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Neighbourhoods and Intersubjectivity

Analogies Between Weyl’s Analyses of the Continuum and Transcendental-Phenomenological Theories of Subjectivity

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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))

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Abstract

This paper shows the interdisciplinary prominence of the concept of neighbourhood (Umgebung) in the work of Weyl. It tracks different appearances of this notion in the contexts of mathematics, physics, and the theory of subjectivity, and describes them as variants of one another. The historical and systematic background of this continuous reliance on (variations of) the concept of neighbourhood is traced back in large part to the interaction between Weyl and the philosopher Fritz Medicus, who introduced Weyl to the work of Fichte. The importance of certain Fichtean concepts, such as (inter)subjectivity and recognition, for Weyl’s analyses of the continuum is shown by unfolding various analogies Weyl provided in his mathematical writings.

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Notes

  1. 1.

    See, for instance, Sieroka (2010b).

  2. 2.

    See, e.g., Tieszen (2000) and Ryckman (2005) regarding Husserl, Eckes (2011) regarding Kant, and Scholz (1995) and Sieroka (2010a) regarding Fichte.

  3. 3.

    Cf., for instance, Debs and Redhead (2007).

  4. 4.

    Cf. Sieroka (2009).

  5. 5.

    See esp. Sigurdsson (1991) for more details on Weyl’s time in Göttingen.

  6. 6.

    In 1946 Weyl wrote retrospectively about his time with Medicus: ‘It was in those years that, not entirely without his influence, I got deeply involved in Fichte and Eckhart, and that later the theory of relativity, the problem of the infinite in mathematics, and finally quantum mechanics became the motivations for my attempts to help clarify the methods of scientific understanding and the theoretical picture of reality as a whole’ (ETH Zurich University Archives, Hs91a: 16).

  7. 7.

    For more details on Weyl’s time and interaction in Zurich, see Sieroka (2007, 2010a). Notably, already before meeting Weyl, Medicus had serious interests in philosophical issues related to the exact sciences. In particular, his doctorate thesis is on Kants transcendentale Aesthetik und die nichteuklidische Geometrie (see Medicus 1899).

  8. 8.

    In Die Freiheit des Willens und ihre Grenzen from 1926 Medicus maintains that ‘Hermann Weyl argued with outmost clarity [against a closed concept of causality in physics …] and with careful attention to the problem of freedom’ (Medicus 1926: 85). In turn, in his PMN (finished the same year and published at the beginning of 1927) Weyl claims that ‘Medicus provides the first philosophical comment on the altered concept of causality which is developing in the physics of our time’ (Weyl 1927: 157).

  9. 9.

    ETH Zurich University Archives, Hs91: 2–3.

  10. 10.

    See, e.g., Weyl (1927: 60, 157). In the later English language edition only the second of these passages occurs (see Weyl 1949: 210). This is because the context of the first passage, where Weyl discusses the concepts of similarity and congruence, has been largely rewritten for that edition.

  11. 11.

    Weyl 1927: 123; see also the English language edition (Weyl 1949: 164).

  12. 12.

    For more details regarding the following sketch of writings and concepts, see Sieroka (2010a: 89–93).

  13. 13.

    Weyl (1913/1997: 17). See also Hilbert (1902).

  14. 14.

    See Weyl (1918a: 80–83); cf. also Scholz (1999), Feferman (2000), as well as Coleman and Korté (2001).

  15. 15.

    Cf. Ryckman (2013).

  16. 16.

    See Weyl (1921a: 177–179), reprinted in Weyl (1968).

  17. 17.

    See Weyl (1923).

  18. 18.

    Weyl (1925a/1988: 5)

  19. 19.

    See Weyl (1925b) and Weyl (1927: 31–34).

  20. 20.

    See Weyl (1925b: 528–34); cf. also Weyl (1927: 42–43). Again, one might wonder why Weyl is not bothered by the fact that Brouwer’s approach presupposes natural numbers.

  21. 21.

    For a detailed discussion of the relevant aspects of Weyl’s physics during that period, see Sieroka (2010a, 2010b).

  22. 22.

    Weyl (1924: 510). This quote stems from the context of Weyl’s agent theory which he held around the mid-1920s. However, related claims can also be found in the context of Weyl’s singularity approach of 1921; and also in 1922 he claims that ‘the matter particle is not even a point in the field space, but nothing spatial (extended) at all. However, it is nested in a spatial neighbourhood from which its field effects originate’ (Weyl in Bovet 1922: 904; an English translation of this text is provided in Weyl 2009: 25–28).

  23. 23.

    Weyl (1921b: 255).

  24. 24.

    Weyl (1924: 510).

  25. 25.

    See Weyl (1920: 122): ‘there remains … a space for autonomous and causally absolutely independent decisions; I consider the elementary quanta of matter to be the place of these decisions.’

  26. 26.

    Fichte 1796/1979: 39, 47) (=Foundations of Natural Right §§3–4).

  27. 27.

    For more details on the reception of Fichte during the first half of the twentieth century see Sieroka (Sieroka 2010a: 141–152).

  28. 28.

    Medicus (1926: 108–109).

  29. 29.

    Medicus (1947/1954: 118, 132).

  30. 30.

    Weyl (1927: 83). See also Sieroka (2018).

  31. 31.

    See, for instance, Weyl (1952).

  32. 32.

    See Weyl (1954: 642).

  33. 33.

    See Goodman (1978).

  34. 34.

    Fichte (1808/1978: 62, 128) (=Addresses to the German Nation).

  35. 35.

    Whereas the political right among the Fichte scholars would usually refer to the Reden, the left-wing readers would usually refer to Fichte’s The Closed Commercial State as masterminding, for instance, socialist five-year plans.

  36. 36.

    See Fichte (1808/1978: 199) (=Addresses to the German Nation). Moreover, Fichte’s claims about the German language must be contextualized to the fact that in 1808 German had been a suitable language in academic and other uplifted and official contexts for about only half a century. Moreover, French was in acute advance again after Napoleonic troops had invaded Berlin in October 1806.

  37. 37.

    Medicus (1930).

  38. 38.

    Regarding religious groups, note that for Cassirer religion is a separate ‘symbolic form’ (see, for instance, Cassirer 1944). This is noteworthy because the aforementioned Goodmanian understanding of symbolic systems is in turn based on a particular interpretation of Cassirer (see Goodman 1978: 1).

  39. 39.

    Medicus (1938).

  40. 40.

    Medicus (1926: 113).

  41. 41.

    Medicus (1947/1954: 118).

  42. 42.

    See, for instance, Weyl (1927: 33).

  43. 43.

    See Weyl (1918a: 70).

  44. 44.

    ‘Every point A determines certain subsystems of points to which the point itself belongs and which are called neighbourhoods of point A … If B is an arbitrary point in a neighbourhood of A, then this neighbourhood is also a neighbourhood of B’ (Hilbert 1902: 234–235).

  45. 45.

    Weyl (1921a: 149–150).

  46. 46.

    A proviso has to be added here though: The reference to love as being poured out implies (in a philosophical technical sense) a notion of (divine) agápe, which fits well with Fichte’s notion of the term. Medicus’s notion from above, however, seems to be nearer to (coequal) philía.

  47. 47.

    Weyl (1918b: 8). A similar phrasing can be found in Weyl (1918a: 72) and also in the English translation of PMN (see Weyl 1949: 123), where it occurs in relation to the analogy of coordinate systems; cf. Sect. 4.5.2 below. Besides, I deliberately put ‘residue’ in brackets here because in the German original the implicit hint at the residue theorem in complex analysis is surely intended. Note that, roughly and metaphorically speaking, a residue is a point which one cannot simply ‘pass over’ (when doing a line integral).

  48. 48.

    Husserl (1913/1977: 91) (=Ideas I: §49). Similarly, one also finds the claim in Husserl that it is the body which marks the origin of one’s orientation in the world – see Husserl 1952: 56, 109 f., 158 f. (=Ideas II).

  49. 49.

    See Weyl (1927: 88–90).

  50. 50.

    Weyl (1927: 89).

  51. 51.

    Weyl (1918b: 3, 82).

  52. 52.

    See Husserl (1950: 90–147) (=Cartesian Meditations §§42–62). I refer to this passage because it is the locus classicus regarding Husserl’s notion on intersubjectivity. I am fully aware though that these meditations were first held as a course of lectures in Paris in 1929 and are thus later than most of the aforementioned writings by Weyl.

  53. 53.

    For more details about the difference between Husserl and Fichte regarding the concepts of ego and intersubjectivity see Tietjen (1980), Kloc-Konkołowicz (2013), and Sieroka (2010a: 115–140).

  54. 54.

    Weyl (1927: 89).

  55. 55.

    See Weyl (1954). An English translation of this paper is given in Weyl (2009: 204–221).

  56. 56.

    However, the analogy of coordinate systems as it occurs in Weyl (1954) differs in some respects from the way it is presented in PMN; for instance, with respect to the existence of an ‘absolute I’, given by the standard basis in which all appearances equal the objects; e.g. \( \overrightarrow{r}={x}_1\left(1,0\right)+{x}_2\left(0,1\right)=\left({x}_1,{x}_2\right) \). For a more detailed discussion of the 1954 version of the analogy, see Bell (2000: 271–272).

  57. 57.

    See Weyl (1954: 642–644).

  58. 58.

    See Weyl (1954: 644–645). Cf., again, also Tietjen (1980) and Kloc-Konkołowicz (2013).

  59. 59.

    See, in particular, ‘the riddle of ego-hood’ (das Rätsel der Ichheit) as discussed in Weyl (1927: 89) and in Weyl (1954: 643).

  60. 60.

    See Weyl (1954: 641, 643): ‘[As compared to Husserl, Fichte] is anything but a phenomenologist, he is a constructivist of the purest sort, who – without looking left or right – goes his own headstrong way of construction ... As for the antagonism between constructivism and phenomenology, on the whole my sympathy is on his side.’ Cf. Breazeale (1996) for a comprehensive discussion of the relation between practical and theoretical philosophy in (esp. the early) Fichte.

  61. 61.

    Medicus (1926: 20, 82–84). Very broadly speaking, one might say that in Kant there are in the end two kinds of ‘the unconditioned’: the thing in itself and spontaneity. Fichte, in contrast, employs only one kind of ‘the unconditioned’ which occurs with different signs though: with an inherent antagonism of real versus ideal action – see Fichte (1794/1997: 211–229) (= Foundations of the Entire Science of Knowledge §§8–19).

  62. 62.

    See Weyl (1927: 157).

  63. 63.

    Weyl (1923: 46).

  64. 64.

    See Scholz (1994) and Coleman and Korté (2001) for more details on Weyl’s notion of ‘nature of a metric’.

  65. 65.

    Remember also how Weyl himself referred to transformations and invariances when introducing idealism by means of the analogy of coordinate systems (‘carrying over the arithmetical appearances to the viewpoint of invariances’; cf. above).

  66. 66.

    As far as the philosophy of science is concerned, see in particular Debs and Redhead (2007). (I might be allowed to add though that I disagree with some of the details of their interpretation of Weyl, especially as it makes him too much of a scientific realist.) Regarding a broader philosophical setup, the most prominent and impressive attempt here is surely Nozick (2001).

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Acknowledgement

I would like to thank the audience at the international workshop ‘Weyl and the Problem of Space: From Science to Philosophy’ for discussion and, in particular, Julien Bernard and Carlos Lobo for all their efforts in organising the workshop and rendering this volume possible. I would also like to thank Jonathan Lorand and Erhard Scholz for their very helpful and much appreciated comments on earlier drafts of this paper.

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  1. Institute of Philosophy, University of Bremen, Bremen, Germany

    Norman Sieroka

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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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Sieroka, N. (2019). Neighbourhoods and Intersubjectivity. 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_4

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