To: Shan A.
From: Geoffrey Klempner
Subject: Zeno's paradoxes of motion
Date: 29th August 2008 12:04

Dear Shan,

Thank you for your email of 15 August, with your University of London Ancient Philosophy essay in response to the question, 'Critically assess Zeno's paradoxes of motion'.

This is a competent piece of work, with which I have few disagreements. In what follows, I will mention issues which you might like to think about further. I would appreciate (in future) if you could include a bibliography of works consulted as well as authors mentioned.

In an exam, you will get credit for mentioning names and giving citations, although it is not necessary to overdo this.

In general, for the Ancient Philosophy paper you need to show that you are aware of where the text (fragments or testimonia) comes from and also, therefore, the question of authenticity. With Zeno we have the testimony of Plato and Aristotle which is pretty solid. Even so, we only have Plato's word for it that Zeno's motive was defending Parmenides.

On the Dichotomy, the first thing that needs to be said is what you mention as an 'other argument' namely that an infinite series can have a finite sum. Is it conceivable that Zeno, or his contemporaries, did not realize this? That seems hard to believe. The point, however, is as you say in the distinction between 'events' and 'tasks'. Granted that there are, e.g. an infinite number of events of Achilles getting ever so slightly closer to the Tortoise, it doesn't follow (as Aristotle indeed points out) that this involves an infinite number of tasks.

However, one can raise the question whether Aristotle is right in going so far as to distinguish 'potential' and 'actual' infinity. Don't we want to say that, given the hypothesis of infinite divisibility of time and space, that there IS an actual infinity of points traversed which occur in finite time? If not, why not?

This leads us to the question whether indeed we are entitled to the assumption that space and time are 'infinitely divisible'. Here, you could refer to Zeno's paradox which deals with the composition of an object out of its parts (as Barnes shows, not a problem if you consistently hold that space is infinitely divisible, or that it is finitely divisible).

The moving rows paradox is also relevant. On the face of it, as you remark, it looks pretty feeble. However, one is led to wonder whether there is an initially plausible assumption about the nature of space and motion which would lead to a genuine sense of paradox in the case of the moving rows. If time and space are both quantized, then we do seem to have a paradox between the idea that a quantum of matter in row B passes a quantum of matter in stationary row A in one quantum of time, but passes two quanta of matter in row C in the same quantum of time. In one quantum of time there can only be room for one event, not two.

Your point about motion being 'the more natural state' relates to Russell's insistence on the reality of relations, in the face of the ontological tradition, going back to the Greeks, were objects and their properties (predicates) are taken as paradigmatically 'real'. Leibniz, who Russell wrote an excellent book about, is one example of a philosopher who attempted a comprehensive description of reality in which all relations are analysable into predicates of 'monads'. It makes Zeno more interesting, if we realize that the move to a genuinely relational ontology is difficult, not easy (not just a matter of asserting that there are truths of the form aRb). In this context, Heraclitus might be seen as the one philosopher who genuinely appreciated the difficulties and challenges of acknowledging the reality of relations.

Regarding the paradox of the Arrow, the nineteenth century French philosopher Henri Bergson gave the most radical critique of what he termed the 'cinematographic' view according to which we can view motion as a sum of infinite instants, so that, in effect, a moving arrow IS at rest. Just as the frames on celluloid only 'move' when the film is projected, so according to the cinematographic view the objects which we observe 'moving' merely appear to move but do not move in reality.

As in Russell's defence of the reality of relations, Bergson in his account of time as 'duree' is concerned to show how radical a shift is needed in order to give a coherent description of spatio-temporal reality. In other words, Zeno's paradoxes are significant precisely because the force this question.

Writing in the 1920's John McTaggart in his 'Nature of Existence' produced what is still regarded by many as the definitive argument against the reality of time (see, e.g. Mellor's 'Real Time'). Physicists happily work with the idea of time as being just another dimension, in addition to the three dimensions of space, so that spatio-temporal objects are, in reality, 'space-time worms'. In a world made of space-time worms, there can be no paradoxes of motion because motion itself is apparent, not real. Zeno would have felt vindicated.

All the best,

Geoffrey