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Do Mathematical Equations Display Social Attributes?

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This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of “mathematical community” is the broadest. We include “schools” of mathematics, circles of correspondence mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

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  1. Alan D. Sokal, “Transgressing the Boundaries-Toward a Transformative Hermeneutics of Quantum Gravity,”Social Text (Spring/Summer 1996): 219–52. Sokal revealed that this article was a hoax designed to parody science studies in his “A Physicist Experiments with Cultural Studies,”Lingua Franca (May/June, 1996): 62–64.

  2. As an example of extreme views, I would cite Bruno Latour’s confused critique of relativity in which he fails to understand whatframe of reference means in physics; I also disagree with his statement in another place: “Since the settlement of a controversy is the cause of Nature’s representation, not the consequence, we cannever (my emphasis-LRG) use the outcome-Nature-to explain how and why a controversy has been settled.” While I differ with Latour’s views, I believe that a more intelligent application of social constructivism can be fruitful, and I attempt to achieve such an application in this article, showing how even equations can be affected by social context. See Bruno Latour, “A Relativistic Account of Einstein’s Relativity,” inSocial Studies of Science 18(1989), pp. 3–44; hisScience in Action: How to Follow Scientists and Engineers Through Society, Harvard University Press, Cambridge, p. 99; also, see Noretta Koertge (ed.),A House Built on Sand: Exposing Postmodernist Myths about Science, Oxford University Press, New York-Oxford, 1998, especially the essays by Alan Sokal and Philip Kitcher.

  3. Loren R. Graham,What Have We Learned About Science and Technology from the Russian Experience?, Stanford University Press, Stanford, 1998.

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  4. lbid., pp. 28–31.

  5. E-mail message from Alan Sokal to Loren Graham, November 17, 1998.

  6. I am grateful to Michael Gordin for this point.

  7. V. A. Fock, “Protiv nevezhestvennoi kritiki sovremennykh fizicheskikh teorii,”Voprosy filosofii (No. 1, 1953).

  8. In dozens of conversations, both with Fock himself and with others, I have been told by many people-both Marxists and non-Marxists-that Fock’s commitment to dialectical materialism was sincere. Some of these conversations with Russians who knew Fock occurred after the fall of the Soviet Union, when there was no longer any political reason to affirm Fock’s Marxism. Two years after the end of the USSR, in 1993, the historian of Russian physics Gennady Gorelik wrote, “There is no doubt that Fock in the nineteen thirties was already sincerely devoted to dialectical materialism.” G. E. Gorelik, “V. A. Fok: filosofiia tiagoteniia i tiazhest’ filosofii,”Priroda (No. 10, 1993), p. 92. E. L. Feinberg went in 1993 even further, stating Fock’s “ ’love’ for diamat was, without question, sincere. I will say even more-maybe people will disdain me, but I also am in agreement with diamat. In itself, it makes sense.” E. L. Feinberg, “Fok govoril to, chto dumal,”ibid., p. 94. For a strong defense by Fock of dialectical materialism as a way of seeing physics, see his reply to my article, “Quantum Mechanics and Dialectical Materialism,” and “Reply” and Paul K. Feyerabend, “Dialectical Materialism and Quantum Theory,”Slavic Review 25 (September, 1966), pp. 381–417.

  9. Planck commented, “The concept of relativity is based on a more fundamental absolute than the erroneously assumed absolute which it has supplanted.” M. Planck,The New Science, Meridian Books, Greenwich, Connecticut, 1959, p. 146. Planck expressed the same idea in early publications, for example,Das Weltbild der Neuen Physik, Leipzig, 1929, p. 18.

  10. Fock, “Protiv nevezhestvennoi kritiki sovremennykh fizicheskikh teorii,” p. 172.

  11. See Gerald Holton, “Introduction: Einstein and the Shaping of Our Imagination,” in Gerald Holton and Yehuda Elkana (eds.),Albert Einstein: Historical and Cultural Perspectives, Princeton University Press, 1982, Princeton, p. xv.

  12. Fock liked to put his viewpoint in French: “(1) La relativité physique n’est pas général; (2) la relativité générale n’est pas physique.” V. A. Fock, “Les principes mécaniques de Galilée et la théorie d’Einstein,” inAtti de convegno sulla relatività générale: Problemi dell’energia e onde gravitazionali, Florence, 1952, p. 12. 13“Es ist nicht überflüssig zu unterstreichen, dass das Verhältnis von Körpern oder Prozessen zum Bezugssystem ebenso objektiv ist (d.h. unabhängig von unserem Bewusstsein) wie überhaupt alle physikalischen und anderen Eigenschaften der Körpern.” V. A. Fock, “Über philosophische Fragen der modernen Physik,”Deutsche Zeitschrift für Philosophie (No. 6, 1955), p. 742.

  13. V. A. Fock,The Theory of Space, Time and Gravitation, trans. N. Kemmer, New York, Pergamon Press, 1959, p. xv.

  14. lbid ., p. xvi.

  15. Ibid., p. 351. See also V. A. Fock, “Poniatiia odnorodnosti, kovariantnosti i otnositel’nosti,”Voprosy filosofii (No. 4, 1955), p. 133.

  16. Furthermore, the form of equation (3) impressed Fock because of its very close similarity with the ordinary wave equation for the propagation of light, being essentially the relativistic d’Alembertiang α,β d/dxα d/dxβ, with the cross-terms vanishing. And the propagation of light was particularly significant to Fock because he saw it as confirmation that space and time were indeed linked objectively in nature, adding extra physical-philosophical relevance to equation (3). See Fock,The Theory of Space, Time and Gravitation, 2nd ed., N.Y., 1964, p. 158, in the section entitled “Properties of Space-Time and Choice of Coordinates.” Here Fock uses the propagation of light to show that “the properties of space-time are objective, they are determined by Nature and do not depend on our choice... . The equation for the propagation of a wave front in free space characterizes not only the properties of the kind of matter being propagated (e.g., of the electromagnetic field) but also thproperties of space-time itself... . Consequently, the concepts of geometry and the notion of time are very closely connected with the law of wave-front propagation in free space.”

  17. V. A. Fock,Teoriia prostranstva, vremeni i tiagoteniia, Gosudarstvennoe izdatel’stvo tekhno-teoreticheskoi literatury, Moscow, 1955; translated into English by N. Kemmer asThe Theory of Space, Time and Gravitation (2nd revised edition), Pergamon Press, New York, 1964; see pp. 4, 192, 263–267, 365–375, 425.

  18. John Wheeler’s response to a paper by Loren Graham, The Einstein Centennial Symposium in Jerusalem, March 14-23, 1979. See Gerald Holton and Yehuda Elkana (eds.),Albert Einstein: Historical and Cultural Perspectives, Princeton University Press, Princeton, 1982, p. 135.

  19. Discussions of Loren Graham and Vladimir Fock, Leningrad, spring 1961.

  20. David Kaiser, “Aψ is just aψ? Pedagogy, Practice, and the Reconstitution of General Relativity, 1942-1975,”Studies in the History and Philosophy of Modern Physics, Vol. 29 (No. 3, 1998), p. 336. I am particularly grateful to David Kaiser for comments and suggestions for my interpretation in this article.

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  21. Siegfried Müller-Markus,Einstein und die Sowjetphilosophie, Vol. II, Dordrecht-Holland, 1966.

  22. “Naravne s Einshteinom,”Poisk (January 29, 1999), p. 8.

  23. Discussed in many sources, and recently in Jed Z. Buchwald,From Maxwell to Microphysics: Aspects of Electromagnetic Theory in the Last Quarter of the Nineteenth Century, University of Chicago Press, Chicago, 1985.

  24. Alan Sokal, e-mail to Loren Graham, January 15, 1999.

  25. In my research on Boris Hessen the distinction helped me in seeing that scholar in a different way than most scholars do; see Loren Graham, “The Socio-Political Roots of Boris Hessen: Soviet Marxism and the History of Science,”Social Studies of Science 15 (1985), 705-722.

  26. Peter Galison,How Experiments End, University of Chicago Press, Chicago and London, 1987, p. 3.

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Correspondence to Loren R. Graham.

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Graham, L.R. Do Mathematical Equations Display Social Attributes?. The Mathematical Intelligencer 22, 31–36 (2000).

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