Formless Structure: An Ontological Problem in Aristotle's Theory of Number

The literature on Aristotle's philosophy of arithmetic is still at a pre-paradigmatic stage. What progress has been made, though, reveals a potential gap in Aristotle's ontology. On Aristotle's standard account, number is an aggregate of discrete units, devoid of any further unifying form or structure. Given this, how can such an account explain how number has particular mathematical properties, such as being odd, prime, or square? The number 4, for instance, is simply four units taken together. Yet, this aggregation alone does not appear to explain why 4 is even or square. Indeed, Aristotle affirms that numbers have attributes apart from quantity (An. Post. 2.13 96a30-b5). Nevertheless, his reduction of number to a mere plurality of units threatens to render such attributes unintelligible within his framework. In what follows, I will reconstruct Aristotle's conception of number and argue that he is, in fact, committed to attributing properties to number beyond mere cardinality. I contend, however, that his account lacks the ontological resources to explain the basis of these properties. As a result, Aristotle's ontology of number fails to provide a satisfactory foundation for arithmetic.