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Mathematics and Mysticism

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L.E.J. Brouwer – Topologist, Intuitionist, Philosopher

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Abstract

Brouwer showed during his study his potential in mathematics by proving a significant decomposition theorem for four-dimensional rotations. During his student years Brouwer suffered from nervous breakdowns and minor physical problems. The cause was his military service, which brought him in a company with little patience for a young clever boy. The two basic exams at the university were passed cum laude. After the last one (1904) Korteweg accepted him as a Ph.D. student. 1904 brought more changes: he married the daughter, Lize, of the widow of a family doctor, bought a cottage, called the hut, in the countryside not far from Amsterdam. The architect was another friend from his student years, Ru Mauve (son of the famous painter). The hut allowed him to work in peace on his ambitious dissertation. In 1905 Brouwer lectured in Delft on “Life, Art, and Mysticism”, a provocative view of a true mystic on life, society, language, …, which contained some elements of his philosophical views that came to underlie his intuitionistic mathematics.

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Notes

  1. 1.

    The Nobel prize winner Van het Hoff had already left Amsterdam for Berlin.

  2. 2.

    Ritter (1898), p. 70 ff.

  3. 3.

    Communication of L. van den Brom.

  4. 4.

    Literally ‘a person who should become a doctor’; a prerequisite for being admitted as a candidate for a doctorate.

  5. 5.

    The equation also occurred in the dissertation of De Vries. A recent book of Willink cites evidence that the role of De Vries was modest, to say the least; cf. Willink (1998).

  6. 6.

    sympatisch uurwerk, Korteweg (1905).

  7. 7.

    Instituut Schreuders.

  8. 8.

    Mannoury (1898a, 1898b, 1900).

  9. 9.

    van Dantzig (1957), p. 7.

  10. 10.

    Peano (1895).

  11. 11.

    The position of privaat docent, similar to Privatdozent in Germany, brought the bearer of the title a nominal fee. Its main attraction was that it enabled one to keep a foothold in academic life, in the hope of a promotion.

  12. 12.

    Brouwer (1946).

  13. 13.

    Wiessing (1960), p. 142.

  14. 14.

    Communicated respectively at the meetings of 27 February 1904, 23 April 1904, 23 April 1904.

  15. 15.

    CW II, p. 22.

  16. 16.

    Brouwer’s theorem in modern formulation reads SO 4≅SU 2×SU 2/±(1,1). Another geometric proof is in Klein (1890), and a similar theorem can be found in Cartan (1914).

  17. 17.

    At that moment Oberlehrer at the Friedrich–Werderschen Oberreal Schule and a Privat Dozent at the Technische Hochschule Berlin.

  18. 18.

    Brouwer to Jahnke, 20 March 1904.

  19. 19.

    The contents of the letter are incorporated in Brouwer (1904).

  20. 20.

    See Freudenthal’s commentary, CW II, p. 22 ff.

  21. 21.

    Jahnke (1904).

  22. 22.

    Caspary (1883).

  23. 23.

    Brouwer to Korteweg, 14 May 1904.

  24. 24.

    Brouwer to Scheltema, 15 November 1903.

  25. 25.

    Draft of a letter to Lorentz 16 February 1918.

  26. 26.

    Brouwer to Scheltema, 4 July 1904.

  27. 27.

    See p. 59. This cottage was referred to as ‘the hut’.

  28. 28.

    Let nobody enter without the knowledge of geometry (Plato).

  29. 29.

    Brouwer to Scheltema, 11 June 1902.

  30. 30.

    Scheltema to Brouwer, 12 June 1902.

  31. 31.

    Most of the information on the courtship and the marriage is from oral communications of Louise Peijpers. Confirmation of the information on Dina Pelswas provided by Mrs. J. Schout-de Waal.

  32. 32.

    See p. 6.

  33. 33.

    In other accounts of the same events the roof is replaced by a tree.

  34. 34.

    Fifteenth century Flemish painter.

  35. 35.

    Cf. the letter from Brouwer to Scheltema, 4 July 1904.

  36. 36.

    Presumably this was one of the uncles, a teacher at the Barlaeus Gymnasium.

  37. 37.

    In this respect Bolland was not an exception. Cf. Wiessing (1960), p. 241.

  38. 38.

    Brouwer to Bolland, 5 March 1904 (Boll. B 1904, 29. Leiden University Library).

  39. 39.

    Zuivere Rede, Bolland (1904).

  40. 40.

    Brouwer to Scheltema, 8 October 1904.

  41. 41.

    One of the lodgers of the Brouwer family went by the name of Lau van der Zee, cf. p. 8. In the Brouwer archive there is a manuscript in Brouwer’s handwriting, signed by Lau van der Zee and subsequently published in Propria Cures. So the identity of this ‘Lau van der Zee’ is well authenticated.

  42. 42.

    The reader may find more about Bolland in a recent biography (Otterspeer 1995) (Dutch).

  43. 43.

    Bolland (1897).

  44. 44.

    Communicated by Mrs. N. Kapteyn-Meerum-Terwogt, the daughter of the above mentioned Meerum-Terwogt. No details of the conversation between Bolland and Brouwer are known.

  45. 45.

    kolonies. Cf. Boersen (1987), Heyting (1994).

  46. 46.

    There is a two-volume biography of Van Eeden (in Dutch) by Fontijn, cf. Fontijn (1990, 1996).

  47. 47.

    Oral communication Louise Peijpers.

  48. 48.

    Studenten-weekblad 6 October 1904.

  49. 49.

    Studenten-weekblad 17 November 1904.

  50. 50.

    Studenten-weekblad, 6 April 1905.

  51. 51.

    Brouwer to Scheltema, 7 April 1905.

  52. 52.

    “If Truth points in the world to the personal life, free from the ties of fear and desire, where the bliss and wisdom and the quiet rejoicing of the timing in upon oneself flourish on modesty, poverty and quiet fulfilment of duty in this life on earth, which is one’s own accomplished karma, then it is Transcendent Truth.”

  53. 53.

    Cf. Dijkstra (1986).

  54. 54.

    Scheltema to Brouwer, 16 May 1905.

  55. 55.

    Korteweg to Brouwer, 13 May 1905.

  56. 56.

    frou-frou, a special kind of wafer.

  57. 57.

    When in 1981 a conference was dedicated to the centenary of Brouwer’s birth, a short biographical article appeared in the Dutch weekly Vrij Nederland, van Dalen (1981). Reading this biography, Louise suddenly realised that Brouwer was not a fool after all. As she regularly communicated with the spirits of the departed, she noted that Brouwer’s spirit had found rest after this public recognition.

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Authors and Affiliations

  1. Department of Philosophy, Utrecht University, Utrecht, Netherlands

    Dirk van Dalen

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  1. Dirk van Dalen
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van Dalen, D. (2013). Mathematics and Mysticism. In: L.E.J. Brouwer – Topologist, Intuitionist, Philosopher. Springer, London. https://doi.org/10.1007/978-1-4471-4616-2_2

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