Zero: The Biography of a Dangerous Idea

by Charles Seife

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.

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.

Irrationality was dangerous to Pythagoras, as it threatened the basis of his ratio-universe.

The irrationals occurred and reoccurred in all sorts of geometrical constructions.

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.

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

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.

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 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.

Parmenides, held that the underlying nature of the universe was changeless and immobile.

The atomists, for example, believed that the universe is made up of little particles called atoms, which are indivisible and eternal.

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.

Aristotle just wished infinity away by stating that it is simply a construct of the human mind.

However, since there is no infinity, there can’t be an endless number of spheres; there must be a last one.

there were only two logical possibilities for the nature of the void, and both implied that the infinite exists. First, there could be an infinite amount of void—thus infinity exists. Second, there could be a finite amount of void, but since void is simply the lack of matter, there must be an infinite amount of matter to make sure that there is only a finite amount of void—thus infinity exists.

Look back through time. Events have happened throughout history, but if there is no such thing as infinity, there cannot be an infinite number of events. Thus, there must be a first event: creation. But what existed before creation?

Syracuse was the richest city on the island of Sicily,

The parabola has a special property: it takes the rays of light from the sun, or any distant source, and focuses them to a point, concentrating all the light’s energy on a very small area.

Archimedes figured out a way to measure the parabola’s area by resorting to the infinite. The first step was to inscribe a triangle inside the parabola. In the two little gaps left, Archimedes inscribed more triangles. This left four gaps, which were filled with more triangles, and so on

axiom of Archimedes,

this axiom says that any number added to itself over and over again can exceed any other number. Zero, clearly, was not included.

But the axiom of Archimedes rejected zero, which is the bridge between the realms of the finite and the infinite, a bridge that is absolutely necessary for calculus and higher mathematics.

The infinite was not needed in the Greek universe.

We don’t have to worry about mixing up the value of the number—its cardinality—with the order in which it arrives—its ordinality—since

since they are essentially the same thing.

As these medieval thinkers imported the philosophy and science of the ancients, they inherited the ancient prejudices: a fear of the infinite and a horror of the void.

When we last saw zero, it was simply a placeholder. It was a blank spot in the Babylonian system of numeration.

The void had an important place in the Hindu religion. Hinduism had started off as a polytheistic religion, a set of tales about warrior gods and battles similar in many ways to the Greek mythos.

As with the yin and yang of the Far East and Zoroaster’s dualism of good and evil in the Near East, creation and destruction were intermingled in Hinduism.

Nothingness was what the world came from,

An important difference between the new Indian number system and the Babylonian style was that Indian numbers were base-10 instead of base-60.

The Indians had borrowed little of Greek geometry. They apparently didn’t have a deep interest in the plane figures that the Greeks loved so much. They never worried about whether the diagonal of a square was rational or irrational, nor did they investigate the conic sections as Archimedes had. But they did learn how to play with numbers.

Unlike the Greeks, the Indians did not see squares in square numbers or the areas of rectangles when they multiplied two different values. Instead, they saw the interplay of numerals—numbers stripped of their geometric significance. This was the birth of what we now know as algebra.

Once numbers shed their geometric significance, mathematicians no longer had to worry about mathematical operations making geometric sense. You can’t remove a three-acre swath from a two-acre field, but nothing prevents you from subtracting three from two. Nowadays we recognize that 2 – 3=–1: negative one.

the Al-jabr in the title (which means something like “completion”) gave us the term algebra.

The Bible told of the creation from the void, while the Greek doctrine rejected the possibility. The Christians, cowed by the power of Greek philosophy, chose Aristotle over their Bible. The Muslims, on the other hand, made the opposite choice.

The Muslims, with their Semitic, Eastern background, believed that God created the universe out of the void—a

Maimonides stated that the act of creation came from nothing.

kabbalism, or Jewish mysticism.

Certain numbers were holy or evil, according to the kabbalists—and

they looked through the Bible for these numbers and for hidden messages found by scanning through it in various ways.

The Hebrew term ein sof, which meant “infinite,” represented the creator aspect of God, the part of the deity that made the universe and that permeates every corner of the cosmos. But at the same time it had a different name: ayin, or “nothing.” The infinite and the void go hand in hand, and are both part of the divine creator. Better yet, the term ayin is an anagram of (and has the same numerical value as) the word aniy, the Hebrew “I.” It could scarcely be clearer: God was saying, in code, “I am nothing.” And at the same time, infinity.

Aristotle still had a firm grip on the church, and its finest thinkers still rejected the infinitely large, the infinitely small, and the void.

Tempier abolished many Aristotelian doctrines that contradicted God’s omnipotence, such as, “God can not move the heavens in a straight line, because that would leave behind a vacuum.” (The rotating spheres caused no problem, because they still occupied the same space. It is only when you move the spheres

in a line that you are forced to have a space to move the heavens into, and you are forced to have a space behind them after they move.)

In the mid-twelfth century the first adaptations of al-Khowarizmi’s Al-jabr were working their way through Spain, England, and the rest of Europe. Zero was on the way, and just as the church was breaking the shackles of Aristotelianism, it arrived.

The man who reintroduced zero to the West was Leonardo of Pisa. The son of an Italian trader, he traveled to northern Africa.

Fibonacci—learned mathematics from the Muslims and soon became a good mathematician in his own right.