# John Forbes Nash - Mathematician

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John Forbes Nash, born 13 June, 1928, in Bluefield, West Virginia, is one of the most celebrated living mathematicians. Given the same name as his father, he has often had to have 'Jr' tacked onto the end of his name, though his own fame is unarguably greater. He has been awarded the von Neumann Prize and the 1994 Nobel Prize for Economics, as well as fellowships of prestigious scientific academies and societies.

However, the fundamental basis of his celebrity will vary according to the interests of who you ask. Most would say he is famous for his contributions to 'game theory'^{1} - with the important concept of 'Nash Equilibria' named after him - others for more esoteric mathematics. A few will know him for (re)inventing the game of Hex. Finally, some would know him for being a real example of the 'mad scientist' - a genius who, between the 1960s and the start of the 1990s, was unable to work because of schizophrenia.

### The Route to Mathematics

In his Nobel autobiography Nash describes Bluefield, the place where he grew up, as:

...a small city in a comparatively remote geographical location in the Appalachians...not a community of scholars or of high technology.

He says his early intellectual stimulation came from the books of his parents and grandparents, in particular an encyclopaedia and the classic *Men of Mathematics* by ET Bell.

Despite this interest in mathematics, which led to Nash taking extra courses at Bluefield College, he initially planned to major in chemical engineering when he went to Carnegie Technical College in Pittsburgh. This was with the intention of becoming an electrical engineer like his father. After only one term, though, he dropped engineering, changing his major to pure chemistry, as he disliked the dullness of courses such as technical drawing. He soon changed subjects again to mathematics, as the chemistry course didn't appear to rate students on their ability to think or to learn facts, but on their co-ordination:

How well one could handle a pipette and perform a titration.^{2}

Thus it was as a mathematics graduate he moved to Princeton for his graduate studies. It was while he was there that Nash invented the popular game of Hex, in 1948, for which he also provided mathematical proofs.

It quickly captivated students of mathematics both at the Institute for Advanced Study and Princeton. The game was commonly called eitherNashorJohn, the latter name referring mainly to the fact that it was often played on the hexagonal tiles of bathroom floors

- fromMathematical Puzzles and Diversionsby Martin Gardner

This mathematical game should not be confused with game theory, which was initially developed by John von Neumann and Oskar Morgenstern as a method of modelling economic theories.

One of Nash's undergraduate Carnegie courses had been in international economics, and it was this that led to his early works on 'the bargaining problem', and later into game theory.

### Game Theory

Game theory is not easy to explain. It is not so much about playing games as trying to understand the basis for making rational decisions where there are two or more 'players' competing (or co-operating) for resources. It can show that the results achieved by people thinking logically about their best choices are, surprisingly, not always the best results that they could achieve. The work produced by Nash has had the most real-life applications of all his work, with great impacts on fields where understanding of strategies of competition and co-operation are involved, such as economics, evolutionary biology and political science. It was this work that led to his 1994 Nobel Prize^{3}.

### Other Celebrated Work

While Nash's work on game theory was ingenious, it has been suggested that this merely expanded on the principles discovered by von Neumann and others. After his initial contributions to game theory, Nash tackled the most difficult problems he could find in geometry and analysis. This later work is considered more mathematically significant, with Nash proving^{4}:

'Every smooth compact manifold can be realised as a sheet of a real algebraic variety' in 1952.

'The highly anti-intuitive C

^{1}isometric embedding theorem' in 1954.'Fundamental existence, uniqueness, and continuity theorems for partial differential equations'.

If you are not a pure mathematician these achievements may leave you cold. If you are a non-mathematician they are likely to simply leave you bemused.

### Schizophrenia and *A Beautiful Mind*

After being highly productive during his twenties, Nash suffered a mental breakdown in 1959, while his wife Alicia, whom he had married in 1957, was pregnant. Nash's schizophrenic delusions led to long periods, sometimes involving involuntary hospitalisation, when he did not publish any mathematical work. This breakdown is controversially covered in the 2001 film *A Beautiful Mind*, which in turn is based on the 1998 unauthorised biography of the same name, written by Sylvia Nasar.

In the film *A Beautiful Mind*, Russell Crowe played Nash. The film showed Nash having many visual hallucinations that told him to do things he wouldn't normally do, and whom he thought of as his closest friends.

However, the real John Nash did not experience visual hallucinations as part of his schizophrenia, rather auditory hallucinations that he was receiving messages from space. Because these messages were received in the same way as his mathematical ideas^{5}, he believed them. The film also shows Nash recovering due to his medication, when in fact he recovered without it^{6}.

It is sad to note that his son also suffers from auditory hallucinations, but Nash is hopeful that, like himself, his son will eventually be able to reject the voices.

These distortions of the true story have caused considerable argument, and even outrage, among those with knowledge of schizophrenia, though to be fair there have been favourable responses to the portrayal of the mental illness too. The changes can be argued as producing a film with a greater impact, and more contentiously, one that is 'more responsible'. If the film had shown schizophrenics recovering without medication then it could be abused by those who refused effective medication because of it. It has been suggested that the film's portrayal of a drug cure could be seen as due to the influence of pharmaceutical companies. This is expanded on in a review of the film from the perspective of a site devoted to the proposition that prescribing drugs is not the best way to deal with schizophrenia.

Although Nash has recovered from delusional thinking, he doesn't think this is necessarily entirely without a potential loss to his mathematical creativity.

So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists. However this is not entirely a matter of joy as if someone returned from physical disability to good physical health. One aspect of this is that rationality of thought imposes a limit on a person's concept of his relation to the cosmos

Without the erratic power of his brilliant madness he may not have broken free of the straitjacket of normality - the ability to have wild new ideas may carry with it the danger of delusions.

### Further Reading

Home Page of John Forbes Nash at Princeton University - not immediately scintillating, but provides contact details and information on work in progress.

John Nash and

*A Beautiful Mind*(note: pdf file) - reviews in more detail the mathematical achievements of Nash, while avoiding probing much of his private life.*Mathematical Puzzles and Diversions*by Martin Gardner, first published 1959 - has the chapter from which the Hex quote was taken.A Brilliant Madness - Nash talks about the 'voices' he heard.

Schizophrenia.com - a site with a bundle of links to articles about schizophrenia.

^{1}Very simplistically: how to play to win.

^{2}Pipettes are calibrated glass tubes that are used to measure precise amounts of liquids. A titration is a procedure in which small, measured amounts of a liquid are added to another, usually of unknown concentration, until a reaction takes place, allowing one to work out the concentration of the test liquid.

^{3}John C Harsanyi, John F Nash Jr and Reinhard Selten received the prize jointly 'for their pioneering analysis of equilibria in the theory of non-cooperative games'.

^{4}All quotes from

*John Nash and 'A Beautiful Mind'*by John Milnor.

^{5}According to Nash himself - see link to 'A Brilliant Madness' in references.

^{6}Discussion of treatments for schizophrenia can be found in the article on Different Approaches to Treating Mental Illnesses.