Truth, Existence and Explanation pp 249-272 | Cite as

# Existence vs. Conceivability in Aristotle: Are Straight Lines Infinitely Extendible?

## Abstract

Aristotle is committed to finitism in mathematics. But there are certain uses of infinity in mathematics which are indispensable, even in the mathematics of his day. So we had better understand Aristotle’s finitism in a way that is compatible with the mathematics of his time (unless we are willing to ascribe complete naivete to him about these mathematics).

In particular, we have to address the issue of infinitely extendible lines, which are used in Greek mathematics – for example, in Euclid’s definition of parallel lines. Aristotle denies that such lines exist: like any other object which exceeds the fixed and finite size of the cosmos, infinitely extendible lines are definitely excluded from Aristotle’s physics. Moreover, due to Aristotle’s immanentism, they are excluded from his mathematics, too.

But how can Aristotle do mathematics without infinitely extendible lines? In the following I will suggest a possible solution, based on an analysis of the procedure of *converse increasing*, which Aristotle introduces and discusses in *Ph*. III 6. This procedure is both infinite – it is infinitely iterable – and does not require the existence of any infinite (or infinitely extendible) magnitudes.

The procedure is interesting, but it must be handled with care. On the one hand, if one fails to acknowledge its subtle mathematical content, one also risks compromising the philosophical interpretation, by ascribing to Aristotle gratuitous naiveties. On the other hand, if one overstates this mathematical content, one risks ascribing to him anachronistic, ‘non-Euclidean’ intents. At the end of the paper I will discuss two cases in point.

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