Mathematical Infinity and the Presocratic Apeiron

Abstract

The Presocratic notion of apeiron, often translated as “unbounded,” has been the subject of interest in classical philosophy. Despite apparent similarities between apeiron and infinity, classicists have typically been reluctant to equate the two, citing the mathematically precise nature of infinity. This paper aims to demonstrate that the properties that Anaximander, Zeno, and Anaxagoras attach to apeiron are not fundamentally different from the characteristics that constitute mathematical infinity. Because the sufficient explanatory mathematical tools had not yet been developed, however, their quantitative reasoning remains implicit. Consequentially, the relationship between infinity and apeiron is much closer than classical scholarship commonly suggests.