Remembering Benoit Mandelbrot

Last week, Benoit Mandelbrot died peacefully of pancreatic cancer at a hospice in Cambridge, Massachusetts.  He was 85 years old.

While it is not unusual or surprising for men of his advanced age to succumb to that kind of grave illness, his loss is nevertheless a profound one that is heartfelt by all who were familiar with his work and his special talent for delivering apparent heresies with a kind face, a soft voice and a childlike glint in his eye.

Yet he was much more than just a genial maverick, he fundamentally changed how we see the world (or at least how we should see the world).  To remember him is not to dwell in the past, but to blaze boldly into the future that he helped build.

Early Life

Mandelbrot was born into an Jewish intellectual family in Warsaw, Poland in 1924.  By 1936, anti-semitism and the spectre of Nazism persuaded the Mandelbrots to emigrate to France, where they joined his uncle Szolem, a prominent mathematician.  When war broke out, young Benoit was forced to flee to the countryside and live under an assumed name.

After the war ended, he resumed his education, but became a refugee of a different sort.  At the time, French mathematics was ruled by the Bourbaki, a group ironically co-founded by his uncle, that stressed formalism and rigorous proof.  Benoit fled his uncle’s École Normale Supérieure and entered the less prestigious École Polytechnique, where he studied under Paul Lévy, a maverick in his own right.

After earning his doctorate, he went to Princeton’s Institute for Advanced Study, where he was John von Neumann’s last student.  Finally, in 1958 he found a home not in academia, but at IBM’s research lab, which at the time was considered intellectual suicide.

It was there that he launched a career that spanned five decades and spawned some of the most consequential and exciting mathematical ideas of the 20th Century.

Fractals

One of the things that Mandelbrot is most famous for is inventing the field of fractal geometry, which uses recurrent patterns to form incredibly complex structures.  It is extensively employed today in computer graphics and other applications.  Every video game enthusiast owes a debt of gratitude to Mandelbrots fractals.

He first became interested in the idea when he pondered the disarmingly difficult question of how long the coast of Britain is.  The answer, of course, is that it depends on what scale you look at.  As you get closer and closer, the roughness of the coast reveals new details and the coast gets longer.  In reality, the length is infinite.

It was a typically perspective for Mandelbrot.  Whereas most mathematicians would see such questions as a nuisance to be overlooked, he saw an unexplored area rich with promise.  Natural phenomena rarely exhibit the smoothness that mathematicians wish they would.  Fractals, however, manage to capture all the complexity with amazing simplicity.

The most famous fractal, of course, is the Mandelbrot Set, which from one basic formula manages to create a structure that is not only infinitely intricate, but also self similar (i.e. the same basic shape recurs at every scale). You can see it animated in the video below.

Chaos

His development of fractal geometry came out of his work on proportional scaling.  To understand the concept, think of a square.  The area increases much faster than its length, but even a child can recognize that a square of any size is still a square.

Mandelbrot found a similar scaling effect in things ranging from “noise” that cause errors in communication lines to the flooding of the Nile river.  This was important because he found that what statisticians normally dismissed as “outliers” actually were interdependent and therefore inherent to systems, not aberrations.

He attributed the pattern to what he called Noah effects and Joseph effects:

Joseph Effects: These are persistent.  Just like in the biblical story, where Joseph predicted seven fat years and seven lean years, events in a time series are highly dependent on what precedes them.

Noah Effects: These create discontinuity.   A storm comes and blows everything away; creating a new fact pattern that will be propagated through Joseph effects.

Taken together, Noah and Joseph effects create a world that is much more capricious than had been assumed before.  While statisticians use relatively tame bell curves to predict future events, Mandelbrot’s work suggested that more volatile power laws fit real world data much better.

Although such thinking made him an outcast in the 1960’s, today the concept is widely accepted.  Power law distributions have been found to regulate everything from word distributions in texts to social networks to e-commerce.  As I wrote before, they can even help us understand Justin Bieber.

Financial Markets

For much of his career, Mandelbrot was mostly viewed as a crackpot, but he did attract some interest from economists.  One of them, Hendrik Houthakker at Harvard, invited him to give a talk about his findings.  When Mandelbrot arrived at Houthakker’s office he remarked how surprised he was to see one of his diagrams on the blackboard.

“What do you mean,” replied Houthakker, “I have no idea what you’re going to talk about.”  It turned out that the graphs on the blackboard had nothing to do with Mandelbrot’s research, but with cotton prices. On that day battle lines were drawn.

At the time, the basic tenets of financial engineering were being formed, which were based on bell curves formed by randomness, not Mandelbrot’s more volatile power laws.  Mandelbrot rushed out a a paper, which was included in Paul Cootner’s 1964 book, The Random Character of Stock Prices, now considered a seminal work in financial circles.

Mandelbrot Cast Aside, but Ultimately Redeemed

While Cootner wrote that Mandelbrot forced economists “to face up in a substantive way to those uncomfortable empirical observations that there is little doubt most of us have had to sweep under the carpet until now.” He then added, “but surely before consigning centuries of work to the ash pile, we should like to have some assurance that all of our work is truly useless.”

In other words, the locomotive was heading down the tracks at full steam and Mandelbrot would be left at the station.  The attractions of financial engineering were too great, the potential profits too gargantuan.

While it might be interesting for traders to discuss Mandelbrot’s findings over a beer after work, his brand of uncertainty did not win clients nor did they create multibillion dollar bonus pools.  Sure, there were some problems, but they tweak the models some more and everything would work out in the end.

At least they hoped it would.  It didn’t.  The recent financial crises has laid bare the lie that Mandelbrot exposed more than 40 years ago.  As with many of his seemingly outlandish ideas, he had been right all along..

Mandelbrot’s Marketing Legacy

On the event of his death, it behooves marketers to understand Mandelbrot’s legacy and not repeat the mistakes of the financial industry.  The world is rough, not smooth; chaotic and not predictable.  It is extreme values that drive trends.  Real markets don’t follow models as we would like them to.

Lately, the increasing role of mathematics has obscured these simple, but inescapable facts.  Many expect ROI metrics to be predictive.  They are not.  They are evaluative and can help us understand our efforts better, but can not replace good judgement.

Mandelbrot’s most prominent disciple, Nassim Taleb, alerted us to the importance of Black Swans.  It is not what our models explain that we should be concerned with, but what they don’t.  That is where both greatness and tragedy dwell.

Therefore, we are compelled to receive the wisdom that Benoit Mandelbrot was kind and courageous enough to share with us.  If you are unfamiliar with his work, you can start with the TED talk below, which he gave in February of this year.

– Greg