  The general aim of this paper is i) to argue for the feasibility and
fruitfulness of the balance between the phenomenological method seeking
intuitive evidence and the axiomatic-deductive method and ii) the importance of
mutual understanding between philosophy and mathematics and of cultivating an
historical self-awareness of their common source in Greek philosophy. To this
end we show how Aristotle's theory of \emph{sunekh\^es, apeiron} and
\emph{topos} as well as their related notions can be given a rigorous
interpretation in terms of modern topology and geometry as well as category
theory. This is facilated by the fact that in Aristotle himself we already find
a good 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.
