John F. Nash, Jr. – Autobiography
My beginning as a legally recognized individual occurred
on June 13, 1928 in Bluefield, West Virginia, in the Bluefield
Sanitarium, a hospital that no longer exists. Of course I can't
consciously remember anything from the first two or three years of
my life after birth. (And, also, one suspects, psychologically, that
the earliest memories have become "memories of memories" and are
comparable to traditional folk tales passed on by tellers and
listeners from generation to generation.) But facts are available
when direct memory fails for many circumstances.
for whom I was named, was an electrical engineer and had come to
Bluefield to work for the electrical utility company there which was
and is the Appalachian Electric Power Company. He was a veteran of
WW1 and had served in France as a lieutenant in the supply services
and consequently had not been in actual front lines combat in the
war. He was originally from Texas and had obtained his B.S. degree
in electrical engineering from Texas Agricultural and Mechanical (Texas A. and M.).
My mother, originally Margaret Virginia Martin, but called
Virginia, was herself also born in Bluefield. She had studied at
West Virginia University and was a school teacher before her
marriage, teaching English and sometimes Latin. But my mother's
later life was considerably affected by a partial loss of hearing
resulting from a scarlet fever infection that came at the time when
she was a student at WVU.
Her parents had come as
a couple to Bluefield from their original homes in western North
Carolina. Her father, Dr. James Everett Martin, had prepared as a
physician at the University of Maryland in Baltimore and came to
Bluefield, which was then expanding rapidly in population, to start
up his practice. But in his later years Dr. Martin became more of a
real estate investor and left actual medical practice. I never saw
my grandfather because he had died before I was born but I have good
memories of my grandmother and of how she could play the piano at
the old house which was located rather centrally in Bluefield.
A sister, Martha, was born about two and a half years later
than me on November 16, 1930.
I went to the standard schools
in Bluefield but also to a kindergarten before starting in the
elementary school level. And my parents provided an encyclopedia,
Compton's Pictured Encyclopedia, that I learned a lot from by
reading it as a child. And also there were other books available
from either our house or the house of the grandparents that were of
Bluefield, a small city in a
comparatively remote geographical location in the Appalachians, was
not a community of scholars or of high technology. It was a center
of businessmen, lawyers, etc. that owed its existence to the
railroad and the rich nearby coal fields of West Virginia and
western Virginia. So, from the intellectual viewpoint, it offered
the sort of challenge that one had to learn from the world's
knowledge rather than from the knowledge of the immediate community.
By the time I was a student in high school I was reading the
classic "Men of Mathematics" by E.T. Bell and I remember succeeding
in proving the classic Fermat theorem about an integer multiplied by
itself p times where p is a prime.
I also did electrical and
chemistry experiments at that time. At first, when asked in school
to prepare an essay about my career, I prepared one about a career
as an electrical engineer like my father. Later, when I actually
entered Carnegie Tech. in Pittsburgh I entered as a student with the
major of chemical engineering.
Regarding the circumstances
of my studies at Carnegie (now Carnegie Mellon U.), I was lucky to
be there on a full scholarship, called the George Westinghouse
Scholarship. But after one semester as a chem. eng. student I
reacted negatively to the regimentation of courses such as
mechanical drawing and shifted to chemistry instead. But again,
after continuing in chemistry for a while I encountered difficulties
with quantitative analysis where it was not a matter of how well one
could think and understand or learn facts but of how well one could
handle a pipette and perform a titration in the laboratory. Also the
mathematics faculty were encouraging me to shift into mathematics as
my major and explaining to me that it was not almost impossible to
make a good career in America as a mathematician. So I shifted again
and became officially a student of mathematics. And in the end I had
learned and progressed so much in mathematics that they gave me an
M. S. in addition to my B. S. when I graduated.
mention that during my last year in the Bluefield schools that my
parents had arranged for me to take supplementary math. courses at
Bluefield College, which was then a 2-year institution operated by
Southern Baptists. I didn't get official advanced standing at
Carnegie because of my extra studies but I had advanced knowledge
and ability and didn't need to learn much from the first math.
courses at Carnegie.
When I graduated I remember that I had
been offered fellowships to enter as a graduate student at either Harvard or Princeton. But
the Princeton fellowship was somewhat more generous since I had not
actually won the Putnam competition and also Princeton seemed more
interested in getting me to come there. Prof. A.W. Tucker wrote a
letter to me encouraging me to come to Princeton and from the family
point of view it seemed attractive that geographically Princeton was
much nearer to Bluefield. Thus Princeton became the choice for my
graduate study location.
But while I was still at Carnegie I
took one elective course in "International Economics" and as a
result of that exposure to economic ideas and problems, arrived at
the idea that led to the paper "The Bargaining Problem" which was
later published in Econometrical. And it was this idea which in
turn, when I was a graduate student at Princeton, led to my interest
in the game theory studies there which had been stimulated by the
work of von Neumann and Morgenstern.
As a graduate student I
studied mathematics fairly broadly and I was fortunate enough,
besides developing the idea which led to "Non-Cooperative Games",
also to make a nice discovery relating to manifolds and real
algebraic varieties. So I was prepared actually for the possibility
that the game theory work would not be regarded as acceptable as a
thesis in the mathematics department and then that I could realize
the objective of a Ph.D. thesis with the other results.
in the event the game theory ideas, which deviated somewhat from the
"line" (as if of "political party lines") of von Neumann and
Morgenstern's book, were accepted as a thesis for a mathematics
Ph.D. and it was later, while I was an instructor at M.I.T., that I wrote up
Real Algebraic Manifolds and sent it in for publication.
I went to M.I.T. in the summer of 1951 as a "C.L.E. Moore
Instructor". I had been an instructor at Princeton for one year
after obtaining my degree in 1950. It seemed desirable more for
personal and social reasons than academic ones to accept the
higher-paying instructorship at M.I.T.
I was on the
mathematics faculty at M.I.T. from 1951 through until I resigned in
the spring of 1959. During academic 1956 - 1957 I had an Alfred P.
Sloan grant and chose to spend the year as a (temporary) member of
the Institute for Advanced Study in Princeton.
period of time I managed to solve a classical unsolved problem
relating to differential geometry which was also of some interest in
relation to the geometric questions arising in general relativity.
This was the problem to prove the isometric embeddability of
abstract Riemannian manifolds in flat (or "Euclidean") spaces. But
this problem, although classical, was not much talked about as an
outstanding problem. It was not like, for example, the 4-color
So as it happened, as soon as I heard in
conversation at M.I.T. about the question of the embeddability being
open I began to study it. The first break led to a curious result
about the embeddability being realizable in surprisingly
low-dimensional ambient spaces provided that one would accept that
the embedding would have only limited smoothness. And later, with
"heavy analysis", the problem was solved in terms of embeddings with
a more proper degree of smoothness.
While I was on my "Sloan
sabbatical" at the IAS in Princeton I studied another problem
involving partial differential equations which I had learned of as a
problem that was unsolved beyond the case of 2 dimensions. Here,
although I did succeed in solving the problem, I ran into some bad
luck since, without my being sufficiently informed on what other
people were doing in the area, it happened that I was working in
parallel with Ennio de Giorgi of Pisa, Italy. And de Giorgi was
first actually to achieve the ascent of the summit (of the
figuratively described problem) at least for the particularly
interesting case of "elliptic equations".
conceivable that if either de Giorgi or Nash had failed in the
attack on this problem (of a priori estimates of Holder continuity)
then that the lone climber reaching the peak would have been
recognized with mathematics' Fields medal (which has traditionally
been restricted to persons less than 40 years old).
must arrive at the time of my change from scientific rationality of
thinking into the delusional thinking characteristic of persons who
are psychiatrically diagnosed as "schizophrenic" or "paranoid
schizophrenic". But I will not really attempt to describe this long
period of time but rather avoid embarrassment by simply omitting to
give the details of truly personal type.
While I was on the
academic sabbatical of 1956-1957 I also entered into marriage.
Alicia had graduated as a physics major from M.I.T. where we had met
and she had a job in the New York City area in 1956-1957. She had
been born in El Salvador but came at an early age to the U.S. and
she and her parents had long been U.S. citizens, her father being an
M. D. and ultimately employed at a hospital operated by the federal
government in Maryland.
The mental disturbances originated
in the early months of 1959 at a time when Alicia happened to be
pregnant. And as a consequence I resigned my position as a faculty
member at M.I.T. and, ultimately, after spending 50 days under
"observation" at the McLean Hospital, travelled to Europe and
attempted to gain status there as a refugee.
I later 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 Probleme de Cauchy pour les E'quations
Differentielles d'un Fluide Generale"; 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".
But after my
return to the dream-like delusional hypotheses in the later 60's I
became a person of delusionally influenced thinking but of
relatively moderate behavior and thus tended to avoid
hospitalization and the direct attention of psychiatrists.
Thus further time passed. Then gradually I began to
intellectually reject some of the delusionally influenced lines of
thinking which had been characteristic of my orientation. This
began, most recognizably, with the rejection of politically-oriented
thinking as essentially a hopeless waste of intellectual effort.
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. For example, a non-Zoroastrian could
think of Zarathustra as simply a madman who led millions of naive
followers to adopt a cult of ritual fire worship. But without his
"madness" Zarathustra would necessarily have been only another of
the millions or billions of human individuals who have lived and
then been forgotten.
Statistically, it would seem improbable
that any mathematician or scientist, at the age of 66, would be able
through continued research efforts, to add much to his or her
previous achievements. However I am still making the effort and it
is conceivable that with the gap period of about 25 years of
partially deluded thinking providing a sort of vacation my situation
may be atypical. Thus I have hopes of being able to achieve
something of value through my current studies or with any new ideas
that come in the future.
Les Prix Nobel 1994.