Zeno left us some horrific paradoxes. He was a mathematician, and the paradoxes depended upon some axioms of mathematics, e.g.
To move from one point in space to another, first one must cover half the distance. Before one can do that, one must cover a half of the half, and so on ad infinitum. One must pass through an infinite number of points, and that is impossible in a finite time.

Achilles races a tortoise, giving it a good head start. According to Zeno, Achilles could never beat the tortoise.

From a common-sense viewpoint, there is no problem. At some point, Achilles' stride-length will EXCEED the current unit (1/2 of the previous unit), and he will beat the tortoise.

I was wondering if modern mathematical theory allows a mathematical proof that Achilles could beat the tortoise.

In a technical sense, Achilles' heel would probably give out from trying to measure 1/2^346 of an inch on the ground. Eventually, though, he will eventually be close enough where he could reach his arms out acros the line, break the ribbon, etc.

One answer to this paradox is the theory that space itself is discrete. According to quantum theory, it's possible that space is discrete, not continuous, so movement can only be defined in increments of a certain (sub-atomic) minimum distance. (For a very technical discussion, see this link: QUANTIZATION OF SPACE-TIME.) Thus, Achilles' movement towards the tortoise is quantized, and when the distances in Zeno's paradox become too small, you're forced to round off to the the nearest quantum distance, thus resolving the paradox.

Imagine Achilles and the tortoise are racing on a stairway, and they can only move in one-step increments. Zeno's infinite series argument breaks down as soon as the series reaches a term less than one (since the racers must move in one-step increments). Rounding the distances to the nearest integer, Achilles will eventually reach the same step as the tortoise, and then he can move ahead of him (using the same argument in reverse).

The same argument will work without recourse to quantum physics. Since Achilles and the tortoise are both macroscopic objects, it's illogical to define their movement in terms of sub-atomic distances (just standing in one place, a person's center of mass constantly shifts by huge sub-atomic distances). We could define a minimum "step" as a measure of distance which would represent significant movement of a person (or tortoise) considered as a whole object. Whatever this basic distance is, Zeno's argument will again break down when the infinite series reaches terms that are smaller than this distance.

This is basically the same as the common sense arguments used by babthrower and WiteoutKing above. Whether you talk about stride-length, or the minimum distance Achilles can move without his heel giving out (nice inside reference, by the way), these are just different ways to quantize movement distance on a macroscopic level. Taking it to the true quantum (sub-atomic) level just guarantees that you're getting the very last word in the argument, since there's no way anybody can split that hair any further.

[This message was edited by silenteuphony on 07-27-02 at 12:11 AM.]