This paper argues that Hegel has much to say to modern mathematical philosophy, although the Hegelian perspective needs to be substantially developed to incorporate within it the extensive advances in post-Hegelian mathematics and its logic. Key to that perspective is the self-referential character of the fundamental concepts of philosophy. The Hegelian approach provides a framework for answering the philosophical problems, discussed by Kurt Gödel in his paper on Bertrand Russell, which arise out of the existence in mathematics of self-referential, non-constructive (...) concepts (such as class). (shrink)

There is at present no intelligible account of what the statements of pure mathematics are about. The philosophy of mathematics is in a mess! Marvin J. Greenberg.

Throughout his life, Hegel showed great interest in physics and mathematics. His most sustained, surviving treatment of Euclidean geometry is his early work ‘Geometrische Studien’, which he completed while he was a private tutor [Hoffmeister] in Frankfurt, shortly before leaving for Jena to join Schelling.GSis not easy reading, but despite that, it seems to me that Hegel presents in it a remarkably erudite as well as interesting and insightful critique of geometry. He investigates some of the themes from the foundations (...) of geometry, in particular from the first book of theElements of Euclid. Like the mathematical philosophies of Kant and Frege, Hegel's understanding of geometry is conceptually based, but unlike them, it is also grounded in the classical Greek philosophy of mathematics, which achieved its definitive expression in Proclus's great commentary onEuclid 1. Much of this classical philosophy of geometry is forgotten nowadays, under the influence of the great modern mathematical philosophers. In my view, it well deserves reconsideration, especially since, as illustrated by Gödel's incompleteness theorems, modern mathematical philosophy has failed in its attempt to ground mathematics within the framework of formal systems.Much ofGShas not survived, and what remains is condensed and fragmentary. It seems that originally, Hegel covered all of the propositions ofEuclid 1rather than just the 14 propositions that are covered in what remains of the originalGS. I have given detailed treatment ofGStogether with related material in Hegel's Jena dissertation elsewhere. The objective of the present paper is to introduce the translation ofGS. (shrink)