Fengyun Liu

Achilles and the Tortoise

The ancient Greek philosopher Zeno has posed a series of “immeasurably subtle and profound” (Bertrand Russel) paradoxes that confused scholars for over two thousand years.

One of the famous paradox is called achilles and the tortoise. Aristotle summarized the paradox succinctly:

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

The argument goes like this: for a faster runner A to catch up a slower runner B ahead of him, he first has to reach the point x where B is. But when A reaches the point x, B is already at another point y ahead of x. This pattern continues so on and so forth, thus the faster runner A can never catch up the slower runner B.

It is hard to accept the conclusion as it is completely against common sense. However, the reasoning seems flawless. What is wrong?

I think we need to admit that the paradox is logically flawless if we assume space is continuous like real numbers, and motion is to visit each number along the axis of real numbers. Otherwise, the paradox would not have stayed more than two thousand years and attracted smart minds like Bertrand Russell.

But what if the space is not continuous like real numbers? Imagine the motion of graphical objects on a computer screen. The paradox simply does not hold in this setting. This is because the space of computer screen is essentially a bitmap, that is a matrix of pixels. Any motion on the screen has to advance at least one pixel.

Is it possibly that the physical space is not continuous in the sense that it cannot be subdivided infinitely? The assumption is logically coherent and it seems can never be contradicted by scientific advances. This is because at any point in human history, there is a measuring limit beyond which science has to be silent.

If making a metaphysical commitment on the nature of space is a price too high to pay, the following proposition is an alternative solution in the same spirit:

Motion covers at least some minimum distance.

This solution has the merit that it does not make any commitment on the nature of space.

Is it compatible with physics? I think so, at least experimentally, as physical rulers always have a measuring limit, we may thus claim the minimum distance to be always below any known measuring limit. If it is compatible with all experiments, then it does not matter if it is at odds with theories, as experiments are the final judge of physical theories.

The shrewd reader would soon point out that the proposition is not falsifiable, thus it is not a scientific proposition. I have to agree with this criticism: the proposition can never be refuted experimentally. However, it’s not bad to hold a belief if it’s certain to be compatible with all experiments and resolves a conceptual paradox.