Benoit Mandelbrot, the most famous mathematician of the last half-century, died on 14 October.
He became famous for developing new branches of mathematics with immediate visual appeal and wide practical application: chaos theory, and fractals.
The "Mandelbrot set" (see picture) exemplifies them. It is "chaotic" in the same way that, for example, turbulent flow of water is. It is "fractal" in being "self-same": put a small segment of it under a magnifying glass, and you see the same "roughness" as you do in the whole shape.
Popular accounts of chaos theory sometimes present it as a story of indeterminism. Actually, it is a story of how simple deterministic relations can produce eerily complicated results.
For anyone who has done A level maths, the Mandelbrot set is defined very simply: all the points z in the complex plane for which the function z-squared + c does not diverge under iteration.
The application of Mandelbrot's ideas to economics has recently been popularised by Nassim Nicholas Taleb's best-seller "The Black Swan": basically, even if economic quantities follow relatively simple mathematical relations, their interactions can produce "weird" results, like the 2008 financial crisis.
In mathematical economics, Mandelbrot stands as the opposite pole to Gerard Debreu. Both men studied mathematics in Paris in the 1940s, and were formed by the same influence: the "Bourbaki" school of French mathematicians.
The "Bourbaki" group, including Benoit Mandelbrot's uncle Szolem, Debreu's teacher Henri Cartan, and others such as Jean Dieudonne, tried to reconstruct mathematics in a more rigorous, abstract, and "top-down" form.
Debreu was enthusiastic, and became the leading figure in developing neo-classical economics into an elaborate mathematical scheme, adapted by many banks in the run-up to 2008 to guide their financial ventures.
Mandelbrot reacted against "Bourbaki". As he put it: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth..."
More: http://bit.ly/mandelbr [1]
http://bit.ly/alejandre [2]
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Bukharin on Determinism and Indeterminism
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George Novack
....For dialectical materialism, reality has developed in a lawful manner and is rationally explicable. The rationality of nature and human history is bound up with matter in motion. The concatenation of cosmic events gives rise to cause-and-effect relations that determine the qualities and evolution of things. The physical preceded and produced the biological, the biological the social, and the social the psychological in a historical series of mutually conditioned stages. The aim of science is to disclose their essential linkages and formulate these into laws that can help pilot human activity.
The rationality, determinism, and causality of the universal process of material development do not exclude but embrace the objective existence and significance of absurdity, indeterminism, and accident.
However, these random features of reality are no more fundamental than regularity. They are not immutable and irremovable aspects of nature and history but relative phenomena which in the course of development can change to the extent of becoming their own opposites. Chance, for example, is the antithesis of necessity. Yet chance has its own laws, which are lodged in the occurrence of statistical regularities. Quantum mechanics and the life insurance business exemplify how individual accidents are convertible into aggregate necessities.
Exceptions are nothing but the least frequent alternatives, and when enough exceptions pile up they give rise to a new rule of operation which supersedes the formerly dominant one. The interplay of chance and necessity through the conversion of the exception into the rule can be seen in the economic development of society. Under tribal life, production for immediate personal consumption is the norm whereas production for exchange is a rare and casual event. Under capitalism, production for sale is the general law; production for one’s own use is uncommon. What was categorically necessary in the first economic system is fortuitous in the second. Moreover, in the transition from one economy to the other the bearers of chance and necessity have changed places, have become transformed into each other.
Social structures that are rational and necessary under certain historical circumstances become absurd and untenable at a further stage of economic development and are scrapped. Thus feudal relations, which corresponded to a given level of the powers of social production, became as anachronistic as Don Quixote and had to give way before the more dynamic forces and more rational forms of bourgeois society.....
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good article here, too:
ReplyDeletehttp://www.marxists.org/archive/shibdas-ghosh/1964/06/26.htm