Why Aristotle's Metaphysics Includes a Philosophy of Mathematics

Abstract

My aim is a new interpretation of Aristotle's philosophy of mathematics. I argue that Aristotle's Metaphysics includes a philosophy of mathematics--presented in Metaphysics M 2-3--for three reasons. First, Aristotle's philosophy of mathematics addresses Metaphysics B's fifth aporia for metaphysical inquiry: the puzzle of the existence and nature of non-sensible substance. Aristotle begins Metaphysics M by saying that he will address non-sensible substance in this book, and we cannot understand MN or its constituent sections if we do not take him--and most commentators do not--at his word. Second, Aristotle's philosophy of mathematics demonstrates the unique explanatory power of his own science of metaphysics. Aristotle says in Metaphysics A 9 that the Platonist philosophy of his day "is mathematics"; but commentators have not so far understood what Platonist views might have motivated this remark, or how the Platonists arrived at these views. I argue that M 2 is a central text in answering both questions, and I employ it (with other passages from the Metaphysics) in reconstructing these lines of thought for the first time. I then argue that Aristotle's criticisms of the Platonists in M 2 show that, and how, a distinctly Aristotelian metaphysics of sensible substance is not only essential to but, to a considerable extent, itself sufficient for a coherent metaphysics of non-sensible substance. A good grasp of these first two aims is, then, I argue, vital to our understanding Aristotle's third aim: a resolution of the familiar puzzle of the ontological status of mathematical entities like numbers and lines (B's fifteenth aporia). Aristotle's philosophy of mathematics proper, which he presents in M 3, is generally regarded as obscure or inadequate, or both. I present a new interpretation, and argue that it is--when correctly understood--a view of remarkable subtlety, scope, elegance and force. In particular, it accounts for the unique precision and accuracy of mathematical truth while showing, in a clear and intuitive way, how it is that we have epistemic access to mathematical entities.