Zero - Charles Seife [17]
Modern mathematicians know that the terms have a limit; the numbers 1, ½, ¼, 1/8, 1/16, and so forth are approaching zero as their limit. The journey has a destination. Once the journey has a destination, it is easy to ask how far away that destination is and how long it will take to get there. It is not that difficult to sum up the distances that Achilles runs: 1 + ½ + ¼ + 1/8 + 1/16 +…+ ½n +…. In the same way that the steps that Achilles takes get smaller and smaller, and closer and closer to zero, the sum of those steps gets closer and closer to 2. How do we know this? Well, let’s start off with 2, and subtract the terms of the sum, one by one. We begin with 2 - 1, which is, of course, 1. Next, we subtract ½, leaving ½. Then remove the next term: subtract ¼, leaving ¼ behind. Subtracting 1/8 leaves 1/8 behind. We’re back to our familiar sequence. We already know that 1, ½, ¼, 1/8, and so forth has a limit of zero; thus, as we subtract the terms from 2, we have nothing left. The limit of the sum 1 + ½ + ¼ + 1/8 + 1/16 +…is 2 (Figure 11). Achilles runs 2 feet in catching up to the tortoise, even though he takes an infinite number of steps to do it. Better yet, look at the time it takes Achilles to overtake the tortoise: 1 + ½ + ¼ + 1/8 + 1/16 +…—2 seconds. Not only does Achilles take an infinite number of steps to run a finite distance, but he takes only 2 seconds to do it.
Figure 11: 1 + ½ + ¼ + 1/8 + 1/16 +…= 2
The Greeks couldn’t do this neat little mathematical trick. They didn’t have the concept of a limit because they didn’t believe in zero. The terms in the infinite series didn’t have a limit or a destination; they seemed to get smaller and smaller without any particular end in sight. As a result, the Greeks couldn’t handle the infinite. They pondered the concept of the void but rejected zero as a number, and they toyed with the concept of the infinite but refused to allow infinity—numbers that are infinitely small and infinitely large—anywhere near the realm of numbers. This is the biggest failure in Greek mathematics, and it is the only thing that kept them from discovering calculus.
Infinity, zero, and the concept of limits are all tied together in a bundle. Greek philosophers were unable to untie that bundle; therefore, they were ill-equipped to solve Zeno’s puzzle. Yet Zeno’s paradox was so powerful that the Greeks tried over and over to explain away his infinities. They were doomed to failure, unarmed with the proper concepts.
Zeno himself didn’t have a proper solution to the paradox, nor did he seek one. The paradox suited his philosophy perfectly. He was a member of the Eleatic school of thought, whose founder, Parmenides, held that the underlying nature of the universe was changeless and immobile. Zeno’s puzzles appear to have been in support of Parmenides’ argument; in showing that change and motion were paradoxical, he hoped to convince people that everything is one—and changeless. Zeno really did believe that motion was impossible, and his paradox was this theory’s chief support.
There were other schools of thought. The atomists, for example, believed that the universe is made up of little particles called atoms, which are indivisible and eternal. Motion, according to the atomists, was the movement of these little particles. Of course, for these atoms to move, there has to be empty space for them to move into. After all, these little atoms had to move around somehow; if there were no such thing as a vacuum, the atoms would be constantly pressed against one another. Everything would be stuck in one