Substantiae sunt sicut numeri. Aristotle on the structure of numbers

Abstract

Aristotle’s contribution to the metaphysics of numbers is often described in terms of a critical response to the Platonist paradigm. Plato, we are told, conceives of numbers as abstract entities entirely distinct from the physical objects around us, while Aristotle takes the more mundane view that numbers are pluralities of physical objects considered in a particular way, a way relevant to mathematics. Without rejecting altogether this familiar picture, this paper aims to show that Aristotle has another major contribution to offer to the history of philosophy of mathematics. In the Metaphysics, he claims that numbers too can be analysed in terms of matter and form (hylomorphism). On the hylomorphic model, a number has both a material component (the units in the number) and a formal one (the structure that keeps the units together). The paper fully explores the motivations behind Aristotle’s hylomorphic conception of numbers, as well as its most significant implications.