Projective Geometry as a Model for Hegel’s Logic

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.

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.