To: Katherine A.
From: Geoffrey Klempner
Subject: Zeno's paradox of infinite divisibility
Date: 9 July 2007 13:09

Dear Katherine,

Thanks for your email of 1 July with your essay for the Ancient Philosophy program, in response to the question, 'Analyze one of Zeno's Paradoxes'.

My impression of your Australian Tax Office's second method of calculating depreciation is that is consistent with the rejection of Zeno's argument from 'complete divisibility'.

With the first method, you cut the value of a capital item into five chunks, and remove the chunks one at a time leaving zero after five years. No problem. With the second method, you reduce the value by a fixed percentage each time, so that the value never reaches zero.

Although a fraction of a cent is not recognized as legal tender, this is not equivalent to 'nothing' as any company accountant will tell you. If I can save my company 0.01 cent per transaction on a billion transactions, I have saved 100,000 Dollars.

It is a simple matter, however, to add the rule that amounts should be rounded down to the nearest cent, so that as soon as the process of reducing by 40 per cent produces a value of less than a cent, then for tax purposes this is equivalent to zero.The challenge posed by Zeno is what happens if we attempt to conceive of a process of subdivision taken to infinity. Surely, at that point, there is nothing left? But adding lots of nothing gives the result, 'nothing'. On the other hand, as Zeno also argues - this is the other side of the dilemma - if the resulting infinite entities do each have a finite size, however small, then putting them together would create an entity of infinite size.

Modern mathematics does not have this problem, because it uses the notion of an 'infinitesimal'. Instead of talking of the 'result' of an infinite process of subdivision, one talks of an infinite series. The criticism you will hear most often against Zeno is that he was writing at a time before mathematics had found sophisticated ways of dealing with the notion of an infinite series.

A similar point applies to Achilles and the Tortoise. In this case we have a series of decreasing fractions, 1/2 + 1/4 + 1/8... whose sum is 1. Again, no problem from a mathematical point of view.

In Zeno's defence, one might point out that he has highlighted a genuine problem with any attempt to talk of an 'actual' infinity, that is to say, outside the context of mathematics where an infinite series can be reduced to a finite rule (the simplest such rule is 'add one').

Suppose there really 'is' an infinite number of parts to an object, not merely in the sense that we can mathematically state a rule, 'divide by 2' which produces an infinite series, but rather in the sense of an 'actually existing' infinite number of parts. Then we really do face an impossible dilemma, just as Zeno said.

I am tempted to rise to the defence of Zeno's 'grain of millet' argument. It is easy enough to state that our senses have a minimum threshold sensitivity, so that above the threshold they are generally reliable but at or below the threshold cease to be reliable. What would Zeno have said in response to that argument?

To admit a threshold, I can imagine Zeno saying, means that we cannot equate a sound with the perception of a sound, or equate light with the perception of light (one can construct a similar argument for light). So now we have sounds that no-one hears and amounts of light that no-one sees.

From a modern perspective, this is no problem because we have a theory according to which sounds are vibrations of air molecules, while light is electromagnetic radiation from the visible spectrum. There is no problem in admitting that sometimes air molecules vibrate in such a way as to be undetectable by the human ear, or that there is electromagnetic radiation from the visible spectrum which is undetectable by the human eye. (Actually, I read somewhere that the eye can under favourable conditions discern a single photon, which is the smallest physical unit of light. However, conditions are not always favourable, and the point is that there is no necessity that our eyes should be this sensitive.)

By contrast, for someone who believes in the reality of sights and sounds, as entities in their own right not explained by some underlying physical process - as most Greeks did, at least until the atomists appeared - the grain of millet does pose a real difficulty, because it forces us to posit an unhearable sound which nevertheless 'exists' as a sound and nothing but a sound, which sounds fair nonsense.

All the best,

Geoffrey