Space, Models and Geometric Fantasies

Umberto Bottazzini
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Umberto Bottazzini: University of Milan, Department of Mathematics

A chapter in Imagine Math 3, 2015, pp 29-41 from Springer

Abstract: Abstract What is space? Kant asked in the Critique of pure reason (1781, 2nd ed. 1787). “Space is not a conception which has been derived from outward experiences”, was his answer, but it “is a necessary representation a priori, which serves for the foundation of all external intuitions”. Thus, “Geometry is a science which determines the properties of space synthetically, and yet a priori”, and “the principles of geometry—for example, that ‘in a triangle, two sides together are greater than the third’—are never deduced from general conceptions of line and triangle, but from intuition, and this a priori, with apodeictic certainty” [1, pp. 39–40]. The geometry Kant was referring to was of course Euclid’s geometry, as exemplified by Kant himself by quoting Euclid’s first axioms repeatedly. By the end of the eighteenth century Kant’s philosophy, and the relevant conception of space and geometry, became predominant in German culture.

Keywords: Riemann Surface; Euclidean Geometry; Hyperbolic Geometry; Extensive Relation; German Culture (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-01231-5_5

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DOI: 10.1007/978-3-319-01231-5_5

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