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Zeno's paradox of the arrow


To: Peter B.
From: Geoffrey Klempner
Subject: Zeno's paradox of the arrow
Date: 22 November 2003 11:47

Dear Peter,

Thank you for your e-mail of 11 November, with your second essay for the Associate award, 'Zeno's Arrow'.

The essay starts well, with the quotes from Plato's 'Parmenides' and Simplicius. I also fully agree with the statement that 'In any attempt to solve the apparent paradox, one must also consider the validity of the method used to find a solution.'

There seemed to be a hint at Henri Bergson's view that time is essentially continuous and not composed of instants (Bergson calls the idea that time is composed of infinitesimal instants the 'cinematographic' view of time.) This is probably the most tangible lead to follow. However, the paragraphs where you discuss the continuity of time were just not clear. If time is regarded as more that the sum of instants of zero duration, why does that not show that Zeno is wrong?

Mathematically, there is no problem with the idea of a magnitude composed of points. From my studies of set theory, I recall that according to Cantor, the set of points on a line is the same size (a 'transfinite number') as the set of infinite sets of integers.

The idea that photography provides a 'real world proof' of the existence of temporal instants looks like a non-starter. However, the example of a camera does seem to be pertinent in view of Bergson's 'cinematographic' criticism. The philosopher who believes that time is a set of instants of zero duration seems to be relying on an *analogy* with the way photography breaks up motion into immobile 'frames'. No physically constructed camera could capture a zero instant of time. However, the philosopher who holds the cinematographic view can say that all there is to time is what *would* be captured by such a physically, but not logically impossible camera.

Quantum mechanics seems a rather heavy weapon to beat Zeno with, yet it is worth remembering that, as he shows in his paradox about the 'Large and Small', Zeno was fully aware of the alternatives, 'Either there exists a smallest size or not'. Applying this dilemma to the Arrow paradox gives just the result that Zeno wants, on *either* alternative:

1. If there is a smallest size with respect to temporal instants, then there is no such thing as motion, because the apparent 'movement' of the arrow is merely the occupation of successive instants by a static arrow. The arrow appears in one position, disappears then reappears in a new position an instant later.

2. If there is no smallest size with respect to temporal instants, then apparent 'movement' is still an illusion. The only difference from case 1. is that the disappearance and reappearance occurs seamlessly.

I didn't follow your argument that 'the Planck scale will not provide the required scenario to prove that Zeno's arrow never moves'. The 'capacity for movement in time' referred to in defining Planck units can be thought of as 'Zeno movement', i.e. the kind of apparent 'movement' that you get in scenario 1. or 2. The Planck/ QM case looks like a confirmation of scenario 1.

Again, I didn't understand your account of Aristotle's criticism or what follows from it. You say, 'If it is assumed that an instant has any period of duration, then Zeno's statements lose their coherency'. But this is just case 1, which clearly vindicates Zeno.

Why is it important (as Bergson and Aristotle thought) to defeat Zeno's conclusion that motion does not occur? Unlike, say, the paradox of the hare and the tortoise, we are not faced here with a situation where we *know* that the conclusion must be false. The hare does catch up with the tortoise in the real world. Here, by contrast, it is no use saying to Zeno, 'But an arrow that you fire from your bow does move' because Zeno's response will be, 'It only looks like movement, but in reality it isn't because what you think of as 'movement' is impossible, for the reasons I have given.'

So, one possible conclusion (unless we follow Bergson or Aristotle) is that the arrow isn't a paradox at all because it describes a situation which we can all agree is in fact the case, without giving up our beliefs about the familiar world around us. (Jonathan Barnes in 'The Presocratic Philosophers' after exhaustively analysing the argument concludes that 'Zeno is vindicated: the moving arrow does not move', p. 279.)

I do feel a residual worry that Zeno might not have intended the 'paradox' in this way, that he really did mean to say, 'Look, if you follow my reasoning you will see that the arrow cannot even leave the bow.' However, I am not aware of any textual evidence to support this extreme interpretation.

Barnes expresses a view which would probably be shared by many philosophers. However, as noted above, Bergson is one example of a dissenting view so it might be worth while to follow this up. (You don't need mention Bergson to in order to make this essay acceptable, this is just a suggestion.) What you do need to do is make your argument clearer. I suspect that in writing this essay you have been handicapped by your assumption that the paradox is a challenge which needs to be met in one way or another.

I need some help in order to make recommendations for your next essay. Has your research on Zeno unearthed topics that you find gripping? Would you like to do something more on the philosophy of time, for example? If so, you might look at Richard Gale Ed. 'The Philosophy of Time' which is the standard collection of readings by historical and contemporary philosophers on various aspects of the philosophy of time.

Let me know and we can discuss this further.

All the best,