Aristotle's Philosophy of Mathematics

Abstract

r he fundamental problem in the philosophy of mathematics, which has persisted from Plato's day until ours, is to provide an account of mathematical truth that is harmonious with our understanding of how we come to know mathematical truths.1 In Physics B2 and Metaphysics M3 Aristotle provided the seeds of a unified philosophy of mathematics. This has not been generally appreciated for two reasons. First, it is commonly assumed that Aristotle thought that the objects in the natural world do not perfectly instantiate mathematical properties: a physical sphere is not truly spherical; a straight edge is not truly straight.2 In consequence, though commentators see Aristotle as railing against a Platonic ontology of geometrical and arithmetical objects, they see him as unable to offer a genuinely alternative epistemology. Mathematicians, according to one influential interpretation of Aristotle, treat objects which are different from all sensible things, perfectly fulfill given conditions and are apprehensible by pure thought.3 This interpretation must view Aristotle as caught in the middle of a conjuring trick: trying to offer an apparently Platonic account of mathematical knowledge while refusing to allow the objects that the knowledge is knowledge of. Second, Aristotle's philosophy of mathematics is often labeled