John Forbes Nash: Three Hundred Words, Thirty Lost Years
In 1948 a professor at the Carnegie Institute of Technology named Richard Duffin was asked to write a letter of recommendation for a twenty-year-old applying to Princeton. He wrote one sentence: "this man is a genius." Two years later the young man produced a paper of 317 words that would reorganise economics, political science, evolutionary biology and the study of nuclear deterrence. Then, for three decades, almost nothing.
Bluefield to Carnegie Tech
Nash was born on 13 June 1928 in Bluefield, West Virginia. He went to the Carnegie Institute of Technology — now Carnegie Mellon — intending to study chemical engineering, and switched to mathematics, the discipline that had been quietly claiming him. Harvard was interested. He chose Princeton, arriving with Duffin's single line of testimony in his file.
Three Hundred and Seventeen Words
The idea arrived almost immediately. In 1950, still a graduate student, Nash published *Equilibrium points in N-person games* — 317 words, one of the shortest consequential papers in the history of the social sciences — and expanded the argument in a further paper in 1951. His doctorate followed from Princeton in 1950.
What he had done was draw a line that nobody had drawn cleanly before, between cooperative games, in which players can form binding agreements, and non-cooperative games, in which they cannot. For the second and far more common case he defined a solution: a Nash equilibrium occurs when nobody can increase their reward by changing strategy, given what everyone else is doing. He then proved such a point always exists.
The power of this is that it does not require anyone to be nice, or coordinated, or even communicative. It describes what stable outcomes look like among rational agents pursuing their own interests in conflict — which is to say, most of human affairs. Economists spent the following forty years rebuilding their discipline around it.
RAND and the Cold War
Nash went to work at the RAND Corporation, where game theory was being applied directly to Cold War strategic policy. The arms race is the canonical non-cooperative game: two players, no binding agreements, an equilibrium that is stable and terrible for both. That his abstraction became the grammar of nuclear strategy is one of the more sobering facts about twentieth-century mathematics.
The Mathematician's Mathematician
Then he did something unusual: he walked away from the field he had founded and did harder work elsewhere. At MIT, his colleague Warren Ambrose challenged him with the embedding problem for manifolds. The Nash embedding theorems of 1954 and 1956 established that every Riemannian manifold can be embedded into some Euclidean space — that any abstractly defined curved geometry can be realised as a surface sitting inside ordinary flat space. John Conway called this "one of the most important pieces of mathematical analysis in this century."
Out of the same period came the Nash–Moser inverse function theorem and, following a suggestion from Louis Nirenberg, the Nash–De Giorgi theorem on parabolic and elliptic partial differential equations. Nash was working, essentially alone, at the highest level of geometric analysis. Princeton's David Gabai has said the embedding theorems "required not only unusual insight but also tremendous technical expertise" and called them "absolutely incredible results."
1959
He was first hospitalised in 1959 and diagnosed with paranoid schizophrenia. He resigned from MIT the same year. He and Alicia Lardé divorced in 1962. What followed were decades in which he was a presence rather than a participant at Princeton — the phantom of Fine Hall, a figure students would see writing on empty blackboards. He began to recover gradually during the 1980s. He and Alicia remarried in 2001.
The Return
In 1994 Nash was awarded a one-third share of the Sveriges Riksbank Prize in Economic Sciences, cited with his co-laureates "for their pioneering analysis of equilibria in the theory of non-cooperative games." He was listed under Princeton, where in 1995 he joined the mathematics department as a senior research mathematician, and where colleagues remembered him as approachable and quietly inspiring to younger mathematicians. Earlier honours had included the John von Neumann Theory Prize in 1978; the Double Helix Medal followed in 2010.
The Abel Prize, and Five Days
In 2015 the Norwegian Academy awarded the Abel Prize — mathematics' most prestigious honour — jointly to Nash and Nirenberg for "striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis." Princeton's Sergiu Klainerman said the award "redressed a historical anomaly" by finally recognising work far deeper than the game theory for which the public knew him. Nash, then eighty-six, was characteristically plain: "The Abel Prize is top-level among mathematics prizes. There's really nothing better."
He remains the only person to have received both the Economic Sciences Prize and the Abel Prize. On 23 May 2015, returning from the ceremony in Norway, Nash and Alicia were killed in a taxi crash in New Jersey.
Why John Is Called a Genius
The word is not a later embellishment; it is in the documentary record from 1948, when Duffin used it and nothing else. What justifies it is range combined with directness. Nash produced foundational results in three fields — game theory, differential geometry, nonlinear partial differential equations — that share almost no technique, and in each case he attacked the problem frontally with machinery he had to build himself. Peter Sarnak put the character of it precisely: "While his publication list is shorter than many mathematicians, the novelty and impact of each of his papers is unique." He did not accumulate. He solved.
The counter-case has to be stated carefully. His research life was short — roughly the decade from 1948 to 1958 — and the illness that began in 1959 did not enrich it but destroyed it. Nothing in the record supports the romantic conflation of madness with insight; the schizophrenia cost him thirty years, his marriage and his position, and the mathematics stopped. His public fame also rests on the least mathematically demanding thing he did: the equilibrium papers are brilliant but brief, and the deeper work in analysis went essentially unrewarded until he was eighty-six, which is what Klainerman meant by a historical anomaly. And the Nobel he received is not a mathematics prize at all. It is possible to admire Nash enormously while noticing that the popular version of him — the tormented seer whose visions were also his gift — inverts what actually happened to him.
Legacy
Nash equilibrium is now infrastructure. It underlies auction design, antitrust analysis, evolutionary biology, arms-control theory and the algorithms that allocate spectrum and match medical residents to hospitals. Among mathematicians the embedding theorems and the PDE work are held in comparable regard. He got both halves of a career the world usually grants to different people, and he got them before he was thirty.
Achievements
- Prize in Economic Sciences in Memory of Alfred Nobel — 1994
- John von Neumann Theory Prize — 1978
- Abel Prize — 2015
- Steele Prize for Seminal Contribution to Research — 1999
- Notable work: Nash equilibrium
- Notable work: Nash blowing-up
- Notable work: Nash embedding theorem
- Notable work: Nash–Moser theorem
- Held posts at Princeton University, Massachusetts Institute of Technology and RAND Corporation
- Fields: game theory, differential geometry and mathematics
