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Economics: Pareto and Bachelier


Fractals and chaos theory in mathematical development
Benoît Mandelbrot Mathematician
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Now those French mathematicians who went to Gottingen were very impressed by one half of the mathematical institute there, the half concerned with fundamental and abstract mathematics. Apparently they didn't pay attention to the fact that Klein, who was the master of this institute, and his successor, Courant, were very careful to also have, in the same institute, professors of numerical methods, also applied mathematics and also theoretical physics. In the mind of Hilbert, Klein and the others, as it was in the mind of Hadamard and Poincaré, all these things together were mathematics. Those mathematicians who came to Germany saw only the abstract part, and picked it. The tradition, the fact that Hadamard and Poincaré were French ordinarily would not matter much. The fact that French mathematics had this break when it was not much followed, when a new tradition was started, I think matters a great deal. One can always regard history with 'ifs'; if Poincaré had lived, if Hadamard had not lost two sons in the war and been a very much less energetic person afterwards, maybe chaos theory would have developed then. But it did not, and as to the idea that fractals enter in this question, the first mention I have of it is an obituary that Hadamard wrote of Poincaré, which again I think is a wonderful piece. I was amazed, and that obituary made me become interested in Kleinian groups for the first time in '77, very deeply. So this strain, this tendency, this sequence of influences disappeared. It did not completely disappear. There were several very notable articles of this sort over the years, and then in the early 60's a new input came, which was Edward Lorenz. Lorenz did not know anything about Poincaré and about Hadamard. He is a brilliant man, a meteorologist, very much hands-on, even though far more mathematical than most meteorologists. He was one of the first to use computers to very great effect. Now I became acquainted with Lorenz in the early '60s when a friend of mine observed with amazement that I was speaking of turbulence in a crazy fashion, and Lorenz was speaking of turbulence in another crazy fashion, and perhaps we had something to tell each other. Well, we met, we liked each other very much but sparks didn't fly, the length of these two approaches took a very much longer time to get realised. But Lorenz' influence was extraordinary. So you see there was a gap of seventy years in one case, from a theory that had been already substantially started by two great men, to the other one. In the case of fractals, the roots, again I find of them in Poincaré in passing, but I wouldn't have seen them there if Hadamard had not told me. They're not developed. Fractals as described then were simply two or three constructions of monstrous character. When I started applying them, there were people saying, "In other words what you are telling us is that the world is ruled by pathology." Well, perhaps it's ruled by pathology in more than one field, but physics is certainly ruled by shapes, parts of physics are ruled by shapes that were viewed as pathological at one time by the mathematicians. But the link between my work and modern chaos theory was totally independent. We very soon got together and fractal dimension was immediately adopted in the mainstream chaos theory circles, because one used measurements of those shapes and that is a basic one. Many developments in fractal geometry occurred in that context. Multifractals are taken up very much by chaos theory and developed in the context of chaotic systems, as well in the context of real space systems, which I was studying. So at this point the two are very close together. In terms of their impact on the public and on education, perhaps I can speak of that in a few minutes. They cannot be separated. But I think that in those terms one part of chaos which is particularly striking is the geometric part, which is both chaos and fractals.

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: Daniel Zajdenweber Bernard Sapoval

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

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.

Duration: 5 minutes, 5 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008