Against the Gods: The Remarkable Story of Risk (6 page)

There is more to that blunt statement than meets the eye. Trade is a
mutually beneficial process, a transaction in which both parties perceive
themselves as wealthier than they were before. What a radical idea! Up
to that point, people who got rich had done so largely by exploitation or
by plundering another's wealth. Although Europeans continued to plunder across the seas, at home the accumulation of wealth was open to the
many rather than the few. The newly rich were now the smart, the
adventuresome, the innovators-most of them businessmen-instead of
just the hereditary princes and their minions.

Trade is also a risky business. As the growth of trade transformed
the principles of gambling into the creation of wealth, the inevitable
result was capitalism, the epitome of risk-taking. But capitalism could
not have flourished without two new activities that had been unnecessary so long as the future was a matter of chance or of God's will. The
first was bookkeeping, a humble activity but one that encouraged the
dissemination of the new techniques of numbering and counting. The
other was forecasting, a much less humble and far more challenging
activity that links risk-taking with direct payoffs.

You do not plan to ship goods across the ocean, or to assemble merchandise for sale, or to borrow money without first trying to determine
what the future may hold in store. Ensuring that the materials you
order are delivered on time, seeing to it that the items you plan to sell
are produced on schedule, and getting your sales facilities in place all
must be planned before that moment when the customers show up and lay their money on the counter. The successful business executive is a
forecaster first; purchasing, producing, marketing, pricing, and organizing all follow.

The men you will meet in the coming chapters recognized the discoveries of Pascal and Fermat as the beginning of wisdom, not just a
solution to an intellectual conundrum involving a game of chance.
They were bold enough to tackle the many facets of risk in the face of
issues of growing complexity and practical importance and to recognize
that these are issues involving the most fundamental philosophical
aspects of human existence.

But philosophy must stand aside for the moment, as the story
should begin at the beginning. Modern methods of dealing with the
unknown start with measurement, with odds and probabilities. The
numbers come first. But where did the numbers come from?

ithout numbers, there are no odds and no probabilities;
without odds and probabilities, the only way to deal with
risk is to appeal to the gods and the fates. Without numbers,
risk is wholly a matter of gut.

We live in a world of numbers and calculations, from the clock we
squint at when we wake up, to the television channel we switch off at
bedtime. As the day proceeds, we count the measures of coffee we put
into the coffeemaker, pay the housekeeper, consult yesterday's stock
prices, dial a friend's telephone number, check the amount of gas in the
car and the speed on the speedometer, press the elevator button in our
office building, and open the office door with our number on it. And the
day has hardly started!

It is hard for us to imagine a time without numbers. Yet if we were
able to spirit a well-educated man from the year 1000 to the present, he
probably would not recognize the number zero and would surely flunk
third-grade arithmetic; few people from the year 1500 would fare
much better.

The story of numbers in the West begins in 1202, when the cathedral of Chartres was nearing completion and King John was finishing
his third year on the throne of England. In that year, a book titled Liber
Abaci, or Book of the Abacus, appeared in Italy. The fifteen chapters of the book were entirely handwritten; almost three hundred years would
pass before the invention of printing. The author, Leonardo Pisano,
was only 27 years old but a very lucky man: his book would receive the
endorsement of the Holy Roman Emperor, Frederick II. No author
could have done much better than that.'

Leonardo Pisano was known for most of his life as Fibonacci, the
name by which he is known today. His father's first name was Bonacio,
and Fibonacci is a contraction of son-of-Bonacio. Bonacio means "simpleton" and Fibonacci means "blockhead." Bonacio must have been
something less than a simpleton, however, for he represented Pisa as
consul in a number of different cities, and his son Leonardo was certainly no blockhead.

Fibonacci was inspired to write Liber Abaci on a visit to Bugia, a
thriving Algerian city where his father was serving as Pisan consul.
While Fibonacci was there, an Arab mathematician revealed to him the
wonders of the Hindu-Arabic numbering system that Arab mathematicians had introduced to the West during the Crusades to the Holy Land.
When Fibonacci saw all the calculations that this system made possiblecalculations that could not possibly be managed with Roman letternumerals-he set about learning everything he could about it. To study
with the leading Arab mathematicians living around the Mediterranean,
he set off on a trip that took him to Egypt, Syria, Greece, Sicily, and
Provence.

The result was a book that is extraordinary by any standard. Liber
Abaci made people aware of a whole new world in which numbers
could be substituted for the Hebrew, Greek, and Roman systems that
used letters for counting and calculating. The book rapidly attracted a
following among mathematicians, both in Italy and across Europe.

Liber Abaci is far more than a primer for reading and writing with the
new numerals. Fibonacci begins with instructions on how to determine
from the number of digits in a numeral whether it is a unit, or a multiple
of ten, or a multiple of 100, and so on. Later chapters exhibit a higher
level of sophistication. There we find calculations using whole numbers
and fractions, rules of proportion, extraction of square roots and roots of
higher orders, and even solutions for linear and quadratic equations.

Ingenious and original as Fibonacci's exercises were, if the book had
dealt only with theory it would probably not have attracted much attention beyond a small circle of mathematical cognoscenti. It commanded an enthusiastic following, however, because Fibonacci filled it with practical applications. For example, he described and illustrated many innovations that the new numbers made possible in commercial bookkeeping,
such as figuring profit margins, money-changing, conversions of weights
and measures, and-though usury was still prohibited in many placeshe even included calculations of interest payments.

Liber Abaci provided just the kind of stimulation that a man as brilliant and creative as the Emperor Frederick would be sure to enjoy.
Though Frederick, who ruled from 1211 to 1250, exhibited cruelty and
an obsession with earthly power, he was genuinely interested in science,
the arts, and the philosophy of government. In Sicily, he destroyed all
the private garrisons and feudal castles, taxed the clergy, and banned
them from civil office. He also set up an expert bureaucracy, abolished
internal tolls, removed all regulations inhibiting imports, and shut down
the state monopolies.

Frederick tolerated no rivals. Unlike his grandfather, Frederick
Barbarossa, who was humbled by the Pope at the Battle of Legnano in
1176, this Frederick reveled in his endless battles with the papacy. His
intransigence brought him not just one excommunication, but two. On
the second occasion, Pope Gregory IX called for Frederick to be
deposed, characterizing him as a heretic, rake, and anti-Christ. Frederick
responded with a savage attack on papal territory; meanwhile his fleet
captured a large delegation of prelates on their way to Rome to join the
synod that had been called to remove him from power.

Frederick surrounded himself with the leading intellectuals of his
age, inviting many of them to join him in Palermo. He built some of
Sicily's most beautiful castles, and in 1224 he founded a university to train
public servants-the first European university to enjoy a royal charter.

Frederick was fascinated with Liber Abaci. Some time in the 1220s,
while on a visit to Pisa, he invited Fibonacci to appear before him. In the
course of the interview, Fibonacci solved problems in algebra and cubic
equations put to him by one of Frederick's many scientists-in-residence.
Fibonacci subsequently wrote a book prompted by this meeting, Liber
Quadratorum, or The Book of Squares, which he dedicated to the Emperor.

Fibonacci is best known for a short passage in Liber Abaci that led to
something of a mathematical miracle. The passage concerns the problem of how many rabbits will be born in the course of a year from an
original pair of rabbits, assuming that every month each pair produces another pair and that rabbits begin to breed when they are two months old. Fibonacci discovered that the original pair of rabbits would have spawned a total of 233 pairs of offspring in the course of a year.

He discovered something else, much more interesting. He had assumed that the original pair would not breed until the second month and then would produce another pair every month. By the fourth month, their first two offspring would begin breeding. After the process got started, the total number of pairs of rabbits at the end of each month would be as follows: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Each successive number is the sum of the two preceding numbers. If the rabbits kept going for a hundred months, the total number pairs would be 354,224,848,179,261,915,075.