Against the Gods. Bernstein Peter L.

algebraic equations we are all familiar with today – equations like a + bx = c– are known as Diophantine equations.

      The centerpiece of the Hindu-Arabic system was the invention of zero —sunya as the Indians called it, and cifr as it became in Arabic.33 The term has come down to us as “cipher,” which means empty and refers to the empty column in the abacus or counting frame.34

      The concept of zero was difficult to grasp for people who had used counting only to keep track of the number of animals killed or the number of days passed or the number of units traveled. Zero had nothing to do with what counting was for in that sense. As the twentieth-century English philosopher Alfred North Whitehead put it,

      The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.35

      Whitehead’s phrase “cultivated modes of thought” suggests that the concept of zero unleashed something more profound than just an enhanced method of counting and calculating. As Diophantus had sensed, a proper numbering system would enable mathematics to develop into a science of the abstract as well as a technique for measurement. Zero blew out the limits to ideas and to progress.

      Zero revolutionized the old numbering system in two ways. First, it meant that people could use only ten digits, from zero to nine, to perform every conceivable calculation and to write any conceivable number. Second, it meant that a sequence of numbers like 1, 10, 100 would indicate that the next number in the sequence would be 1000. Zero makes the whole structure of the numbering system immediately visible and clear. Try that with the Roman numerals I, X, and C, or with V, L, and D – what is the next number in those sequences?

      The earliest known work in Arabic arithmetic was written by al-Khowârizmî, a mathematician who lived around 825, some four hundred years before Fibonacci.36 Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “al-Khowârizmî” fast. That’s where we get the word “algorithm,” which means rules for computing.37 It was al-Khowârizmî who was the first mathematician to establish rules for adding, subtracting, multiplying, and dividing with the new Hindu numerals. In another treatise, Hisâb al-jabr w’ almuqâbalah, or “Science of transposition and cancellation,” he specifies the process for manipulating algebraic equations. The word al-jabr thus gives us our word algebra, the science of equations.38

      One of the most important, surely the most famous, early mathematician was Omar Khayyam, who lived from about 1050 to about 1130 and was the author of the collection of poems known as the Rubaiyat.39 His haunting sequence of 75 four-line poems (the word Rubaiyat defines the poetic form) was translated in Victorian times by the English poet Edward Fitzgerald. This slim volume has more to do with the delights of drinking wine and taking advantage of the transitory nature of life than with science or mathematics. Indeed, in number XXVII, Omar Khayyam writes:

      Myself when young did eagerly frequent

      Doctor and Saint, and heard great Argument

      About it and about; but evermore

      Came out by the same Door as in I went.

      According to Fitzgerald, Omar Khayyam was educated along with two friends, both as bright as he: Nizam al Mulk and Hasan al Sabbah. One day Hasan proposed that, since at least one of the three would attain wealth and power, they should vow that “to whomsoever this fortune falls, he shall share it equally with the rest, and preserve no preeminence for himself.” They all took the oath, and in time Nizam became vizier to the sultan. His two friends sought him out and claimed their due, which he granted as promised.

      Hasan demanded and received a place in the government, but, dissatisfied with his advancement, left to become head of a sect of fanatics who spread terror throughout the Mohammedan world. Many years later, Hasan would end up assassinating his old friend Nizam.

      Omar Khayyam asked for neither title nor office. “The greatest boon you can confer on me,” he said to Nizam, “is to let me live in a corner under the shadow of your fortune, to spread wide the advantages of science and pray for your long life and prosperity.” Although the sultan loved Omar Khayyam and showered favors on him, “Omar’s epicurean audacity of thought and speech caused him to be regarded askance in his own time and country.”

      Omar Khayyam used the new numbering system to develop a language of calculation that went beyond the efforts of al-Khowârizmî and served as a basis for the more complicated language of algebra. In addition, Omar Khayyam used technical mathematical observations to reform the calendar and to devise a triangular rearrangement of numbers that facilitated the figuring of squares, cubes, and higher powers of mathematics; this triangle formed the basis of concepts developed by the seventeenth-century French mathematician Blaise Pascal, one of the fathers of the theory of choice, chance, and probability.

      The impressive achievements of the Arabs suggest once again that an idea can go so far and still stop short of a logical conclusion. Why, given their advanced mathematical ideas, did the Arabs not proceed to probability theory and risk management? The answer, I believe, has to do with their view of life. Who determines our future: the fates, the gods, or ourselves? The idea of risk management emerges only when people believe that they are to some degree free agents. Like the Greeks and the early Christians, the fatalistic Muslims were not yet ready to take the leap.

      By the year 1000, the new numbering system was being popularized by Moorish universities in Spain and elsewhere and by the Saracens in Sicily. A Sicilian coin, issued by the Normans and dated “1134 Annoy Domini,” is the first known example of the system in actual use. Still, the new numbers were not widely used until the thirteenth century.

      Despite Emperor Frederick’s patronage of Fibonacci’s book and the book’s widespread distribution across Europe, introduction of the Hindu-Arabic numbering system provoked intense and bitter resistance up to the early 1500s. Here, for once, we can explain the delay. Two factors were at work.

      Part of the resistance stemmed from the inertial forces that oppose any change in matters hallowed by centuries of use. Learning radically new methods never finds an easy welcome.

      The second factor was based on more solid ground: it was easier to commit fraud with the new numbers than with the old. Turning a 0 into a 6 or a 9 was temptingly easy, and a 1 could be readily converted into a 4, 6, 7, or 9 (one reason Europeans write 7 as 7). Although the new numbers had gained their first foothold in Italy, where education levels were high, Florence issued an edict in 1229 that forbade bankers from using the “infidel” symbols. As a result, many people who wanted to learn the new system had to disguise themselves as Moslems in order to do so.40

      The invention of printing with movable type in the middle of the fifteenth century was the catalyst that finally overcame opposition to the full use of the new numbers. Now the fraudulent alterations were no longer possible. Now the ridiculous complications of using Roman numerals became clear to everyone. The breakthrough gave a great lift to commercial transactions. Now al-Khowârizmî’s multiplication tables became something that all school children have had to learn forever after. Finally, with the first inklings of the laws of probability, gambling took on a whole new dimension.

      The algebraic solution to the epigram about Diophantus is as follows. If x was his age when he died, then:

33

See Hogben, 1968, pp. 244–246.

34

The Arabic term survives even in Russian, where it appears as tsifra, which is the word for number.

35

From Newman, 1988a, p. 433.

36

The background material on al-Khowârizmî is primarily from Muir, 1961, and Hogben, 1968.

37

Hogben, 1968, p. 243.

38

See Hogben, 1968, Chapter VI, for an extended and stimulating discussion of the development of algebra and the uses of zero.

39

The background material on Omar Khayyam is from Fitzgerald.

40

Hogben, 1968, p. 245.