Abstract
This paper deals with the transformation of the concept of space in the post-Kantian philosophy of geometry from the second half of the nineteenth century to the early twentieth century. Kant famously characterized space and time as forms of intuitions, which lie at the foundations of the apodictic knowledge of mathematics. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of apodictic certainty at that time. However, such later scientific developments as non-Euclidean geometries and the general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space not so much as a source of knowledge, but as an open question for empirical research.
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Notes
- 1.
Several examples are discussed in Friedman [2, Ch. 1].
- 2.
See [8, Ch. 5]. Helmholtz addressed two different questions. The first concerned the two-dimensionality of vision. At the time Helmholtz was writing, the dominant view endorsed by Johannes Müller, among others, was that a two-dimensional, spatial representation is primitively given in vision. In this view, only the perceptions of depth and of distance (i.e., the kind of perceptions that presuppose three-dimensionality) have to be learned. By contrast, Helmholtz sought to derive all spatial representations from the association of nonspatial sensations. The second question concerned the singularity of vision. Helmholtz called nativist Müller, Ewald Hering, and all those who derived the singularity of vision from the supposition of an anatomical connection between the two retinas.
- 3.
For a discussion of Helmholtz’s claims about the “transcendental” status of space, see Biagioli [10].
- 4.
- 5.
For an introductory account of the discovery of non-Euclidean geometry and its prehistory, see Engel and Stäckel [12].
- 6.
Sartorius von Waltershausen [15, p. 81] reported that even Gauss made an attempt to test the Euclidean hypothesis during his geodetic work. However, this interpretation is controversial and it was only after Bolyai’s and Lobachevsky’s works that the question arose whether the geometry of space could be non-Euclidean (see [16]).
- 7.
For a comprehensive account of nineteenth-century philosophy of geometry, see Torretti [18].
- 8.
On the discussion of Helmholtz’s view in neo-Kantianism, see Biagioli [19].
- 9.
- 10.
On the development of group theory from Galois to Klein, see Wussing [24].
- 11.
- 12.
See Ryckman [29, Chaps. 5 and 6].
- 13.
On the development of Cassirer’s thought from neo-Kantianism to the philosophy of symbolic forms, see Ferrari [31].
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Biagioli, F. (2017). Space as a Source and as an Object of Knowledge: The Transformation of the Concept of Space in the Post-Kantian Philosophy of Geometry. In: Wuppuluri, S., Ghirardi, G. (eds) Space, Time and the Limits of Human Understanding. The Frontiers Collection. Springer, Cham. https://doi.org/10.1007/978-3-319-44418-5_1
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