Zero (5 page)

Figure 9: The Parthenon, the chambered nautilus, and the golden ratio

Zero had no place within the Pythagorean framework. The equivalence of numbers and shapes made the ancient Greeks the masters of geometry, yet it had a serious drawback. It precluded anyone from treating zero as a number. What shape, after all, could zero be?

It is easy to visualize a square with width two and height two, but what is a square with width zero and height zero? It's hard to imagine something with no width and no height—with no substance at all—being a square. This meant that multiplication by zero didn't make any sense either. Multiplying two numbers was equivalent to taking an area of a rectangle, but what could the area of a rectangle with zero height or zero width be?

Nowadays the great unsolved problems in mathematics are stated in terms of conjectures that mathematicians are unable to prove. In ancient Greece, however, number-shapes inspired a different way of thinking. The famous unsolved problems of the day were geometric: With only a straightedge and compasses, could you make a square equal in area to a given circle? Could you use those tools to trisect an angle?
*
Geometric constructions and shapes were the same thing. Zero was a number that didn't seem to make any geometric sense, so to include it, the Greeks would have had to revamp their entire way of doing mathematics. They chose not to.

Even if zero were a number in the Greek sense, the act of taking a ratio with zero in it would seem to defy nature. No longer would a proportion be a relationship between two objects. The ratio of zero to anything—zero divided by a number—is always zero; the other number is completely consumed by the zero. And the ratio of anything to zero—a number divided by zero—can destroy logic. Zero would punch a hole in the neat Pythagorean order of the universe, and for that reason it could not be tolerated.

Indeed, the Pythagoreans had tried to squelch another troublesome mathematical concept—the irrational. This concept was the first challenge to the Pythagorean point of view, and the brotherhood tried to keep it secret. When the secret leaked out, the cult turned to violence.

The concept of the irrational was hidden like a time bomb inside Greek mathematics. Thanks to the number-shape duality, to the Greeks counting was tantamount to measuring a line. Thus, a ratio of two numbers was nothing more than the comparison of two lines of different lengths. However, to make any sort of measurement, you need a standard, a common yardstick, to compare to the size of the lines. As an example, imagine a line exactly a foot long. Make a mark, say, five and a half inches from one end, dividing the foot into two unequal parts. The Greeks would figure out the ratio by dividing the line into tiny pieces, using, for example, a standard or yardstick half an inch long. One line segment contains 11 of those pieces; the other contains 13. The ratio of the two segments, then, is 11 to 13.

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 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 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.