## 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 O

_{k}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.).

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## 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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