For everything in the universe to be governed by ratios, as the Pythagoreans hoped, everything that made sense in the universe had to be related to a nice, neat proportion. It literally had to be rational. More precisely, these ratios had to be written in the form a/b, where a and b were nice, neat counting numbers like 1, 2, or 47. (Mathematicians are careful to note that b is not allowed to be zero, for that would be tantamount to division by zero, which we know to be disastrous.) Needless to say, the universe is not really that orderly. Some numbers cannot be expressed as a simple ratio of a/b. These irrational numbers were an unavoidable consequence of Greek mathematics.
The square is one of the simplest figures of geometry, and it was duly revered by the Pythagoreans. (It had four sides, corresponding to the four elements; it symbolized the perfection of numbers.) But the irrational is nestled within the simplicity of the square. If you draw the diagonal—a line from one corner to the opposite corner—the irrational appears. As a concrete example, imagine a square whose sides are one foot long. Draw the diagonal. Ratio-obsessed people like the Greeks naturally looked at the side of the square and the diagonal and asked themselves: what is the ratio of the two lines?
The first step, again, is to create a common yardstick, perhaps a tiny ruler half an inch long. The next step is to use that yardstick to divide each of the two lines into equal segments. With a half-inch yardstick we can divide the foot-long side of the square into 24 segments, each half an inch long. What happens when we measure the diagonal? Using the same yardstick, we see that the diagonal gets…well, almost 34 segments, but it doesn’t come out quite evenly. The 34th segment is a wee bit too short; the half-inch ruler juts out a little beyond the corner of the square. We can do better. Let’s divide the line into even smaller segments, using, say, a ruler one-sixth of an inch long. The side of the square is partitioned into 72 segments, while the diagonal comes out to more than 101 but fewer than 102 segments. Again, the measurement is not quite perfect. What happens when we try really small segments, measuring in bits a millionth of an inch each? The side of the square gets 12 million bits, and the diagonal gets a tad less than 16,970,563 bits. Once again, our ruler doesn’t fit both lines exactly. No matter what ruler we choose, our measurement never seems to come out right.
In fact, no matter how tiny you make the bits, it is impossible to choose a common yardstick that will measure both the side and the diagonal perfectly: the diagonal is incommensurable with the side. However, without a common yardstick, it is impossible to express the two lines in a ratio. For a square of size one, this means that we cannot choose counting numbers a and b such that the diagonal of the square can be expressed as a/b. In other words, the diagonal of that square is irrational—and nowadays we recognize that number as the square root of two.
This meant trouble for the Pythagorean doctrine. How could nature be governed by ratios and proportions when something as simple as a square can confound the language of ratios? This idea was hard for the Pythagoreans to believe, but it was incontrovertible—a consequence of the mathematical laws that they held so dear. One of the first mathematical proofs in history was about the incommensurability/irrationality of the square’s diagonal.
Irrationality was dangerous to Pythagoras, as it threatened the basis of his ratio-universe. To add insult to injury, the Pythagoreans soon discovered that the golden ratio, the ultimate Pythagorean symbol of beauty and rationality, was an irrational number. To keep these horrible numbers from ruining the Pythagorean doctrine, the irrationals were kept secret. Everyone in the Pythagorean brotherhood was already tight-lipped—nobody was allowed even to take written notes—and the incommensurability of the square root of two became the deepest, darkest secret of the Pythagorean order.
However, irrational numbers, unlike zero, could not easily be ignored by the Greeks. The irrationals occurred and reoccurred in all sorts of geometrical constructions. It would be hard to keep the secret of the irrational hidden from a people so obsessed with geometry and ratios. One day someone was going to let the secret out. This someone was Hippasus of Metapontum, a mathematician and member of the Pythagorean brotherhood. The secret of the irrationals would cause him great misfortune.
The legends are very hazy and contain contradictory stories about the betrayal and ultimate fate of Hippasus. Mathematicians to this day tell of the hapless man who revealed the secret of the irrational to the world. Some say that the Pythagoreans tossed Hippasus overboard, drowning him, a just punishment for ruining a beautiful theory with harsh facts. Ancient sources talk about his perishing at sea for his impiety, or alternatively, say that the brotherhood banished him and constructed a tomb for him, expelling him from the world of human beings. But whatever Hippasus’s true fate was, there is little doubt that he was reviled by his brothers. The secret he revealed shook the very foundations of the Pythagorean doctrine, but by considering the irrational an anomaly, the Pythagoreans could keep the irrationals from contaminating their view of the universe. Indeed, over time the Greeks reluctantly admitted the irrationals to the realm of numbers. The irrationals didn’t kill Pythagoras. Beans did.
Just as the legends of Hippasus’s murder are hazy, so too are the legends of Pythagoras’s end. Nevertheless, they all imply that the master died in a bizarre way. Some say that Pythagoras starved himself, but the most common versions all say that beans were his undoing. One day, according to a version of the legend, his house was set ablaze by his enemies (who were angry at not being considered worthy to be admitted into Pythagoras’s presence), and the brothers in the house scattered in all directions, running for their lives. The mob slaughtered Pythagorean after Pythagorean. The brotherhood was being destroyed. Pythagoras himself fled for his life, and he might have gotten away had he not run smack into a bean field. There he stopped. He declared that he would rather be killed than cross the field of beans. His pursuers were more than happy to oblige. They cut his throat.
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 be
liefs 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.
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 position for eternity, unable to move. Thus, the atomic theory required that the universe be filled with emptiness—an infinite void. The atomists embraced the concept of the infinite vacuum—infinity and zero wrapped into one. This was a shocking conclusion, but the indivisible kernels of matter in atomic theory got around the problem of Zeno’s paradoxes. Because atoms are indivisible, there is a point beyond which things could not be divided. Zeno’s hair-splitting doesn’t go on ad infinitum. After a number of strides, Achilles would be taking tiny steps that can’t get any smaller; eventually he would have to hurdle an atom that the tortoise doesn’t. Achilles would finally catch up to the elusive turtle.
Another philosophy vied with the atomic theory, and instead of posing such bizarre concepts as the infinite vacuum, it turned the universe into a cozy nutshell. There was no infinity, no void—just beautiful spheres that surrounded the earth, which was naturally placed at the very center of the universe. This was the Aristotelian system, which was later refined by the Alexandrian astronomer Ptolemy. It became the dominant philosophy in the Western world. And by rejecting zero and infinity, Aristotle explained away Zeno’s paradoxes.
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