From: Geoffrey Klempner

Subject: What do Zeno's paradoxes show about motion?

Date: 17 November 2006 12:18

Dear Dennis,

Thank you for your email of 8 November, with your essay in response to the University of London Pre-Socratics and Plato 2005 Examination question, 'What do Zeno's paradoxes show us about the nature of motion?'

I do think that it is an excellent strategy to take with any philosopher whose arguments seem questionable or paradoxical, to attempt to describe a plausible theory or position which the arguments might have been intended to attack. In other words to approach the arguments dialectically.

This is an application of what Donald Davidson calls the 'principle of charity'. There is greater chance of correctly understanding what a philosopher is trying to tell us if we find an interpretation which makes his arguments look gripping rather than mere 'sophisms'.

However, in this case we have to deal with written evidence about Zeno's intentions which must be taken seriously. In Plato's dialogue Parmenides, the character Zeno says about his 'book':

'In reality the book is a sort of defence of Parmenides’ argument against those who try to make fun of it by showing that, if there is a One, many absurd and contradictory consequences follow for his argument. This book is a retort against those who believe in plurality; it pays them back in their own coin, and with something to spare, by seeking to show that, if anyone examines the matter thoroughly, yet more absurd consequences follow from their hypothesis of plurality than from that of the One.'

It is pretty clear from this that Plato's 'Zeno' thinks he is attacking the hypothesis of pluralism as such, and not a particular view about the nature of numbers. You can save your interpretation by saying that Zeno made the false assumption that pluralism can only be accounted for in Pythagorean terms. This makes his arguments still look interesting and so satisfying Russell.

This title of this essay, however, does not focus on the interpretation of Zeno but rather on what his paradoxes 'show us'. So let us look at the question from this point of view.

If you are tempted by a certain view of numbers which looks pretty implausible today - in fact, hard to even imagine - then you will find yourself unable to resist Zeno's logic. Admittedly, it is not difficult for the beginner in maths to feel very perplexed by ideas like the finite sum of an infinite series, or Cantor's definition of an infinite set as one which maps onto a proper subset. As you clearly show, these are the mathematical tools we need to deal with Zeno. But is that all?

Here is a possible theory of motion. There is no such thing as motion because what we perceive as motion is in fact a succession of static states. The paradox of the arrow is no paradox, because the arrow is always 'at rest'. First it is at rest here, then there, then there, until it 'reaches' its target.

One question to ask would be if this describes a logically possible world. Could the laws of physics be sufficiently different from what they are in this world so that no 'motion' actually occurred, although it appeared to? Why not?

Let's have a look at The Dichotomy.

Object A occupies a finite series of static positions until it reaches half way to its destination, and then until it reaches half way again, and so on. There are two possibilities: either there is a unit of position or not. If there is a unit of position then there will be a final 'half-way' point such that only one unit separates this point from the destination. In which case, there is only one more 'jump' to make. No paradox.

On the other hand, if there is not an ultimate unit of position, i.e. if positions equivalent to points on a line - i.e. the series of real numbers - then the 'theory' that motion is a series of static states becomes indistinguishable from the view that motion is real and continuous. Again no paradox.

The problems arise with the hybrid theory which attempts to combine infinite divisibility with the naive idea that we can form the notion of the result of equally dividing an object, or a distance, into parts, conceived as each having a finite size. This becomes apparent in Zeno's arguments against infinite divisibility. Either the smallest parts have finite size, in which case an infinite number of parts produces an object of infinite extent, or the smallest parts have no size, in which case putting any number of parts together produces an object of zero extent.

The final paradox of the moving is interesting in the light of the possible theory that the universe is ultimately composed of a finite number of 'positions', which as we have seen is immune to the first three arguments. I would not agree that the moving rows create a problem for this theory. All you need to do is distinguish absolute and relative position. Absolute motion can only be from one position to the adjacent position. But this happily allows for relative motion which 'jumps' over intermediate positions.

All the best,

Geoffrey