Continuity and Mathematical Ontology in Aristotle

In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next I argue briefly that Aristotle intends for his discussion of continuity to apply to pure mathematical objects such as lines and figures, as well as to extended bodies. I show that this leads him to a difficulty, for it does not at first appear that the distinction between continuity and contiguity can be preserved for abstract mathematicals. Finally, I present a solution according to which Aristotle's understanding of continuity can only be saved if he holds a certain kind of mathematical ontology.