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Projective Geometry as a Model for Hegel’s Logic
Version 1
: Received: 23 October 2023 / Approved: 24 October 2023 / Online: 25 October 2023 (05:28:22 CEST)
A peer-reviewed article of this Preprint also exists.
Redding, P. Projective Geometry as a Model for Hegel’s Logic. Logics 2024, 2, 11-30. Redding, P. Projective Geometry as a Model for Hegel’s Logic. Logics 2024, 2, 11-30.
Abstract
Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.
Keywords
projective geometry; Greek music theory; Hegel’s logic; Plato’s cosmology; Aristototle’s syllogistic
Subject
Arts and Humanities, Philosophy
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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He had read Lazare Carnot's work on the metaphysics of calculus, which had been an important work on that topic around the turn of the nineteenth century. Carnot has an interesting take on "impossible numbers" (negative numbers, irrationals, imaginary numbers, etc.) that play a necessary role in physics. Carnot thought that some number types, such as the natural numbers, had a straightforward application to the world, but other more problematic number types, such a irrationals, infinitesimals, and so on, functioned internally to the mathematics, being used in science so as to allow different parts of the theories to be connected up. Carnot was one of the mathematicians who revived projective geometry in the nineteenth century, and I don't know if he actually applied this idea to the ideas of "points at infinity" as used within projective geometry, but it would be consistent, I think, with his approach to infinitesimals, that Hegel was familiar with.