Against the Gods: The Remarkable Story of Risk (8 page)

The new system of numbering had a galvanizing effect on intellectual activity in lands to the west. Baghdad, already a great center of
learning, emerged as a hub of mathematical research and activity, and
the Caliph retained Jewish scholars to translate the works of such pioneers of mathematics as Ptolemy and Euclid. The major works of mathematics were soon circulating throughout the Arab empire and by the
ninth and tenth centuries were in use as far west as Spain.

Actually, one westerner had suggested a numbering system at least
two centuries earlier than the Hindus. About 250 AD, an Alexandrian mathematician named Diophantus wrote a treatise setting forth the advantages of a system of true numbers to replace letters substituting for numbers.6

Not much is known about Diophantus, but the little we do know is amusing. According to Herbert Warren Turnbull, a historian of mathematics, a Greek epigram about Diophantus states that "his boyhood lasted 1/6th of his life; his beard grew after 1/12th more; he married after 1 /7th more, and his son was born five years later; the son lived to half his father's age, and the father died four years after his son." How old was Diophantus when he died?7 Algebra enthusiasts will find the answer at the end of this chapter.

Diophantus carried the idea of symbolic algebra-the use of symbols to stand for numbers-a long way, but he could not quite make it all the way. He comments on "the impossible solution of the absurd equation 4 = 4x + 20."8 Impossible? Absurd? The equation requires x to be a negative number: -4. Without the concept of zero, which Diophantus lacked, a negative number is a logical impossibility.

Diophantus's remarkable innovations seem to have been ignored. Almost a millennium and a half passed before anyone took note of his work. At last his achievements received their due: his treatise played a central role in the flowering of algebra in the seventeenth century. The 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.9 The term has come down to us as "cipher," which means empty and refers to the empty column in the abacus or counting frame.*

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 twentiethcentury 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.10

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 alKhowarizmi, a mathematician who lived around 825, some four hundred years before Fibonacci." Although few beneficiaries of his work
are likely to have heard of him, most of us know of him indirectly. Try
saying "al-Khowarizmi" fast. That's where we get the word "algorithm," which means rules for computing.12 It was al-Khowarizmi who
was the first mathematician to establish rules for adding, subtracting,
multiplying, and dividing with the new Hindu numerals. In another
treatise, Hisab al-jabr w'almugabalah, 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. 13

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.14 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:

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-Khowarizmi 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 v-). 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.15

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-Khowarizmi'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: