“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’?
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.
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.”
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.