Zero (10 page)

Brahmagupta's mistake did not last for very long. In time the Indians realized that 1 ÷ 0 was infinite. “This fraction of which the denominator is a cipher, is termed an infinite quantity,” writes Bhaskara, a twelfth-century Indian mathematician, who tells of what happens when you add a number to 1 ÷ 0. “There is no alteration, though many be inserted or extracted; as no change takes place in the infinite and immutable God.”

God was found in infinity—and in zero.

By the seventh century, the West had withered with the fall of Rome, but the East was flourishing. India's growth was eclipsed by another Eastern civilization. As the star of the West sank below the horizon, another star was rising: Islam. Islam would take zero from India—and the West would eventually take it from Islam. Zero's rise to preeminence had to begin in the East.

One evening in 610
AD
, Mohammed, a thirty-year-old native of Mecca, fell into a trance on Mount Hira. According to legend, the angel Gabriel told him, “Recite!” Mohammed did, and his divine revelations started a wildfire. A decade after Mohammed's death in 632, his followers had captured Egypt, Syria, Mesopotamia, and Persia. Jerusalem, the holy city of the Jews and the Christians, had fallen. By 700, Islam stretched as far as the Indus River in the East and Algiers in the West. In 711 the Muslims captured Spain, and they advanced as far as France. In the East they defeated the Chinese in 751. Their empire stretched farther than even Alexander could have imagined. Along the way to China, the Muslims conquered India. And there the Arabs learned about Indian numerals.

The Muslims were quick to absorb the wisdom of the peoples that they conquered. Scholars started translating texts into Arabic, and in the ninth century Caliph al-Mamun founded a great library: the House of Wisdom at Baghdad. It was to become the center of learning in the Eastern world—and one of its first scholars was the mathematician Mohammed ibn-Musa al-Khowarizmi.

Al-Khowarizmi wrote several important books, like
Aljabr wa'l muqabala,
a treatise on how to solve elementary equations; the
Al-jabr
in the title (which means something like “completion”) gave us the term algebra. He also wrote a book about the Hindu numeral system, which allowed the new style of numbers to spread quickly through the Arab world—along with
algorithms,
the tricks for multiplying and dividing Hindu numerals quickly. In fact, the word
algorithm
was a corruption of al-Khowarizmi's name. Though the Arabs took the notation from India, the rest of the world would dub the new system Arabic numerals.

The very word
zero
smacks of its Hindu and Arabic roots. When the Arabs adopted Hindu-Arabic numerals, they also adopted zero. The Indian name for zero was
sunya,
meaning “empty,” which the Arabs turned into
sifr.
When some Western scholars described the new number to their colleagues, they turned
sifr
into a Latin-sounding word, yielding
zephirus,
which is the root of our word zero. Other Western mathematicians didn't change the word so heavily and called zero
cifra,
which became
cipher.
Zero was so important to the new set of numbers that people started calling all numbers ciphers, which gave the French their term
chiffre,
digit.

However, when al-Khowarizmi was writing about the Hindu system of numbers, the West was still far from adopting zero. Even the Muslim world, with its Eastern traditions, was heavily contaminated by the teachings of Aristotle, thanks to the conquests of Alexander the Great. However, as Indian mathematicians had made quite clear, zero was the embodiment of the void. Thus, if the Muslims were to accept zero, they had to reject Aristotle. That was precisely what they did.

A twelfth-century Jewish scholar, Moses Maimonides, described the Kalam—the beliefs of Islamic theologians—with horror. He noted that instead of accepting Aristotle's proof of God, the Muslim scholars turned to the atomists, Aristotle's old rivals, whose doctrine, though out of favor, managed to survive the ravages of time. The atomists, remember, held that matter was composed of individual particles called atoms, and if these particles were able to move about, there had to be a vacuum between them, otherwise the atoms would be bumping into one another, unable to get out of one another's way.

The Muslims seized upon the atomists' ideas; after all, now that zero was around, the void was again a respectable idea. Aristotle hated the void; the atomists required it. 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.

Zero was an emblem of the new teachings, of the rejection of Aristotle and the acceptance of the void and the infinite. As Islam spread, zero diffused throughout the Muslim-controlled world, everywhere conflicting with Aristotle's doctrine. Islamic scholars battled back and forth, and in the eleventh century a Muslim philosopher, Abu Hamid al-Ghazali, declared that clinging to Aristotelian doctrine should be punishable by death. The debate ended shortly thereafter.

It's no surprise that zero caused such discord. The Muslims, with their Semitic, Eastern background, believed that God created the universe out of the void—a doctrine that could never be accepted where people shared Aristotle's hatred of the void and of the infinite. As zero spread through the Arab lands, the Muslims embraced it and rejected Aristotle. The Jews were the next in line.

For millennia the center of Jewish life had been planted firmly in the Middle East, but in the tenth century an opportunity for Jews arose in Spain. Caliph Abd al-Rahman III had a Jewish minister who imported a number of intellectuals from Babylonia. Soon a large Jewish community flourished in Islamic Spain.

Early medieval Jews, both in Spain and in Babylon, were wed firmly to Aristotle's doctrines. Like their Christian counterparts, they refused to believe in the infinite or the void. However, just as Aristotelian philosophy conflicted with Islamic teachings, it conflicted with Jewish theology. This is what drove Maimonides, the twelfth-century rabbi, to write a tome to reconcile the Semitic, Eastern Bible with the Greek, Western philosophy that permeated Europe.

From Aristotle, Maimonides had learned to prove God's existence by denying the infinite. Reproducing the Greek arguments faithfully, Maimonides contended that the hollow spheres that twirled about the earth had to be moved by something, say, the next sphere out. But the next sphere out had to be moved by something—the next sphere in line. However, since there cannot be an infinite number of spheres (because infinity was impossible), something had to be moving the outermost sphere. That was the prime mover: God.

Maimonides' argument was, indeed, a “proof” of God's existence—something incredibly valuable in any theology. Yet at the same time, the Bible and other Semitic traditions were full of the ideas of the infinite and the void, ideas that the Muslims already embraced. Just like Saint Augustine 800 years earlier, Maimonides tried to reshape the Semitic Bible to fit into Greek doctrine: doctrine that had an unreasonable fear of the void. But unlike the early Christians, who had freed themselves to interpret parts of the Old Testament as metaphor, Maimonides was unwilling to Hellenize his religion completely. Rabbinic tradition compelled him to accept the biblical account of the universe's creation from the void. This, in turn, meant contradicting Aristotle.

Maimonides argued that there were flaws in Aristotle's proof that the universe had always existed. After all, it conflicted with the Scriptures. This, of course, meant that Aristotle had to go. Maimonides stated that the act of creation came from nothing. It was
creatio ex nihilo,
despite the Aristotelian ban on the vacuum. With that stroke the void moved from sacrilege to holiness.

For the Jews, the years after Maimonides' death became the era of nothing. In the thirteenth century a new doctrine spread: kabbalism, or Jewish mysticism. One centerpiece of kabbalistic thought is gematria—the search for coded messages within the text of the Bible. Like the Greeks, the Hebrews used letters from their alphabet to represent numbers, so every word had a numerical value. This could be used to interpret the hidden meaning of words. For instance, Gulf War participants might have noticed that Saddam has the following value: samech (60) + aleph (1) + daled (4) + aleph (1) + mem (600) = 666—a number that Christians associate with the evil Beast that appears during the Apocalypse. (Whether “Saddam” has two daleds or one would make no difference to the kabbalists, who often used alternate spellings of words to make sums come out right.) Kabbalists thought that words and phrases with the same numerical value were mystically linked. For instance, Genesis 49:10 states, “The scepter shall not depart from Judah…until Shiloh come.” The Hebrew phrase for “until Shiloh come” has a value of 358, exactly the same for the Hebrew word
meshiach,
messiah. Hence, the passage presages the coming of the Messiah. 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. A recent bestseller,
The Bible Code,
purported to find prophecies by this method.

The kabbalah was much more than number crunching; it was a tradition so mystical that some scholars say that it bears a striking resemblance to Hinduism. For instance, the kabbalah seized upon the idea of the dual nature of God. 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.

As the Jews pitted their Western sensibilities against their Eastern Bible, the same battle was under way in the Christian world. Even as the Christians battled the Muslims—during Charlemagne's reign in the ninth century and during the Crusades in the eleventh, twelfth, and thirteenth centuries—warrior-monks, scholars, and traders began to bring Islamic ideas back to the West. Monks discovered that the astrolabe, an Arabic invention, was a handy tool for keeping track of time in the evening, helping them keep their prayers on schedule. The astrolabes were often inscribed with Arabic numerals.

The new numbers didn't catch on, even though a tenth-century pope, Sylvester II, was an admirer of them. He probably learned about the numerals during a visit to Spain and brought them back with him when he returned to Italy. But the version he learned did not have a zero—and the system would have been even less popular if it had. Aristotle still had a firm grip on the church, and its finest thinkers still rejected the infinitely large, the infinitely small, and the void. Even as the Crusades drew to a close in the thirteenth century, Saint Thomas Aquinas declared that God could not make something that was infinite any more than he could make a scholarly horse. But that implied that God was not omnipotent—a forbidden thought in Christian theology.

In 1277 the bishop of Paris, Étienne Tempier, called an assembly of scholars to discuss Aristotelianism, or rather, to attack it. 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.) God could make a vacuum if he wanted. All of a sudden the void was allowed, because an omnipotent deity doesn't need to follow Aristotle's rules if he doesn't want to.

Tempier's pronouncements were not the final blow to Aristotelian philosophy, but they certainly signaled that the foundations were crumbling. The church would cling to Aristotle for a few more centuries, but the fall of Aristotle and the rise of the void and the infinite were clearly beginning. It was a propitious time for zero to arrive in the West. In the mid-twelfth century the first adaptations of al-Khowarizmi's
Aljabr
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.