Though the Indian number system was useful for everyday tasks like addition and multiplication, the true impact of Indian numbers was considerably deeper. Numbers had finally become distinct from geometry; numbers were used to do more than merely measure objects. 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. Though this mind-set prevented the Indians from contributing much to geometry, it had another, unexpected effect. It freed the Indians from the shortcomings of the Greek system of thought—and their rejection of zero.
Figure 15: An algorist versus an abacist
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. However, this was not at all obvious to the ancients. Many times they solved equations only to get a negative result and concluded that their answer had no meaning. After all, if you are thinking in geometric terms, what is a negative area? It simply didn’t make any sense to the Greeks.
To the Indians, negative numbers made perfect sense. Indeed, it was in India (and in China) that negative numbers first appeared. Brahmagupta, an Indian mathematician of the seventh century, gave rules for dividing numbers by each other, and he included the negatives. “Positive divided by positive, or negative by negative, is affirmative,” he wrote. “Positive divided by negative is negative. Negative divided by affirmative is negative.” These are the rules that we recognize today: divide two numbers and the answer is positive if the numbers’ signs are the same.
Just as 2 – 3 was now a number, so was 2 – 2. It was zero. Not just a mere placeholder zero that represents an empty space on the abacus, but zero the number. It had a specific value, a fixed place on the number line. Since zero was equal to 2 – 2, then it had to be placed between one (2 – 1) and negative one (2 – 3). Nothing else made sense. No longer could zero sit to the right of nine, just as it does on the top of the computer keyboard; zero had a position in the number line that was all its own. A number line without zero could no more exist than a number system without two. Zero had finally arrived.
However, even the Indians thought that zero was a pretty bizarre number, for all the usual reasons. After all, zero multiplied by anything is zero; it sucks everything into itself. And when you divide with it, all hell breaks loose. Brahmagupta tried to figure out what 0 ÷ 0 and 1 ÷ 0 were, and failed. “Cipher divided by cipher is naught,” he wrote. “Positive or negative divided by cipher is a fraction with that for a denominator.” In other words, he thought 0 ÷ 0 was 0 (he was wrong, as we will see), and he thought that 1 ÷ 0 was, well, we don’t really know, because he doesn’t make a whole lot of sense. Basically, he was waving his hands and hoping that the problem would go away.
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.
The Arab Numeral
Does man forget that We created him out of the void?
—THE KORAN
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.
I Am That I Am: Nothing
Nothingness is being and being nothingness…. Our limited mind can not grasp or fathom this, for it joins infinity.
—AZRAEL OF GERONA
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, everywher
e 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.
Zero’s Triumph
…a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions.
—PIERRE-SIMON LAPLACE
Christianity initially rejected zero, but trade would soon demand it. The man who reintroduced zero to the West was Leonardo of Pisa. The son of an Italian trader, he traveled to northern Africa. There the young man—better known as Fibonacci—learned mathematics
from the Muslims and soon became a good mathematician in his own right.
Fibonacci is best remembered for a silly little problem he posed in his book, Liber Abaci, which was published in 1202. Imagine that a farmer has a pair of baby rabbits. Babies take two months to reach maturity, and from then on they produce another pair of rabbits at the beginning of every month. As these rabbits mature and reproduce, and those rabbits mature and reproduce, and so on, how many pairs of rabbits do you have during any given month?
Well, during the first month, you have one pair of rabbits, and since they haven’t matured, they can’t reproduce.
During the second month you still have only one pair.
But at the beginning of the third month, the first pair reproduces: you’ve got two pairs.
At the beginning of the fourth month, the first pair reproduces again, but the second pair is not mature enough: three pairs.
The next month the first pair reproduces, the second pair reproduces, since it has reached maturity, but the third pair is too young. That is two additional pairs of rabbits: five in all.
The number of rabbits goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…; the number of rabbits you have in any given month is the sum of the rabbits that you had in each of the two previous months. Mathematicians instantly realized the importance of this series. Take any term and divide it by its previous term. For instance, 8/5 = 1.6; 13/8 = 1.625; 21/13 =1.61538…. These ratios approach a particularly interesting number: the golden ratio, which is 1.61803….
Pythagoras had noticed that nature seemed to be governed by the golden ratio. Fibonacci discovered the sequence that is responsible. The size of the chambers of the nautilus and the number of clockwise grooves to counterclockwise grooves in the pineapple are governed by this sequence. This is why their ratios approach the golden ratio.
Though this sequence is the source of Fibonacci’s fame, Fibonacci’s Liber Abaci had a much more important purpose than animal husbandry. Fibonacci had learned his mathematics from the Muslims, so he knew about Arabic numerals, including zero. He included the new system in Liber Abaci, finally introducing Europe to zero. The book showed how useful Arabic numerals were for doing complex calculations, and the Italian merchants and bankers quickly seized upon the new system, zero included.
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