Modern incarnations of the Aristotelian concepts of Continuum and Topos

TL;DR

This paper explores how Aristotelian concepts like Continuum and Topos can be rigorously interpreted using modern mathematical frameworks such as topology, geometry, and category theory, fostering dialogue between philosophy and mathematics.

Contribution

It demonstrates the reinterpretation of Aristotle's notions within modern mathematical structures, promoting a balanced approach between intuition and formal logic.

Findings

Aristotle's sunekh extsuperscript{es} and topos can be modeled with topology and category theory.

Modern mathematics embodies Aristotelian intuitions and concepts.

A dialogue between philosophy and mathematics enhances understanding of foundational ideas.

Abstract

The aim of this paper is i) to argue for the feasibility and fruitfulness of a balance between the phenomenological method seeking intuitive evidence and the axiomatic-deductive method and ii) that there should be a mutual understanding between philosophy and mathematics and a cultivation of a historical self-awareness with regards to their common source in Greek philosophy. To this end we show how Aristotle's theory of \emph{sunekh\^es, apeiron} and \emph{topos} and related notions can be given a rigorous interpretation in terms of modern topology and geometry as well as category theory. This is facilitated by the fact that in Aristotle himself we already find a balance between intuition and formal logic. We also show how these powerful Aristotelian intuitions and concepts are found incarnated in diverse domains of modern mathematics.