 |
The talk begins with a historical introduction of a forgotten alternative to the standard Cantor-Dedekind point-based conception of the linear continuum via a reconstruction of Veronese's conception of the absolute non-Archimedean i.e. infinitesimal-enriched continuum. It is shown in what sense Veronese's theory could (and indeed that it should) be construed as a non-Archimedean variation and an expansion of the Aristotelian interval-based conception. This will allow us to overcome the “great struggle” between the Cantorians and the Aristotelians in an unexpected way – by showing how the two theories are reconciled as merely relative continua within the richer structure of Veronese's absolute continuum which turns out to be a maximally inclusive non-Archimedean ordered field. Furthermore, it is also shown how such absolute continua allow us to bridge the gap between the continuous and the discrete (i.e. between points and lines and, more generally, between arithmetic and geometry) by providing a non-Archimedean version of the Cantor-Dedekind axiom. This is followed by a short examination of the varieties of non-Archimedean arithmetic continua introduced by Levi-Civita, Robinson, and Conway. Finally, we pledge for a revisiting of infinitesimalism by sketching out some of the prospects of working with absolute continua in philosophy of mathematics, philosophy of physics and even mathematics and physics proper.
