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Fractals and chaos theory in mathematical development


The inevitability of mathematical development
Benoît Mandelbrot Mathematician
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The first man in chaos theory was unquestionably Henri Poincaré, and I also think one of the first men for fractals. It is actually a very strange and enigmatic figure from the early 19th century named Bolzano played a part in it, but he didn't publish it and his papers were published only in the 1920s. They had no influence on history. The Cantor set which is a very essential fractal was introduced by Cantor and also by two other people, Derra and Smith, at about the same time - as a pure counter-example of analysis, just as a set which contradicted some simple and natural ways of thinking about the world; it had, how to say, no physical reality whatsoever. But almost at the same time Poincaré was describing some limit sets of some Kleinian groups and observing them made of a kind of dust, which he characterised, and that dust indeed was something Cantor had said. So if you put the creation of chaos in the hands of Poincaré, which I think is fair, he was also the creator of the notion of fractals, and of the notion that fractals were not simply an invention of mathematicians, but something useful. However, both of these influences were totally not continued, except again for this very important work of Hadamard in 1897 or '98. Then it was interrupted, and I think, if you allow me to say a few words about my view of the development of sciences, I wondered very much why it was interrupted. Here are the two men who were effectively the most influential and the second most influential intellectual in science in France at the time, whose work has no progeny. Some people, for example, the abstract mathematicians with whom I had so many disagreements throughout my life, had the very strong feeling of 'The March of History', the inevitable 'March of History', in which mathematics was destined to become more and more pure, removed from reality and abstract. That idea of mathematics is actually very recent. It started shortly after World War I. Its sources are multiple. One of the sources was in the work of Hilbert who was insisting on axiomatics; one source was in the works of the Polish school of mathematics, which had nothing to do with the inevitable 'March of History', and everything to do with political events in that part of the world where I was born. In 1919 a very forceful person named Sierpinski, whose name I gave to some basic shapes, decided he was going to create a Polish school of mathematics that would not be a distant suburb and colony of Paris, Gottingen or Cambridge. Therefore he decided to concentrate on topics that the others were not interested in, very abstract topics indeed. Then World War I had an extraordinary dreadful effect on France in many ways, because so many young Frenchmen were either killed or, equally bad, after spending four years in trenches, incapable of the terrifyingly sustained effort that science requires. Therefore there was a break. After a few years, young Frenchmen decided to go and seek inspiration, not among their teachers and their contemporaries in France, but in Gottingen or Berlin, among German mathematicians. Germany did not have the policy of sending scientists to the front to be killed which was very much more noble in this case of France, if you want - much more democratic, yet extraordinarily destructive. So Germany had more variety of activities in mathematics than France had.

Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal geometry - the maths of the shapes found in nature.

Listeners: Bernard Sapoval Daniel Zajdenweber

Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.

Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.

Duration: 4 minutes, 35 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008