Zero: The Biography of a Dangerous Idea

by Charles Seife

There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.

Nothing can be created from nothing. —LUCRETIUS, DE RERUM NATURA

In that universe there is no such thing as nothing. There is no zero. Because of this, the West could not accept zero for nearly two millennia.

Zero’s absence would stunt the growth of mathematics, stifle innovation in science, and, incidentally, make a mess of the calendar.

Before they could accept zero, philosophers in the West would have to d...

Around him stood the members of a cult, a secret brotherhood that he had betrayed.

For revealing that secret, the great Pythagoras himself sentenced Hippasus to death by drowning. To protect their number-philosophy, the cult would kill. Yet as deadly as the secret that Hippasus revealed was, it was small compared to the dangers of zero.

The leader of the cult was Pythagoras, an ancient radical.

And to Pythagoras the connection between shapes and numbers was deep and mystical. Every number-shape had a hidden meaning, and the most

beautiful number-shapes were sacred.

They contained a number-shape that was the ultimate symbol of the Pythagorean view of the universe: the golden ratio.

The importance of the golden ratio comes from a Pythagorean discovery that is now barely remembered. In modern schools, children learn of Pythagoras for his famed theorem: the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

It was known more than 1,000 years before Pythagoras’s time.

(The discordant tritone, for instance, was dubbed the “devil in music” and was rejected by medieval musicians.)

To Pythagoras, playing music was a mathematical act.

squares and triangles, lines were number-shapes, so dividing a string into two parts was the same as taking a ratio of two numbers.

The harmony of the monochord was the harmony of mathematics—and the ha...

Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature w...

This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets.

This is what Pythagoras meant when he insisted, “All is number.”

ratios were the keys to understanding nature,

One of these means yielded the most “beautiful” number in the world: the golden ratio.

Achieving this blissful mean is a matter of dividing a line in a special way: divide it in two so that the ratio of the small part to the large part is the same as the ratio of the large part to the whole

the most aesthetically pleasing,

The golden ratio was favored by artists and nature alike and seemed to prove the Pythagorean assertion that music, beauty, architecture, nature, and the very construction of the cosmos were all intertwined and inseparable.

The supernatural link between aesthetics, ratios, and the universe became one of the central and long-lasting tenets of Western civilization.

Zero had no place within the Pythagorean framework.

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 ...

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.

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.

And the ratio of anything to zero—a number divided by zero—can destroy logic.

Thus, a ratio of two numbers was nothing more than the comparison of two lines of different lengths.

These irrational numbers were an unavoidable consequence of Greek mathematics.

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.

incommensurability of the square root of two became the deepest, darkest secret of the Pythagorean order.

irrational numbers, unlike zero, could not easily be ignored by the Greeks.

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.

The Greeks had learned about zero because of their obsession with the night sky.

Around 500 BC the placeholder zero began to appear in Babylonian writings;

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.

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.

Pythagoras’s doctrine became the centerpiece of Western philosophy: all the universe was governed by ratios and shapes;

Aristotle and later philosophers would insist that there could not be an infinite number of nested spheres.

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.

According to Zeno, nothing in the universe could move.