# Why can't mathematics explain Zeno's Paradox?

Mathematics has no trouble at all dealing with Zeno’s paradoxes. Let us start with the simplest of Zeno’s paradoxes:

If the tortoise is ahead of Achilles, Achilles can never overtake the tortoise, since, by the time Achilles gets to a position previously occupied by the tortoise, the tortoise has already moved on.

The solution: Whenever Achilles is occupying a point that the tortoise *has already *occupied, Achilles is indeed behind the tortoise. This is a tautology and in no way paradoxical. So the paradox must therefore be that in order for Achilles to catch up to the tortoise, he must traverse an infinite series of distances in a finite amount of time. But there is nothing paradoxical about this, given that any finite number can be represented as the sum of an infinite series (for example, 1 can be represented as the sum of ½ and ¼ and 1/8 and so on) and given, consequently, that any case of an object’s moving at a finite rate involves its traversing infinitely many finite distances in a finite amount of time.

Each of Zeno’s other paradoxes is structurally similar; so—no problem, either for mathematics or for philosophy.