A Mathematical Genius

It is said that geniuses are born and not evolved. Srinivasa Aiyanagar Ramanujan, the Indian mathematician, was one of such geniuses who despite his inherent disadvantages, was born into a poor family, did not have access to good library facilities, not much of guidance, etc., was able nonetheless to make very significant contributions in mathematics. So is John Nash Jr. who burst into the world of mathematics with his 27-page Ph.D. thesis (Princeton University) in which he had developed his theory of “Nash Equilibrium”, when he was only twenty-two years of age. It is said that George Friedrich Bernhard Riemann’s thesis was only ten page long.

Great minds have great ways of impacting the world. Nash’s contribution was in the theory of games. The great John von Neumann had launched the theory of games with Oskar Morgenstein; their work dealt with what is called the cooperative game theory (zero sum); in other words “one man’s gain is another man’s loss”. Nash accomplished the much more difficult task of formulating a theory of non-cooperative games which allows more than two parties in the game and none of them has to be a loser in the relative sense. The game theory found applications in many practical fields as diverse as economics and the cold war politics. Regarding application of the game theory to auctions, Simon Singh (www.nytimes.com/books/98/06/14/reviews/980614.14singht.htm) described, “On December 5, 1994, Vice President Al Gore announced the opening of what was to be called the greatest auction ever. The Government was setting off licenses that would allow companies to set up mobile phone networks using particular sections of the radio spectrum. Three months later, when bidding had concluded, the Treasury was $7billion richer, and the auction was proclaimed a resounding success….that it (auction) was organized according to an innovative and revolutionary principle and was a concrete vindication of game theory, one of the most important developments in modern economics”.

According to Robert Solov, a Nobel laureate in economics at MIT, “It wasn’t until Nash that game theory came alive for economists”. One of the extraordinary talents that the geniuses usually have is that they look at things the way they were never looked at before. According to Sylvia Nasar, Nash’s biographer, (www.nytimes.com/books/98/06/14/reviews/nasar-nash.htm), “One of his (Nash’s) teachers told John’s mother that her son was having trouble in math. ‘He could see ways to solve problems that were different from his teacher’s’, Mrs. Legg (Nash’s sister) said laughing”. Nearly two hundreds years earlier on, Carl Gauss (1777-1855), the great German mathematician and Riemann’s Professor, had a similar problem with his school teacher whom he had described sarcastically as “a poet among the mathematicians and a mathematician among the poets”. R.J. Duffin, Nash’s graduate Professor, recalls “Mr. Nash as a tall, slightly awkward student who came to him one day and described a problem he thought he had solved. Professor Duffin realized with some astonishment that Mr. Nash, without knowing it, had independently proved Brower’s famed theorem”. In his letter of recommendation for Nash, Professor Duffin had written just one line: “This man is a genius”.

He could see the solution to a difficult problem, like Ramanujan, before developing the actual proof for it. Ramanujan’s proofs were frequently shoddy, inelegant, and non-rigorous but his results were seldom wrong. David Goodstein (www.nytimes.com/books/98/06/14/reviews/nasar-nash.htm) told an episode according to which, “While an MIT colleague named Warren Ambrose wrote a meticulous proof on the blackboard, Nash could be heard muttering ‘hack, hack’ from the back of the room. Perhaps out of revenge, Ambrose asked Nash, ‘Is it possible to embed any Riemann manifold in a Euclidean space?’….Having satisfied himself that it was a problem worthy of his talents, Nash went after it with relentless tenacity, producing at the end of two years what a Princeton mathematician called ‘one of the most important pieces of mathematical analysis in this century’. Mathematicians consider the Nobel winning work trivial by comparison”. After finishing his Ph.D. work at Princeton, Nash got a teaching job at MIT. According to Sylvia Nasar, “By the mid 1950’s (he was born in 1928), Mr. Nash was phenomenally productive. When he got tired of mathematicians, he would wander over to the economics department to talk to Mr. Solov and another Nobel laureate, Paul Samuelson”. He married Alicia Larde, an El-Salvadoran, in 1957, a Physics student at MIT who took advanced calculus from him. She became pregnant in the fall of 1957, with their son, John Charles Martin Nash.

When the going was so good and Nash was in the prime of his creative work, heavens fell on him. He was struck by paranoid schizophrenia soon after the birth of their son. Schizophrenia is such a terrible mental disease that some people call it the ‘cancer of the mind’. Sylvia Nasar has written Nash’s biography and called it ‘A Beautiful Mind’. The Beautiful Mind was afflicted with the cancer of the mind. He started hearing voices and experiencing the other symptoms of the disease. He was seized by the fear that the people wanted to break into his office to steal his ideas. According to George F. Will (John Nash’s Remission, Newsweek, January 14, 2002, p.68), “He (Nash) felt that he was simultaneously the epicenter of the universe, yet controlled by a psyche other than his own. He was both abject petitioner and a figure of gigantic political or religious – but secret importance. One moment he was furtive, then he was Emperor of Antarctica”. At first, his family tried to keep his affliction under covers but things went from bad to worse. He resigned his job at MIT in 1959, and in 1963, his wife divorced him but she let him live at her house. Nasar said of her, “The loyal wife who stood by him when she no longer was her wife”.

He was hospitalized several times over a period of some thirty years, and then miraculously remission occurred. He started coming out of the shell, communicating with people, and picking up the work in which he got interested. He started going to Princeton on daily rounds. Freeman Dyson, a Nobel laureate in Physics, used to see Nash every day at the institute and out of respect for him, he would greet Nash without receiving any response. According to Nasar, “On one of those gray mornings, sometime in the late 1980’s, he (Dyson) said his usual good morning to Nash. I see your daughter is in the news again today, Nash said to Dyson, whose daughter Esther is a frequently quoted authority on computers. Dyson, who had never heard Nash speak, said later: ‘I had no idea he was aware of her existence….Slowly, he just somehow woke up”.

In 1994, Nash received the Nobel Prize in economics for the work that he had done in his youthful years. Nasar described the events that led to his winning the award, as follows: “By mid – 1985, the prize committee was evidently actively considering an award for game theory…Five years later, the committee was making discreet inquiries not just about Mr. Nash’s contribution but about his state of mind. There is no formal rule that a recipient must travel to Stockholm to accept the prize in person, give a Nobel lecture there or deliver a few profundities and words of gratitude to the king at the banquet. And there is certainly no rule that the recipient must hold a university post or have maintained an active career beyond the prize winning contribution….To most young game theorists, Mr. Nash was a demigod. But Professor Kuhn played a particular role. A noted game theorist himself, he made it clear to the committee that it would be a grave injustice if Mr. Nash’s illness cost him the prize”.

He was also the recipient of the 1978 John von Neumann Theory Prize, together with Carlton E. Lemke, for their outstanding contributions to the theory of games. Part of the citation for this award reads: “..the distinctive and complementary contributions of John Forbes Nash Jr. and Carlton Edward Lemke to this theory have impacted areas of Operations Research and Management Science and other sciences, far beyond the formal limits of game theory….”. He had narrowly missed winning the Fields Medal (as prestigious as Nobel Prize. There is no Nobel Prize for mathematics.). The somewhat belated recognition culminated into a movie film called A Beautiful Mind based on Nasar’s book. The movie is directed by Ron Howard and the cast includes Russell Crowe (in the role of Nash Jr.), Ed Harris, Jennifer Connelly, Adam Goldberg, and Paul Bettany. The script is written by Akiva Goldsman based on Nasar’s book. The movie was released the first week of January, 2002.

It is said that a hair thin line demarcates the boundary between a genius and a mad man. It was surely true of Nash Jr.; another great philosopher that comes to mind who had the same malady is Frederich Nietsche. I end my essay with a verse from John Dryden:

Great wits are sure to madness near allied,

And thin partitions do their bounds divide.