Many infinities, or only one?

Giovanna Giardina; Marco Panza; D. C. Struppa

doi:10.35352/gioenia.v56i386.108

Giovanna Giardina DISUM, University of Catania, Italy http://orcid.org/0000-0002-3009-8171
Marco Panza Departments of Philosophy and Mathematics, Chapman University, Orange (CA), USA; CNRS, IHPST (CNRS and University of Paris 1 Panthéon-Sorbonne), France http://orcid.org/0000-0003-4131-7103
D. C. Struppa The Donald Bren Presidential Chair in Mathematics, Chapman University, Orange (CA), USA http://orcid.org/0000-0002-3664-1729

DOI: https://doi.org/10.35352/gioenia.v56i386.108

Keywords: Aristotle, Potential and actual infinity, Projective Geometry, Infinitesimal calculus, Set Theory, Measure Theory

Abstract

In this short note we first account for Aristotle's views on infinity, by clarifying the way his notion of potential infinity should be understood, in the light of his notion of entelechy. We then present four distinct ways in which mathematicians attempted to tame the notion of actual infinity, and we ask the question of whether they are indeed four different ways, or whether they ultimately are variations on the same concept. We suggest that what mathematicians are doing is indeed to find a way to construct a form of actual infinity that subsumes, into itself, the potential infinity, and observe how the presence of the former brings to a different view about potential infinity than Aristotle's.