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Work on thermodynamics in Geneva


Further work on the Power-Law Distribution
Benoît Mandelbrot Mathematician
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These two years I think were very interesting and very disappointing, because I became very aware of the difficulty of the enterprise. It was not something as simple as Piaget thought somebody with tools was going to put them into a nice box, but where to put that? Most of the ideas were not in themselves well-defined, and therefore how can you have laws about things that are ill-defined? Now, as a matter of fact, that was the occasion of a paper of mine in 1956, which I forgot for a long time, but now have found again and that brought back a stream of thinking. I became interested in the following bizarre question: given that all those concepts in psychology, in economics, and so on, were extremely ill-defined, if defined at all, given that how could there could be any law? So the question was not about how to explain observed laws in these fields, but how could there be any law. I wrote this paper, which had the following theme: it said, before explaining one must understand against which possible alternative one is explaining. If the power-law distribution were found to be incorrect, could it take any other form? Would I have to explain it's power-law and not Gaussian or something? And so my argument went as follows - it was much developed several years later, but it was very well written down in '56. Imagine you have quantities that are ill-defined in which somehow different people report them in different ways. Take genera and species in taxonomy: some naturalists would put a big genus with many species; others would make several genera with few species. Very difficult to say one is right, one was wrong. Today of course we can, but we are dealing with old-fashioned taxonomies. So there were cases where some were simply making them bulky, and others small; cases where one simply put a genus just because - somehow for symmetry and for beauty, a bit at random. Now given all that how is it that this man Willis in 1922 found some very specific distribution of numbers of species within genera, a very well defined power-law distribution, which was a very great mystery - now that power-law distribution had been explained by Yule as being the effect of evolution. But to me it was not enough, because given that evolution is very poorly represented by the idea of species and genus, could there have been any other distribution, any other law applicable? I argued not; that the only distribution which is in a certain sense invariant under what I called filters, which are consistently adding, splitting, in averaging, in moving, in taking the largest - all kinds of, how should I say, observational filters, the only distribution that could ever have been obtained were power-law distributions. Now this is entered very concisely in this paper, and that indeed is to a very great extent the idea which was underlying what physicists shortly afterwards started doing in physics; that somehow you must have small ensembles, bigger ensembles, and the same rules should apply to all these various groupings. And the rules not invariant under this renormalization would not be observed because they would not be robust in a certain sense. Then I started that in a totally different environment, let me give you an idea about how this applies. When statisticians deal with the distribution of firms sizes, - rather, when economists deal with that, they respect their colleagues in statistics who tell them that you must take the average size of a firm, and you have reports in this industry, that industry, that the average size firm is so and so. This average very often is totally meaningless. The most extreme case is that of the gun dealers in the US. At one time to become a gun dealer in the US, you had to pay twenty dollars. Therefore anybody who wanted to buy a single gun and could do it a bit ahead would become a gun dealer and then buy from the wholesaler or the maker. Then the US, to control guns, put a rather high, say two thousand, three thousand dollars fee to just become a gun dealer. Almost all the gun dealers did not renew their licenses. Therefore the number of guns sold by big dealers moved a little bit. So, say, one quarter were sold by one chain before " and after that, a bit more. But the average gun size was a numerator, which was more or less the same; the denominator had changed in the nature of one to ten, or to three or to four, I don't know. But the idea of an average size was meaningless. The average size of a city - are you going to take the definition of a commune in France, or a township, or of a school district in America? It's a very elusive notion. So there is underlying all those, how to say, harder sciences, softer because they are just harder - I don't call them soft, I call them hard sciences. The idea that somehow we don't have those solid numbers to deal with. Take income. The initial data at which Pareto was looking were collected before income tax existed. Nobody had any reason for hiding income. Now my income is something extraordinarily ill-defined. There's some money that I am spending that is not income, and other money, which we declare as income even though I don't think it should be. Though I am a good citizen and I pay income tax according to the law I don't think that the number at the end, which I sign every year has any meaning whatsoever. How does it come about that any distribution could apply to it? It's something that is fundamental. It's not a matter of explaining ill-known quantities, but very specific ones. Now, this idea again came first in the context of taxonomy and later on I generalised very broadly.

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: 6 minutes, 30 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008