http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Nash.html
John Forbes Nash
Born: 13 June 1928 in Bluefield, West Virginia, USA
John F Nash's father, also called John Forbes Nash so we shall refer to him
as John Nash Senior, was a native of Texas. John Nash Senior was born in 18
92 and had an unhappy childhood from which he escaped when he studied electr
ical engineering at Texas Agricultural and Mechanical. After military servic
e in France during World War I, John Nash Senior lectured on electrical engi
neering for a year at the University of Texas before joining the Appalacian
Power Company in Bluefield, West Virginia. John F Nash's mother, Margaret Vi
rginia Martin, was known as Virginia. She had a university education, studyi
ng languages at the Martha Washington College and then at West Virginia Univ
ersity. She was a school teacher for ten years before meeting John Nash Seni
or, and the two were married on 6 September 1924.
Johnny Nash, as he was called by his family, was born in Bluefield Sanatoriu
m and baptised into the Episcopal Church. He was [2]:-
... a singular little boy, solitary and introverted ...
but he was brought up in a loving family surrounded by close relations who s
howed him much affection. After a couple of years Johnny had a sister when M
artha was born. He seems to have shown a lot of interest in books when he wa
s young but little interest in playing with other children. It was not becau
se of lack of children that Johnny was behaved in this way for Martha and her cousins played the usual childhood ga
mes cutting patterns out of books, playing hide-and-seek in the attic, playi
ng football. However while the others played together Johnny played by himse
lf with toy airplanes and matchbox cars.
His mother responded by enthusiastically encouraging Johnny's education, bot
h by seeing that he got good schooling and also by teaching him herself. Joh
nny's father responded by treating him like an adult, giving him science boo
ks when other parents might give their children colouring books.
Johnny's teachers at school certainly did not recognise his genius, and it w
ould appear that he gave them little reason to realise that he had extraordi
nary talents. They were more conscious of his lack of social skills and, bec
ause 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 th
e time he was about twelve years old he was showing great interest in carryi
ng out scientific experiments in his room at home. It is fairly clear that h
e learnt more at home than he did at school.
Martha seems to have been a remarkably normal child while Johnny seemed diff
erent from other children. She wrote later in life (see [2]):-
Johnny was always different. [My parents] knew he was different. And they kn
ew he was bright. He always wanted to do thinks his way. Mother insisted I d
o things for him, that I include him in my friendships. ... but I wasn't too
keen on showing off my somewhat odd brother.
His parents encouraged him to take part in social activities and he did not
refuse, but sports, dances, visits to relatives and similar events he treate
d as tedious distractions from his books and experiments.
Nash first showed an interest in mathematics when he was about 14 years old.
Quite how he came to read E T Bell's Men of mathematics is unclear but cert
ainly this book inspired him. He tried, and succeeded, in proving for himsel
f results due to Fermat which Bell stated in his book. The excitement that N
ash 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 a
s well as science courses, in particular studying chemistry which was a favo
urite topic. He began to show abilities in mathematics, particularly in prob
lem solving, but still with hardly any friends and behaving in a somewhat ec
centric manner, this only added to his fellow pupils view of him as peculiar
. He did not considered a career in mathematics at this time, however, which
is not surprising since it was an unusual profession. Rather he assumed tha
t he would study electrical engineering and follow his father but he continu
ed to conduct his own chemistry experiments and was involved in making explo
sives which led to the death of one of his fellow pupils. [2]:-
Boredom and simmering adolescent aggression led him to play pranks, occasion
ally ones with a nasty edge.
He caricatured classmates he disliked with weird cartoons, enjoyed torturing
animals, and once tried to get his sister to sit in a chair he had wired up
with batteries.
Nash won a scholarship in the George Westinghouse Competition and was accept
ed by the Carnegie Institute of Technology (now Carnegie-Mellon University)
which he entered in June 1945 with the intention of taking a degree in chemi
cal engineering. Soon, however, his growing interest in mathematics had him
take courses on tensor calculus and relativity. There he came in contact wit
h John Synge who had recently been appointed as Head of the Mathematics Depa
rtment and taught the relativity course. Synge and the other mathematics pro
fessors quickly recognised Nash's remarkable mathematical talents and persua
ded him to become a mathematics specialist. They realised that he had the ta
lent to become a professional mathematician and strongly encouraged him.
Nash quickly aspired to great things in mathematics. He took the William Low
ell Putnam Mathematics Competition twice but, although he did well, he did n
ot make the top five. It was a failure in Nash's eyes and one which he took
badly. The Putnam Mathematics Competition was not the only thing going badly
for Nash. Although his mathematics professors heaped praise on him, his fel
low students found him a very strange person. Physically he was strong and t
his saved him from being bullied, but his fellow students took delight in ma
king fun of Nash who they saw as an awkward immature person displaying child
ish 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 a melting ice cre
am cone melting on top of his castoff clothing, walking on his roommate's sl
eeping body to turn off the light.
Another wrote:-
He was extremely lonely.
And a third fellow student wrote:-
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 r
eacted by making fun of the fact that he was attracted to boys and humiliate
d him. They played cruel pranks on him and he reacted by asking his fellow s
tudents to challenge him with mathematics problems. He ended up doing the ho
mework of many of the students.
Nash received a BA and an MA in mathematics in 1948. By this time he had bee
n accepted into the mathematics programme at Harvard, Princeton, Chicago and
Michigan. Now he felt that Harvard was the leading university and so he wan
ted to go there, but on the other hand their offer to him was less generous
than that of Princeton. Nash felt that Princeton were keen that he went ther
e while he felt that his lack of success in the Putnam Mathematics Competiti
on meant that Harvard were less enthusiastic. He took a while to make his de
cision, while he was encouraged by Synge and his other professors to accept
Princeton. When Lefschetz offered him the most prestigious Fellowship that P
rinceton had, Nash made his decision to study there.
In September 1948 Nash entered Princeton where he showed an interest in a br
oad range of pure mathematics: topology, algebraic geometry, game theory and
logic were among his interests but he seems to have avoided attending lectu
res. Usually those who decide not to learn through lectures turn to books bu
t this appears not to be so for Nash who decided not to learn mathematics "s
econd-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 wa
y.
In 1949, while studying for his doctorate, he wrote a paper which 45 years l
ater was to win a Nobel prize for economics. During this period Nash establi
shed the mathematical principles of game theory. 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' ele
ction 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 t
he things that we predict about people.
Milnor, who was a fellow student, describes Nash during his years at Princet
on in [6]:-
He was always full of mathematical ideas, not only on game theory, but in ge
ometry and topology as well. However, my most vivid memory of this time is o
f the many games which were played in the common room. I was introduced to G
o and Kriegspiel, and also to an ingenious topological game which we called
Nash in honor of the inventor.
In fact the game "Nash" was almost identical to Hex which had been invented
independently by Piet Hein in Denmark.
Here are three comments from fellow students:-
Nash was out of the ordinary. If he was in a room with twenty people, and th
ey were talking, if you asked an observer who struck you as odd it would hav
e been Nash. It wasn't anything he consciously did. It was his bearing. His
aloofness.
Nash was totally spooky. He wouldn't look at you. he'd take a lot of time an
swering a question. If he thought the question was foolish he wouldn't answe
r at all. He had no effect. It was a mixture of pride and something else. He
was so isolated but there really was underneath it all a warmth and appreci
ation of people.
A lot of us would discount what Nash said. ... I wouldn't want to listen. Yo
u didn't feel comfortable with the person.
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 reg
arding gravity. After explaining complicated mathematics to Einstein for abo
ut an hour, Einstein advised him to go and learn more physics. Apparently a
physicist did publish a similar idea some years later.
In 1950 Nash received his doctorate from Princeton with a thesis entitled No
n-cooperative Games. In the summer of that year he worked for the RAND Corpo
ration where his work on game theory made him a leading expert on the Cold W
ar conflict which dominated RAND's work. He worked there from time to time o
ver the next few years as the Corporation tried to apply game theory to mili
tary and diplomatic strategy. Back at Princeton in the autumn of 1950 he beg
an to work seriously on pure mathematical problems. It might seem that someo
ne who had just introduced ideas which would, one day, be considered worthy
of a Nobel Prize would have no problems finding an academic post. However, N
ash's work was not seen at the time to be of outstanding importance and he s
aw that he needed to make his mark in other ways. We should also note that i
t was not really a move towards pure mathematics for he had always considere
d himself a pure mathematician. He had already obtained results on manifolds
and algebraic varieties before writing his thesis on game theory. His famou
s theorem, that any compact real manifold is diffeomorphic to a component of
a real-algebraic variety, was thought of by Nash as a possible result to fa
ll back on if his work on game theory was not considered suitable for a doct
oral thesis. He said in a recent interview:-
I developed a very good idea in pure mathematics. I got what became Real Alg
ebraic Manifolds. I could have published that earlier, but it wasn't rushed
to publication. I took some time in writing it up. Somebody suggested that I
was a prodigy. Another time it was suggested that I should be called "bug b
rains", because I had ideas, but they were sort of buggy or not perfectly so
und. So that might have been an anticipation of mental problems. I mean, tak
ing it at face value.
In 1952 Nash published Real Algebraic Manifolds in the Annals of Mathematics
. The most important result in this paper is that two real algebraic manifol
ds are equivalent if and only if they are analytically homeomorphic. Althoug
h publication of this paper on manifolds established him as a leading mathem
atician, not everyone at Princeton was prepared to see him join the Faculty
there. This was nothing to do with his mathematical ability which everyone a
ccepted as outstanding, but rather some mathematicians such as Artin felt th
at they could not have Nash as a colleague due to his aggressive personality
.
Halmos received the following letter in early 1953 from Warren Ambrose relat
ing to Nash (see for example [2]):-
There's no significant news from here, as always. Martin is appointing John
Nash to an Assistant Professorship (not the Nash at Illinois, the one out of
Princeton by Steenrod) and I'm pretty annoyed at that. Nash is a childish b
right guy who wants to be "basically original," which I suppose is fine for
those who have some basic originality in them. He also makes a damned fool o
f himself in various ways contrary to this philosophy. He recently heard of
the unsolved problem about imbedding a Riemannian manifold isometrically in
Euclidean space, felt that this was his sort of thing, provided the problem
were sufficiently worthwhile to justify his efforts; so he proceeded to writ
e to everyone in the math society to cheek on that, was told that it probabl
y was, and proceeded to announce that he had solved it, module details, and
told Mackey he would like to talk about it at the Harvard colloquium. Meanwh
ile he went to Levinson to inquire about a differential equation that interv
ened and Levinson says it is a system of partial differential equations and
if he could only [get] to the essentially simpler analog of a single ordinar
y differential equation it would be a damned good paper - and Nash had only
the vaguest notions about the whole thing. So it is generally conceded he is
getting nowhere and making an even bigger ass of himself than he has been p
reviously supposed by those with less insight than myself. But we've got him
and saved ourselves the possibility of having gotten a real mathematician.
He's a bright guy but conceited as Hell, childish as Wiener, hasty as X, obs
treperous as Y, for arbitrary X and Y.
Ambrose, the author of this letter, and Nash had rubbed each other the wrong
way for a while. They had played silly pranks on each other and Ambrose see
ms not to have been able to ignore Nash's digs in the way others had learned
to do. It had been Ambrose who had said to Nash:-
If you're so good, why don't you solve the embedding theorem for manifolds.
From 1952 Nash had taught at the Massachusetts Institute of Technology but h
is teaching was unusual (and unpopular with students) and his examining meth
ods were highly unorthodox. His research on the theory of real algebraic var
ieties, Riemannian geometry, parabolic and elliptic equations was, however,
extremely deep and significant in the development of all these topics. His p
aper C1 isometric imbeddings was published in 1954 and Chern, in a review, n
oted that it:-
... contains some surprising results on the C1-isometric imbedding into an E
uclidean space of a Riemannian manifold with a positive definite C0-metric.
Nash continued to develop this work in the paper The imbedding problem for R
iemannian manifolds published in 1956. This paper contains his famous deep i
mplicit function theorem. After this Nash worked on ideas that would appear
in his paper Continuity of solutions of parabolic and elliptic equations whi
ch was published in the American Journal of Mathematics in 1958. Nash, howev
er, was very disappointed when he discovered that E De Giorgi has proved sim
ilar 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 with his work on parab
olic and elliptic equations was still unpublished when the Committee made th
eir decisions he did not make it. One imagines that the Committee would have
expected him to be a leading contender, perhaps even a virtual certainty, f
or a 1962 Fields' Medal but mental illness destroyed his career long before
those decisions were made.
During his time at MIT Nash began to have personal problems with his life wh
ich were in addition to the social difficulties he had always suffered. Coll
eagues said:-
Nash was always forming intense friendships with men that had a romantic qua
lity. He was very adolescent, always with the boys. He was very experimental
- mostly he just kissed.
He met Eleanor Stier and they had a son, John David Stier, who was born on 1
9 June 1953. Eleanor was a shy girl, lacking confidence, a little afraid of
men, didn't want to be involved. She found in Nash someone who was even less
experienced than she was and found that attractive. [2]:-
Nash was looking for emotional partners who were more interested in giving t
han receiving, and Eleanor, was very much that sort.
Nash did not want to marry Eleanor although she tried hard to persuade him.
In the summer of 1954, while working for RAND, Nash was arrested in a police
operation to trap homosexuals. He was dismissed from RAND.
One of Nash's students at MIT, Alicia Larde, became friendly with him and by
the summer of 1955 they were seeing each other regularly. He also had a spe
cial friendship with a male graduate student at this time Jack Bricker. Elea
nor found out about Alicia in the spring of 1956 when she came to Nash's hou
se and found him in bed with Alicia. Nash said to a friend:-
My perfect little world is ruined, my perfect little world is ruined.
Alicia didn't seem too upset at discovering that Nash had a child with Elean
or and deduced that since the affair had been going on for three years, Nash
was probably not serious about her. In 1956 Nash's parents found out about
his continuing affair with Eleanor and about his son John David Stier. The s
hock may have contributed to the death of Nash's father soon after but even
if it did not Nash may have blamed himself. In February of 1957 Nash married
Alicia; by the autumn of 1958 she was pregnant but, a couple of months late
r near the end of 1958, Nash's mental state became very disturbed.
At a New Year's Party Nash appeared at midnight dressed only with a nappy an
d a sash with "1959" written on it. He spent most of the evening curled up,
like the baby he was dressed as, on his wife's lap. Some described his behav
iour as stranger than usual. On 4 January he was back at the university and
started to teach his game theory course. His opening comments to the class w
ere:-
The question occurs to me. Why are you here?
One student immediately dropped the course! Nash asked a graduate student to
take over his course and vanished for a couple of weeks. When he returned h
e walked into the common room with a copy of the New York Times saying that
it contained encrypted messages from outer space that were meant only for hi
m. For a few days people thought he was playing an elaborate private joke.
Norbert Wiener was one of the first to recognize that Nash's extreme eccentr
icities and personality problems were actually symptoms of a medical disorde
r. After months of bizarre behaviour, Alicia had her husband involuntarily h
ospitalised at McLean Hospital, a private psychiatric hospital outside of Bo
ston. Upon his release, Nash abruptly resigned from M.I.T., withdrew his pen
sion, and went to Europe, where he intended to renounce his U.S. citizenship
. Alicia left her newborn son with her mother, and followed the ill Nash. Sh
e then had Nash deported - back to the United States.
After their return, the two settled in Princeton where Alicia took a job. Na
sh's illness continued, transforming him into a frightening figure. He spent
most of his time hanging around on the Princeton campus, talking about hims
elf 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 J
ersey where he endured insulin-coma therapy, an aggressive and risky treatme
nt, five days a week for a month and a half. A long sad episode followed whi
ch included periods of hospital treatment, temporary recovery, then further
treatment. Alicia divorced Nash in 1962. Nash spent a while with Eleanor and
John David. In 1970 Alicia tried to help him taking him in as a boarder, bu
t he appeared to be lost to the world, removed from ordinary society, althou
gh he spent much of his time in the Mathematics Department at Princeton. The
book [2] is highly recommended for its moving account of Nash's mental suff
erings.
Slowly over many years Nash recovered. He delivered a paper at the tenth Wor
ld Congress of Psychiatry in 1996 describing his illness; it is reported in
[3]. 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, a
nd 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 some
thing like telephone calls in my head, from people opposed to my ideas. ...T
he delirium was like a dream from which I seemed never to awake.
Despite spending periods in hospital because of his mental condition, his ma
thematical 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 man
iacal characteristics, delirium and symptoms of schizophrenia.
In the 1990s Nash made a recovery from the schizophrenia from which he had s
uffered since 1959. His ability to produce mathematics of the highest qualit
y did not totally leave him. He said:-
I would not treat myself as recovered if I could not produce good things in
my work.
Nash was awarded (jointly with Harsanyi and Selten) the 1994 Nobel Prize in
Economic Science for his work on game theory. In 1999 he was awarded the Ler
oy P Steele Prize by the American Mathematical Society:-
... for a seminal contribution to research.
Article by: J J O'Connor and E F Robertson