Read Against the Gods: The Remarkable Story of Risk Online
Authors: Peter L. Bernstein
Against the Gods: The Remarkable Story of Risk (7 page)
The Fibonacci series is a lot more than a source of amusement. Divide any of the Fibonacci numbers by the next higher number. After 3, the answer is always 0.625. After 89, the answer is always 0.618; after higher numbers, more decimal places can be filled in.*
Divide any number by its preceding number. After 2, the answer is always 1.6. After 144, the answer is always 1.618.
In one of its more romantic manifestations, the Fibonacci ratio defines the proportions and shape of a beautiful spiral. The accompanying illustrations demonstrate how the spiral develops from a series of squares whose successive relative dimensions are determined by the
Fibonacci series. The process begins with two small squares of equal size.
It then progresses to an adjacent square twice the size of the first two,
then to a square three times the size of the first two, then to five times,
and so on. Note that the sequence produces a series of rectangles with
the proportions of the golden mean. Then quarter-circle arcs connect
the opposite corners of the squares, starting with the smallest squares
and proceeding in sequence.
Construction of an equiangular spiral using Fibonacci proportions.
Begin with a 1-unit square, attach another 1-unit square, then a 2-unit square then a 2unit square where it fits, followed by a 3-unit square where it fits and, continuing in the
same direction, attach squares of 5, 8, 13, 21, and 34 units and so on.
(Reproduced with permission from Fascinating Fibonaccis, by Trudy Hammel Garland; copyright
1987 by Dale Seymour Publications, P.O. Box 10888, Palo Alto, CA 94303.)
This familiar-looking spiral appears in the shape of certain galaxies,
in a ram's horn, in many seashells, and in the coil of the ocean waves that
surfers ride. The structure maintains its form without change as it is
made larger and larger and regardless of the size of the initial square with
which the process is launched: form is independent of growth. The
journalist William Hoffer has remarked, "The great golden spiral seems
to be nature's way of building quantity without sacrificing quality. "2
Some people believe that the Fibonacci numbers can be used to make
a wide variety of predictions, especially predictions about the stock market; such predictions work just often enough to keep the enthusiasm
going. The Fibonacci sequence is so fascinating that there is even an
American Fibonacci Association, located at Santa Clara University in
California, which has published thousands of pages of research on the
subject since 1962.
Fibonacci's Liber Abaci was a spectacular first step in making measurement the key factor in the taming of risk. But society was not yet
prepared to attach numbers to risk. In Fibonacci's day, most people still
thought that risk stemmed from the capriciousness of nature. People
would have to learn to recognize man-made risks and acquire the
courage to do battle with the fates before they would accept the techniques of taming risk. That acceptance was still at least two hundred
years in the future.
We can appreciate the full measure of Fibonacci's achievement
only by looking back to the era before he explained how to tell the difference between 10 and 100. Yet even there we shall discover some
remarkable innovators.
Primitive people like the Neanderthals knew how to tally, but they
had few things that required tallying. They marked the passage of days
on a stone or a log and kept track of the number of animals they killed.
The sun kept time for them, and five minutes or a half-hour either way
hardly mattered.
The first systematic efforts to measure and count were undertaken
some ten thousand years before the birth of Christ.' It was then that
humans settled down to grow food in the valleys washed by such great
rivers as the Tigris and the Euphrates, the Nile, the Indus, the Yangtse,
the Mississippi, and the Amazon. The rivers soon became highways for
trade and travel, eventually leading the more venturesome people to
the oceans and seas into which the rivers emptied. To travelers ranging
over longer and longer distances, calendar time, navigation, and geography mattered a great deal and these factors required ever more precise
computations.
Priests were the first astronomers, and from astronomy came mathematics. When people recognized that nicks on stones and sticks no
longer sufficed, they began to group numbers into tens or twenties,
which were easy to count on fingers and toes.
Although the Egyptians became experts in astronomy and in predicting the times when the Nile would flood or withdraw, managing or
influencing the future probably never entered their minds. Change was
not part of their mental processes, which were dominated by habit, seasonality, and respect for the past.
About 450 BC, the Greeks devised an alphabetic numbering system
that used the 24 letters of the Greek alphabet and three letters that subsequently became obsolete. Each number from 1 to 9 had its own letter, and the multiples of ten each had a letter. For example, the symbol
"pi" comes from the first letter of the Greek word "penta," which represented 5; delta, the first letter of "deca," the word for 10, represented
10; alpha, the first letter of the alphabet, represented 1, and rho represented 100. Thus, 115 was written rho-deca-penta, or p&7r. The
Hebrews, although Bemitic rather than Indo-European, used the same
kind of cipher-alphabet system.4
Handy as these letter-numbers were in helping people to build
stronger structures, travel longer distances, and keep more accurate
time, the system had serious limitations. You could use letters only with great difficulty-and almost never in your head-for adding or subtracting or multiplying or dividing. These substitutes for numbers provided nothing more than a means of recording the results of calculations
performed by other methods, most often on a counting frame or abacus. The abacus-the oldest counting device in history-ruled the
world of mathematics until the Hindu-Arabic numbering system
arrived on the scene between about 1000 and 1200 AD.
The abacus works by specifying an upper limit for the number of
counters in each column; in adding, as the furthest right column fills up,
the excess counters move one column to the left, and so on. Our concepts of "borrow one" or "carry over three" date back to the abacus.5
Despite the limitations of these early forms of mathematics, they
made possible great advances in knowledge, particularly in geometrythe language of shape-and its many applications in astronomy, navigation, and mechanics. Here the most impressive advances were made
by the Greeks and by their colleagues in Alexandria. Only the Bible has
appeared in more editions and printings than Euclid's most famous
book, Elements.
Still, the greatest contribution of the Greeks was not in scientific
innovation. After all, the temple priests of Egypt and Babylonia had
learned a good bit about geometry long before Euclid came along.
Even the famous theorem of Pythagoras-the square of the hypotenuse
of a right triangle is equal to the sum of the square of the other two
sides-was in use in the Tigris-Euphrates valley as early as 2000 BC.
The unique quality of the Greek spirit was the insistence on proof.
"Why?" mattered more to them than "What?" The Greeks were able
to reframe the ultimate questions because theirs was the first civilization
in history to be free of the intellectual straitjacket imposed by an allpowerful priesthood. This same set of attitudes led the Greeks to
become the world's first tourists and colonizers as they made the
Mediterranean basin their private preserve.
More worldly as a consequence, the Greeks refused to accept at
face value the rules of thumb that older societies passed on to them.
They were not interested in samples; their goal was to find concepts that would apply everywhere, in every case. For example, mere measurement would confirm that the square of the hypotenuse of a right
triangle is equal to the sum of the squares of the other two sides. But
the Greeks asked why that should be so, in all right triangles, great and
small, without a single exception to the rule. Proof is what Euclidean
geometry is all about. And proof, not calculation, would dominate the
theory of mathematics forever after.
This radical break with the analytical methodologies of other civilizations makes us wonder again why it was that the Greeks failed to
discover the laws of probability, and calculus, and even simple algebra.
Perhaps, despite all they achieved, it was because they had to depend on
a clumsy numbering system based on their alphabet. The Romans suffered from the same handicap. As simple a number as 9 required two
letters: IX. The Romans could not write 32 as III II, because people
would have no way of knowing whether it meant 32, 302, 3020, or
some larger combination of 3, 2, and 0. Calculations based on such a
system were impossible.
But the discovery of a superior numbering system would not occur
until about 500 AD, when the Hindus developed the numbering system
we use today. Who contrived this miraculous invention, and what circumstances led to its spread throughout the Indian subcontinent,
remain mysteries. The Arabs encountered the new numbers for the first
time some ninety years after Mohammed established Islam as a proselytizing religion in 622 and his followers, united into a powerful nation,
swept into India and beyond.
