Robin Herbert

Zeno of Elea(1), the 5th century philosopher, used an elegant argumentum ad absurdum to demonstrate that although the idea of travelling along a continuous path seems at first reasonable, that it could be shown to lead to a contradiction. You should forget the childish “Zeno claims we can’t move” caricature.

The resolution that has been attributed to him is that motion itself is an illusion based on some different underlying reality.

Today we might not go quite so far and say rather that our perceptions are based on an underlying reality which works in a different way than we imagine. No sensible person ever thought that our everyday perceptions are all that there is to reality and so why should any sensible person have a reason to reject this solution?

Zeno said that to pass a distance, some object should first have to pass half-way to that distance. And of course it should then half to pass the half-way of the half-way and so on. This would continue on to infinity and then there should be no start of the journey.(2)

In fact this is perfectly reasonable. Consider, for simplicity, a point on a line and imagine it can move one direction or another. What is the first place it can move to? Suppose there was some “first” place that it could go to. But then there would be two distinct places on the line that had no space between them. But for the line to be continuous it must be, at least in this usage, infinitely divisible. So no continuous (that is to say infinitely divisible) path can have a “next” value. Our best mathematical description of continuity, the real number, the number we use to measure space and time and other things and which underlies all our maths and science also has this feature, that there is no “next” number for any number, for example no smallest value greater than zero.

To put it one more way, the concepts of “moving” and “infinitely divisible” clash because the idea of making progress and the idea of “infinite amounts of space between” are completely incompatible. It holds no challenge for science.

So really there is not a paradox, merely the dissonance between our everyday intuitions and the complexities of nature, but the principle Zeno demonstrates is well understood and completely unproblematic for mathematics and science.

So why the “resolutions”? I don’t know. I don’t see what there ever was to resolve except a faulty intuition which was surely Zeno’s point in the first place.

Let’s look at the leading contender for a resolution and ask if it really resolves anything.

The “convergent series” is often suggested as a resolution, but nobody says quite how. The sum of the series doesn’t give Zeno more information. He describes a series of positions along a finite length, halving and halving infinitely many times. Why on earth should we suppose that Zeno didn’t realise that the sum of those parts should not equal the original whole? That makes no sense. Using calculus to tell Zeno what is already apparent from the example does not add anything to it.

And if the convergent series implies that there is a first step then it actually creates a paradox, a “next” place on a continuum which breaks the whole definition of a continuous space. What the calculus says is that just because there is “no first step” does not imply that there is anything missing. Zeno didn’t say anything was missing, his point was different.

The key is that Zeno’s Dichotomy paradoxes are paradoxes of *motion*. A lecturer at my University used to flinch and correct students when they said that a series “converged”. He hated the wrong impression this gave that “converging” is something that the series does. Of course it is not, we say that the series is convergent, meaning that we can arrive at a finite sum of the terms using calculus but it will not do that in time. Even if you could try to calculate the sum of a convergent infinite series in time, it would never even in principle reach the sum found by calculus. So it would in fact support Zeno.

I don’t want to make a big fuss about this, it is clear. Zeno did just what he said, he took the concepts of moving and a continuous path and showed that it led to a contradiction. He proposed that motion was an illusion but we don’t have to go that far and say that the intuitive notion we have of moving through a continuous path is just a perceptual construction and the mechanism underneath, well we leave that to the professionals – the physicists.

Footnotes:

- Of course we have few records of the time, for convenience I am using “Zeno” to whoever it was who authored the argument in question and whoever authored the resolution of it mentioned here.
- This is quite airtight. People have been trying to find a resolution to the paradox for 2,500 years and the best they have found is the “convergent series” response which , as I show, turns out to actually support Zeno.