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What Is the Answer to Zeno’s Paradox?

The Greek philosopher Zeno wrote a book of paradoxes nearly 2,500 years ago. “Achilles and the Tortoise” is the easiest to understand, but it’s devilishly difficult to explain away. For those who haven’t already learned it, here are the basics of Zeno’s logic puzzle, as we understand it after generations of retelling:

Achilles, the fleet-footed hero of the Trojan War, is engaged in a race with a lowly tortoise, which has been granted a head start. Achilles’ task initially seems easy, but he has a problem. Before he can overtake the tortoise, he must first catch up with it. While Achilles is covering the gap between himself and the tortoise that existed at the start of the race, however, the tortoise creates a new gap. The new gap is smaller than the first, but it is still a finite distance that Achilles must cover to catch up with the animal. Achilles then races across the new gap. To Achilles’ frustration, while he was scampering across the second gap, the tortoise was establishing a third. The upshot is that Achilles can never overtake the tortoise. No matter how quickly Achilles closes each gap, the slow-but-steady tortoise will always open new, smaller ones and remain just ahead of the Greek hero.

It’s tempting to dismiss Zeno’s argument as sophistry, but that reaction is based on either laziness or fear. Laziness, because thinking about the paradox gives the feeling that you’re perpetually on the verge of solving it without ever doing so—the same feeling that Achilles would have about catching the tortoise. Fear, because being outwitted by a man who died before humans conceived of the number zero delivers a significant blow to one’s self-image. But what if your 11-year-old daughter asked you to explain why Zeno is wrong? Would you just tell her that Achilles is faster than a tortoise, and change the subject? That would be pretty weak. Zeno assumes that Achilles is running faster than the tortoise, which is why the gaps are forever getting smaller. But it doesn’t answer the question.

Let’s see if we can do better. I consulted a number of professors of philosophy and mathematics. Most of them insisted you could write a book on this (and some of them have), but I condensed the arguments and broke them into three parts.

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