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Mathematics: art and science

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Notes

  1. The dissertation was by L. Kronecker, cf.Werke, 5 Vol., Teubner, Leipzig, 1895-1930, Vol. 1, p. 73. The opponent was G. Eisenstein. The source I am aware of for the name and the opinion of the opponent is a footnote by E. Lampe to a lecture by P. du Bois-Reymond,Was will die Mathematik und was will der Mathematiker?, published posthumously by E. Lampe in Jahresbericht der Deutschen Mathematiker Vereinigung 19 (1910), 190–198.

  2. For a discussion of a number of such opinions see A. Pringsheim,Ueber den Wert und angeblichen Unwert der Mathematik, Jahresbericht der Deutschen Mathematiker Vereinigung 13 (1904), 357–382.

  3. Letter to F. W. Bessel, 18 November 1811. See G. F. Auwers Verlag,Briefwechsel zwischen Gauss und Bessel Leipzig 1880, p. 156.

  4. Actually the beginnings of group theory can already be traced to some earlier work, notably by Lagrange, which was in part familiar to Galois. The latter’s standpoint was, however, so general and abstract and, in addition, so sketchily described that it was assimilated only slowly. For historical information on the theory of equations and the beginnings of group theory see, for example, N. Bourbaki,Eléments ďhistoire des mathématiques, Hermann éd., Paris, 1969, third and fifth articles.

  5. F. J. Dyson,Mathematics in the physical sciences, Scientific American 211 September (1964), 129–146.

  6. See B. L. van der Waerden’s historical introduction in “Sources in Quantum Mechanics”,Classics of Science, Vol. 5, Dover Publications, New York, 1967, especially pp. 36-38. Cf. also Dirac’s remarks on the introduction of non-commutativity in quantum mechanics inloc cit. [7].

  7. E. P. Wigner,The unreasonable effectiveness of mathematics in the natural sciences, Communication on Pure and Aplied Mathematic 13 (1960), 1–14 Among the many aspects of this interaction, the one that appears most remarkable to me is that the mathematical formalism sometimes leads to basic, new and purely physical ideas. One well-known example is the discovery of the positron: In 1928 P. A. M. Dirac set up quantum mechanic relativistic equations for the movement of the electron. These equations also allowed a solution with the same mass as the electron, but with the opposite electrical charge. All attempts to explain these solutions satisfactorily, or to eliminate them by some suitable modification of the equation, were unsuccessful. This led Dirac eventually to conjecture the existence of a particle with the necessary properties, which was later established by Anderson. For this see P. A. M. Dirac, “The development of quantum theory” (J. R. Oppenheimer Memorial Prize acceptance speech), Gordon and Breach, New York, 1971. A newer and even more comprehensive example would be the use of irreducible representations of the special unitary group SU(3) in three complex variables, which led to the so-called “eightfold way”. One of the first successes of this theory was quite striking, namely, the discovery of the particle: Nine baryons were assigned, through consideration of two of their characteristic quantum numbers, to nine points of a very specific mathematical configuration consisting of 10 points in a plane [the 10 weights of an irreducible 10-dimensional representation of SU(3)]; this led M. Geli’man to conjecture that there should also be a particle corresponding to the tenth point, which would then possess certain well-defined properties. Such a particle was observed some two years later. A further development along these lines led to the theory of “quarks”. For the beginnings of this theory see F. J. Dyson,loc. cit. [5] and M. Gell'man and Y. Ne’eman,The Eightfold Way, W. A. Benjamin, New York, 1964.

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  8. See a number of papers in L. Euler’sOpera Omnia, espe- cially 1.2, 62-63, 285, 461, 576; 1.3, 5.2. I want to thank A. Weil for pointing this out to me. Here is an example (translated from Latin by Weil),loc. cit. pp. 62-63, published in 1747: “Nor is the author disturbed by the authority of the greatest mathematicians when they sometimes pronounce that number-theory is altogether useless and does not deserve investigation. In the first place, knowledge is always good in itself, even when it seems to be far removed from common use. Secondly, all the aspects of the truth which are accessible to our mind are so closely related to one another that we dare not reject any of them as being altogether useless. Moreover, even if the proof of some proposition does not appear to have any present use, it usually turns out that the method by which this problem has been solved opens the way to the discovery of more useful results. “Consequently, the present author considers that he has by no means wasted his time and effort in attempting to prove various theorems concerning integers and their divisors. Actually, far from being useless, this theory is of no little use even in analysis. Moreover, there is little doubt that the method used here by the author will turn out to be of no small value in other investigations of greater import.”

  9. G. H. Hardy,A Mathematician’s Apology, Cambridge University Press, 1940; new printing with a foreword by C. P. Snow, pp. 139-140. 10. P. Valéry,Degas, danse, dessin, A. Vollard éd., Paris,

  10. P. Valéry,Degas, danse, dessin, A. Vollard éd., Paris, 1936;Oeuvres II, La Pléiade, Gallimard éd., Paris, 1966, pp. 1163-1240, especially pp. 1207–1209.

  11. The following excerpt from a letter from C. F. Gauss to Olbers, written on 3 September 1805, shortly after Gauss had solved a problem (the “sign of the Gaussian Sums”) he had been working on for years, can serve as an example: “Finally, just a few days ago, success—but not as a result of my laborious search, but only by the grace of God I would say. Just as it is when lightning strikes, the puzzle was solved; I myself would not be able to show the threads which connect that which I knew before, that with which I had made my last attempt, and that by which it succeeded.” See Gauss,Gesammelte Werke, Vol. 101, pp. 24-25. Here one must also mention H. Poincaré’s description of some of his fundamental discoveries on automorphic functions. H. Poincaré, “Ľinvention mathématique” inScience et Méthode, E. Flammarion éd., Paris, 1908, Chap. III.

  12. J. v. Neumann, “The mathematician” in Robert B. Heywood,The Works of the Mind, University of Chicago Press, 1947, pp. 180–187.Collected Works, 6 Vol., Pergamon, New York, 1961, Vol. I, pp. 1-9.

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  13. H. Poincaré,La Valeur de la Science, E. Flammarion, Paris, 1905, Chap. 5, p. 139. Actually, this chapter is the printed version of a lecture which Poincaré had delivered at the First International Congress of Mathematicians, Zurich, 1897.

  14. Loc. cit. [13] p. 147.

  15. W. Kandinsky,Rückblick 1901-1913, H. Walden éd., 1913. New printing by W. Klein Verlag, Baden-Baden, 1955, See pp. 20–21.

  16. J. v. Neumann, “The role of mathematics in the science and in society”, address to Princeton Graduate Alumni, June 1954. Cf.Collected Works, 6 Vol., Pergamon, New York, 1961, Vol. VI, pp. 477–490.

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  17. Cf. G. H. Hardy,loc. cit. [9] pp. 123–124.

  18. G. Darboux,La vie et ľOeuvre de Charles Hermite, Revue du mois, 10 January 1906, p. 46.

  19. See L. White,The locus of mathematical reality: An anthropological footnote, Philosophy of Science 14 (1947), 189–303; also in J. R. Newman,The World of Mathematics, 4 Vol., Simon and Schuster, New York, 1956, Vol. 4., pp. 2348-2364.

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  20. H. Poincaré,loe. cit. [13] p. 262.

  21. A. Einstein,Vier Vorlesungen über Relativitätstheorie, held in May 1921 at Princeton University, Fr. Vieweg und Sohn, Braunschweig, 1922, p. 1. English translation in:The Meaning of Relativity, Princeton University Press, Princeton, 1945.

  22. Cf. L. Königsberger,Die Mathematik eine Geistes-oder Naturwissenschaft?, Jahresbericht der Deutschen Mathematiker-Vereinigung 23 (1914), 1–12.

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  23. In a letter of 2 July 1830, to A. M. Legendre, cf. C. G. J. Jacobi,Gesammelte Werke, G. Riemer, Berlin, 1881-1891, Vol. 1, pp. 453-455. Since this statement is sometimes misquoted, we prefer to give here its original context: “Mais M. Poisson n’aurait pas dû reproduire dans son rapport une phrase peu adroite de feu M. Fourier, où ce dernier nous fait des reproches, à Abel et à moi, de ne pas nous être occupés de préférence du mouvement de la chaleur. Il est vrai que M. Fourier avait ľopinion que le but principal des mathématiques était ľutilité publique et ľexplication des phénomènes naturels; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est ľhonneur de ľesprit humain et que sous ce titre une question de nombres vaut autant qu’une question du système du monde.”

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Borel, A. Mathematics: art and science. The Mathematical Intelligencer 5, 9–17 (1983). https://doi.org/10.1007/BF03026504

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