Hegel and Modern Topology

TL;DR

The paper investigates how fundamental ideas from modern topology, logic, and category theory can be philosophically illuminated by Hegel's Science Logic, while showing how contemporary mathematical concepts can concretely illustrate Hegelian deductions. It surveys topics from the logic of Being and the transition to Essence, through measure, topology, and sheaf theory, to higher topos theory, ∞-groupoids, and modern categorical logic, proposing a mutual enrichment between philosophy and mathematics. The approach uses Hegelian motifs (synthesis, negation, dialectic, germ/point, the One) as interpretive tools for topological and categorical constructions, such as quotients, completions, sheafification, and higher structures. The work aims to motivate rigorous cross-disciplinary dialogue and topos-inspired formalisms as a framework for reconciling philosophical notions with rigorous mathematical practice, with potential applications to physics, biology, and cognitive science.

Abstract

In this paper we sketch how some fundamental concepts of modern topology (as well as logic and category theory) can be understood philosophically in the light of Hegel's Science Logic as well how modern topological concepts can provide concrete illustrations of many of the concepts and deductions that Hegel used. Also these modern concepts can in turn be very powerful hermeneutic tools permitting a more rigorous and thorough grasp of Hegelian concepts. This paper can be seen as a continuation of our paper \cite{pro} where we argued that the prototypes of many fundamental notions of modern topology were already found in Aristotle's Physics. More generally it is hoped that this note makes a case for the possibility of a rigorous enriching interaction and mutual support between philosophy on one hand and modern logic and mathematics on the other. This paper is obviously meant only as a preliminary sketch and to offer some motivation for exploring in a more detailed and thorough way the subjects discussed.