Zero - Charles Seife [16]
Zeno was born around 490 BC, at the beginning of the Persian wars—a great conflict between East and West. Greece would defeat the Persians; Greek philosophy would never quite defeat Zeno—for Zeno had a paradox, a logical puzzle that seemed intractable to the reasoning of Greek philosophers. It was the most troubling argument in Greece: Zeno had proved the impossible.
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 a foot…a sixteenth of a foot…a thirty-second of a foot…smaller and smaller distances, but Achilles never catches up. The tortoise is always ahead (Figure 10).
Everybody knows that, in the real world, Achilles would quickly run past the tortoise, but Zeno’s argument seemed to prove that Achilles could never catch up. The philosophers of his day were unable to refute the paradox. Even though they knew that the conclusion was wrong, they could never find a mistake in Zeno’s mathematical proof. The philosophers’ main weapon was logic, but logical deduction seemed useless against Zeno’s argument. Each step along the way seemed airtight, and if all the steps are correct, how could the conclusion be wrong?
Figure 10: Achilles and the tortoise
The Greeks were stumped by the problem, but they did find the source of the trouble: infinity. It is the infinite that lies at the heart of Zeno’s paradox: Zeno had taken continuous motion and divided it into an infinite number of tiny steps. Because there are an infinite number of steps, the Greeks assumed that the race would go on forever and ever, even though the steps get smaller and smaller. The race would never finish in finite time—or so they thought. The ancients didn’t have the equipment to deal with the infinite, but modern mathematicians have learned to handle it. The infinite must be approached very carefully, but it can be mastered, with the help of zero. Armed with 2,400 years’ worth of extra mathematics, it is not hard for us to go back and find Zeno’s Achilles’ heel.
The Greeks did not have zero, but we do, and it is the key to solving Zeno’s puzzle. It is sometimes possible to add infinite terms together to get a finite result—but to do so, the terms being added together must approach zero.* This is the case with Achilles and the tortoise. When you add up the distance that Achilles runs, you start with the number 1, then add ½, then add ¼, then add 1/8, and so