Though the brotherhood was scattered and the leader was dead, the essence of the Pythagorean teachings lived on. It was soon to become the basis of the most influential philosophy in Western history—the Aristotelian doctrine that would live for two millennia. Zero would clash with this doctrine, and unlike the irrational, zero could be ignored. The number-shape duality in Greek numbers made it easy; after all, zero didn’t have a shape and could thus not be a number.
Yet it was not the Greek number system that prevented zero’s acceptance—nor was it lack of knowledge. The Greeks had learned about zero because of their obsession with the night sky. Like most ancient peoples, the Greeks were stargazers, and the Babylonians were the first masters of astronomy: they had learned how to predict eclipses. Thales, the first Greek astronomer, learned how to do this from the Babylonians, or perhaps through the Egyptians. In 585 BC he was said to have predicted a solar eclipse.
With Babylonian astronomy came Babylonian numbers. For astronomical purposes the Greeks adopted a sexagesimal number system and even divided hours into 60 minutes, and minutes into 60 seconds. Around 500 BC the placeholder zero began to appear in Babylonian writings; it naturally spread to the Greek astronomical community. During the peak of ancient astronomy, Greek astronomical tables regularly employed zero; its symbol was the lowercase omicron, o, which looks very much like our modern-day zero, though it’s probably a coincidence. (Perhaps the use of omicron came from the first letter of the Greek word for nothing, ouden.) The Greeks didn’t like zero at all and used it as infrequently as possible. After doing their calculations with Babylonian notation, Greek astronomers usually converted the numbers back into clunky Greek-style numerals—without zero. Zero never worked its way into ancient Western numbers, so it is unlikely that the omicron is the mother of our 0. The Greeks saw the usefulness of zero in their calculations, yet they still rejected it.
So it was not ignorance that led the Greeks to reject zero, nor was it the restrictive Greek number-shape system. It was philosophy. Zero conflicted with the fundamental philosophical beliefs of the West, for contained within zero are two ideas that were poisonous to Western doctrine. Indeed, these concepts would eventually destroy Aristotelian philosophy after its long reign. These dangerous ideas are the void and the infinite.
The Infinite, the Void, and the West
So, naturalists observe, a flea
Hath smaller fleas that on him prey,
And these have smaller yet to bite ’em,
And so proceed ad infinitum….
—JONATHAN SWIFT, “ON POETRY: A RHAPSODY”
The infinite and the void had powers that frightened the Greeks. The infinite threatened to make all motion impossible, while the void threatened to smash the nutshell universe into a thousand flinders. By rejecting zero, the Greek philosophers gave their view of the universe the durability to survive for two millennia.
Pythagoras’s doctrine became the centerpiece of Western philosophy: all the universe was governed by ratios and shapes; the planets moved in heavenly spheres that made music as they turned. But what lay beyond these spheres? Were there more and more spheres, each larger than its neighbor? Or was the outermost sphere the end of the universe? Aristotle and later philosophers would insist that there could not be an infinite number of nested spheres. With the adoption of this philosophy, the West had no room for infinity or the infinite. They rejected it outright. For the infinite had already begun to gnaw at the roots of Western thought, thanks to Zeno of Elea, a philosopher reckoned by his contemporaries to be the most annoying man in the West.
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 on, with the terms getting smaller and smaller, getting closer and closer to zero; each term is like a step along a journey where the destination is zero. However, since the Greeks rejected the number zero, they couldn’t understand that this journey could ever have an end. To them, the numbers 1, ½, ¼, 1/8, 1/16, and so forth aren’t approaching anything; the destination doesn’t exist. Instead, the Greeks just saw the terms as simply getting smaller and smaller, meandering outside the realm of numbers.