The changing role of continuity and discontinuity in the history of philosophy and mathematics

Abstract

The aim of this article is to highlight the inevitability of employing discreteness and continuity as primitive (indefinable) modes of explanation in the history of philosophy and mathematics. It embodies the general challenge to account for the coherence of what is unique. Gödel emphasises the coherence of ‘primitive concepts’. Greek philosophy already discovered the spatial whole and/or parts relation with its infinite divisibility. During and after the medieval era philosophers toggled between an atomistic appreciation of the continuum and its opposite, for example found in the thought of Leibniz who postulated his law of continuity (lex continui). The discovery of incommensurability (irrational numbers) by the Greeks caused the first foundational crisis of mathematics, as well as its geometrisation. Leibniz and Newton did not resolve the problems surrounding the limit concept and soon it induced the third foundational crisis of mathematics. It caused Frege and the ‘continuum theoreticians’ to assign priority to the continuum – discreteness is a catastrophe. Recently Smooth Infinitesimal Analysis appreciated what is ‘continuous’ as constituting ‘an unbroken or uninterrupted whole’. Intuitionistic mathematics once more proceeded from an emphasis on the whole and/or parts relation. In spite of alternating attempts to understand continuity exclusively, either in arithmetical or in spatial terms, the history of philosophy and mathematics undeniably confirms that the co-conditioning role of these two modes of explanation remains a constant element in reflections on continuity and discontinuity. (The role of continuity and discontinuity within the disciplines of physics and biology will be discussed in a separate article.)