Modern incarnations of the Aristotelian concepts of Continuum and Topos

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.