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Husserl and Hilbert

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Essays on Husserl's Logic and Philosophy of Mathematics

Part of the book series: Synthese Library ((SYLI,volume 384))

Abstract

The paper examines Husserl’s (1859–1938) phenomenology and Hilbert’s (1862–1943) view of the foundations of mathematics against the backdrop of their lifelong friendship. After a brief account of the complementary nature of their early approaches, the paper focuses on Husserl’s Formale und transzendentale Logik (1929) viewed as a response to Hilbert’s “new foundations” developed in the 1920s. While both Husserl and Hilbert share a “mathematics first,” nonrevisionist approach toward mathematics, they disagree about the way in which the access to it should be construed: Hilbert wanted to reach it and show it consistent by his formalism on the basis of sensuous signs, Husserl held that there should be a reduction to elementary judgements about individuals. Husserl’s reduction does not establish the consistency of mathematics but he claims it is important for the considerations of truth.

I wish to thank Øystein Linnebo, Matti Eklund, Besim Karakadilar, Mitsuhiro Okada, Volker Peckhaus, and Sören Stenlund for valuable feedback on earlier versions of this paper.

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Notes

  1. 1.

    Peckhaus 1990, 2016–208; Husserl Archive Mitteilungsblatt 34, 12, see also Hill & Da Silva 2013.

  2. 2.

    Peckhaus 1990, 208–210.

  3. 3.

    The most recent details of Husserl’s and Hilbert’s friendship and how it also extended to their families have been documented in the Husserl Archive Leuven Mitteilungsblatt 36, 2013.

  4. 4.

    Husserl 1994, 119.

  5. 5.

    Husserl 1974. Henceforth cited as FTL. English translations refer to (Husserl 1969) unless otherwise indicated.

  6. 6.

    FTL, §89b.

  7. 7.

    cf. Ferreirós 2007, 31. Soon after the move to Göttingen, Husserl’s wife, Malvine, reported that in Göttingen a „ganz anderer Zug im geistigen Leben der Universität als in Halle <herrsche>, u. besonders sind es die Mathematiker (Klein u. Hilbert), die Edmund in ihren Kreis ziehen u. ihn <…> anregen.” David Hilbert and Edmund Husserl developed a „tiefe achtungsvolle Freundschaft” which, according to Husserl’s wife, was a consequence of the „gleichen Ethos einer restlosen Hingabe an sein Werk” (Husserl Archive Mitteilungsblatt 2013, 15).

  8. 8.

    Rowe 1989, 198.

  9. 9.

    Cited from Rowe 1989, 212.

  10. 10.

    Husserl 1975, henceforth cited as Prolegomena, §69, translation modified.

  11. 11.

    Mahnke 1977 [1923], 75.

  12. 12.

    Prolegomena, §71.

  13. 13.

    Loc. cit.

  14. 14.

    Hilbert 1900a, 1092–1093.

  15. 15.

    Hilbert 1950, 15.

  16. 16.

    Hilbert 1900a, 1094.

  17. 17.

    Awodey & Reck 2001, 11–20.

  18. 18.

    Hilbert 1900a, 1095.

  19. 19.

    Hilbert 1900b, 1104.

  20. 20.

    Sieg 1999, 12.

  21. 21.

    Op. cit., 23, Sieg 2013, 115.

  22. 22.

    Mahnke 1977, 76–77.

  23. 23.

    Op. cit., 465.

  24. 24.

    Op. cit., 464–465.

  25. 25.

    Op. cit., 465.

  26. 26.

    Op. cit., 467.

  27. 27.

    Rowe 1989, 199–200.

  28. 28.

    Op. cit., 211.

  29. 29.

    Sieg 2013, 106.

  30. 30.

    Op. cit., 316.

  31. 31.

    Sieg 2009, 450.

  32. 32.

    In Husserl’s words: “Eine axiomatisch definierte Mannigfaltigkeit kann die Eigenschaft haben, daβ jedes ihrer Objekte operativ bestimmbar ist, und zwar eindeutig. D. h. jedes Objekt, das für sie als existierend definiert ist (in die Sphäre der Existenz gehört, welche die Axiome umschreiben), ist durch die zugrunde liegenden oder eine endliche Zahl willkürlich anzunehmender bestimmter Existenzen unmittelbar oder mittelbar zu bestimmen, und zwar eindeutig. Eine solche Mannigfaltigkeit ist eine mathematische und ist definit (d.h. ihr Axiomensystem ist definit). […] Relativ definit ist ein Axiomensystem, wenn es zwar für sein Existential gebiet keine Axiome mehr zuläβt, aber es zuläβt, daβ weiteres Gebiet dieselben und dann natürlich auch neue Axiome gelten. Neue Axiome, denn die bloβ alten Axiome bestimmen ja nur das alte Gebiet. Relativ definit ist die Sphäre der ganzen, der gebrochenen Zahlen, der rationalen Zahlen, ebenso der diskreten Doppelreihenzahlen (komplexen Zahlen). Absolut definit nenne ich eine Mannigfaltigkeit, wenn es keine andere Mannigfaltigkeit gibt, welche dieselben Axiome hat wie sie (alle zusammen). Kontinuierliche Zahlenreihe, kontinuierliche Doppelzahlenreihe” (Schuhmann & Schuhmann 2001, 101–102).

  33. 33.

    Hilbert 1900, 1094.

  34. 34.

    Op. cit., 103.

  35. 35.

    Op. cit., 102.

  36. 36.

    Husserl 1950, §72.

  37. 37.

    Mahnke 1977, 80.

  38. 38.

    Hence, it has given a rise to several competing interpretations. See for example Lohmar (1989), Da Silva (2000), Centrone (2010), Hartimo (2007), Recently Mitsuhiro Okada has defended a computational view of Husserl’s completeness (Okada 2013).

  39. 39.

    This paper was originally written in 2013 and represents my thoughts about the matter then. More papers on the topic have been published since, most notably, Da Silva (2015) and Hartimo and Okada (2015), and most recently in Hartimo (2016).

  40. 40.

    Mahnke 1977, 77.

  41. 41.

    Within a few months after Husserl’s Definitheit lectures Hilbert showed Husserl his so called Memoir, the second foundations to geometry, on which Husserl took detailed notes (cf. Hartimo 2008). Husserl’s interest in it, like Hilbert’s, shows his unprejudiced interest in different kinds of axiomatic systems. Husserl was also well aware about the set theoretical paradoxes that plagued Hilbert’s school. Zermelo’s version of ‘Russell’s paradox’ has been found written down by Husserl (Husserl 1979, 399). Hilbert also showed Husserl his correspondence with Frege about the nature of the axioms in geometry. Husserl’s comment to the exchange is that Frege does not understand Hilbert’s axiomatic foundations of geometry (Husserl 1970, 447–451). Husserl was also aware of the contents of Hilbert’s 1905 lectures thanks to Dietrich Mahnke, who sent the lecture notes for him. In that connection Husserl expressed the wish, „recht viel aus Hilberts Darstellungen zu lernen, wie es ja eigentlich selbstverständlich ist” (Husserl Archive Leuven Mitteilungsblatt 2013, 15).

  42. 42.

    Mahnke 1977, 78.

  43. 43.

    Op. cit., 79.

  44. 44.

    Loc. cit.

  45. 45.

    Op. cit., 80.

  46. 46.

    Op. cit., 80–81.

  47. 47.

    Op. cit., 81.

  48. 48.

    Op. cit., 82.

  49. 49.

    FTL, 11.

  50. 50.

    During his Freiburg years Husserl considered himself as an old friend of Hilbert’s household. Husserl visited Hilbert about which he reported to Heidegger that Hilbert’s reception had been very friendly, „Sehr freundschaftlich kam uns Hilbert entgegen” (Husserl to Heidegger 9.5.1928, Husserl-Archive Mitteilungsblatt 26, 2013, 16).

  51. 51.

    FTL, §69.

  52. 52.

    Hilbert 1927, 475.

  53. 53.

    FTL, §70a.

  54. 54.

    Op. cit., §71.

  55. 55.

    Op. cit., §13.

  56. 56.

    For a nice exposition of Husserl’s conception of logic in FTL, see Cavaillès 1970, 386–409.

  57. 57.

    Op. cit., §82.

  58. 58.

    Op. cit., §51.

  59. 59.

    Op. cit., §52.

  60. 60.

    Op. cit., §18.

  61. 61.

    Op. cit., §89a.

  62. 62.

    Op. cit., §89a.

  63. 63.

    Op. cit., §89b.

  64. 64.

    Loc. cit.

  65. 65.

    FTL, §89 a–b.

  66. 66.

    Op. cit., §90.

  67. 67.

    Op. cit., §89b.

  68. 68.

    Op. cit., §70a.

  69. 69.

    ‘Categorial intuition’ is Husserl’s term for perception of formal structures, typically states of affairs. From Husserl’s remarks it is difficult to say what everything could be an object of categorial intuition. Dietrich Mahnke and Oskar Becker discussed this matter and disagreed about it: Becker held that categorial intuition is restricted to human consciousness and that we cannot intuit transfinite elements. Mahnke thought that this is not the case, and that the consciousness in question is an ideal consciousness (Mancosu & Ryckman 2010, 350–355). Given Husserl’s overall non-revisionist attitude Mahnke’s view about the matter seems to be closer to Husserl’s intentions.

  70. 70.

    Ebbinghaus 2007, 156.

  71. 71.

    FTL, §34.

  72. 72.

    Hilbert, 1927, 475.

  73. 73.

    FTL, §70.

  74. 74.

    FTL, §82.

  75. 75.

    FTL, §82.

  76. 76.

    FTL, §82.

  77. 77.

    FTL, §19.

  78. 78.

    Sieg 2013, 17.

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  1. Norwegian University of Life Sciences, P.O. Box 5003, 1432, Aas, Norway

    Mirja Hartimo

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  1. Institut für Philosophie, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany

    Stefania Centrone

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Hartimo, M. (2017). Husserl and Hilbert. In: Centrone, S. (eds) Essays on Husserl's Logic and Philosophy of Mathematics. Synthese Library, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1132-4_11

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