*This is a guest post by Matthew Duncombe, a fellow Cambridge ancient philosophy student. I hope it (and future posts!) will expand the range of topics on the blog. *

Aristotle’s answer to Zeno’s basic paradox of progress, the runner, is known to invoke a distinction between ‘actual’ and ‘potential’ infinities. Below, I ponder for your amusement, whether the solution based on the distinction of actual and potential infinities leads to incoherence. The paradox, with which I imagine most people are familiar, is as follows. Suppose that in a finite period of time, *T*, a runner, Bolt, traverses a finite distance, AB. At some point within *T*, *t1*, Bolt* *will ‘touch’ (*haptesthai* in Aristotle’s presentation) the mid-point of AB, namely *a1*. At some time after *t1*, Bolt will ‘touch’ the midpoint between *a1* and B, namely *a2*, and so on. In general, for any *aN *there is a point, *aM*, midway between *aN* and B which Bolt will ‘touch’ at some point in *T. *That is an infinite number of points. So (P1) if Bolt runs from A to B, then Bolt completes and infinite number of tasks. (P2) It is impossible to complete an infinite number of tasks. (C) Therefore, Bolt cannot run from A to B.[1]

Aristotle’s objection is that the argument trades on an ambiguity in the notion of ‘infinity’ in the argument (*Phys.* 263b3-9). There is an innocuous sense of infinity, potential infinity, which is the sense of ‘infinite’ in (P1), and a vicious sense of infinity, actual infinity, which is the sense operative in (P2) (*Phys*. 206a9-b2). The difference can be seen with the following case: Imagine another runner, Colt, who runs exactly the same course as Bolt, but who takes his hat off as he ‘touches’ point *aN*, and puts it back on as he touches point *aM*. Is he wearing his hat when he reaches B? It cannot be off, because each time he took it off, he put it back on again. But it cannot be on, because every time he put it on, he took it off. So it is neither on, nor off. So reaching B is impossible. This is because Colt has *actualized* each of the infinite number of *potential* points on the course AB. But Bolt did not actualize them, so there is no contradiction in Bolt’s case.[2]

To put the point another way: (AI) There is an actual infinity of tasks to complete running from AB just in case, between any two actual or potential tasks, there is a third actual task. (PI) There is *potential* infinity of tasks just in case between any two *actual* or *potential* tasks on AB there a third *potential* task. But (PI) does not entail (AI). Problem solved.

But: (M) If *x* is potentially *F* then it is possible that* x *is actually *F*. (PI) does entail that on course AB there is an infinite number of potential tasks to be completed to run from A to B. So the course AB is traversable only by completing an infinite number of potential tasks. But that is to say that AB is traversable only by it being possible to complete an infinite number of actual tasks. But Aristotle agreed that it is impossible to traverse AB if one must complete an infinite number of actual tasks.

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