Benoit Mandelbrot

Submitted by martin on 19 October, 2010 - 7:19 Author: Martin Thomas
Mandelbrot set

Benoit Mandelbrot, the most famous mathematician of the last half-century, died on 14 October.

He became famous for developing new branches of mathematics with immediate visual appeal and wide practical application: chaos theory, and fractals.

The "Mandelbrot set" (see picture) exemplifies them. It is "chaotic" in the same way that, for example, turbulent flow of water is. It is "fractal" in being "self-same": put a small segment of it under a magnifying glass, and you see the same "roughness" as you do in the whole shape.

Popular accounts of chaos theory sometimes present it as a story of indeterminism. Actually, it is a story of how simple deterministic relations can produce eerily complicated results.

For anyone who has done A level maths, the Mandelbrot set is defined very simply: all the points z in the complex plane for which the function z-squared + c does not diverge under iteration.

The application of Mandelbrot's ideas to economics has recently been popularised by Nassim Nicholas Taleb's best-seller "The Black Swan": basically, even if economic quantities follow relatively simple mathematical relations, their interactions can produce "weird" results, like the 2008 financial crisis.

In mathematical economics, Mandelbrot stands as the opposite pole to Gerard Debreu. Both men studied mathematics in Paris in the 1940s, and were formed by the same influence: the "Bourbaki" school of French mathematicians.

The "Bourbaki" group, including Benoit Mandelbrot's uncle Szolem, Debreu's teacher Henri Cartan, and others such as Jean Dieudonne, tried to reconstruct mathematics in a more rigorous, abstract, and "top-down" form.

Debreu was enthusiastic, and became the leading figure in developing neo-classical economics into an elaborate mathematical scheme, adapted by many banks in the run-up to 2008 to guide their financial ventures.

Mandelbrot reacted against "Bourbaki". As he put it: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth..."


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