dealing with randomness. Mandelbrot proved me wrong.
He speaks an unusually precise and formal French, much like that spoken
by Levantines of my parents' generation or Old World aristocrats.
This made it odd to hear, on occasion, his accented, but very standard, colloquial
American English. He is tall, overweight, which makes him look
baby-faced (although I've never seen him eat a large meal), and has a
strong physical presence.
From the outside one would think that what Mandelbrot and I have in
common is wild uncertainty, Black Swans, and dull (and sometimes less
dull) statistical notions. But, although we are collaborators, this is not
what our major conversations revolve around. It is mostly matters literary
and aesthetic, or historical gossip about people of extraordinary intellectual
refinement. I mean refinement, not achievement. Mandelbrot could
tell stories about the phenomenal array of hotshots he has worked with
over the past century, but somehow I am programmed to consider scientists'
personae far less interesting than those of colorful erudites. Like me,
Mandelbrot takes an interest in urbane individuals who combine traits
THE A E S T H E T I C S OF RANDOMNESS 2 5 5
generally thought not to coexist together. One person he often mentions is
Baron Pierre Jean de Menasce, whom he met at Princeton in the 1950s,
where de Menasce was the roommate of the physicist Oppenheimer. De
Menasce was exactly the kind of person I am interested in, the embodiment
of a Black Swan. He came from an opulent Alexandrian Jewish merchant
family, French and Italian-speaking like all sophisticated Levantines.
His forebears had taken a Venetian spelling for their Arabic name, added
a Hungarian noble title along the way, and socialized with royalty. De
Menasce not only converted to Christianity, but became a Dominican
priest and a great scholar of Semitic and Persian languages. Mandelbrot
kept questioning me about Alexandria, since he was always looking for
such characters.
True, intellectually sophisticated characters were exactly what I looked
for in life. My erudite and polymathic father—who, were he still alive,
would have only been two weeks older than Beno?t M.—liked the company
of extremely cultured Jesuit priests. I remember these Jesuit visitors
occupying my chair at the dining table. I recall that one had a medical degree
and a PhD in physics, yet taught Aramaic to locals in Beirut's Institute
of Eastern Languages. His previous assignment could have been teaching
high school physics, and the one before that was perhaps in the medical
school. This kind of erudition impressed my father far more than scientific
assembly-line work. I may have something in my genes driving me away
from bildungsphilisters.
Although Mandelbrot often expressed amazement at the temperament
of high-flying erudites and remarkable but not-so-famous scientists, such
as his old friend Carleton Gajdusek, a man who impressed him with his
ability to uncover the causes of tropical diseases, he did not seem eager to
trumpet his association with those we consider great scientists. It took me
a while to discover that he had worked with an impressive list of scientists
in seemingly every field, something a name-dropper would have brought
up continuously. Although I have been working with him for a few years
now, only the other day, as I was chatting with his wife, did I discover that
he spent two years as the mathematical collaborator of the psychologist
Jean Piaget. Another shock came when I discovered that he had also
worked with the great historian Fernand Braudel, but Mandelbrot did not
seem to be interested in Braudel. He did not care to discuss John von Neuman
with whom he had worked as a postdoctoral fellow. His scale was inverted.
I asked him once about Charles Tresser, an unknown physicist I
met at a party who wrote papers on chaos theory and supplemented his re2
5 6 THOSE GRAY SWANS OF EXTREMISTAN
searcher's income by making pastry for a shop he ran near New York City.
He was emphatic: "un homme extraordinaire," he called Tresser, and
could not stop praising him. But when I asked him about a particular famous
hotshot, he replied, "He is the prototypical bon élève, a student with
good grades, no depth, and no vision." That hotshot was a Nobel laureate.
THE PLATONICITY OF TRIANGLES
Now, why am I calling this business Mandelbrotian, or fractal, randomness?
Every single bit and piece of the puzzle has been previously mentioned
by someone else, such as Pareto, Yule, and Zipf, but it was
Mandelbrot who a) connected the dots, b) linked randomness to geometry
(and a special brand at that), and c) took the subject to its natural conclusion.
Indeed many mathematicians are famous today partly because he
dug out their works to back up his claims—the strategy I am following
here in this book. "I had to invent my predecessors, so people take me seriously,"
he once told me, and he used the credibility of big guns as a
rhetorical device. One can almost always ferret out predecessors for any
thought. You can always find someone who worked on a part of your argument
and use his contribution as your backup. The scientific association
with a big idea, the "brand name," goes to the one who connects the dots,
not the one who makes a casual observation—even Charles Darwin, who
uncultured scientists claim "invented" the survival of the fittest, was not
the first to mention it. He wrote in the introduction of The Origin of
Species that the facts he presented were not necessarily original; it was the
consequences that he thought were "interesting" (as he put it with characteristic
Victorian modesty). In the end it is those who derive consequences
and seize the importance of the ideas, seeing their real value, who win the
day. They are the ones who can talk about the subject.
So let me describe Mandelbrotian geometry.
The Geometry of Nature
Triangles, squares, circles, and the other geometric concepts that made
many of us yawn in the classroom may be beautiful and pure notions, but
they seem more present in the minds of architects, design artists, modern
art buildings, and schoolteachers than in nature itself. That's fine, except
that most of us aren't aware of this. Mountains are not triangles or pyraTHE
A E S T H E T I C S OF RANDOMNESS 2 5 7
mids; trees are not circles; straight lines are almost never seen anywhere.
Mother Nature did not attend high school geometry courses or read the
books of Euclid of Alexandria. Her geometry is jagged, but with a logic of
its own and one that is easy to understand.
I have said that we seem naturally inclined to Platonify, and to think
exclusively in terms of studied material: nobody, whether a bricklayer or a
natural philosopher, can easily escape the enslavement of such conditioning.
Consider that the great Galileo, otherwise a debunker of falsehoods,
wrote the following:
The great book of Nature lies ever open before our eyes and the true
philosophy is written in it. . . . But we cannot read it unless we have
first learned the language and the characters in which it is written. . . .
It is written in mathematical language and the characters are triangles,
circles and other geometric figures.
Was Galileo legally blind? Even the great Galileo, with all his alleged
independence of mind, was not capable of taking a clean look at Mother
Nature. I am confident that he had windows in his house and that he ventured
outside from time to time: he should have known that triangles are
not easily found in nature. We are so easily brainwashed.
We are either blind, or illiterate, or both. That nature's geometry is not
Euclid's was so obvious, and nobody, almost nobody, saw it.
This (physical) blindness is identical to the ludic fallacy that makes us
think casinos represent randomness.
Fractality
But first, a description of fractals. Then we will show how they link to
what we call power laws, or scalable laws.
Fractal is a word Mandelbrot coined to describe the geometry of the
rough and broken—from the Latin fractus, the origin of fractured. Fractality
is the repetition of geometric patterns at different scales, revealing
smaller and smaller versions of themselves. Small parts resemble, to some
degree, the whole. I will try to show in this chapter how the fractal applies
to the brand of uncertainty that should bear Mandelbrot's name: Mandelbrotian
randomness.
The veins in leaves look like branches; branches look like trees; rocks
2 5 8 THOSE GRAY SWANS OF EXTREMISTAN
look like small mountains. There is no qualitative change when an object
changes size. If you look at the coast of Britain from an airplane, it resembles
what you see when you look at it with a magnifying glass. This character
of self-affinity implies that one deceptively short and simple rule of
iteration can be used, either by a computer or, more randomly, by Mother
Nature, to build shapes of seemingly great complexity. This can come in
handy for computer graphics, but, more important, it is how nature
works. Mandelbrot designed the mathematical object now known as the
Mandelbrot set, the most famous object in the history of mathematics. It
became popular with followers of chaos theory because it generates pictures
of ever increasing complexity by using a deceptively minuscule recursive
rule; recursive means that something can be reapplied to itself
infinitely. You can look at the set at smaller and smaller resolutions without
ever reaching the limit; you will continue to see recognizable shapes.
The shapes are never the same, yet they bear an affinity to one another, a
strong family resemblance.
These objects play a role in aesthetics. Consider the following applications:
Visual arts: Most computer-generated objects are now based on some
version of the Mandelbrotian fractal. We can also see fractals in architecture,
paintings, and many works of visual art—of course, not consciously
incorporated by the work's creator.
Music: Slowly hum the four-note opening of Beethoven's Fifth Symphony:
ta-ta-ta-ta. Then replace each individual note with the same fournote
opening, so that you end up with a measure of sixteen notes. You will
see (or, rather, hear) that each smaller wave resembles the original larger
one. Bach and Mahler, for instance, wrote submovements that resemble
the larger movements of which they are a part.
Poetry: Emily Dickinson's poetry, for instance, is fractal: the large resembles
the small. It has, according to a commentator, "a consciously
made assemblage of dictions, metres, rhetorics, gestures, and tones."
Fractals initially made Beno?t M. a pariah in the mathematical establishment.
French mathematicians were horrified. What? Images? Mon
dieu! It was like showing a porno movie to an assembly of devout Eastern
Orthodox grandmothers in my ancestral village of Amioun. So Mandelbrot
spent time as an intellectual refugee at an IBM research center in
upstate New York. It was a f * * * you money situation, as IBM let him do
whatever he felt like doing.
THE A E S T H E T I C S OF RANDOMNESS 2 5 9
But the general public (mostly computer geeks) got the point. Mandelbrot's
book The Fractal Geometry of Nature made a splash when it came
out a quarter century ago. It spread through artistic circles and led to studies
in aesthetics, architectural design, even large industrial applications.
Beno?t M. was even offered a position as a professor of medicine! Supposedly
the lungs are self-similar. His talks were invaded by all sorts of artists,
earning him the nickname the Rock Star of Mathematics. The computer
age helped him become one of the most influential mathematicians in history,
in terms of the applications of his work, way before his acceptance
by the ivory tower. We will see that, in addition to its universality, his
work offers an unusual attribute: it is remarkably easy to understand.
A few words on his biography. Mandelbrot came to France from Warsaw
in 1936, at the age of twelve. Owing to the vicissitudes of a clandestine
life during Nazi-occupied France, he was spared some of the conventional
Gallic education with its uninspiring algebraic drills, becoming largely
self-taught. He was later deeply influenced by his uncle Szolem, a prominent
member of the French mathematical establishment and holder of a
chair at the Collège de France. Beno?t M. later settled in the United States,
working most of his life as an industrial scientist, with a few transitory
and varied academic appointments.
The computer played two roles in the new science Mandelbrot helped
conceive. First, fractal objects, as we have seen, can be generated with a
simple rule applied to itself, which makes them ideal for the automatic activity
of a computer (or Mother Nature). Second, in the generation of visual
intuitions lies a dialectic between the mathematician and the objects
generated.
Now let us see how this takes us to randomness. In fact, it is with probability
that Mandelbrot started his career.
A Visual Approach to Extremistan/Mediocristan
I am looking at the rug in my study. If I examine it with a microscope, I
will see a very rugged terrain. If I look at it with a magnifying glass, the
terrain will be smoother but still highly uneven. But when I look at it from
a standing position, it appears uniform—it is almost as smooth as a sheet
of paper. The rug at eye level corresponds to Mediocristan and the law of
large numbers: I am seeing the sum of undulations, and these iron out.
This is like Gaussian randomness: the reason my cup of coffee does not
2 6 0 THOSE GRAY SWANS OF EXTREMISTAN
jump is that the sum of all of its moving particles becomes smooth. Likewise,
you reach certainties by adding up small Gaussian uncertainties: this
is the law of large numbers.
The Gaussian is not self-similar, and that is why my coffee cup does
not jump on my desk.
Now, consider a trip up a mountain. No matter how high you go on
the surface of the earth, it will remain jagged. This is even true at a height
