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Self-similarity

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Fractals and the importance of proper description (Part 2)
Benoît Mandelbrot Mathematician
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To my mind, if you reduce a variety of phenomena to a few underlying invariances - that is, the small parts are like the whole with certain additional restrictions - you have done far more than curve-fitting, than describing; far less perhaps than fully explaining and reducing. A certain quality in between which is very valuable. In advanced fields of physics - as well as in parts of my work that are close to physics and to which the basic rules can be brought to bear - this intermediate theory does not deserve to be put forward because there is an underlying theory that explains it also. In the fields in which most of my work occurred the intermediate theory is very often the only realistic one. That is - take economics: one can try to explain all the enormous mess of economic phenomena by simple rules, but for all practical purposes if one is demanding that the rules have to be as strong, as convincing, as those of physics are, economics has no rules. There's no law like E = mc2 or like acceleration times mass equals force. Such laws do not exist in economics. In fact, as I said and keep saying, the very concepts, which these formulas represent are very often illndefined. Therefore it is perhaps an illusion to put a reduction in the centre of everything. In turbulence the wish to explain is always as strong as ever and I certainly share the hope that turbulence will be explained in a certain old-fashioned way very soon. However, I observe that every second or third year in my life someone has come with a theory that didn't last for very long because it had some very drastic weaknesses. You are describing what Thomas Kuhn named a scientific revolution in which the goalposts, and the situation to explain, have changed. Well, I don't like the word revolution because it's a big word and I tend to avoid them. It's Thomas Kuhn's word? Yes, but the fact that much of my work in life has consisted of setting goals which were different from the goals of the community I was visiting is very strong. The fact that I was visiting those communities, that even though for me I was always pursuing a certain very clear goal in different forms. Seen by others I was coming to a field then leaving, then coming again and leaving, and in several cases my hosts kindly, or unkindly, described me as being a sudden apparition and sudden dis-apparition for reasons that they could not understand. Indeed the logic was that of my process and not the logic of their field. The reason why I influenced those fields to varying degrees and sometimes very strongly is simply because I was asking questions they were not asking. And I was asking them in terms which were perhaps more ruthless than the terms they're used to because I did not know about the circumstances of what they were dealing with. But the overall solution is that, as in the whole of fractal geometry, this desire to introduce an intermediary system which is not the ultimate explanation, but which is more than ordinary description, has been the principle procedure. I found that in the fields that are closest to engineering, to practical questions, this intermediary structure was particularly essential and also, surprisingly, particularly difficult to defend. And I would like to comment upon that, for example, in hydrology. In hydrology I brought the notions of infinite dependence and the like. Hydrologists were terrified by them, but they did not primarily ask me to justify their use by experiment and by analysis, but they were very - at least many of them were very concerned about how to explain it; how does it fit in the grand view of things? And I realised to what extent in a certain sense physics and mathematics had intimidated some rather practical engineers; that instead of saying, "Is it true? Is it useful? Is it coherent? Is it amusing? Is it elegant?" they were asking, "Is it explainable by the criteria of physics?" And the criteria of physics, of course, when ultimately one names them, are those of Newton, Einstein, and the quantum mechanics pioneers. These are criteria that physics very rarely met even at its core, and to impose these criteria in the fields where I was working seemed to me quite pointless.

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, 55 seconds

Date story recorded: May 1998

Date story went live: 24 January 2008