A Beautiful Mind

[This review originally appeared in the May 2000 issue of the MAA’s College Mathematics Journal (Vol. 31, No. 3, 240–244).]

The beautiful mind of the title is that of John Nash, the only mathematician ever to win a Nobel Prize (in Economics, 1994). Sylvia Nasar, an economics correspondent for the New York Times, has written a very readable, well-researched biography of Nash that gives fascinating glimpses of mathematicians’ lives and the mathematical community at Princeton and MIT in the 1950s and thereafter. The following partial list of academics who play a role in this biography is impressive: E. Artin, K. Arrow, A. Borel, F. Browder, E. Calabi, L. Carleson, P. Cohen, A. Einstein, D. Gale, A. Grothendieck, L. Hormander, H. Kuhn, S. Lefschetz, N. Levinson, J. Milnor, O. Morgenstern, J. Moser, D.J. Newman, L. Nirenberg, R. Palais, G.-C. Rota, P. Samuelson, A. Selberg, L. Shapley, I.M. Singer, D. Spencer, N. Steenrod, J. von Neumann, and N. Wiener. Most of those who are still living were interviewed, often extensively, for the book.

Before proceeding I should point out that if you’re interested only in Nash’s mathematics you should skip the Nasar book and just read “A Nobel Prize for John Nash” by John Milnor (The Mathematical Intelligencer 17:3 (1995) 11–17). Milnor, a Fields Medalist and former colleague and roommate of Nash, gives an expository summary of some of Nash’s most important work in noncooperative game theory (for which Nash won the Nobel), geometry (Nash’s embedding theorems), and analysis. The game theory section of Milnor’s article includes an elegant proof of an existence theorem for Nash equilibria which uses the Brouwer Fixed Point Theorem. It also contains a “Commentary” discussing the significance of Nash’s equilibrium-point theory for understanding competitive situations in the real world. The penultimate chapter of the Nasar book describes an interesting application of Nash equilibria (and other ideas like backward induction) in recent FCC auctions of airwave rights for use by wireless devices like cell phones, pagers, and fax machines.

For me the most fascinating part of Nash’s story isn’t his mathematics or even his winning a Nobel Prize. Rather, it’s his descent, in the late 1950s, into decades of paranoid schizophrenia, followed by an extremely rare “spontaneous recovery” in the late 1980s. As with alcoholics the term recovery here may not be appropriate; Nasar points out that no one knows if Nash’s remission is permanent. In fact, Swedish concern about Nash’s possible behavior in Nobel Prize ceremonies even played a role in the close votes by both the Nobel Prize committee and the full body of the Royal Swedish Academy of Sciences.

The chapters describing “The Prize” and the political intrigues leading to the votes are absorbing. They describe hostility towards mathematics by economists as well as hostility towards economics by natural scientists, particularly physicists, and mathematicians. Lars Garding, a mathematician in the Academy who considered Nash to be a genius, even said that Nash’s Nobel Prize was for “a very small thing”. The statutes of the Nobel Prize require that details of the highly secret selection process not be divulged for 50 years after an award, so the author here relies mostly on confidential sources. In the rest of the book the author depends heavily on interviews and e-mails with Princeton and MIT faculty as well as with Nash’s family and friends. As John Milnor pointed out in his review of the book, “John Nash and ‘A Beautiful Mind’” (Notices of the American Mathematical Society 45:10 (November 1998) 1329–1332), it was written without Nash’s consent or cooperation. Milnor does note, however, that the book is “carefully documented”.

In the remainder of this review I’ll discuss the three parts of the story of Nash’s life that the author calls genius, madness, and awakening. I’ll then give a nuts and bolts description of the book itself, and conclude with a bottom line as to whether or not the book is worth reading.

Nash’s upbringing was traditional, although by his undergraduate years at Carnegie Institute of Technology he was already a socially underdeveloped, eccentric loner who dazzled his mathematics professors. At Carnegie, Nash attended the only economics course he ever took, one on international trade (not bad for a Nobel Prize winner in economics).

In 1948 Lefschetz recruited Nash to graduate school at Princeton, which had become, as Harald Bohr called it, “the mathematical center of the universe.” In addition to the outstanding mathematics faculty at the university, Einstein, von Neumann (whom the graduate students were all afraid of), Gödel and others were a mile away at the Institute for Advanced Study. With brilliant graduate students like John Milnor, John Tate, Serge Lang, Harold Kuhn, David Gale and others the atmosphere was very competitive, an “exotic little mathematical hothouse.” Even the mathematical cliques centered around faculty were ranked: topology (Lefschetz, Fox, Steenrod) at the top, next analysis (Bochner), then algebra (Artin), while logic (Church) and game theory (Tucker) were not as highly regarded.

Intellectually, Nash thrived at Princeton, producing a 27-page thesis in game theory in two years. He avoided attending classes and reading mathematics, but picked others’ brains or discovered things from scratch. At the time Norman Steenrod said, “More than any other student I have ever known, Nash believes in learning a subject by doing research in it.” Nash was cocky and arrogant but he impressed faculty by his mathematical audacity, for example with his theorem that any smooth compact \(n\)-dimensional manifold can be embedded as a component of a real algebraic variety in \(\mathbb{R}^{2n+1}\). Nash’s self-confidence even led him to make an appointment with Einstein in his first term to discuss an idea about gravity, friction, and radiation. Einstein ended the discussion by remarking, “You had better study some more physics, young man.”

Nash hoped to stay on at Princeton as a faculty member, but his difficult personality antagonized Artin and others. In 1952 he accepted an instructorship at MIT, where he remained until his first involuntary commitment in 1959. For the first few years his behavior there was not too eccentric (by MIT standards), but the department chair did upbraid him for putting the equivalent of Fermat’s Last Theorem on a final exam. Also, Nash lost a summer job at RAND in 1954 when he was arrested for indecent behavior in a men’s room. At MIT, Nash’s superiority and arrogance in putting down other mathematicians earned him the nickname Gnash. Nash responded, “G obviously stands for genius. In fact, there are few geniuses these days here at MIT. Me, of course, and also Norbert Wiener. Even Norbert may no longer be a genius, but there is evidence that he once was.”

Nash’s time at MIT included a secret life with a mistress and an illegitimate son. He eventually married a physics major, Alicia, who he once said would make a fine wife because she watched so much television that she wouldn’t require much attention from him. Ironically Alicia turned out to be the heroine of the story, taking care of him in later years during his episodic illness, even after their divorce.

By early 1959, Nash was behavior was sufficiently irrational and paranoiac that he was relieved of teaching for the spring semester, even as the department was recommending him for tenure. One of his more bizarre acts was to turn down a prestigious appointment at the University of Chicago because, he wrote chair A. A. Albert, he was scheduled to become Emperor of Antarctica.

MIT faculty and students were torn as to whether Nash was actually insane or just very eccentric, illustrating that there is a fine line between genius and madness. After Alicia finally had Nash committed to a local mental hospital that spring, Harvard professor George Mackey visited Nash there. Mackey asked Nash how he, as a mathematician devoted to reason and logic, could believe that aliens from outer space were recruiting him to save the world. Nash replied that his ideas about supernatural beings came to him the same way that his mathematical ideas did, so he took them seriously.

Nash’s nervous breakdown in 1959 was the beginning of several decades of illness alternating with periods of normality. In his autobiographical essay, written after winning the Nobel Prize, Nash referred to these “interludes, as it were, of enforced rationality”, and he stated that “rational thought imposes a limit on a person’s concept of his relation to the cosmos.”

In late 1959, when Nash was out of the hospital and on a trip to Europe, he threw away his passport as a result of “non-local thinking” and a desire to become a citizen of the world. Forcibly deported from both Switzerland and France, Nash returned to the Princeton area where eventually his mother and sister had him committed to Trenton State Hospital to receive insulin shock treatments.

In later years antipsychotic drugs helped reduce (not increase, as some people believed) Nash’s zombie-like behavior, and even enabled him to write a paper on Cauchy’s problem for the general Navier-Stokes equations while in a hospital. But sometimes while not incarcerated Nash would stop taking his medication because, “If I take drugs I stop hearing the voices.”

In recent years Nash has been a caretaker of his and Alicia’s son Johnny, who is also a mathematician and has paranoid schizophrenia. Nash made a telling remark about Johnny that might apply to himself: “I don’t think of my son … as entirely a sufferer. In part, he is simply choosing to escape from the world.”

What factors triggered Nash’s mental illness? Two hypothesized by the author are:

1. His work at RAND for four years, with “its worship of the rational life and quantification, its geopolitical obsessions, and its weirdly compelling mix of Olympian detachment, paranoia, and megalomania.”
2. Nash’s attempt while at MIT to prove the Riemann Hypothesis, which Enrico Bombieri has described as “not just a problem. It is the problem.” Regarding the Riemann Hypothesis, Nasar asserts, “Nash’s compulsion to scale this most difficult, most dangerous peak proved central to his undoing.”

Nash himself identified two different factors, in his Plenary Lecture to the World Congress of Psychiatry in Madrid in 1996: the stress of teaching and its regimentation; his attempts, starting in 1957, to resolve the contradictions in quantum theory, which were “possibly overreaching and psychologically destabilizing.”

Nash’s obsession with rationality led him to numerology and attempts to apply mathematics such as differential topology and algebraic geometry to politics, human relationships, the stock market, and so on. For example, when Hironaka won a Fields Medal for work on resolution of singularities Nash sent him a message asking if he could resolve the singularity in \(N^5+I^5+X^5+O^5+N^5 = 0.\)

At times Nash’s delusions appear almost humorous; his fears of world annihilation, for example, were not only of Armageddon, the Apocalypse, and the Final Day of Judgment, but also of the Day of Resolution of Singularities. At other times the delusions are painfully absurd and pathetic, such as the following message Nash left on a blackboard when he was known as the Phantom of Fine Hall (at Princeton): “Mao-Tse-Tung’s Bar Mitzvah was 13 years, 13 months and 13 days after Brezhnev’s circumcision.”

Nash’s recovery from paranoid schizophrenia was gradual; such a recovery is so rare that in his 1996 World Congress of Psychiatry lecture Nash was introduced as “a symbol of hope”. What accounted for this miraculous remission after nearly three decades of mental illness? No one knows, although it is known that no new treatment caused it.

Nash himself believes that he willed his own recovery, in a manner “analogous to the role of willpower in effective dieting; if one makes an effort to ‘rationalize’ one’s thinking then one can simply recognize and reject the irrational hypotheses of delusional thinking.” Nash also resolved to avoid anything related to politics, religion, and teaching, and focused instead on mathematics and computers.

Alicia, Nash’s wife, believes that other factors in his gradual recovery were “being taken care of and not too much pressure” and “living a quiet life” (as they and son Johnny now do in Princeton Junction). Nash is now humbler, more modest, more self-aware, and, as Alicia put it, “a very fine person”. He has retained a sense of humor, as shown in some poignant remarks he made at a small party in Fine Hall on the afternoon of the Nobel announcement: “First, he hoped that getting the Nobel would improve his credit rating because he really wanted a credit card. Second, he said that one is supposed to say that he is glad that he is sharing the prize (with Harsanyi and Selten), but he wished he had won the whole thing because he really needed the money badly.” Even at the height of his madness Nash seemed to have kept a sense of humor; in 1959 his son was known as Baby Epsilon for a long time before being named Johnny.

As promised here are several nuts and bolts on the book itself. Given that the author is a journalist and not a mathematician I was surprised to find only one mathematical error, and this may have been just a typesetting error. On page 230 she misquotes Bell’s Men of Mathematics by saying that the zeta function of \(s\) is \(1+1/2s+1/3s+1/4s+\dots\) (the divergence of this harmonic series would somewhat trivialize the Riemann Hypothesis!).

Elsewhere she does an excellent job of explaining the mathematics or she relies on explanations from experts like John Milnor. The twenty-six black and white photos, mostly of Nash and his relatives, are intriguing but I was disappointed to find only two photos with other mathematicians in them. (Where is Paul Halmos when we need him?) The nineteen page index and forty five pages of reference notes are both good, as they should be for a biography like this one.

The bottom line — is Nash’s biography worth reading? My answer is yes if you’re fascinated by Nash’s story and the mathematical culture surrounding him, and if you’re not bored by the author’s extensive discussion of schizophrenia, both in general and as applied to Nash himself.

I have one caveat, though. I found myself getting a little depressed with Nash’s character midway through the story. So, if you do start the book be sure to make it to the more upbeat concluding chapters.

You might be hearing more of John Nash in the media in the future as he may become the first mathematician to be the focus of a feature-length Hollywood film. According to a recent article by Nasar, “... a major Hollywood producer has bought the rights to his (Nash’s) life story for a high-six-figure sum” (page 17 of “Nobel Economics: Spending the Check,” New York Times, Money and Business section, 12/5/99). Nasar points out that Nash now has some financial security, what with the $200,000 he received and invested from the Nobel prize and with his $25,000-a-year part-time research position at Princeton. Since Nash earlier suffered years of grinding poverty, an economic upturn certainly is appropriate for a Nobel laureate in economics.

Peter Ross (pross@scu.edu) was an undergraduate at MIT when John Nash taught there. He completed his Ph.D. at the University of California at Berkeley, after teaching in India as a Peace Corps Volunteer (1963–65) and working on new “new math” projects for the government (1969–74). He taught in the Department of Mathematics and Computer Science at Santa Clara University from 1982 until he retired in 2010. Since 1985 he has been writing Media Highlights for the College Mathematics Journal, often highlighting articles on the history of mathematics.