Benoit Mandelbrot – A True Philosopher Prophet

October 19, 2010 — Ben Ramalingam

“Why is geometry often described as cold and dry? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree.”  

Benoit Mandelbrot, 1924-2010

Benoit Mandelbrot, one of the most influential and original mathematicians of the past century, died last Thursday aged 85. Mandelbrot’s contribution to science was to try to describe natural phenomena as they actually are, as opposed to using idealised shapes such as circles, spheres, squares and straight lines common to Euclidian geometry. As one thinker put it, ‘[before Mandelbrot] geometry was concerned with abstract perfection almost non-existent in the real world.’

But how exactly did he give scientific validity to the famous words of Gertrude Stein: ‘There is no straight line in nature’?

In the 1960s Mandelbrot, then a research fellow with IBM, undertook mathematical analysis of electronic “noise” which caused interference with electronic transmissions of signals. It was here that he first discovered one of the core properties of fractals which he would go on to identify in a wide range of other phenomena. 

Although the nature of [electronic transmission] errors was not understood, IBM scientists noted that the blips occurred in clusters; a period of no errors would be followed by a period with many. Examining these clusters, Mandelbrot noticed that they formed a pattern and that the closer they were examined, the more complex the pattern seemed to become. An hour might pass with no errors, while the next hour might pass with several errors. However, if one of the hours that contained errors was divided into 20-minute sections, there would be 20 minutes with no errors, then 20 minutes with many errors. On any scale of magnification, Mandelbrot found, the proportion of error-free transmission to error-ridden transmission remained constant. In other words the electronic interference exhibited “self-similarity” at every scale of magnification: each small part, when magnified, reproduced exactly the larger portion.

The same phenomena of “self similarity” was also present in other phenomena. Mandelbrot’s now-famous analysis of cotton prices showed that while “daily, monthly and yearly pricing of cotton was random, the curves of daily monthly and yearly price changes were identical.”  Such phenomena could not be explained using existing tools, leading Mandelbrot to develop ‘fractal geometry’ as a means by which to systematically explore self-similar phenomena. He argued that  most traditional mathematical and classical geometric models were ill-suited to natural forms and processes. In his own words: 

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line”

All of these phenomena and many others – stock market prices, earthquakes, planetary motion, blood vessels, even inequality (as Bill Easterly recently argued) – could be modelled using fractals. You can see more examples in Mandelbrot’s popular TED 2010 talk: 

Mandelbrot’s own view of his contribution is short and sweet:

In the whole of science, the whole of mathematics, smoothness was everything… What I did was open up roughness for investigation.”

In the last 40 or so years, Mandelbrot’s work on roughness has left a mark on fields as diverse as statistical physics, cosmology, meteorology, hydrology, geomorphology, anatomy, taxonomy, neurology, linguistics, information technology, computer graphics, and mathematics itself.

His most recent book, The Misbehaviour of Markets, which was published some five years ago and updated after the financial crisis, looks in detail at the fractal nature of risk and reward in financial markets. Co-authored with the Europe editor of the Wall Street Journal, the central message of the book is that modern financial theory is deeply flawed.

Finance, they argue, is built on assumptions of predictability and normal distribution of events (hm, reminiscent of any other sectors?) and as a result financiers vastly underestimate the likelihood of major crises and catastrophes.

Overall, the authors concur with the opinion of Wassily Leontief, a Harvard economist and 1973 Nobel Prize winner: “In no field of empirical enquiry has so massive and sophisticated a statistical machinery been used with such indifferent results.”

This takes on a whole new resonance on this side of the biggest crisis since the 1930s. As Mandelbrot presciently put it in the first edition, in 2005:

the financiers and investors of the world are, at the moment, like mariners who heed no weather warnings…. it is frightening because there are so many people of great brilliance and extraordinary greed who work there. They don’t understand the market, but they understand the numbers… “

And an interview he gave at that time reinforced the point:

A stockbroker wrote me a very plaintive letter asking why I was giving stockbrokers such a hard time. His argument was that what he did was right 98 percent of the time. Why bother about the events that occur in the rest of the time? The answer is that those events are the ones that really count… It is quite clear that some portfolios that were declared to be free of risk turned out not to be. They are very good for 90 percent or more of the time, but at the critical moment, they fail. They are just dreadful. Given the inter-connectedness of things, they may lead to very, very embarrassing complications for the whole world.”

Mandelbrot also gave a fascinating interview to the FT last year, where he takes his axe to the efficient market hypothesis:

It is perhaps unsurprising that the 2008 bestseller The Black Swan, on the importance of low frequency, high impact events in shaping the course of world history, was dedicated to Mandelbrot. (Nassim Nicholas Taleb also wrote an excellent essay on the Misbehaviour of Markets that can be downloaded here.)

To get a bit more into Mandelbrot’s way of thinking, it’s really worth taking a look at the transcript of this 2008 interview with PBS, A Radical Mind.

 This contains one of Aid on the Edge’s favourite quips, which gives us some insight into the nature of the man: “I abandon problems when a constituency gets created around them.”

A true maverick to the end.