To: Max W.
From: Geoffrey Klempner
Subject: Zeno's paradoxes of motion
Date: 30th September 2011 14:17

Dear Max,

Thank you for your email of 23 September, with your essay for the University of London Plato and the Presocratics module, in response to the question, 'What do Zeno's paradoxes of motion demonstrate about the nature of motion?'

This is an excellent piece of work, well researched and well argued.

Coincidentally, earlier this week I posted a reply which I wrote in 2005 to Pat, another University of London student, on Electronic Philosopher http://electronicphilosopher.com. In my letter, I quote from the Pathways Presocratics program, where I consider Aristotle's own response to the Stadium paradox, based on his distinction between 'actual' and 'potential' infinity. The idea seems to be that the infinity, or infinities, that we discover when we analyse motion are merely constructed by us, not 'actually there' in reality. Although I don't state this in the program unit, pursuing this line would take us to an intuitionist reconstruction of mathematics, and in particular the mathematics of motion.

That would be a remarkable conclusion. However, the challenge of Zeno's paradoxes, and the Stadium in particular, is to see how *little* we can give up, how few of our pre-reflective assumptions we have to throw overboard, in order to resolve the paradox. At the limit, if we don't have to give up anything, then that is equivalent to stating that the so-called paradoxes are merely fallacies, albeit instructive fallacies.

Interestingly, the question you are responding to is slightly different from Pat's question: 'What, if anything do Zeno's paradoxes tell us about motion?' Your question apparently makes a more substantial claim. There is something that Zeno's paradoxes 'tells us' about motion, and moreover, it is something that the paradoxes successfully demonstrate. But what is that? I don't want to make too much of this, but a point I've made before is that it's always a good strategy to consider the precise wording of the question!

At one point in your essay you refer to early Greek atomism. There is a problem, however, in that we have no direct evidence that the atomists made any claims about the nature of space or time as such. The atoms are exemplars of Parmenidean Being, separated by non-Being. We know that there exist (by the 'ou mallon' or 'no more reason' principle) atoms of every possible shape, and, moreover, size (which would imply the existence of atoms the size of planets!). However, there is nothing to stop us identifying different spatial positions on an atom (say, a apple-sized atom). In other words, space and time remain continuous. (There is an analogy here to a point you repeat from Grunbaum, that quantum mechanics does not imply that space and time as such are 'quantized'.)

I think we have to be a bit creative here and consider the possibility that in the paradoxes of motion Zeno was responding to an idea which we don't find explicitly in the available fragments, to the effect that space and time themselves are parceled up into discrete units.

This would make sense of the Stadium paradox. Imagine that there is a smallest distance d and a smallest time t. Then it would be logically impossible for a body B to travel 2d in time t. Proof: Assume the opposite. It follows that the body B travels distance d in time t/2. But there is no such interval as t/2. Contradiction.

So far, so good. But now consider body B and body B' travelling in opposite directions at the logically maximum speed d per t, then B and B' approach one another at speed 2d per t. Contradiction. Could this have been Zeno's point? However, the argument can be resisted by a defender of spatio-temporal atomism. All one has to say is that the logical speed limit of d per t applies to absolute motion, not relative motion.

Interestingly, the spatio-temporal atomist has a defence against the analogous argument that there could be no slower speed than d per t, which would in effect mean that only one speed was possible, in obvious contradiction to our experience. (I.e. just as there is no interval t/2, so there can be no distance d/2.) This would be to say that when a body moves at an apparently 'slower' speed, it's movement is in fact a series of stops and starts, each movement taking place at exactly d per t.

If we look at Zeno's paradoxes relating to the composition of a body out of parts, there is indeed an explicit dichotomy, where we are invited to consider both possibilities, either that there are smallest parts or that there are no smallest parts. It is tempting to apply this to the paradoxes of motion, even though no explicit mention is made of the two alternative possibilities. Zeno saw, or rather thought he saw, a fatal flaw in both hypotheses, the hypothesis of discreteness and the hypothesis of continuity.

As the subsequent developments in mathematics show, Zeno was wrong about continuity. Say, if you like, that he 'demonstrates' the possibility of infinitesimals, and the finite sum of an infinite series, this clearly wasn't the intention. The intention (as stated by Plato at the beginning of his dialogue Parmenides, surely a reliable authority) was to show that any attempt to describe a changeable, plural universe leads to insoluble paradoxes.

All the best,

Geoffrey