John Nash, Mathematician, is Awarded the Abel Prize

by Duane Nichols on April 11, 2015

John Nash of Bluefield, WV is 86

‘Beautiful mind’ John Nash adds Abel Prize to his Economics Nobel Prize

Excerpt from an Article by Phillip Ball, Nature Journal, March 25, 2015

Although some consider the Abel Prize to be the ‘Nobel of mathematics’, its winners are hardly ever household names. But this year’s prize, announced on March 25th, includes a notable exception: John Nash, the subject of the 2001 film A Beautiful Mind starring Russell Crowe — and a previous winner of an Economics Nobel prize. (He was raised in Bluefield, WV. This surely makes him the first person to win not-quite-Nobels in two completely different fields.)

Nash, who spent most of his career at Princeton University in New Jersey, got the award for work on partial differential equations. Nash’s contributions to this field are widely considered more profound than the research in “game theory” that earned him an Economics Nobel.

“Partial differential equations lie at the foundation of many areas, both within and beyond mathematics, ranging from geometry to physics,” says mathematician Robert Kohn of the Courant Institute in New York City.

Nash, now 86, showed in the 1950s that the extrinsic and intrinsic approaches to mathematical manifolds are equivalent. He proved that a Riemannian manifold can always be ‘embedded’ as a subset of a (possibly much higher-dimensional) Euclidean space. In proving this theorem, Nash devised new methods for solving partial differential equations. “It was an unknown area,” he said in a 2011 interview. “I didn’t realize that, and I said, this doesn’t seem so difficult”.

John Nash developed the concept of the “Nash equilibrium” to graphically display the “game theory” condition as a mathematical representation.  The ecologist Garrett Hardin extended this concept as a “commonize cost vs. privatize profit game” with reference to resource allocations.  When natural resources are concerned, the gaining of private profit also results in costs to the community or state, called the “tragedy of the commons.”


ARTICLE: The “Tragedy of the Commons” by Professor Garrett Hardin

Reference: Science 13 December 1968: Vol. 162 no. 3859 pp. 1243-1248

The author was professor of biology, University of California, Santa Barbara. This is an excerpt of an article based on a presidential address presented before the meeting of the Pacific Division of the American Association for the Advancement of Science at Utah State University, Logan, 25 June 1968.

ABSTRACT — The tragedy of the commons can be considered in relation to environmental issues such as sustainability. The commons dilemma stands as a model for a great variety of resource problems in society today, such as water, forests, and non-renewable energy sources as fossil fuels.

The Pollution Problem –

In a reverse way, the tragedy of the commons reappears in problems of pollution. Here it is not a question of taking something out of the commons, but of putting something in — sewage, or chemical, radioactive, and heat wastes into water; noxious and dangerous fumes into the air, and distracting and unpleasant advertising signs into the line of sight. The calculations of utility are much the same as before. The rational man finds that his share of the cost of the wastes he discharges into the commons is less than the cost of purifying his wastes before releasing them. Since this is true for everyone, we are locked into a system of “fouling our own nest,” so long as we behave only as independent, rational, free-enterprisers.

The air and waters surrounding us cannot readily be fenced, and so the tragedy of the commons as a cesspool must be prevented by different means, by coercive laws or taxing devices that make it cheaper for the polluter to treat his pollutants than to discharge them untreated. We have not progressed as far with the solution of this problem as we have with the first. Indeed, our particular concept of private property, which deters us from exhausting the positive resources of the earth, favors pollution. The owner of a factory on the bank of a stream — whose property extends to the middle of the stream, often has difficulty seeing why it is not his natural right to muddy the waters flowing past his door. The law, always behind the times, requires elaborate stitching and fitting to adapt it to this newly perceived aspect of the commons.

PBS Special on John Nash, Mathematician

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