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

- 3.
Ryckman (2005, pp. 73–74).

- 4.
See Friedman (1997).

- 5.
Ryckman (2005, Ch. 5).

- 6.
See Scholz (2004).

- 7.
Weyl (1923, p. 1).

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

- 10.
Weyl (1921/1952, pp. 96–97).

- 11.
Weyl (1921/1952, p. 97).

- 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.
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.
Weyl (1923, Vorrede).

- 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.
Lie (1893, pp. 437–471).

- 17.
Weyl (1923, p. 43).

- 18.
Weyl (1923, p. 45).

- 19.
Weyl (1923, p. 46).

- 20.
Weyl (1923, p. 49).

- 21.
Weyl (1923, pp. 44–45).

- 22.
See Biagioli (2014b) for a more thorough discussion of Helmholtz’s stance towards the Kantian theory of space.

- 23.
Königsberger (1903, pp. 126–138).

- 24.
- 25.
Scholz (2013).

- 26.
Here dated 1866. However, this paper first appeared in 1868.

- 27.
- 28.
Helmholtz (1921/1977, p. 153).

- 29.
- 30.
- 31.
- 32.
Lie (1893, pp. 437–471).

- 33.
In the following, I refer to the reprinted version of Klein’s review in

*Mathematische Annalen*(1898). - 34.
Klein (1890, p. 572).

- 35.
Klein (1898, p. 595).

- 36.
- 37.
Klein (1898, p. 593).

- 38.
Klein (1898, p. 597).

- 39.
Klein (1898, p. 599).

- 40.
Klein (1898, p. 598).

- 41.
Helmholtz himself drew attention to the relevance of his psychological investigations to his mathematical considerations in Helmholtz (1870/1977, p. 15).

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

- 46.
Schlick in Helmholtz (1921/1977, p. 167).

- 47.
- 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.
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.
Weyl (1918, p. 73).

- 51.
Klein (1890, p. 572).

- 52.
Ibid.

- 53.
Schlick (1918, p. 301).

- 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.
Weyl (1921/1952, pp. 147–148).

- 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.
See Sect. 3.2.

- 58.
Weyl (1918, pp. 70–71).

- 59.
Weyl (1921/1952, p. 8).

- 60.
von Helmholtz (1878a/1977, p. 158).

- 61.
See Scholz (2005).

## References

Beltrami, Eugenio. 1869. Teoria fondamentale degli spazi a curvatura costante. In

*Opere Matematiche 1*, 406–429. Milano: Hoepli. 1902.Biagioli, Francesca. 2014a. Hermann Cohen and Alois Riehl on geometrical empiricism.

*HOPOS: The Journal of the International Society for the History of Philosophy of Science*4: 83–105.———. 2014b. What does it mean that ‘space can be transcendental without the axioms being so’? Helmholtz’s claim in context.

*Journal for General Philosophy of Science*45: 1–21.———. 2016.

*Space, number, and geometry from Helmholtz to Cassirer*. Cham: Springer.Birkhoff, Garrett, and Mary Katherine Bennett. 1988. Felix Klein and his

*Erlanger Programm*(Please reintroduce capital letters as shown in my original manuscript. This and the other instances marked below are German nouns, which are always written with capital letters, whether they occur in a title or in a sentence). In*History and philosophy of modern mathematics*, ed. William Aspray and Philip Kitcher, 144–176. Minneapolis: University of Minnesota Press.Boi, Luciano, Livia Giancardi, and Rossana Tazzioli, eds. 1998.

*La découverte de la géométrie non euclidienne sur la pseudosphère: Les lettres d’Eugenio Beltrami à Joules Hoüel; (1868–1881)*. Paris: Blanchard.Coleman, Robert A., and Herbert Korté. 2001. Hermann Weyl: Mathematician, physicist, philosopher. In

*Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work*, ed. Erhard Scholz, 161–386. Basel: Birkhäuser.DiSalle, Robert. 1993. Helmholtz’s empiricist philosophy of mathematics: Between laws of perception and laws of nature. In

*Hermann von Helmholtz and the foundations of nineteenth-century science*, ed. David Cahan, 498–521. Berkeley: University of California Press.———. 2008.

*Understanding space-time: The philosophical development of physics from Newton to Einstein*. Cambridge: Cambridge University Press.Erdmann, Benno. 1877.

*Die Axiome der Geometrie: Eine philosophische Untersuchung der Riemann-Helmholtz’schen Raumtheorie*. Leipzig: Voss.Friedman, Michael. 1997. Helmholtz’s

*Zeichentheorie*and Schlick’s*Allgemeine Erkenntnislehre*: Early logical empiricism and its nineteenth-century background.*Philosophical Topics*25: 19–50.Hawkins, Thomas. 1984. The

*Erlanger Programm*of Felix Klein: Reflections on its place in the history of mathematics.*Historia Mathematica*11: 442–470.Helmholtz, Hermann v. 1868. Über die Tatsachen, die der Geometrie zugrunde liegen. In

*Helmholtz (1921/1977)*, 38–55.———. 1870. Über den Ursprung und die Bedeutung der geometrischen Axiome. In

*Helmholtz (1921/1977),*1–24.———. 1878a. Die Tatsachen in der Wahrnehmung. In

*Helmholtz (1921/1977),*109–152.———. 1878b. The origin and meaning of geometrical axioms. Part 2.

*Mind*3: 212–225.———. 1883.

*Wissenschaftliche Abhandlungen*. Vol. 2. Leipzig: Barth.———. 1887. Zählen und Messen, erkenntnistheoretisch betrachtet. In Helmholtz (1921/1977), 70–97.

———. 1921/1977.

*Schriften zur Erkenntnistheorie*, ed. Paul Hertz and Moritz Schlick. Berlin: Springer. English edition: Helmholtz, Hermann von. 1977.*Epistemological writings*, Trans. Lowe, Malcom F., ed. Robert S. Cohen and Yehuda Elkana. Dordrecht: Reidel.Hyder, David. 2009.

*The determinate world: Kant and Helmholtz on the physical meaning of geometry*. Berlin: De Gruyter.Klein, Felix. 1871. Über die sogenannte Nicht-Euklidische Geometrie.

*Mathematische Annalen*4: 573–625.———. 1872.

*Vergleichende Betrachtungen über neuere geometrische Forschungen*. Erlangen: Deichert.———. 1890. Zur Nicht-Euklidischen Geometrie.

*Mathematische Annalen*37: 544–572.———. 1898. Gutachten, betreffend den dritten Band der Theorie der Transformationsgruppen von S. Lie anlässlich der ersten Vertheilung des Lobatschewsky-Preises.

*Mathematische Annalen*50: 583–600.Königsberger, Leo. 1902–1903.

*Hermann von Helmholtz*, vols. 3. Braunschweig: Vieweg.Land, Jan Pieter Nicolaas. 1877. Kant’s space and modern mathematics.

*Mind*2: 38–46.Lenoir, Timothy. 2006. Operationalizing Kant: Manifolds, models, and mathematics in Helmholtz’s theories of perception. In

*The Kantian legacy in nineteenth-century science*, ed. Michael Friedman and Alfred Nordmann, 141–210. Cambridge, MA: The MIT Press.Lie, Sophus. 1893.

*Theorie der Transformationsgruppen*. Vol. 3. Leipzig: Teubner.Riemann, Bernhard. 1854/1867. “Über die Hypothesen, welche der Geometrie zu Grunde liegen.”

*Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen*13: 133–152.Rowe, David E. 1989. Klein, Hilbert, and the Göttingen mathematical tradition.

*Osiris*5: 186–213.———. 1992. Klein, Lie, and the Erlanger Programm. In

*1830–1930: A century of geometry, epistemology, history and mathematics*, ed. Luciano Boi, Dominique Flament, and Jean- Michel Salanskis, 45–54. Berlin: Springer.Ryckman, Thomas. 2005.

*The reign of relativity: Philosophy in physics, 1915–1925*. New York: Oxford University Press.Schlick, Moritz. 1918.

*Allgemeine Erkenntnislehre*. Berlin: Springer.Scholz, Erhard. 2001. Weyl’s Infinitesimalgeometrie, 1917–1925. In

*Hermann Weyl’s Raum-Zeit-Materie and a general introduction to his scientific work*, ed. Erhard Scholz, 48–104. Basel: Birkäuser.———. 2004. Hermann Weyls analysis of the problem of space and the origin of gauge structures.

*Science in Context*17: 165–197.———. 2005. Local spinor structures in V. Fock’s and H. Weyl’s work on the Dirac equation (1929). In

*Géométrie au XXe siècle, 1930–2000: histoire et horizons*, ed. Joseph Kouneiher, Philippe Nabonnand, and Jean-Jacques Szczeciniarz, 284–301. Montréal: Presses internationales Polytechnique.———. 2013. The problem of space in the light of relativity: The views of H. Weyl and E. Cartan. E-Print: https://arxiv.org/abs/1310.7334, accessed on March 8, 2019)

Sigurdsson, Skuli. 1994. Unification, geometry and ambivalence: Hilbert, Weyl and the Göttingen community. In

*Trends in the historiography of science*, ed. Kostas Gavroglu, Jean Christianidis, and Efthymios Nicolaidis, 355–367. Dordrecht: Springer.Torretti, Roberto. 1978.

*Philosophy of geometry from Riemann to Poincaré*. Dordrecht: Reidel.Weyl, Hermann. 1918.

*Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis*. Leipzig: Veit.———. 1921/1952.

*Raum-Zeit-Materie: Vorlesungen über allgemeine Relativitätstheorie*, 4th ed. Berlin: Springer. English edition: Weyl, Hermann.*Space-time-matter*. Trad. Henry L. Brose. New York: Dover, 1952.———. 1923.

*Mathematische Analyse des Raumproblems*. Berlin: Springer.———. 1949.

*Philosophy of mathematics and natural science*. Princeton: University Press. Trad. Helmer, Olaf. An expanded English version of*Philosophie der Mathematik und Naturwissenschaft*, München, Leibniz Verlag, 1927.———. 2015.

*L’analyse mathématique du problème de l’espace*, ed. Éric Audureau and Julien Bernard. Aix-en-Provence: Presses universitaires de Provence.

## 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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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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