Achilles, the swiftest runner in Greek legend, agrees to race a tortoise and generously gives it a head start. Common sense says he wins easily. An argument made about 2,500 years ago says he can never catch up at all, and for centuries nobody could say exactly what was wrong with it.
The argument runs like this. By the time Achilles reaches the spot where the tortoise began, the tortoise has crawled a little farther ahead. By the time he covers that new gap, the tortoise has moved on again, a smaller distance this time but still ahead. Every time Achilles arrives where the tortoise was, the tortoise has already left. There are infinitely many of these gaps to close, one after another, and, the argument insists, you cannot complete an infinite number of tasks. So Achilles is doomed to forever approach the tortoise without passing it.
The puzzle comes from Zeno of Elea, a Greek philosopher who devised a set of paradoxes meant to show that motion itself is an illusion. This one, Achilles and the tortoise, is the most famous. Everyone watching a real race knows the conclusion is false. The hard part is naming the exact flaw.
The resolution needed the proper mathematics of the infinite. The key idea is that an infinite list of numbers can still add up to a finite total. The gaps Achilles must close shrink in a steady proportion, and if you sum that endless dwindling series, a half plus a quarter plus an eighth and on forever, it does not grow without bound. It settles on a finite distance, the exact point where Achilles draws level. He closes infinitely many gaps, yes, but they all fit inside a finite stretch of track and a finite amount of time.
Zeno’s mistake was assuming that infinitely many steps must take infinitely long. They need not, so long as the steps shrink fast enough. It is a tidy ending to a two-thousand-year argument, though it is worth remembering how long the puzzle stood. A conclusion can be obviously wrong and still take the human race twenty centuries to explain properly.
