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

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Summary

Introduction

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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