Zero: The Biography of a Dangerous Idea

by Charles Seife

According to Zeno, nothing in the universe could move. Of course, this is a silly statement; anyone can refute it by walking across the room. Though everybody knew that Zeno’s statement was false, nobody could find a flaw in Zeno’s argument. He had come up with a paradox. Zeno’s logical puzzle baffled Greek philosophers—as well as the philosophers who came after them. Zeno’s riddles plagued mathematicians for nearly two thousand years. In his most famous puzzle, “The Achilles,” Zeno proves that swift Achilles can never catch up with a lumbering tortoise that has a head start. To make things more concrete, let’s put some numbers on the problem. Imagine that Achilles runs at a foot a second, while the tortoise runs at half that speed. Imagine, too, that the tortoise starts off a foot ahead of Achilles. Achilles speeds ahead, and in a mere second he has caught up to where the tortoise was. But by the time he reaches that point, the tortoise, which is also running, has moved ahead by half a foot. No matter. Achilles is faster, so in half a second, he makes up the half foot. But again, the tortoise has moved ahead, this time by a quarter foot. In a flash—a quarter second—Achilles has made up the distance. But the tortoise lumbers ahead in that time by an eighth of a foot. Achilles runs and runs, but the tortoise scoots ahead each time; no matter how close Achilles gets to the tortoise, by the time he reaches the point where the tortoise was, the tortoise has moved. An eighth of...