To: Alistair L.
From: Geoffrey Klempner
Subject: Zeno's paradoxes and the nature of motion
Date: 12th November 2010 13:11

Dear Alistair,

Thank you for your email of 4 November, with your essay for the University of London BA Greek Philosophy: Plato and the Presocratics module, in response to the question, 'What do Zeno's paradoxes demonstrate about the nature of motion?'

I think it was a good decision to look at all of Zeno's paradoxes and not just those which specifically address the concept of motion. I agree with you that they are all, in different ways, relevant to the question.

The basis of your argument is an important distinction between the 'reality' of motion, in the physical world, and our 'models' of it. We deal with things in motion including our own bodies at the most basic level without having to reflect on it. It is when we bring our 'higher mental faculties' into play, in an attempt to coherently model our own movements or the motions that we perceive, that the trouble starts.

Following on from this distinction, there are two sorts of consequences that one might seek to draw from Zeno's paradoxes regarding motion: consequences for the nature of motion itself, which would in effect be a priori knowledge of the physical world; and consequences for the nature of our models of motion, which might lead to a Kantian-style claim to the effect that our knowledge of external reality is constrained by conditions which apply to any possible model of physical motion. (In stating this distinction, I am merely drawing out the implications of what you say in your essay. I am not claiming that the distinction is necessarily valid, or even that I fully understand it.)

Parmenides had set the example for philosophical arguments which have direct, a priori necessary consequences for the nature of reality. Zeno, who as you report say himself as following in Parmenides' footsteps would have attempted nothing less. Whereas we (post Kant) are much more inclined to talk of limits on the way we model reality, on what is or could be an acceptable physical theory.

Your 'short answer' to the question is, No, Zeno's paradoxes don't demonstrate anything about the nature of motion. 'What Zeno does is demonstrate quite important questions about the nature of our models of motion and in doing so questions our models of space, time and reality in general... there is still the ever-present gap between our models and reality itself and while the investigation of Zeno's paradoxes has led to better justification of these models there is no guarantee that they are what reality is.'

I am not sure that I fully understand this claim. Let's go back to the distinction I described earlier. Physical motion, as a thing in itself, is something out there which we discover through action and sense perception. Although we may be confident that it exists, we only 'know what it is' to the extent that we are able to describe models, formulate theories, in short, do the maths. According to the maths, the sum of an infinite series can be finite. Again, according to the maths, the rational numbers cannot be put in 1-1 mapping with the real numbers (*if*, as you say, you accept Cantor's proof). Both of these results are arguably needed in order to deal with Zeno's paradoxes, and that is an important discovery -- something the Greeks didn't know.

How do our models still fall short? You say at one point, 'The concept of the limit of a convergent infinite series is a useful mathematical tool and provides a definition that solves problems like those in these two paradoxes but the limit is not actually in the series so if you do take Achilles as traversing an infinite number of smaller and smaller distances on his way to catch the tortoise the last step from the end of the infinite series to the limit (position of the tortoise) is not part of the model.'

But according to the model there is no 'last step', no 'end of the infinite series', so long as we follow Zeno and have Achilles approach the tortoise step-by-step. The solution, surely, is to distinguish between actions that Achilles does, and the series of distances that he traverses. He traverses an infinite series of distances but this does not require an infinite series of actions.

I agree that one can't just say, 'If you accept the maths, then there's nothing more to say about the reality.' Space (and time) *might* be quantized. So what if they were? All the problems Zeno raises would disappear. The arrow *is* at rest, in effect, because what we term 'motion' is merely the result of a cinematic illusion (Bergson opposed his 'duree' to what he termed the erroneous 'cinematic model' of movement and perception). The arrow is at rest here, then here, then here. The only residual difficulty would be with the moving rows, but that's easy enough to deal with if you distinguish absolute motion (an object occupying successive 'slots' as in a Lego brick) from merely relative motion.

All the best,

Geoffrey