Abstract
In analyzing the problem of space from 1917 to 1923, Hermann Weyl confronted with the philosophical underpinnings of the theories of space. Weyl endorsed the distinction between the question of the essence of space and the question of its objective representation, a distinction that many philosophers, such as Ernst Cassirer, inherited from Immanuel Kant’s philosophy. However, Weyl aimed to offer a reliable alternative to Kant’s transcendental idealism of space and time, by means of mathematics and symbolic construction. The consequences of this move will be analyzed in Weyl’s reflection on the epistemology of science after the 1920s and in his late works, with emphasis on his “Why is the World Four-Dimensional?” (1955): a signature of the fact that the problem of space had open questions that engaged the mathematical physicist throughout his entire life.
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Notes
- 1.
- 2.
Thus, according to Weyl (1923), the nature of the metric (1) is its being a non-singular quadratic differential form, namely it is an orthogonal group Ok where k is the signature of the corresponding form (1). For the generalization of the quadratic form (1) to its automorphism group, see Scheibe (2001), p. 453.
- 3.
Even if in the 1920s Weyl uses the expression “Wesen des Raumes”, he makes clear that he wants possibly to include time as a fourth dimension. This also happens in Weyl (1955b).
- 4.
That in his late writings Weyl changes his mind with respect to embracing Husserl’s phenomenology is pointed out by J. Bell (2004). He also states that Weyl’s late works are closer to Cassirer, even if it must be recalled that Husserl himself in The Crisis assumes a position, which is closer to Cassirer’s doctrine. That Weyl abandons Husserl’s phenomenology emerges in a clear way in Weyl (1955a). For a comparison between Husserl and Weyl, see Feist (2004).
- 5.
On the philosophical underpinnings of Weyl’s notion of space, see Bernard (2015).
- 6.
In the 1955 paper on the dimensionality of the world, Weyl changes his perspective with respect to his early-1920s writings, where he did not present the problem of justifying the Pythagorean metric through physics. For instance, about the physical laws and the problem of dimensionality he says: “Hence in these laws there is no reason to be found for the Creator’s whim to fashion a 4-dimensional world as the scene of our activities. Of course, our present knowledge of the laws of the physical world is incomplete, and one day it might strike a deeper level on which dimensionality ceases to be indifferent, but at the moment this is merely a hope and not a fact” (see Weyl 1955b, p. 211). Also notice the difference with the position expressed in the fourth edition of Space, Time, Matter (see Sect. 7.6 below).
- 7.
The question of the dimensionality of space and arguments related to its explanation are discussed in De Bianchi & Wells (2015). An interesting study concerning the electromagnetic generation of the Lorentz signature of the metric of space-time is Itin and Hehl (2004). Weyl (1921) provides a theorem showing that the space-time metric is already fully determined by the inertial and causal structure of space-time. For a study on the causal theory of space-time and its history, see (Winnie 1977). Important contributions are also Sklar (1974, 1977).
- 8.
In (Weyl 1955a), it is argued that if besides physical space one recognizes an intuitive one endowed with an Euclidean structure, this does not necessarily contradict our physical insight, because the latter also holds to the validity of Pythagoras theorem in any infinitely small neighborhood of a point O in which the self is momentarily located.
- 9.
This reference is present also in the fourth edition, see (Weyl 1922a).
- 10.
According to Ryckman (2005, p. 155), Weyl followed upon the Helmoltz-Lie tradition when searching for the uniqueness of the quadratic metric determination in an n-dimensional differentiable manifold M, by treating congruence through a continuous group of motions.
- 11.
Translation is mine, the original German text reads: „Raum und Zeit sind, wie Kant sagt, Formen unserer Anschauung. Die Koordinaten sind dazu da, die Stellen im Kontinuum von Raum und Zeit voneinander zu unterscheiden. Sie spielen die gleiche Rolle wie die Namen, durch welche Personen voneinander unterschieden und nennbar gemacht werden, oder wie eine willkürliche Nummerierung der Objekte in einem aus diskreten Elementen bestehenden Objecktbereich”
- 12.
The date is probably 1948, because in the manuscript Weyl mentions that he is writing just 30 years after he presented his theory unifying electromagnetic and gravitational potential, which was in Raum-Zeit-Materie (1918).
- 13.
For Weyl’s application of the Lorentz group to quantum mechanics, see Weyl (1931b, p. 147).
- 14.
Even though Weyl is aware of a certain affinity with them. This awareness is mostly based on Hilbert’s interpretation of Kant’s transcendental ideal of the pure reason with respect to the systematic unity of science, see Weyl (1930, p. 28).
- 15.
For further details, see Coleman and Korté (1984).
- 16.
- 17.
For a clear account of Weyl’s symbolic construction, see (Majer 1998).
- 18.
I thank Julien Bernard for pressing me in highlighting this point.
- 19.
According to Weyl, a continuous deformation, a one-to-one continuous transformation does not affect local values.
- 20.
According to Weyl (1940, p. 82), the topological scheme is bounded only by certain axioms and wherever axioms occur, they ultimately serve to describe the range of variables in explicitly constructed functional relations.
- 21.
In a note to Weyl (1940), p. 83 Pesic recalls that, according to Aristotle, substance (ousia) denotes the common essence (say of a biological genus), whereas accident (sumbebekos) denotes a quality of an individual member of that genus that does not specifically reflect its underlying essence. Pesic reminds of Aristotle distinction because he thinks that it applies to Weyl’s terminology. However, I argue that it is not the case.
- 22.
For a reconstruction of causal topology and Weyl’s notion of space-time structure as essence, see Ryckman (2005, pp. 155 ff.).
References
Barrow, J., and F. J. Tipler. 1986. The anthropic cosmological principle. Oxford: Clarendon Press.
Bell, J. 2004. Hermann Weyl’s later philosophical views: his divergence from Husserl. In Husserl and the sciences, ed. R. Feist, 173–185. Ottawa: University of Ottawa Press.
Bergmann, P.G. 1992. The riddle of gravitation. New York: Courier.
Bernard, J. 2015. Becker–Blaschke problem of space. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 52: 251–266.
Cassirer, E. 1923. Substance and function and Einstein’s theory of relativity. Chicago: The Open Court Publishing Company.
Coleman, R.A., and H. Korté. 1984. Constraints on the nature of inertial motion arising from the universality of free fall and the conformal causal structure of spacetime. The Journal of Mathematical Physics 25 (12): 3513–3526.
De Bianchi, S., and J.D. Wells. 2015. Explanation and the dimensionality of space. Synthese 192 (1): 287–303.
Feist, R. 2004. Husserl and Weyl: Phenomenology, mathematics, and physics. In Husserl and the sciences, ed. R. Feist, 153–172. Ottawa: University of Ottawa Press.
Husserl, E. [1913]1983. Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy. First Book. Trans F. Kersten. The Hague: Martinus Nijhoff.
———. [1936] 1970. The Crisis of European Sciences and Transcendental Phenomenology: An Introduction to Phenomenological Philosophy. Ed. and Trans. D. Carr, 1954; reprint. Evanston: Northwestern University Press.
Itin, Y., and F.W. Hehl. 2004. Is the Lorentz signature of the metric of spacetime electromagnetic in origin? Annals of Physics 312: 60–83.
Kant, I. [1787]1998. Critique of Pure Reason. Ed. P. Guyer. Cambridge: Cambridge University Press.
Lobo, C. 2009. Mathématicien philosophe et philosophe mathématicien, introduction et annotation de la correspondance Husserl-Weyl et Becker-Weyl. Annales de phénoménologie 8: 205–226; 227–252.
Majer, U. 1998. Knowledge by symbolic constructions. In Philosophy and the many faces of science, ed. D. Anapolitanos, A. Baltas, and S. Tsinorema, 40–64. Lanham: Rowman & Littlefield.
Mancosu, P., and T. Ryckman. 2002. Mathematics and phenomenology: The correspondence between O. Becker and H. Weyl. Philosophia Mathematica 10: 130–202.
McCall, S. 2006. Philosophical consequences of the twins paradox. In The ontology of spacetime, ed. D. Dieks, 191–204. Amsterdam: Elsevier.
Mittelstaedt, P., and P.A. Weingartner. 2005. Laws of nature. Berlin: Springer.
Ryckman, T. 2005. The reign of relativity. Oxford: Oxford University Press.
Scheibe, E. 1957. Über das Weylsche Raumproblem. Journal für Mathematik 197 (3/4): 162–207. (Dissertation Göttingen 1955).
———. 1988. Hermann Weyl and the nature of spacetime. In Exact sciences and their philosophical foundations: Exakte Wissenschaften und ihre philosophische Grundlegung. Vorträge des internationalen Hermann-Weyl-Kongresses, ed. Wolfgang Deppert and Kurt Hübner, 61–82. Frankfurt/M/Bern: Peter Lang Verlag.
———. 2001. Between rationalism and empiricism. Selected papers in the philosophy of physics. New York/Berlin/Heidelberg: Springer.
Scholz, E. 2004. Hermann Weyl’s analysis of the “problem of space” and the origin of gauge structures. Science in Context 17: 165–197.
———. 2011. H. Weyl’s and E. Cartan’s proposals for infinitesimal geometry in the early 1920s. Boletim da Sociedada Portuguesa de Matemàtica Número Especial A. da Mira Fernandes, 225–245.
———. 2012. Leibnizian traces in H. Weyl’s “Philosophie der Mathematik und Naturwissenschaft”. In New essays on Leibniz reception, ed. R. Krömer and Y. Chin-Drian, 203–216. Basel: Birkhäuser.
Sieroka, N. 2007. Weyl’s ‘agens theory’ of matter and the Zurich Fichte. Studies in History and Philosophy of Science 38: 84–107.
Sklar, L. 1974. Space, time, and spacetime. Berkeley: University of California Press.
———. 1977. Facts, conventions and assumptions. In Foundations of space-time theories, volume VIII of Minnesota Studies in the Philosophy of Science, ed. J. Earman, C.N. Glymour, and J.J. Stachel, 206–274. Minneapolis: University of Minnesota Press.
Torretti, R. 1983. Causality and spacetime structure. In Physics, philosophy and psychoanalysis: Essays in Honor of Adolf Grünbaum, ed. R.S. Cohen, 273–294. Dordrecht: Reidel.
Weinert, F. 2005. The scientist as philosopher: Philosophical consequences of great scientific discoveries. Springer.
Weyl, H. 1918. Reine Infinitesimalgeometrie. Mathematische Zeitschrift 2: 384–411.
———. 1921. Zur Infinitesimalgeometrie: Einordnung der projektiven und konformen Auffassung. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen; Mathematisch-physikalische Klasse, 99–112.
———. 1922a. Space-Time-Matter (from the 4th German edition), London.
———. 1922b. Das Raumproblem. Jahresbericht der Deutschen Mathematikervereinigung 31: 205–221.
———. 1922c. Die Einzigartigkeit der Pythagoreischen Maßbestimmung. Mathematische Zeitschrift 12: 114–146.
———. 1923. Mathematische Analyse Des Raumproblems. Berlin: Springer.
———. 1930. Weyl levels of infinity. In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, 17–32. New York: Dover.
———. 1931a. Geometrie und Physik. Die Naturwissenschaften 19: 49–58.
———. 1931b. Theory of groups and quantum mechanics. New York: Dover.
———. 1940. The mathematical way of thinking. In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, vol. 2012, 67–84. New York: Dover.
———. 1948. ETH-Bibliothek, University Archives, Hs 91a:31, Similarity and congruence: A chapter in the epistemology of science.
———. 1949. Philosophy of mathematics and natural science. Vol. 2, 35–37. Princeton: Princeton University Press.
———. 1955a. Insight and Reflection. In Mind and nature, selected writings on philosophy, mathematics, and physics, ed. P. Pesic, vol. 2009, 204–221. Princeton: Princeton University Press.
———. 1955b. Why is the world four-dimensional? In Levels of infinity, selected writings on mathematics and philosophy, ed. P. Pesic, vol. 2012, 203–216. New York: Dover.
Winnie, J.A. 1977. The causal theory of space-time. In Foundations of space-time theories, volume VIII of Minnesota Studies in the Philosophy of Science, ed. J. Earman, C.N. Glymour, and J.J. Stachel, 134–205. Minneapolis: University of Minnesota Press.
Zeeman, E.C. 1964. Causality implies the Lorentz group. Journal of Mathematical Physics 5 (4): 490–493.
Acknowledgement
The research leading to this chapter has been made possible thanks to the fellowship “Research in Paris 2013” offered by the Ville de Paris and to the FP7-COFUND program Beatriu de Pinós (grant n. 2013BP-B00101). The research has been made possible also thanks to the projects 2014 SGR 1410 sponsored by the AGAUR and HAR2014-57776 sponsored by MINECO. I am grateful to Monica Bussmann and to the Staff at the ETH in Zurich, who assisted me in visiting the archives in October 2014. I am very thankful to Julien Bernard and Carlos Lobo who invited me to present the earlier draft of this paper at the workshop Weyl and the Problem of Space, From Science to Philosophy (Konstanz, 27-29 May 2015).
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De Bianchi, S. (2019). From the Problem of Space to the Epistemology of Science: Hermann Weyl’s Reflection on the Dimensionality of the World. 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_7
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