. 9, p. 470.
Rapports du Congrès de Physique de 1900, vol. 1, pp. 22-3; reproduced in Science and Hypothesis, chs. ix and x; for the passage concerned, see The Foundations of Science, pp. 147-8.
Poincaré, Oeuvres, vol. 9, p. 570; The Foundations of Science, p. 505.
The Foundations of Science, pp. 123–5.
Ibid., p. 102.
In contrast, geometry and kinematics have as their subject matter space and time—and, one may say, “ moving points”-- not empirically given objects at all, but mathematical idealizations.
Howard Stein, “Yes, but . . . —Some Skeptical Remarks on Realism and Anti-Realism,” Dialectica 43 (1989), pp. 47-65; for the passage quoted, see p. 56.
I have been asked, in the course of discussion of this paper (after an oral presentation), to state more exactly what I see as Poincaré’s “mistake”; and also what my stand would be on the question whether Poincaré or Einstein was truly the author of the special theory of relativity. On the latter point, in particular, it was remarked—assuming Einstein were to be given the preference—that it would seem odd to call someone the author of a theory on the grounds that he was the first one to “take it seriously.” Let me attempt to clarify my view by some further comment here.
The basic mistake that I ascribe to Poincaré is that of seeing the significance of theoretical work as residing essentially and exclusively in its function in organizing knowledge (putative as well as real): that is, organizing the “real generalizations”—which count as presently claimed knowledge, although it is always possible that they may later fail experimental test. He recognizes that in forming those (lower-level) “hypotheses” that he calls “real generalizations” one goes beyond the actual data (and, as I have remarked in the text, even “corrects”—i.e., formally, contradicts—some of the data); this he takes to be a legitimate predictive activity. But he does not consider the “hypotheses” he calls “neutral,” or those of very high level that are “definitions in disguise,” to have any analogous legitimate function: cf. the discussion in the text of his figure of science as a library, and the theorist as drawing up the catalogue.
A consequence, in Poincaré’s specific theoretical work, of this general philosophic doctrine—which doctrine I think one can here see him obeying with some consistency—is that he regards the theories he is concerned to develop as “convenient—or more-or-less convenient—fictions.” Indeed, I am inclined to venture the psychological hypothesis that Poincaré, whose confidence in his own mathematical powers was very great indeed, had some diffidence about trespassing on the domain of physical prediction; and that this theory of the fictitiousness, or “indifference,” of his own physical theories was for him liberating: that it made it possible for him to speculate quite boldly, because of—as I should maintain—the fiction (!) that those speculations were in principle sterile exercises.
And this is the crucial difference, as I see it, between Poincaré’s relation to the special theory of relativity and Einstein’s. Both of them discovered this theory—and did so independently. So far as its mathematical structure is concerned, Poincaré’s grasp of the theory was in some important respects superior to Einstein’s. But Einstein “took the theory seriously” in the sense that he looked to it for NEW INFORMATION about the physical world—that is, in Poincaré’s language, he regarded it as “fertile”: as a source of new “real generalizations”—of empirically testable consequences. And in doing so, Einstein attributed physical significance to the basic notions of the theory itself in a way that Poincaré did not. Furthermore, this point of view led Einstein to seek—and successfully—consequences of the theory that (a) transform conventional notions (I here mean the word “conventional” in its ordinary sense; but I mean also to suggest the transformation of what in his own special sense Poincaré calls “conventions”) and (b) lead to novel predictions. Of the latter, Whittaker’s notorious discussion credits Einstein with the relativistic formula for the transverse Doppler effect; but everyone at the present day will agree that the discovery of the inertia of energy was far worthier of citation. And I have explained in the text above how close Poincaré himself really was to this discovery—which yet he failed to make. That Poincaré did not regard such new consequences as a desideratum for a theory, indeed that in a certain sense he viewed it as unreasonable and illegitimate for a theory to be expected to have such consequences, is something I have tried to show in citations of his own words. In his actual practice, he does not suggest or look for such new consequences. And, indeed, in the memoir of 1906, when he comes to discuss how the Newtonian law of gravitational attraction might be modified so as to fit in the framework of the theory he has developed--that is, when he proposes possible Lorentz-invariant forms of a law of gravitation--he concludes as follows:
[T]he first question that presents itself is that of knowing whether [these forms of the law] are compatible with the astronomical observations; the divergence with the law of Newton is of the order of [the square of the magnitude of the velocities of the gravitating bodies—on a scale on which the velocity of light is unity], that is to say 10,000 times smaller than if it were of the order of the velocities themselves, that is to say [than] if the propagation were made with the velocity of light ceteris non mutatis. [The point here is that the divergence will be 10,000 times smaller than that which had been calculated by Laplace on the hypothesis that gravitation is propagated with the velocity of light: Laplace’s result had seemed to exclude such propagation, as incompatible with the observations.] It is therefore allowable to hope that the divergence will not be too great. But only a deeper discussion will be able to settle the matter. (—“La dynamique de l’électron,” in Poincaré, Oeuvres, vol. 9, p. 550.)
This is all perfectly reasonable; but the one thing one misses is something that is conspicuous in Einstein’s parallel reflection upon his general theory of relativity, roughly ten years later. Einstein too, of course, is concerned to show that his new theory agrees with Newton’s to a good enough approximation; but for him it is crucial to try to find detectable deviations from the predictions of Newton. And so he is led to the famous three initial empirical tests of general relativity; and first of all to the extra motion of the perihelion of a planet, of detectable magnitude in the case of Mercury.
Perhaps this attempt at clarification has grown too long, and too circumstantial, to be clear. Let me cite in evidence one more item, however: a piece of testimony; but the testimony of Einstein, concerning a face-to-face discussion he had with Poincaré at the first Solvay Conference, in Brussels, in 1911. The statement occurs in a letter from Einstein to H. Zangger, dated November 15, 1911; I quote it from the admirable book of Abraham Pais, ‘Subtle is the Lord...’ The Science and the Life of Albert Einstein (Oxford University Press, 1982), p. 170; but my translation differs somewhat from that of Pais. Einstein writes: “Poincaré was (towards the relativity-theory) simply opposed in general [einfach allgemein ablehnend], [and] showed for all his acuteness little understanding of the situation.”
If you are still inclined to ask, How is this possible–on the part of a man who was one of the independent authors of that theory?–if this seems bizarre, paradoxical–then I refer you to my introductory remarks, in which I have called the case strange, and uniquely so! But how it was possible is just what I have tried to shed light on in this paper.
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Communicated by Carlo Rovelli.
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Stein, H. Physics and Philosophy Meet: the Strange Case of Poincaré. Found Phys 51, 69 (2021). https://doi.org/10.1007/s10701-021-00460-x
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DOI: https://doi.org/10.1007/s10701-021-00460-x