While establishing the claims of Prince John and Princess Alicia to the thrones of the Romanovs, Hapsburgs, Bourbons and, presumably, Saxe-Coburg-Windsor, Nasar takes time out to ogle Nash's physique. At age 20, she writes, (page 67) :
" He had the build, if not the bearing of an athlete, 'a very strong, very masculine body', one fellow graduate student recalled. He was, moreover, 'handsome as a god', according to another student."
Lest we think that Nasar is merely quoting citing 50-year old recollections of former class-mates, she adds:
" His high forehead, somewhat protruding ears, distinctive nose, fleshy lips and small chin gave him the look of an English aristocrat."
Imagine what Daumier would have done with that recipe ! In between she casts a few side glances at other mathematicians, telling us that (page 71) " John Milnor ..was..tall, lithe, with a baby face and the body of a gymnast. Milnor was only a freshman but he was already the department's golden boy." And that (page 73) Emil Artin, refugee mathematician from Germany, " looked like a 1920's German matinee idol. "
Back to Nash. On page 149, she reminds us that " Nash was built like a Greek God. " The legs surface again in a few places, even, on page 385, at the age of 70. Nasar's personal opinions, free from the protective coloration of quotation marks, emerge, resplendent in shameless nakedness when , on page 196, she calls him A genius with a penis. !Here is the exact quote:
" It was his good looks, however, that made Alicia's heart beat faster, ' A genius with a penis. Isn't that what we all want?' an actress once quipped, and the quip captures the combination of brains, status and sex appeal that made Nash so irresistible."
I guess I'm bemused, and more than a little flattered, to learn that mathematicians have any sex appeal at all. It makes me want to consider going back to doing mathematics full time. I doubt that I'd be able to reconcile the time spent fighting off all the women eager to get at me, with the long hours required for my research.
The topic of 'schizophrenia' is treated at some length in A Beautiful Mind . Previously I stated that 'schizophrenia' is known abroad as 'the American diagnosis.' My source for this comes from the 50's, (the period of Nash's first hospitalization) but I'm sure it's still true today. Here is the reference:
"...a given patient might be diagnosed quite differently from one country to another...The English call almost any kind of emotional trouble 'neurosis', said Henri Ellenberger, the great historian of psychiatry, in the mid-1950s. 'The French apply the diagnosis of feeblemindedness very liberally.' As for the Swiss, 'The French say that the Swiss diagnose schizophrenia is '90 percent of the psychotics and 50 percent of the normal"'
But nobody used the diagnosis of schizophrenia more often than the Americans. Schizophrenia was the great foible of American psychiatry. ..In one study, 46 American psychiatrists and 205 British psychiatrists watched a videotape of 'patient F', a young man from Brooklyn who had a hysterical paralysis of one arm and a mood fluctuation associated with alcohol abuse. Afterward, 69 percent of the Americans diagnosed 'schizophrenia', 2 percent of the British" ( pg. 296, Edward Shorter, "A History of Psychiatry", John Wiley and Co, 1997)
Sylvia Nasar's theories of schizophrenia are no worse than most fantasies current in modern psychiatry. Since she is a painstaking referencer, one need only consult her footnotes to learn the sources of her ideas. For the most part they come from the books cited in her bibliography. ( I. Gottesman, " Schizophrenic Genesis: The Origins of Madness", W.H. Freeman, 1991 ; G. Winokur and M. Tsuang, "The Natural History of Mania, Depression and Schizophrenia" American Psychiatric Press, 1996 ; L. Sass, "Madness and Modernism", Basic Books, 1992; E. Fuller-Torrey, "Surviving Schizophrenia, A Family Manual", Harper & Row,1988 )
These writers emphasis the genetic theories of schizophrenia and neuroleptic drug regimes in use today.
Although Nasar does not regard psychiatry with my acerbic skepticism, she should not be charged with accepting all of its dogmas at face value. Here is her description of psychiatry as practiced at McLean's hospital at the time of John Nash's first incarceration:
" Fagi [Levinson] recalled that Alicia's pregnancy was thought to be the culprit. 'It was the height of the Freudian period-all things were explained by fetus envy.' [Paul] Cohen said : 'His psychoanalysts theorized that his illness was brought on by latent homosexuality'. .....Freud's now discredited theory linking schizophrenia to repressed homosexuality had such currency at McLean that for many years any male with a diagnosis of schizophrenia who arrived at the hospital in an agitated state was said to be suffering from 'homosexual panic.'" ( pg. 259)
This, together with the above citation from Edward Shorter, implies that American doctors considered the density of repressed homosexuality in American society to be greater than that of England by a factor of 35 ! - which , combined with the oft-quoted figure of 1% for the percentage of schizophrenics in the human race, plus the ratio of 6 to one for the population of the United States over that of the United Kingdom , implies that........
Mazel Tov!
If Nasar's picture of the symptoms and causes of schizophrenia lacks coherence, this is only because the contemporary psychiatric identifier , 'schizophrenia', is not coherent. Drawing from the books of such psychiatric authorities, Nasar spreads the following list of 'schizoid' symptoms across the pages of A Beautiful Mind :
...The 'schizoid state' is 'characterized by a sense of meaningless and futility' .
...John Nash was exceptional because ' Men of scientific genius, however eccentric, rarely become truly insane' ,
... schizophrenia has a genetic basis and 'tends to run in families' ( Most of the literature in defense of this assertion can be traced to a single research finding, the Copenhagen twins study done in 1995 , a remarkable example of shoddy science. See Peter Breggin, "Toxic Psychiatry", St. Martin's Press, 1991, pg. 97 )
... Schizophrenia 'leads to a lifelong pattern of social isolation and indifference to the attitudes of others' .
... consistent with her reactionary , even monarchist tone, she explains on page 271 , more or less, that radical political activity ' has long been a hallmark of a developing schizophrenic consciousness. ' This assertion, if true, stands in stark contradiction to the one just above it.
... 'voices... are the most characteristic delusion of schizophrenia.' . What this statement says is that when a patient tells a psychiatrist that he's hearing voices, the psychiatrist is likely to write down a diagnosis of 'schizophrenia'.
....schizophrenics are 'insensitive to physical pain' .
Finally, on page 258, we are provided with a bargain - basementful of symptoms:
"...simultaneously grandiose and persecutory beliefs - tense, suspicious behavior - relative coherence of speech - ( relative to what? To other mental diseases? To what one ought to expect? To normal people?) - blankness of facial expression - extreme detachment of voice - reserve to the point of muteness ......"
By now we begin to realize that the word 'schizophrenia' is a grab-bag into which one is welcome to throw anything that may be considered abnormal. This conclusion is evaded by Nasar's psychiatric experts through the use of traditional political loopholes of like the following:
" Symptoms vary so much between individuals and over time for the same individual that the notion of a 'typical case' is virtually non-existent"
" ...self-contradiction is also characteristic of schizophrenia, every symptom being matched by a 'counter-symptom..." , and so forth.
Finally, as if to exculpate psychiatry from its vagaries , a former psychiatrist of Nash at McLean Hospital suggests , (page 318, footnote 36) , that Nash may not have been suffering from 'paranoid schizophrenia' at all, but 'bipolar disorder': " The quality of ...two papers [written between 1965 and 1967] - the first of which geometer Mikhail Gromov called 'amazing' - constitutes the single strongest reason for questioning Nash's diagnosis of paranoid schizophrenia. "
Being both everything and nothing, 'schizophrenia' can be conveniently employed to say everything and nothing. Nasar's account of modern psychiatric dogma is competently done, though I feel that she takes too much of it at face value.
Unfortunately, just when one is tempted to credit her with a measure of common sense, she then abandons her doctrinal moorings to wander far out to sea with numerous theories, all of a sentimental, ad hoc or silly character, about the causes for Nash's insanity and its long-postponed remission in the late 80's. Most of these theories are her own, some were proposed by Nash's colleagues. Nash's own theories shed an interesting light on his character. Among them we find:
.... teasing in elementary school ( page 188)
.... the stress of teaching ( page 125. Proposed by Nash himself)
.... fear of being drafted into the Korean war ( page 126)
.... the horrible stories his father made of about what would happen if the Japanese invaded West Virginia. ( page 36)
.... McCarthyism at MIT ( page 154)
.... dismissal from RAND after his arrest ( page 188)
.... because Emilio di Giorgi published his research on parabolic partial differential equations a few months earlier than his own (page 220. This theory was proposed by mathematician Gian Carlo-Rota )
.... agonizing too much over the contradictions of quantum theory (page 221. Another Nash theory )
.... the rejection of his amorous advances by young logician Paul Cohen ( page 243)
.... his failure to win the Bocher Prize ( page 243)
.... the humiliation he was exposed to after the presentation of his proof of the Riemann Hypothesis ( Quote:, page 232: " Nash's compulsion to scale this most difficult, most dangerous peak proved central to his undoing." )
Apparently unaware of what she's doing, Sylvia Nasar systematically follows each account of every misfortune or setback suffered by Nash with a statement to the effect that it was probably the cause of his mental collapse. It must be admitted that, given that 'amateur' and 'professional' psychiatry overlap are indistinguishable in our own day, her personal list of theories stands up fairly well against 'fetus envy', 'homosexual panic' and the like.
Nasar's views on the causes of his remission are grouped together in the 20 pages starting at page 335. Here she claims that "Princeton functioned as a therapeutic community." The assertion is not unreasonable , although 20 years does seem like a long time for therapy to reveal its benefits ( Favorable as it might appear in comparison with the length of time for a classical psycho-analysis. ) There exist mathematics departments that are dangerous for certain kinds of people, functioning as negative potential wells, in which they risk getting stuck in stable regimes for decades, sometimes their entire lives. The department at Stanford University apparently had that effect on Ivan Streletsky. His 20 years sojourn in its doldrums led to a major tragedy in the early 80's with the brutal murder of his thesis adviser .
This Twilight Zone phenomenon is known to everyone who has spent a fair amount of time wandering about the mathematics community, yet I've never seen it described in published accounts . My own encounters with it have been at U. Pennsylvania, M.I.T. and U.C. Berkeley. It's very easy, by the way, to get typed as one of these individuals . They acquire a reputation as fixtures hanging around in the lounges and libraries. They may show up at lectures and colloquia where they tend to ask questions that have no connection with the subject matter. The public reaction to their remarks on these occasions hovers between a suppressed laugh and an embarrassed silence.
Beyond making fun of them, few persons take much interest in their situation. Nobody does anything for them. They may live with their families, like Nash did, or have some other form of guaranteed income. Basically, provided they don't run interference with the grinding of the wheels of the great theoremizing engine, they're allowed to rot. Sitting alone in the departmental reading rooms, they appear to be off in a world of their own, perhaps with a math book or article on their lap or the table before them, perhaps with nothing at all.
An unhappy young man fitting this description, whom I got to know , installed himself as the "phantom of the MSRI ( Math Sciences Research Institute)" during the years 1983 to 87 when I lived in Berkeley. At that time he was in his 20's. When I came back for a visit in 1996 he was still there, essentially unchanged. The staff and administration of the MSRI are very aware of his presence. I doubt that anyone has ever suggested constructive ways to help him. It's not their style. High-principled decisions to " respect others' privacy" are sometimes the easiest way out for those who just don't give a damn. Gangrene no doubt sets in in other disciplines as well, ( I recall a few 'phantoms' around the NE Conservatory in the 80's. Of course there will always be administrators who think that anyone who isn't paying money to the institution is a phantom. ) , but I suspect that it's only in Mathematics where such situations are allowed to continue indefinitely without anything being done about them.
John Nash fit this category, with the very important difference that he had something to recover back into. Sylvia Nasar is certain that the "gentle manner" of his ex-wife Alicia Larde, "played a substantial role in his recovery." Alicia adds her own comments: (page 342) " Did the way he was treated help him get better? Oh, I think so. He had his room and board, his basic needs taken care of, and not too much pressure. That's what you need: being taken care of and not too much pressure."
The issue is controversial, isn't it? Since Nash's marriage, the birth of their son, and Nash's first mental breakdown all occurred in the late 50's, one could argue, ( with as little scientific foundation), that it was the marriage itself that caused his illness, and that it took him thirty years to recover from it! Strongly motivated persons often thrive in situations in which basic needs are guaranteed. Those lacking such motivation may instead go to seed , turning perhaps to alcoholism and self-justifying abusiveness or violence through a lack of meaningful work. Still others may coast along for 20 years or more under such a regimen, getting neither better nor worse. All 3 of these scenarios appear to have been at work in Nash's case.
These speculations of Nasar's are at least reasonable. On page 336, however, she makes the bizarre assertion that Nash's dabblings in Cabbalistic numerology kept his grey matter intact :
" The immense effort [ of calculating and writing down numerological messages] may have played a role in preventing Nash's mental capacities from deteriorating."
Her remaining suggestions are brought together in a single paragraph on page 353. All reveal a peculiar shallowness of perspective:
" ..... high social class .... " ( A constant refrain, although no evidence is provided anywhere in the biography to show that Nash's social class is higher than anyone else's . )
" ... high IQ ... " ( Has anyone established a meaningful correlation between IQ and recovery from mental illness? )
"... high achievement ... " ( ditto. Our unlucky friends, Robert Schumann, Vaslav Nijinsky, Jonathan Swift, Vincent van Gogh, Friedrich Nietzsche, Antonio Salieri , John Ruskin ,Georg Cantor, Kurt Goedel ..... )
" ... no schizophrenic relatives ... " ( what about grandfather Alexander Nash? ( page 26) : " a strange and unstable individual, a ne'er-do-well, a drinker and philanderer who either abandoned his wife and three children ... or, more likely, was thrown out." )
"... disease acquired in 3rd decade .. " (Although Nasar's documentation of Nash's borderline pathology in his 20's is quite thorough.)
This somewhat arbitrary list is followed by ruminations on the role of hospitalization, psychogenic drugs, and even Nash's rejection of them. Everything and its opposite somehow contributed to his spontaneous remission. Nash's own theory appears to be the most sensible:
" I emerged from irrational thinking" , he said in 1996 , "ultimately, without medicine other than the natural hormonal changes of aging. "
"Take a hecatonicosihedrigon and multiply by four
(A sexicosihedrigon plus half as many more:
Put in some polyhedrigons whose gaps suggest a minus
And you'll have a polyhedral-perpendodicahedrinus!"
-Luran W. Sheldon, NY Times, 1910, in honor of William Sidis's lecture on 4-dimensional geometry at Harvard at the age of 11
Despite this claim to moral relativism, ' importance' is the most abused word in mathematical discourse . As part of their professional equipment, mathematicians are required to have a sense of what is 'important' in the history of mathematics, in contemporary mathematics, and certainly in their own field. Sensible people do not work for very long on a difficult problem they don't consider 'important'. Why the properties of prime numbers should be considered more 'important' than the mathematical prodigy William Sidis's encyclopedic knowledge of the properties of trolley car transfers 5 is not easy to put into words.(The Prodigy, Amy Wallace; E.P. Dutton, 1986)
The uses and interpretations of the word "important" vary so much from one person to the next that, were mathematicians as perspicacious as they claim they would recognize how arbitrary it's has become, and discard it as meaningless. Here, for example, is a short list of statements made by his colleagues about the value of Nash's research:
"The embedding theorem .... is one of the most important pieces of mathematical analysis in this century. " John Conway. ( Although not an analyst Conway is a major figure in mathematics . I suspect him of making this comment because it sounded at the time like the right thing to say.)
" [Nash's work on parabolic differential equations] .... which many mathematicians regard as Nash's most important work . " ( Sylvia Nasar. She doesn't tell us who these mathematicians are, but they presumably work in the field of partial differential equations.)
" The concept of a Nash equilibrium n-tuple is perhaps the most important idea in non-cooperative game theory. " Economist P. Ordeshook. ( Notice the 'universal qualifier' "perhaps" ! Lots of people feel that Game Theory is "not important".Besides, 'non-cooperative game theory' is built around Nash equilibrium, so the assertion is circular )
" Let me describe an important application [ of the manifold-real varieties paper of 1951] " John Milnor.
( The application, the Artin-Mazur Theorem , is important to the theory of dynamical systems. One can find mathematicians who think that the field of dynamical systems is "not of much importance". )
" During these three years [ 1945-48, when he was still a teen-ager] , Nash completed an important piece of work on bargaining. " ( Harold Kuhn, mathematical economist at Princeton. )
" It gives me great pleasure to chair this seminar on the importance of John Nash's work on the occasion of the first Nobel award that recognizes the central importance of game theory in current economic theory." ( ditto. How "important" is current economic theory?)
" In the short period of 1950-53 John Nash published four brilliant papers in which he made at least three fundamentally important contributions to game theory. " (John Harsanyi, co-recipient of the Nobel prize in economics, 1994.)
Without going into the details, everyone of these statements attributes a different meaning to the word "important". One is reminded of the story about the student employed by Thomas Kuhn's to proof-read the original manuscript of his " Structure of Scientific Revolutions". He told Kuhn that the word "paradigm" was being used in 64 different ways!
At the same time, I recognize that, because it is the very pillar, the spine of the entire enterprise the valuator, or validator "important " cannot be dropped from the meta-vocabulary of mathematics: what sane person would spend weeks, months, even years, on a mathematics problem unless he ( and, increasingly, she ) thought of it as important? The word determines careers, causes quarrels, ruins friendships, alienates the profession from the public, and vice-versa .
One example: after Louis DeBranges of Purdue proved the 'Bieberbach Conjecture' around 1986, he barn-stormed the nation, ranting and bullying audiences at math conference for not giving him the credit he deserved. No one , he fumed , could imagine that anything coming out of Purdue might be "important"! He finally descended into that black Slough of Despond, his own "proof" of the Riemann Hypothesis.
When a lecture presented in a research department is poorly attended, there are usually two reasons for it: the first is that it's on a topic the department deems "unimportant". In 1997 I attended a lecture given by a woman whose very name is revered in applied mathematics, Olga Ladyshenksaya . Only a devoted handful were present at her talk. Applied mathematics is considered of 'no importance' at Berkeley.The second reason is that it's topic is too specialized , perhaps with an esoteric vocabulary that only a few can understand. Some people may be sitting in anyway, though uncomprehending : the subject is 'important'!
A life in mathematics conditions people to apply the powerful law of contradiction to many situations, even where inappropriate. It is easy to over-generalize but many mathematicians strong in computational ability and the gift of pattern recognition will be lacking in the kind of judgment that weighs alternatives, decision-making of the sort that is done by doctors, judges, politicians, etc. in their professional activity. The same criticism applies of course to their perceptions of the value, meaning, worth, or 'importance' of their own fields . Such notions can be inflexible, biased, often unreliable.
It is easier to say what is important in applied mathematics, because one can speak of its effect on the field of application. A new way of computing solutions to the Navier-Stokes equation will be deemed an 'important' advance if these solutions lead to a deeper understanding of hydrodynamics, oceans, clouds, tornadoes, jet streams, etc. Thus the concept of the fractal is very important from the viewpoint of applied mathematics, though pure mathematicians see it as just slightly above the obvious.
If one now defines pure mathematics as applied mathematics in which the field of application is mathematics itself, it becomes easier to decide matters of relative importance. Observe that discoveries of applied mathematics are not directly evaluated . It is in the effect of such discoveries on the field of application that their importance resides. Ingenuity of reasoning, prolonged labor of computation, high levels of abstraction, extensive learning count for little . "By their fruits shall you know them" is the only reliable criterion for judging the "importance" of advances in any scientific field.
Take the two well-known theorems of Fermat. His last theorem is a conjecture. It might have been made by anyone, but because it was made by Fermat, mathematicians took a look at it. Formulating it required no particular insight; it's the kind of conjecture anyone might make after reading a book on number theory. Its' importance is incontestable: it lay the ground for 4 centuries of incredible effort.
Fermat's little theorem is also "important". This states that , where p is a prime and a is any positive integer. It can be proven by any talented high school student who's learned something about congruences in his course on the "new math". It is also among the most frequently employed tools in Number Theory. Today it is the foundation stone for the theory and practice of 'unbreakable' crytographic codes.
"Nash equilibrium" is really just an insight. It is even easier to prove than Fermat's Little Theorem. But it gave economists the feeling that they were doing something useful. They therefore considered it very important; and to many people, Sylvia Nasar among them, a Nobel Prize proves that a result is important.
Freudian psycho-analysis, the very paradigm of a pseudo-science, is likewise important because it shaped world civilization for a century, The movie Jurassic Park was extremely important for Paleontology because it led to increased funding for dinosaur research. The flap of that butterfly's wing in the Amazon was also very important because it caused all the dreadful floods in China we've been reading about.
The adjective 'important' when applied to mathematics ought to be given essentially the same meanings that it has in other sciences:
I. Tool-making: methodologies for problems solving
II. Discoveries
III. Solutions of outstanding unsolved problems
IV. Introduction of new concepts
V. Practical applications
I. In terms of problem-solving methods , Nash's and diGiorgio's papers on higher dimensional parabolic partial differential equations are indeed important.
II. Discoveries in mathematics are like Vaughn Jones' discovery of the 'Jones polynomial' in Knot Theory, or Euler's discovery of the properties of the number e = 2.718281828....., or Feigenbaum's discovery of the universality of the 'Feigenbaum constant' . Nash has, to date, not made any discoveries of this nature .
III. John Nash is unquestionably one of the great problem-solvers of history. The isometric embedding theorem is an 'achievement' comparable, in mathematical terms, to batting 65 home runs. Surprisingly, it does not require that much of a command of mathematics to read his papers . Nash did not cultivate an encyclopedic knowledge of mathematics, or even of his area. He was rather like a mountain climber who, rising to a certain height, decides that he needs oxygen or an ice-ax, goes back to base camp to get them, then returns to the assault. Nash only learned what he needed to attain his objectives. It's amazing how much he did with so little, particularly when he is compared to those who know enormous amounts of mathematics but never do any notable research.
IV. Nor did Nash introduce any new concepts: things like 'transfinite number', 'category', 'derivatives', 'fractal' , 'topos', 'group', ' manifold' , etc. These are ideas that illuminate the whole domain of mathematics . Another example is the idea of 'probability' developed by Pascal and Fermat. The closest that Nash ever came to inventing a concept is the 'non-cooperative game', a variant on the 'cooperative game'.
In arguing that Nash was (is) or was (is) not a genius, one needs to look at all five of the above categories and judge him separately relative to the requirements of each. This already shows how unprofitable the label of 'genius' is, mathematics being a field so rich in its diverse aspects that it is very difficult to judge the worth of discoveries without relating them to the purpose for which they were intended. Although it may be possible, in the abstract , to separate brilliance from the results of brilliance, it remains a futile exercise that is best carried out by journalists and their public, who will always be in need of miracle-workers and wizards to inspire them in their journey through this trackless wilderness of earthly existence.
Mathematics is a monumental ediface. Some are the architects, others work on its construction. Still others design the wallpaper, decor, furniture, etc. If the word 'genius' be applied to all of them, what term can we reserve for the Master Builders: Pythagorus, Brahmagupta, Archimedes, Descartes, Newton, Laplace. Gauss, Riemann, Galois, Jacobi, Kowalewski, Hilbert , Grothendieck ...?
John Nash's remarkable gifts have been attested to by everyone qualified to understand them. Nor ought one discount his heroic accomplishment in rising above a prolonged ordeal of terrible suffering to return to the world of rationality, however fragile that world itself may seem. Unrelieved flattery is not complimentary, and its real , often hidden intent may yield destructive consequences.