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Cultures of Creativity

Mathematics and Physics

Published online by Cambridge University Press:  28 February 2024

Arthur I. Miller

Extract

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The cultures here in question are those of mathematics and of physics that I shall interpret with the goal of exploring different modes of creativity. As case studies I will consider two scientists who were exemplars of these cultures, the mathematician Henri Poincaré (1854-1912) and the physicist Albert Einstein (1879-1955). The modes of creativity that I will compare and contrast are their notions of aesthetics and intuition. In order to accomplish this we begin by studying their introspections.

Type
Research Article
Information
Diogenes , Volume 45 , Issue 177 , March 1997 , pp. 53 - 72
Copyright
Copyright © 1997 Fédération Internationale des Sociétés de Philosophie / International Federation of Philosophical Societies (FISP)

References

Notes

1. H. Poincaré, "L'Invention mathématique," in Science et méthode, Paris, 1908, pp. 43-63.

2. E. Toulouse, Enquète médico-psychologique sur la supériorité intellectuelle: Henri Poincaré, Paris, 1910.

3. Among these materials, which had disappeared since Poincaré's death in 1912, figure manuscripts and letters that considerably augmente our knowl edge of the history of physics and mathematics, but also of the history of sci ence in France. These archives have been placed at the Université Nancy-II, publication center of several volumes of correspondence and manuscripts.

4. For biographical details as well as further discussion of Toulouse's psycholog ical data see A.I. Miller, Insights of Genius: Imagery and Creativity in Science and Art, New York, 1996.

5. E. Toulouse, (note 2 above), p. 136.

6. Ibid., pp. 145-146 and 187, my emphasis.

7. Ibid., p. 200.

8. But I am not claiming that Poincaré considered visual imagery to be ancillary to scientific research. Quite the contrary. For example, Poincaré's pioneering research on dynamical systems was ingeniously carried out using topological concepts in two and three dimensions in lieu of rigorous and far too difficult investigations of the exact solutions of certain differential equations. Such rig orous investigations, concluded Poincaré, would have obscured essential points. He was right. See H. Poincaré, "Sur le problème des trois corps et les équations de la dynamique," in Acta Mathematica, vol. 13,1890, pp. 27-40.

9. H. Poincaré, "Du rôle de l'intuition et de la logique en mathématiques," in La valeur de la science, 1905; all references are from the 1970 edition, Paris, p. 33.

10. Ibid., p. 32, emphasis in the original.

11. Ibid., p. 40.

12. H. Poincaré, (note 1 above), p. 47.

13. H. Poincaré, (note 8 above), p. 22.

14. H. Poincaré, (note 1 above), p. 56.

15. For discussion of language-like and analog representations see A.I. Miller, (note 2 above), chapters 8 and 9.

16. H. Poincaré, (note 1 above), p. 58.

17. Ibid., p. 49.

18. Ibid., p. 43.

19. Ibid., p. 49.

20. Ibid., p. 56.

21. H. Poincaré, "Prefatory to Translation," in The Value of Science, trans. G.B. Hal sted, New York, 1958, p. 8.

22. See, for example, M. Boden, Creative Mind: Myths and Mechanisms, London, 1990; and R. Penrose, The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford, 1989.

23. For more details see A.I. Miller, "Scientific Creativity: A Comparative Study of Henri Poincaré and Albert Einstein," in Creativity Research Journal, vol. 5, pp. 385-418.

24. H. Poincaré, (note 1 above), p. 51.

25. Ibid.

26. See A.I. Miller, (note 23 above), p. 418.

27. H. Poincaré, (note 1 above), p. 52.

28. W. Hildesheimer, Mozart, trans. M. Faber, New York, p. 238.

29. H. Poincaré, "L'avenir des mathématiques," in Science et méthode, (note 1 above), p. 30.

30. For discussion of this episode see A.I. Miller, "Why did Poincaré not Formu late Special Relativity in 1905?," in J.L. Greffe, G. Heinzmann and K. Lorenz (eds.), Henri Poincaré: Science and Philosophy, Berlin, pp. 69-100. Some scholars interpret the situation to the effect that both men arrived at the special theory of relativity and, consequently, Poincaré ought to share the accolades with Einstein. Existing archival and primary sources render the co-discovery claim insupportable.

31. Einstein discussed scientific creativity beginning in 1916 with his colleague at the University of Berlin, the Gestalt psychologist Max Wertheimer. But Wertheimer's subsequent published analysis of Einstein's discovery of special relativity turns out to be a reconstruction according to the guidelines of Gestalt psychology and is lacking in historicity. See A.I. Miller, Imagery in Sci entific Thought: Creating 20th Century Physics, Cambridge, 1986, chapter 5.

32. L. Boltzmann, "Lectures on the Principles of Mechanics," in B. McGuiness (ed.), Ludwig Boltzmann: Theoretical Physics and Philosophical Problems, trans. R. Foulkes, Boston, 1974, p. 225.

33. J. Renn and R. Schulmann (eds.), Albert Einstein, Mileva Maric: The Love Letters, trans. by S. Smith, Princeton, 1992.

34. J. Stachel, D.C. Cassidy, and R. Schulmann (eds.), The Collected Papers of Albert Einstein: Volume I, The Early Years, 1879-1902, Princeton, 1987, p. 284-285.

35. Ibid., pp. 332-333.

36. These issues are discussed in A.I. Miller, Albert Einstein's Special Theory of Rela tivity : Emergence (1905) and Early Interpretation (1905-1911), MA, 1981.

37. A. Einstein, "Prinzipien der Forschung," (How I see the World) in Mein Welt bild, Berlin, 1934, p. 109.

38. For further discussion see A.I. Miller, (note 3 above).

39. J. Hadamard, The Psychology of Invention in the Mathematical Field, New York, 1954, pp. 142-143.

40. A. Einstein, "Autobiographical Notes," in P.A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist, IL, 1949, pp. 2-94.

41. Ibid., p. 9.

42. Ibid.

43. A. Einstein, "Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung," in Physikalische Zeitschrift, vol. 10, 1909, pp. 817-825.

44. C. Seelig, Albert Einstein: Eine dokumentarische Biographie, Zürich, 1954, p. 89.

45. A. Einstein, "Über einen die Erzeugung und Verwandlung des Lichtes betref fenden heuristischen Gesichtspunkt," in Annalen der Physik, vol. 17, 1905, pp. 132-148.

46. A. Einstein, "Zur Elektrodynamik bewegter Körper," in Annalen der Physik, vol. 17, 1905, pp. 891-921.

47. A. Einstein, (note 40 above), p. 53.

48. J. Ishiwara, Einstein Kyôzyu-Kôen-roku, Tokyo, 1971.

49. A.I. Miller, "Albert Einstein's 1907 Jahrbuch Paper: The First Step from SRT to GRT," in J. Eisenstaedt and A.J. Kox (eds.), Studies in the History of General Rel ativity, Boston, 1992, pp. 319-335, quote on p. 325.

50. N. Stangos (ed.), Concepts of Modern Art, London, 1991, p. 130.

51. From an interview of 2 December 1933 of Picasso with Daniel-Henry Kahn weiler, in Le Point, vol. XLII, October 1952, p. 24; reproduced in Huit entretiens avec Picasso, Paris, 1988.

52. H.B. Chipp (ed.), Theories of Modern Art: A Source Book by Artists and Critics, Los Angeles, 1968, p. 548.