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Intuition and Conceptual Construction in Weyl’s Analysis of the Problem of Space

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Weyl and the Problem of Space

Part of the book series: Studies in History and Philosophy of Science ((AUST,volume 49))

Abstract

Hermann Weyl adopted the Kantian definition of space as a form of intuition and referred to Edmund Husserl’s phenomenological approach for the philosophical characterization of space in the introduction to Raum-Zeit-Materie (1918) and other writings from the same period (1918–1923). At the same time, Weyl emphasized that subjective factors are completely excluded from the mathematical construction of physical reality in Albert Einstein’s general theory of relativity, with the sole exception of the setting of a coordinate system, which for Weyl is what remains of the original perspective of the self in becoming aware of one’s own intuitions. This paper reconsiders Weyl’s philosophical position as a possible response to the earlier debate on the relation between intuition and conceptual construction in the foundation of geometry, key figures of which, besides Husserl, included Hermann von Helmholtz, Felix Klein, and Moritz Schlick.

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Notes

  1. 1.

    I refer to Scholz (2013) for a survey of different problems of space from the classical problem posed by Helmholtz to Weyl’s and Cartan’s problem of reformulating the older criteria for determining the structure of space in the light of differential geometry and the general theory of relativity (i.e., the modern or relativistic problem of space).

  2. 2.

    On Helmholtz’s empiricist philosophy of mathematics, see esp. DiSalle (1993).

  3. 3.

    Ryckman (2005, pp. 73–74).

  4. 4.

    See Friedman (1997).

  5. 5.

    Ryckman (2005, Ch. 5).

  6. 6.

    See Scholz (2004).

  7. 7.

    Weyl (1923, p. 1).

  8. 8.

    Helmholtz (1870). We have already mentioned that Helmholtz had posed the classical problem of space as the problem of establishing the necessary and sufficient conditions for obtaining a Riemannian metric of constant curvature in Helmholtz (1868). However, it was only after his correspondence with Beltrami, in 1869, that he became aware of the fact that his previous characterization of space included non-Euclidean manifolds of constant curvature. The relevant correspondence between Beltrami and Helmholtz is now available in Boi et al. (1998, pp. 204–205).

  9. 9.

    See, for example, Erdmann’s (1877) then popular exposition.

  10. 10.

    Weyl (1921/1952, pp. 96–97).

  11. 11.

    Weyl (1921/1952, p. 97).

  12. 12.

    Weyl (1921/1952, pp. 138–148). The question whether the group-theoretical view provides a suitable interpretation of Helmholtz’s thought experiments, as Weyl assumes, is discussed in the next section.

  13. 13.

    A French translation of this text, along with notes drawing the relevant comparisons with the original version of Weyl’s lectures, has been made available by Audureau and Bernard (Weyl 2015)

  14. 14.

    Weyl (1923, Vorrede).

  15. 15.

    For a detailed discussion of Weyl’s approach to the problem of space and its development from 1921 to 1923, see Coleman and Korté (2001).

  16. 16.

    Lie (1893, pp. 437–471).

  17. 17.

    Weyl (1923, p. 43).

  18. 18.

    Weyl (1923, p. 45).

  19. 19.

    Weyl (1923, p. 46).

  20. 20.

    Weyl (1923, p. 49).

  21. 21.

    Weyl (1923, pp. 44–45).

  22. 22.

    See Biagioli (2014b) for a more thorough discussion of Helmholtz’s stance towards the Kantian theory of space.

  23. 23.

    Königsberger (1903, pp. 126–138).

  24. 24.

    Königsberger (1903, p. 34). The English translation of this and other passages is found in Hyder (2009, pp.140–146).

  25. 25.

    Scholz (2013).

  26. 26.

    Here dated 1866. However, this paper first appeared in 1868.

  27. 27.

    In Helmholtz (1883, p. 660). I quoted from the English translation of the same passage in Helmholtz (1921/1977, pp. 162–163).

  28. 28.

    Helmholtz (1921/1977, p. 153).

  29. 29.

    Helmholtz (1921/1977, p. 143). This passage was added in the full paper “Die Tatsachen in der Wahrnehmung,” but not in the shortened version of the paper that appeared in Helmholtz (1883).

  30. 30.

    Arguably, this was one of the main reasons for Helmholtz’s influence on the renewal of Kant’s transcendental philosophy proposed by different directions of neo-Kantianism (Biagioli 2014a, 2016, Ch. 1–2).

  31. 31.

    The latter aspect has been emphasized by several historical studies (see esp. Hawkins 1984; Rowe 1992). Cf. Birkhoff and Bennett (1988) for the opposing view that the Erlangen Program was very influential.

  32. 32.

    Lie (1893, pp. 437–471).

  33. 33.

    In the following, I refer to the reprinted version of Klein’s review in Mathematische Annalen (1898).

  34. 34.

    Klein (1890, p. 572).

  35. 35.

    Klein (1898, p. 595).

  36. 36.

    Klein’s solution to this problem is found in Klein (1890). This is now known as a distinct problem of space, which is called “Clifford-Klein” or “the problem of the form of space” (see Torretti 1978, p. 151).

  37. 37.

    Klein (1898, p. 593).

  38. 38.

    Klein (1898, p. 597).

  39. 39.

    Klein (1898, p. 599).

  40. 40.

    Klein (1898, p. 598).

  41. 41.

    Helmholtz himself drew attention to the relevance of his psychological investigations to his mathematical considerations in Helmholtz (1870/1977, p. 15).

  42. 42.

    Arguably, Schlick commented on Helmholtz’s most philosophical papers (i.e., Helmholtz 1870, 1878a) while leaving to Paul Hertz the comments on Helmholtz’s mathematical papers (Helmholtz 1868, 1887). It might be objected that such a division obscures the connection between the philosophical and the mathematical considerations in Helmholtz’s work. In the following, I suggest that this partly depends on Schlick’s own attempt to clarify the different aspects of Helmholtz’s notion of space.

  43. 43.

    Helmholtz examples include such propositions as: Between two points only one straight line is possible; through any three points a plane can be placed; through any point only one line parallel to a given line is possible (von Helmholtz 1878a/1977, p. 128).

  44. 44.

    Schlick contrasts the received interpretation via a projective metric with Poincaré’s identification of a purely qualitative geometry as his development of “analysis situs,” which became known as “topology” (Schick in Helmholtz 1921/1977, pp. 172–173). It might be added that even more recent axiomatic interpretations of Helmholtz’s distinction differ slightly, although most interpreters identify Helmholtz’s general characterization of space as a differentiable, three-dimensional manifold of constant curvature (Cf. Torretti 1978; Lenoir 2006).

  45. 45.

    Schlick in Helmholtz (1921/1977, p. 172–173).

  46. 46.

    Schlick in Helmholtz (1921/1977, p. 167).

  47. 47.

    On the development of this tradition from Riemann to Klein and Hilbert, see Rowe (1989). For further details about Weyl’s relationship to the Göttingen community, see Sigurdsson (1994).

  48. 48.

    A letter dated December 28, 1920, in which Weyl informs Klein about the group-theoretical treatment of metrical space in the fourth edition of Raum-Zeit-Materie is found in Klein’s Nachlass. An extract of this letter is quoted by Scholz (2001, p. 87).

  49. 49.

    Weyl distanced himself from his earlier approach in the course of his mathematical analysis of the problem of space – which was largely based on the set-theoretic account of analysis – and later, more explicitly, in Weyl (1949, p. 54), on account of the unjustified limits that Brower’s intuitionism would impose on mathematical practice.

  50. 50.

    Weyl (1918, p. 73).

  51. 51.

    Klein (1890, p. 572).

  52. 52.

    Ibid.

  53. 53.

    Schlick (1918, p. 301).

  54. 54.

    Ryckman (2005, pp. 113–114). Weyl’s quotation, in the English translation provided by Ryckman, reads: “In Schlick’s opinion, the essence [Wesen] of the process of cognition is exhausted by [the semiotic character of cognition]. To the reviewer, it is incomprehensible how anyone, who has ever striven for insight [Einsicht], can be satisfied with this. To be sure, Schlick also speaks of ‘acquaintance’ [‘Kennen’, in opposition to cognizing, Erkennen] as the mere intuitive grasping of the given; but he says nothing of its structure, also nothing of the grounding connections between the given and the meanings giving it expression. To the extent that he ignores intuition, in so far as it ranges beyond the mere modalities of sense experience, he outrightly rejects self-evidence [die Evidenz] which is still the sole source of all insight.”

  55. 55.

    Weyl (1921/1952, pp. 147–148).

  56. 56.

    In the literature, a closer connection between Riemann and Helmholtz as the proponents of an empiricist view has been emphasized by DiSalle (2008, p. 91): “The empiricist view […] was that dynamical principles – principles involving time as well as space – could force revision of the spatial geometry that had been originally assumed in their development. We might say that this view acknowledges the possibility, at least, that space-time is more fundamental than space.”

  57. 57.

    See Sect. 3.2.

  58. 58.

    Weyl (1918, pp. 70–71).

  59. 59.

    Weyl (1921/1952, p. 8).

  60. 60.

    von Helmholtz (1878a/1977, p. 158).

  61. 61.

    See Scholz (2005).

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Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 715222). Prior to this, research leading to this paper was carried out within the project “Mathematical and Transcendental Method in Ernst Cassirer’s Philosophy of Science”, funded by the Marie Curie Actions in co-funding with the Zukunftskolleg at the University of Konstanz. I wish to thank Julien Bernard, Carlos Lobo, Silvia De Bianchi, Paola Cantù, Thomas Ryckman and Georg Schiemer for helpful comments and discussions of this material.

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  1. Department of Philosophy, University of Vienna, Vienna, Austria

    Francesca Biagioli

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  1. Assistant Professor in Philosophy, Centre Gilles Gaston Granger (CGGG) UMR 7304, Aix-Marseille-University, Marseille, France

    Julien Bernard

  2. Collège International de Philosophie, Paris, France, Centro de Filosofia das Ciencias Universidade de Lisboa, Lisboa, Portugal

    Carlos Lobo

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Biagioli, F. (2019). Intuition and Conceptual Construction in Weyl’s Analysis of the Problem of Space. In: Bernard, J., Lobo, C. (eds) Weyl and the Problem of Space. Studies in History and Philosophy of Science, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-11527-2_12

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