Aristotle’s Actual Infinities
In histories of thought about the infinite, Aristotle is constantly said to have rejected any form of actual infinite, and to have allowed quantities to be at most potentially infinite. Aristotle does reject actual infinites in spatial magnitude: nothing is infinitely big or infinitely small. But in the central case of plurality, the evidence for potentialism is much weaker. This paper argues that Aristotle had no principled objection to the idea that there are actually infinitely many things. One part of the argument concerns the distinction in Aristotle between plurality (πλῆθος) and number (ἀριθμός). Another part concerns the meaning of phrases like ‘infinite by division’, arguing that such phrases do not refer to how many times something has been divided, but rather to how small something is. The argument of this paper, if successful, affects how we should think about the metaphysics of parts in Aristotle’s theory of the continuum.