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The four-thirds conjecture and proof that mathematics is still alive
Benoît Mandelbrot Mathematician
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To continue my image of picking up diamonds, emeralds and rubies it's as if the next generation had to bring shovels to dig. At this stage we are in the stage of enormous machines which grind huge amounts of data-rich soil to extract new results, with extreme difficulty and extreme technical complication. It's a field that at the same time keeps its attractiveness for children because I get all the time letters from adolescents who tell me that they've been using the computer to make this conjecture, and say, "Well, I hope it is new." The first years, sometimes it is new; today it's very rarely new, but for them it is still new. For them it represents the fact that someone with a little computer and pleasure in looking at shapes, a pleasure in playing with them, can make observations which are absolutely not implicit in the construction. This is in sharp contrast to the property of a circle, for example, or ellipse, in which although the proofs are rather difficult sometimes, most theorems in Euclid are rather simple to state, and even if you are good with your hands, rather obvious in a certain sense by inspection, but are difficult to prove. In this other case, nothing is implicit in the definition. From the definition to the observation a child can make, there's an infinitely great jump in difficulty, a very short distance in actual time performed, and again very often one ends up with questions for which there is absolutely no answer at this point. The four thirds hypothesis I use all the time for this purpose, to show that mathematics is alive, that it's not a technique that must built, or a field that must be built entirely upon old work and that is entirely cumulative. There are ways of getting to the frontier of mathematics by, how should I say, short cuts that have not been explored by anyone else, which are not long steps of reasoning. And I think that one of the most characteristic features of fractals, perhaps a unique feature among branches of mathematics, is the presence of such very simple constructions and conjectures that lead to impossibly difficult statements. To me there is no simpler way of proving the existence and the life of mathematics than the fact that many people try to prove it and the difficulty of mathematics. The fact that so many people fail to prove it; and also the ease of mathematics - the fact that many of these conjectures are in fact proven rather quickly by specialists. So every configuration is present in these cases.

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: 3 minutes, 27 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008