Abstract
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.
Highlights
In this paper I argue that Aristotle’s understanding of mathematical continuity constrains the mathematical ontology he can consistently hold
While the idea that continua are composed of infinitely many points is the present day orthodoxy, the Aristotelian understanding of continua as non-punctiform and infinitely divisible was the reigning theory for much of the history of western mathematics, and there is renewed interest in it from current mathematicians and philosophers of mathematics
In examining Aristotle’s views about continuity, I was led to the somewhat surprising conclusion that Aristotle’s particular way of accounting for the non-punctiform nature of continuity cannot be separated from his mathematical ontology
Summary
In this paper I argue that Aristotle’s understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. Having claimed that something is not a continuum if the boundaries of the parts are two and not one, Aristotle goes on to say that “[T]his definition makes it plain that continuity belongs to things that naturally in virtue of their mutual contact form a unity. Since according to Aristotle the point is not divisible, Bostock argues it is consistent with the proposed definition of continuity that that parts of continua are not all further divisible.
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