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The Mysterious Genius: John F. Nash

The Mysterious Genius: John F. Nash

John Forbes Nash, Jr. is an American mathematician who is known for his very revolutionary "Nash Equilibrium" and Game Theory. His works in Game Theory, Differential Geometry, and Partial Differential Equations have provided insight into the forces that govern chance and events inside complex systems in daily life. His theories are used in market economics, computing, evolutionary biology, artificial intelligence, accounting, politics and military theory. Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi.

 

Nash was born on June 13, 1928, in Bluefield, West Virginia.His father, after whom he is named, was an electrical engineer with an incisive mind. His mother, Virginia, also intelligent, was a former teacher who had large dreams for her son, pushing him to read at 4, learn Latin, and skip a grade at school. Johnny Nash, as he was called by his family, was a singular little boy, solitary and introverted.He seems to have shown a lot of interest in books when he was young but little interest in playing with other children.However while the others played together, Johnny played by himself with toy airplanes and matchbox cars.

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Johnny's teachers at school certainly did not recognise his genius, and it would appear that he gave them little reason to realise that he had extraordinary talents. They were more conscious of his lack of social skills and, because of this, labelled him as backward. Although it is easy to be wise after the event, it now would appear that he was extremely bored at school. By the time he was about twelve years old he was showing great interest in carrying out scientific experiments in his room at home. It is fairly clear that he learnt more at home than he did at school.

Martha, Johnny's sister, seems to have been a remarkably normal child while Johnny seemed different from other children. She wrote later in life :

    "Johnny was always different. (My parents) knew he was different. And they knew he was bright. He always wanted to do things his way. Mother insisted I do things for him, that I include him in my friendships. She wanted me to get him dates. She was right. But I wasn't too keen on showing off my somewhat odd brother."

Nash first showed an interest in mathematics when he was about 14 years old. Quite how he came to read E T Bell's classic "Men of Mathematics" is unclear but certainly this book inspired him. He tried, and succeeded, in proving for himself the classic Fermat theorem about an integer multiplied by itself p times where p is a prime.The excitement that Nash found here was in contrast to the mathematics that he studied at school which failed to interest him.

He entered Bluefield College in 1941 and there he took mathematics courses as well as science courses, in particular studying chemistry, which was a favourite topic. He began to show abilities in mathematics, particularly in problem solving, but still with hardly any friends and behaving in a somewhat eccentric manner, this only added to his fellow pupils view of him as peculiar. He did not consider a career in mathematics at this time. Rather he assumed that he would study electrical engineering and follow his father.

Nash attended Carnegie Institute of Technology (now Carnegie Mellon University) with a full scholarship, the George Westinghouse Scholarship and initially majored in Chemical Engineering. He switched to Chemistry, and eventually to Mathematics due to his growing interest in mathematics. There he came in contact with John Synge who had recently been appointed as Head of the Mathematics Department and taught the relativity course. Synge and the other mathematics professors quickly recognised Nash's remarkable mathematical talents and persuaded him to become a mathematics specialist. They realised that he had the talent to become a professional mathematician and strongly encouraged him.

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Although his mathematics professors singled him out as a potential star, his fellow students found him a very strange person. Physically he was strong and this saved him from being bullied, but his fellow students took delight in making fun of Nash who they saw as an awkward immature person displaying childish tantrums. One of his fellow students wrote:

"He was a country boy unsophisticated even by our standards. He behaved oddly, playing a single chord on a piano over and over, leaving an ice cream cone melting on top of his cast-off clothing in the lounge, walking on his roommate's sleeping body to turn off the light, pouting when he lost a game of bridge."

Another wrote:

"He was extremely lonely."

And a third fellow student wrote:

"He was the butt of people's jokes because he was different. We tormented poor John. We were very unkind. We were obnoxious. We sensed he had a mental problem."

He showed homosexual tendencies, climbing into bed with the other boys who reacted by making fun of the fact that he was attracted to boys and humiliated him. They played cruel pranks on him and he reacted by asking his fellow students to challenge him with mathematics problems. He ended up doing the homework of many of the students.

After graduating in 1948 with bachelor of science and master of science degrees in mathematics, he accepted a scholarship to Princeton University, the mathematical centre of the universe, where he pursued his graduate studies in Mathematics. In Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures. Usually those who decide not to learn through lectures turn to books but this appears not to be so for Nash, who decided not to learn mathematics "second-hand" but rather to develop topics himself. In many ways this approach was successful for it did contribute to him developing into one of the most original of mathematicians who would attack a problem in a totally novel way.

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Nash earned a doctorate in 1950 with a 28-page dissertation on non-cooperative games. The thesis, which was written under the supervision of Albert W. Tucker, contained the definition and properties of what would later be called the "Nash equilibrium". It's a crucial concept in non-cooperative games, and won Nash the Nobel prize in economics in 1994. P Ordeshook wrote:
"The concept of a Nash equilibrium n-tuple is perhaps the most important idea in noncooperative game theory. ... Whether we are analysing candidates' election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people."
Milnor, who was a fellow student, describes Nash during his years at Princeton in:

    "He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. However, my most vivid memory of this time is of the many games which were played in the common room. I was introduced to Go and Kriegspiel, and also to an ingenious topological game which we called Nash in honor of the inventor."

He had ideas and was very sure they were important. He went to see Einstein not long after he arrived in Princeton and told him about an idea he had regarding gravity, friction and radiation. The friction he was thinking of was the friction that a particle, say a photon, might encounter as it moved through space due to its fluctuating gravitational field interacting with other gravitational fields. After explaining complicated mathematics to Einstein for about an hour, Einstein advised him to go and learn more physics. Apparently a physicist did publish a similar idea some years later.

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In the summer of 1950, after receiving doctorate, he worked for the RAND Corporation where his work on game theory made him a leading expert on the Cold War conflict which dominated RAND's work. He worked there from time to time over the next few years as the Corporation tried to apply game theory to military and diplomatic strategy. Back at Princeton in the autumn of 1950 he began to work seriously on pure mathematical problems.

Nash did ground-breaking work in the area of real algebraic geometry. His work in mathematics includes the Nash embedding theorem, which shows that any abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space. He also made significant contributions to the theory of nonlinear parabolic partial differential equations and to singularity theory.

From 1951 Nash had taught at the Massachusetts Institute of Technology but his teaching was unusual and unpopular with students; his examining methods were highly unorthodox.

Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958. Nash, however, was very disappointed when he discovered that Ennio De Giorgi had proved similar results by completely different methods. The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it.

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During his time at MIT, he met Alicia Lopez-Harrison de Lardé. Alicia graduated from M.I.T., having majored in physics. They married in February 1957 at a Catholic ceremony, although Nash was an atheist. Nash experienced the first symptoms of mental illness in early 1959, when his wife was pregnant. He resigned his position as member of the M.I.T. mathematics faculty in the spring of 1959. Nash's wife admitted Nash to the McLean Hospital for schizophrenia in 1959. In his own words, he states,

"I spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release. And it did happen that when I had been long enough hospitalized that I would finally renounce my delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return to mathematical research. In these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le problème de Cauchy pour les équations différentielles d'un fluide général"; the idea that Prof. Hironaka called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data"."

In Princeton, John's illness continued, transforming him into a frightening figure. He spent most of his time hanging around on the Princeton campus, talking about himself in the third person as Johann von Nassau, writing nonsensical postcards and making phone calls to former colleagues. They stoically listened to his endless discussions of numerology and world political affairs. Her husband's worsening condition depressed Alicia more and more.

In January 1961 the despondent Alicia, John's mother, and his sister Martha made the difficult decision to commit him to Trenton State Hospital in New Jersey where he endured insulin-coma therapy, an aggressive and risky treatment, five days a week for a month and a half. A long sad episode followed which included periods of hospital treatment, temporary recovery, then further treatment. Alicia divorced Nash in 1962.

Slowly over many years Nash recovered. He delivered a paper at the tenth World Congress of Psychiatry in 1996 describing his illness. He was described in 1958 as the:

"... most promising young mathematician in the world ..."

but he soon began to feel that:

"... the staff at my university, the Massachusetts Institute of Technology, and later all of Boston were behaving strangely towards me. ... I started to see crypto-communists everywhere ... I started to think I was a man of great religious importance, and to hear voices all the time. I began to hear something like telephone calls in my head, from people opposed to my ideas. ...The delirium was like a dream from which I seemed never to awake."

Despite spending periods in hospital because of his mental condition, his mathematical work continued to have success after success. He said:

"I would not dare to say that there is a direct relation between mathematics and madness, but there is no doubt that great mathematicians suffer from maniacal characteristics, delirium and symptoms of schizophrenia."

In the 1990s Nash made a recovery from the schizophrenia from which he had suffered since 1959. His ability to produce mathematics of the highest quality did not totally leave him. He said:

"I would not treat myself as recovered if I could not produce good things in my work."

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In 1994, at the age of 66, John Nash received the Nobel Prize in Economics in Stockholm, Sweden, for his work on Game Theory. Thirty-eight years after their divorce, Alicia and John remarried. Nash has returned to an office at Princeton, where he continues to explore mathematics, the world in which he first succeeded, the world that carried him during his debilitating illness, and the world that has embraced him again.

 

 

 

 


Written by Dhiraj Sarmah and Krishna Pandit.
Undergraduate students, Department of Mathematics.
Cotton College, Guwahati-01

Courtesy :  1. "A Beautiful Mind" by Sylvia Nasar

2. "A Brilliant Madness"  documentary

 

To buy the book "A Beautiful Mind", Click here.


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