Part 6 (1/2)

One point which should particularly be noticed is the linearity hypothesis in Nash's theoreenstern theory of numerical utility; the claim that it is possible to measure the relative desirability of different possible outcomes by a real-valued function which is linear with respect to probabilitiesMy own belief is that this is quite reasonable as a normative theory, but that it may not be realistic as a descriptive theory

Evidently, Nash's theory was not a finished answer to the proble competitive situations In fact, it should be emphasized that no simple mathematical theory can provide a coy of the players and the mechanism of their interaction 20

Nevertheless, decades later, econoard this ”failure” of an experiment as a very hile one Casual as the experiment was in one sense, it became a model for a new method of economic research, one that had never before been tried in the two hundred years since Ada was that even if the experih to sho people's brains work, watching the way people played games could draw researchers' attention to ele or implicit threats - that couldn't be derived axiomatically21 By the time the experiment was run the relationshi+p between Nash and Milnor had becoina Avenue apartment

Milnor says now that Nash made a sexual overture toward him ”I was very naive and very ho people talked about then”22 But what Nash felt toward Milnorclose to love A dozen years later, in a letter to Milnor, Nash wrote: ”Concerning love, I know a conjugation: amo, amas, amat, amamus, amatis, amant Perhaps amas is also the imperative, love! Perhaps one must be very masculine to use the imperative” But what Nash felt toward Milnorclose to love A dozen years later, in a letter to Milnor, Nash wrote: ”Concerning love, I know a conjugation: amo, amas, amat, amamus, amatis, amant Perhaps amas is also the imperative, love! Perhaps one must be very masculine to use the imperative”23

CHAPTER 19

Reds Spring 1953 Spring 1953

Now, the thing I think would interest the coreatly, if you could possibly explain to themDoctor how you can account for ould seee of communists at MIT?

-ROBERT L K L KUNZIG, Counsel HUAC, April 22, 1953

THE C COLD W WAR proar daddy of the MIT mathematics department, but McCarthy ism - which blamed the setbacks in that war on sinister conspiracies and domestic subversion - threatened to devour it proar daddy of the MIT mathematics department, but McCarthy ism - which blamed the setbacks in that war on sinister conspiracies and domestic subversion - threatened to devour it

While Nash and his graduate student friends were shooting each other down and playing gaators were fanning out around Ca individuals under surveillance, and questioning neighbors, colleagues, students, and even children1 Their targets, as Nash and everyone else at MIT would learn in early 1953, included the chairman and the deputy chairman of the MIT mathematics department, as well as a tenured full professor of mathematics, Dirk Struik - all three one-tie cell of the Communist Party All three were subpoenaed by the House Un-Aets, as Nash and everyone else at MIT would learn in early 1953, included the chairman and the deputy chairman of the MIT mathematics department, as well as a tenured full professor of mathematics, Dirk Struik - all three one-tie cell of the Communist Party All three were subpoenaed by the House Un-Ae and everyone in the e and everyone in the mathematics department felt the threat

At the time, Nash was no doubt farcomplications of his personal life - than with the possible repercussions for himself of the persecution of his benefactors Nevertheless, the whole episode was a warning that the world he and other ressional committee could destroy your career, just as your draft board could send you halfway around the world

The whole thing had begun as a farce3 McCarthy's original list of communists, announced in February 1950, was studded with acade the father of Nash's friend Lloyd Shapley, Harvard astronomy professor Harlow Shapley, whom McCarthy incorrectly identified to reporters as ”Howard shi+pley, astrologer” But as the red hunt gathered inal list of communists, announced in February 1950, was studded with acade the father of Nash's friend Lloyd Shapley, Harvard astronomy professor Harlow Shapley, whom McCarthy incorrectly identified to reporters as ”Howard shi+pley, astrologer” But as the red hunt gathered momentum, the entire scientific community felt vulnerable Princeton's Solomon Lefschetz would be identified as a possible coative body felt vulnerable Princeton's Solomon Lefschetz would be identified as a possible coative body4 Within a year, Robert Oppenheimer, head of the Manhattan Project, one of the most revered scientists in America and the director of the Institute for Advanced Study, would be humiliated by the McCarthyites Within a year, Robert Oppenheimer, head of the Manhattan Project, one of the most revered scientists in America and the director of the Institute for Advanced Study, would be humiliated by the McCarthyites

When the subpoenas were issued, nobody kne MIT would handle the s and suspensions5 ”McCarthyis threat to these schools,” Zipporah Levinson, Norovern money into them The threat was that the research money would dry up It was a bread-and-butter issue” ”McCarthyis threat to these schools,” Zipporah Levinson, Norovern money into them The threat was that the research money would dry up It was a bread-and-butter issue”6 Martin and Levinson were certain that they were about to lose their jobs and wind up blacklisted for good, like soa pluators had their eye on the three Browder boys - sons of former Communist Party head Earl Browder, who had all studied or were studying mathematics at MIT and were scholarshi+p recipients, as well Martin and Levinson were certain that they were about to lose their jobs and wind up blacklisted for good, like soa pluators had their eye on the three Browder boys - sons of former Communist Party head Earl Browder, who had all studied or were studying mathematics at MIT and were scholarshi+p recipients, as well7 ”MIT was turned topsy-turvy,” Mrs Levinson recalled ”The faculty debated and debated how to prove that MIT was patriotic There was strong pressure to name names”8 As it turned out, Karl Compton, the president of the university and an outspoken liberal as a supporter of the Chinese revolution and a critic of Chiang Kai-shek, may have felt that he himself would soon be subpoenaed He hired a white-shoe Boston law firm, Choate, Hall & Steward, to defend Martin, Levinson, and the others for a minimal fee As it turned out, Karl Compton, the president of the university and an outspoken liberal as a supporter of the Chinese revolution and a critic of Chiang Kai-shek, may have felt that he himself would soon be subpoenaed He hired a white-shoe Boston law firm, Choate, Hall & Steward, to defend Martin, Levinson, and the others for a minimal fee9 By April, when Martin and Levinson were forced to testify, By April, when Martin and Levinson were forced to testify, The Tech The Tech was running daily stories and anti-McCarthy senti daily stories and anti-McCarthy sentih on campus10 There is no evidence that the FBI ever questioned Nash or any other students or faculty in the department, or asked for depositions, in an effort to establish a link between Levinson's and Martin's Communist Party membershi+p and classified defense research - a link that probably never existed, given that both left the party soon after the end of the war The graduate students and junior faculty in the department stood on the sidelines and watched lives and careers ruined and ho people had prospects, jobs, optiroup - didn't want to be too friendly They were scared They distanced themselves”11 Martin and several others named their former associates Norman Levinson refused to name anyone who had not been previously named ”Ted and Izzy Amadur hemmed and hawed Norman knew that Ted Martin and Izzy would cooperate They spilled all the names Norman said he'd talk freely about the party but that he wouldn't name names The lawyer told Norman, no you don't have to say any naive any nahtened performance Levinson's testimony, by contrast, demonstrated the qualities of intellect and character that made him such a force in the mathematics community In a series of forceful and eloquent answers to direct questioning, he ed at one and the saave a pathetic, frightened performance Levinson's testimony, by contrast, demonstrated the qualities of intellect and character that made him such a force in the mathematics community In a series of forceful and eloquent answers to direct questioning, he ed at one and the same time to defend the youthful idealism that led him into the party, attack the intellectual poverty of communism, and, implicitly, call into question the committee's assumption that coainst the hounding of forainst the blacklisting of Browder's oldest son, Felix, who had finished his PhD and was unable to obtain an academic post him into the party, attack the intellectual poverty of communism, and, implicitly, call into question the committee's assumption that coainst the hounding of forainst the blacklisting of Browder's oldest son, Felix, who had finished his PhD and was unable to obtain an academic post

Thanks to MIT's support and the compromises they struck, Levinson and the others kept their jobs But the whole dispiriting affair, which had been preceded by months of harassment and threats, left deep scars on everyone involved Martin, in particular, was shattered and deeply depressed, and was unable, nearly forty-five years later, to talk about it13 Levinson's younger daughter, a student in junior high school, suffered a breakdown and was diagnosed with manic depression Levinson and his wife bla harassed by the FBI Levinson's younger daughter, a student in junior high school, suffered a breakdown and was diagnosed with manic depression Levinson and his wife bla harassed by the FBI14 And those on the periphery, ostensibly unaffected, learned a lesson, naranted was dangerously fragile and vulnerable to forces beyond its control And those on the periphery, ostensibly unaffected, learned a lesson, naranted was dangerously fragile and vulnerable to forces beyond its control

Nash took no part in the heated discussions araduate students over the morality of the overnment15 Any discussion of ry, frightening, turbulent time would supply him with some of the prosecutory demons that came to haunt him later Any discussion of ry, frightening, turbulent time would supply him with some of the prosecutory demons that came to haunt him later16

CHAPTER 20

Geometry

There are two kinds of mathematical contributions: work that's important to the history of mathematics and work that's simply a triumph of the human spirit

-PAUL J C J COHEN, 1996

IN THE SPRING OF 1953, Paul Halo, received the following letter froue of Nash's: 1953, Paul Halo, received the following letter froue of Nash's: There's no significant news fro John Nash to an assistant Professorshi+p (not the Nash at Illinois, the one out of Princeton by Steenrod) and I'uy ants to be ”basically original,” which I suppose is fine for those who have soinality in them He also makes a damned fool of himself in various ways contrary to this philosophy He recently heard of the unsolved proble a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently hile to justify his efforts; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a systeet] to the essentially sile ordinary differential equation it would be a dauest notions about the whole thing So it is generally conceded he is getting nowhere and er ass of himself than he has been previously supposed by those with less insight than ot hiotten a real uy but conceited as hell, childish as Wiener, hasty as X, obstreperous as Y, for arbitrary X and Y1

Ambrose had every reason to be both skeptical and annoyed

Ambrose was a moody, intense, somewhat frustrated mathematician in his late thirties, full, as his letter indicates, of black humor2 He was a radical and nonconforave a lecture on ”Why I a deot himself beaten up and jailed for his efforts He was also a jazz fanatic, a personal friend of Charlie Parker, and a fine trumpet player He was a radical and nonconforave a lecture on ”Why I a deot himself beaten up and jailed for his efforts He was also a jazz fanatic, a personal friend of Charlie Parker, and a fine trumpet player3 Handsome, solidly built, with a boxer's broken nose - the consequence of an accident in an elevator! - he was one of the most popular members of the department He and Nash clashed from the start Handsome, solidly built, with a boxer's broken nose - the consequence of an accident in an elevator! - he was one of the most popular members of the department He and Nash clashed froive an impression of stupidity: ”I'm a simple man, I can't understand this” Robert Aumann recalled: ”Ambrose came to class one day with one shoelace tied and the other untied 'Did you know your right shoelace is untied?' we asked 'Oh, ht the other must be tied by considerations of synored Nash's putdowns and jibes Ambrose did not Soon a tit-for-tat rivalry was under way As, for detail His blackboard notes were so dense that rather than atte theraph them5 Nash, who disliked laborious, step-by-step expositions, found ly argu a seminar, Nash would mutter, ”Hack, Hack,” from the back of the room Nash, who disliked laborious, step-by-step expositions, found ly argu a seminar, Nash would mutter, ”Hack, Hack,” froet of several pranks ”Sen that Nash posted one day ”The seminar will meet weekly Thursdays at 2 PM PM in the Common Room” Thursday at 2:00 in the Common Rooht his graduate course in analysis was the hour that Araduate course in analysis7 On another occasion, after Ambrose delivered a lecture at the Harvard e bouquet of red roses delivered to the podiu her bows On another occasion, after Ambrose delivered a lecture at the Harvard e bouquet of red roses delivered to the podiu her bows8 Ambrose needled back He wrote ”fuck Myself” on the ”To Do” list that Nash kept hanging over his desk on a clipboard9 It was he who nickna remarks about other mathematicians It was he who nickna re a discussion in the common room, after one of Nash's diatribes about hacks and drones, Aood, why don't you solve the e problem for manifolds?” - a notoriously difficult problem that had been around since it was posed by Rie a discussion in the common room, after one of Nash's diatribes about hacks and drones, Aood, why don't you solve the e problem for manifolds?” - a notoriously difficult problem that had been around since it was posed by Riemann11 So Nash did

Two years later at the University of Chicago, Nash began a lecture describing his first really big theore statement spoke volumes about who he was He was a rand sche problems In the taxonomy of mathematicians, there are probleed to the first group He was not a gaist, or mathematical physicist But he zeroed in on areas in these fields where essentially nobody had achieved anything The thing was to find an interesting question that he could say soist, or mathematical physicist But he zeroed in on areas in these fields where essentially nobody had achieved anything The thing was to find an interesting question that he could say soe, Nash wanted to be certain that solving the problelory He not only quizzed various experts on the proble to Felix Browder, another Moore Instructor, clai before he actually had13 When a mathematician at Harvard confronted Nash, recalled Browder, ”Nash explained that he wanted to find out whether it orth working on” When a mathematician at Harvard confronted Nash, recalled Browder, ”Nash explained that he wanted to find out whether it orth working on”14 ”The discussion ofto the air around him ”The precise question that A: Is it possible to embed any Riemannian manifold in a Euclidean space?”15 It's a ”deep philosophical question” concerning the foundations of geometry that virtually every mathematician - from Riemann and Hilbert to Elie-Joseph Cartan and Hereometry for the past century had asked hi Schlafli in the 1870s, had evolved naturally froression of other questions that had been posed and partly answered beginning in the mid-nineteenth century The question, first posed explicitly by Ludwig Schlafli in the 1870s, had evolved naturally froression of other questions that had been posed and partly answered beginning in the mid-nineteenth century17 First mathematicians studied ordinary curves, then surfaces, and finally, thanks to Rieures of nineteenth-century her dimensions Riemann discovered examples of manifolds inside Euclidean spaces But in the early 1950s interest shi+fted to e role that distorted space and time relationshi+ps had in Einstein's theory of relativity First mathematicians studied ordinary curves, then surfaces, and finally, thanks to Rieures of nineteenth-century her dimensions Riemann discovered examples of manifolds inside Euclidean spaces But in the early 1950s interest shi+fted to e role that distorted space and time relationshi+ps had in Einstein's theory of relativity

Nash's own description of the eraphy hints at the reason he wished tothe probleh classical, was notproblem It was not like, for exa involves portraying a geo it a subset of - some space in some dimension Take the surface of a balloon You can't put it on a blackboard, which is a two-dimensional space But you can make it a subset of spaces of three or htly more complicated object, say a Klein bottle A Klein bottle looks like a tin can whose lid and bottom have been removed and whose top has been stretched around and reconnected through the side to the bottom If you think about it, it's obvious that if you try that in three-di intersects itself That's bad frohborhood in the iular, and attempts to calculate various attributes like distance or rates of change in that part of the object tend to blow up But put the same Klein bottle into a space of four dier intersects itself Like a ball embedded in three-space, a Klein bottle in four-space becomes a perfectly well-behaved manifold calculate various attributes like distance or rates of change in that part of the object tend to blow up But put the same Klein bottle into a space of four dier intersects itself Like a ball embedded in three-space, a Klein bottle in four-space becomes a perfectly well-behaved manifold

Nash's theorem stated that any kind of surface that embodied a special notion of smoothness can actually be embedded in Euclidean space He showed that you could fold theit nobody would have expected Nash's theorem to be true In fact, everyone would have expected it to be false ”It showed incredible originality,” said Mikhail Groeometer whose book Partial Differential Relations Partial Differential Relations builds on Nash's work He went on: builds on Nash's work He went on: Many of us have the power to develop existing ideas We follow paths prepared by others Butco Psychologically the barrier he broke is absolutely fantastic He has coed the perspective on partial differential equations There has been some tendency in recent decades to move from harmony to chaos Nash says chaos is just around the corner19

John Conway, the Princeton matheame of Life, called Nash's result ”one of the most important pieces of mathematical analysis in this century”20 It was also, one must add, a deliberate jab at then-fashi+onable approaches to Rieae to von Neuhly abstract and conceptual description of suchGerman mathematician who came to know Nash well in the mid-1950s, put it, ”Nash didn't like that style of mathematics at all He was out to show that this, to his mind, exotic approach was completely unnecessary since any such h dimensional Euclidean space”21 Nash's more important achievement may have been the powerful technique he invented to obtain his result In order to prove his theorely insur a certain set of partial differential equations that were i methods

That obstacle cropped up in many mathematical and physical proble to Ambrose's letter, pointed out to Nash, and it is a difficulty that crops up in many, many problems - in particular, nonlinear probleiven is some function, and one finds estimates of derivatives of a solution in teriven function Nash's solution was re an equation, the thing that is given is some function, and one finds estimates of derivatives of a solution in teriven function Nash's solution was remarkable in that the a priori a priori estimates lost derivatives nobody kne to deal with such equations Nash invented a novel iterative uesses - for finding roots of equations, and co to counteract the loss of derivatives estimates lost derivatives nobody kne to deal with such equations Nash invented a novel iterative uesses - for finding roots of equations, and co to counteract the loss of derivatives23 Newman described Nash as a ”very poetic, different kind of thinker”24 In this instance, Nash used differential calculus, not geoebraic rowths of nineteenth-century calculus The technique is now referred to as the Nash-Moser theoreinator In this instance, Nash used differential calculus, not geoebraic rowths of nineteenth-century calculus The technique is now referred to as the Nash-Moser theoreinator25 Jurgen Moser was to sho Nash's technique could be modified and applied to celestial mechanics - thethe stability of periodic orbits Jurgen Moser was to sho Nash's technique could be modified and applied to celestial mechanics - thethe stability of periodic orbits26 Nash solved the problem in two steps He discovered that one could embed a Rienored smoothness27 One had, so to speak, to crue and interesting result, but a mathematical curiosity, or so it seemed One had, so to speak, to crue and interesting result, but a mathematical curiosity, or so it see without wrinkles, e in which the smoothness of the manifold could be preserved Mathe without wrinkles, e in which the smoothness of the raphical essay, Nash wrote: So as it happened, as soon as I heard in conversation at MIT about the question of ean to study it The first break led to a curious result about the ely low-dimensional ambient spaces provided that one would accept that the e would have only limited smoothness And later, with ”heavy analysis,” the probleree of smoothness29

Nash presented his initial, ”curious” result at a se of 1953, at around the sa letter to Halmos Emil Artin was in the audience He made no secret of his doubts

”Well, that's all well and good, but what about the eet it”