Aristotle’s solution to Zeno’s paradox of the runner

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.

Potential Solutions

Option 1: Deny the modal premise (M). Specifically, say that Aristotle’s notion of ‘potentiality’ is not related to possibility in that way. I can see that Aristotle would reject (M) if (M) were a bi-conditional: (M’) x is potentially F iff it is possible that x is actually F. In this case, the true claim, ‘It is possible that I am actually a light-switch’ entails ‘I am potentially a light-switch’, a claim which Aristotle would presumably deny, because I am not a mere lump of matter. But the simple conditional (M) would not have such unacceptable consequences. Moreover, it is hard to see how (M) could be rejected by Aristotle, since there must be some link between potentiality and actuality, and that link cannot be cashed out in material terms: (M’’) If x is potentially F then it is sometimes true that x is actually F, would not be sufficient. Some lump of matter could be potentially a light-switch, and never actually be one. The link must be cashed out in counter-factual terms.

Option 2: Choose the right modal logic. Aristotle does not seem to have realized that ‘impossibility’ is infectious. Say that the accessibility relation R holds between worlds w and w’ iff w’ is possible given the facts of w. Thus, any world that can access an impossible world (i.e. a world which contains a true contradiction) is itself an impossible world, because a world where it is possible that there is a true contradiction is an impossible world. So we could try to say that in (M) the ‘possible’ should be construed so that the actual world cannot access the impossible world, that is, Colt’s world.

This seems to get closer to identifying the problem with Aristotle’s argument: Aristotle certainly seems not to have seen that impossibility is infectious, nor that ‘potential impossibility’ is suspect. We could also make a case that for Aristotle’s (M) the accessibility relation is a non-Euclidean relation (i.e. it’s not the case that every possible world can access every other). Some wood is potentially a table, potentially a chair and potentially a bookcase. So it is possible that the wood is a table, chair or bookcase. But a table is not a potential chair, nor a chair a potential bookcase etc. Hence, under Aristotle’s notion of potentiality, the possible world where there is a table is not accessible by the possible world where there is a bookcase. Aristotle’s accessibility relation looks reflexive (wood is potential wood) and symmetric (a chair is potential wood, wood is a potential chair), but not transitive or Euclidean. Unfortunately, for Aristotle’s solution to Zeno work, it looks like Bolt’s world must be accessible by Colt’s world. And as long as accessibility is symmetric, the impossibility of Colt’s world infects Bolt’s world.

[1] This presentation is based on Barnes, J. (1979) The Presocratic Philosophers, London: Routledge. p. 262.

[2] This presentation owes a debt to Nick Denyer’s lectures on the Physics. Its main advantage, as far as I can see, is that it makes the actual and the potential infinite different in a way that is relevant to the solution of Zeno’s paradox: actual infinite series lead to contradiction, potential ones do not. Barnes’ construal of the difference, for example, fails to do this.

  1. Word Matt!

    I think Aristotle should and would deny (M). If some collection of Fs (e.g. actions) is potentially infinite, that is to say that it is possible that, for any actualised number n of Fs, it is possible for there to be n+1 Fs. It is not to say that it is possible for there to be an infinite number of Fs.

    Still, I’ve no sympathy at all for the Aristotelian solution. Consider the paradox of the snail race. Recall that snails leave a continuous trail behind them as they move (well, sort of). So if Lola the snail is to win the snail-race, then to get to the end of the race-course, she has to leave a speck of snai-trail at every intermediary point on the course. Each of those is an actual task; so an ordinary task necessitates completing an actual infinity of tasks. And if actual infinities are banned, then the ordinary task is impossible. So snails can’t move.


    • Matthew Duncombe
    • April 21st, 2011

    Hi Tim,

    Thanks very much for that. I’ll have to think a bit more about (M). There are some remarks which suggest that Aristotle did reject M, but its not clear whether it is for the reasons you suggest. And, certainly, rejecting finitism is a better way to resolve the paradoxes!



    • Dan
    • March 21st, 2012

    There is a much simpler and more convincing way of resolving this “paradox”, which is to observe that points exist beyond the finish line, which must be halfway to somewhere else. The false assumption in the argument is that the finish line is a true boundary, but a true boundary of reality would not have a position. It is true that it wouldn’t be reachable, but that is because it would be nothing. It would not be something that exists, since it would be negated of space, time, and energy. Just my two cents.

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