Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

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

Comments (2)

Comment 1
Received: 27 October 2023
Commenter:
The commenter has declared there is no conflict of interests.
Comment: Thank you for your recent work, Dr. Redding. I must admit that I wasn’t able to fully understand your paper, but using “projective geometry” as a model for understanding Hegel’s logic seems like an intellectually stimulating approach. I also found it interesting that a certain portion of Hegel’s texts can be interpreted to anticipate Popper’s falsificationism. However, I was most intrigued by the statement that “within Hegel’s projective approach, the meaningfulness of such aperspectival views ‘from infinity is dependent on their relation to the limited views from the finite points of view located within space and time.” I roughly take it to mean that the truth of reality beyond our sensible realm is somehow tied to our consciousness. Although Hegel deserved some criticisms from Russell (e.g., for being an obscurantist), I believe Hegel marks both the culmination and beginning of philosophy.
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Comment 2
Received: 30 October 2023
Commenter:
Commenter's Conflict of Interests: I am one of the author
Comment: Many thanks for your comments. Often Hegel's approach to mathematical issues is not taken particularly seriously. But although principally a humanistic thinker, he had been deeply interested in both mathematics and its application in modern science, especially mechanics. His thoughts are, I believe, very interesting.

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.
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