>From * The Chronicle Review* dated January 25, 2002

PHOTOFEST

AS AN UNDERGRADUATE mathematics major at Princeton in the early 1980s, I have many memories of John Nash. A thin, raincoat-clad, umbrella-carrying specter in the bowels of Fine Hall, pacing the solemn and too quiet hallways outside the mathematics library, brushing along the badly lit walls, which were (and perhaps still are) decorated with eerie and garish paintings of imagined planetary landscapes. Or, instead, chain-smoking, lying flat on his back, on a bench outside the library doors, eyes fixed on the ceiling.

I had heard the stories and wondered if they were true -- that he wrote the cryptic numerologico-political remarks left on blackboards, that he was a once-promising, even famous mathematician who, on the verge of publishing the solution of a long-outstanding problem, had been scooped by another mathematician and so had been driven to a nervous breakdown, convinced that he had been spied upon all these long years. "He sees little green men" is what I was told. When I saw him in the math department and in the library, I would nod, sometimes say hello, never sure if he recognized me from one day to the next. And I wondered -- I'm sure like many a budding mathematician -- just how close any one of us might be to his fate.

It was, thus, with amazement that several years later I heard on the news that John Nash had received the Nobel prize in economics. Like many others, I raced through Sylvia Nasar's award-winning biography of Nash, gripped by the twists and turns of his improbable story, which Hollywood saw full of cinematic promise.

The true story seemed ready-made for the big screen. A driven, arrogant, and socially awkward intellectual with eyes only for academic stardom, Nash was disdainful of pedagogical convention. His singular outlook led him to mathematical discoveries that reinvented the subject of game theory, which has become a mathematical pillar of economics and sociology, and later to breakthroughs that recast modern geometry, as well as the equations that describe the turbulent flow of fluids.

In the 1950s, as a consultant on nuclear strategies at the top-secret Rand Institute and a regular visitor to Princeton and the Institute for Advanced Study, Nash hobnobbed with the great scientists of the day. But, while his scientific career rocketed upward, his personal life lurched along a chaotic path of confused and seemingly conflicted sexual identity (leading to the loss of his security clearance), the fathering of an illegitimate child, and finally a difficult but faithful marriage to Alicia Larde, a South American physicist who had been drawn to both Nash's handsome appearance and his seemingly assured intellectual status in the scientific firmament.

All of this happened before Nash turned 31. Then, just as quickly, it became a career cut short by the sudden and completely debilitating onset of paranoid schizophrenia, leading to a 35-year wandering about in an emotional and psychological desert, in and out of institutions, subjected to shock treatments and mind-numbing drug therapies. Unable to work or to think, harassed by the demons of a cold war-tinged hallucinatory nightmare, he survived through the support of friends and, most important, his long-suffering wife, who, almost single-handedly, raised their child while working numerous jobs and managing Nash's illness.

Then came Nash's slow but steady re-awakening, simultaneous with a growing recognition of the import of his work, which culminated in the Nobel. A hubris-laden hero; a life begun, lost, and regained; creativity entwined with madness; redemption by the love of a good woman. Oscar, here we come!

So it was with a little shock, and much dismay, that I sat
through the movie, *A Beautiful Mind*, squirming amid the
conflation of fact, fiction, and fantasy, and the reappearance
of all the old mathematician/scientist stereotypes. The
robotic graduate student who speaks to women using language
from a high-school sex-ed book; the suggestion that the
paranoid delusions helped and even inspired Nash's work,
roasting once again that favorite chestnut of madness equals
genius (especially in mathematics); and the cliche of
mathematician as code-cracker. And then a postscript that
leads anyone not knowing the story to believe that all that
preceded was true, from the weird pen-giving ceremony at
Princeton (what was that?!) to the now avuncular John Nash
happily teaching freshman calculus there.

PERHAPS THE STAGE is better-suited than the screen to the telling of a mathematical story. The best plays create an entire world within the imagination from a sharp script and the necessarily few and relatively subtle hints that even the most well-appointed production might provide. The reliance on language rather than spectacle to build a self-consistent world mirrors a science whose chief tool is the finely chiseled argument for which technology can appear only as a servant in liege to logic. Be it mathematics or drama, even the most magical computer visualization will never replace a beautifully crafted, cogent argument. The simpler scale, the immediacy, even the smaller budgets hint that the play is to the movie as mathematics is to the big science of the laboratory or engineering center.

David Auburn's Pulitzer Prize-winning *Proof* gives a real sense
of the process of mathematical discovery and argument, while
still packing them in on Broadway. That play is the best of a
collection that includes Michael Frayn's popular
quantum-mechanical and uncertainty-driven drama *Copenhagen* and
even a number-theoretic musical, *Fermat's Last Tango*, which
brought to life the intellectual dance of problem solving.

Somehow, when mathematics goes to Hollywood, all hell breaks loose. Hyperbole and exaggeration come with the change in scale and the attendant need, and desire, to appeal to the broadest possible audience. Facts morph subtly, and sometimes less so, into fiction and fantasy. Perhaps these are the wages of fame, the price paid for exposure on the big screen.

The real story of Nash's life is rife with poetry, irony, and metaphor that could have, should have, fueled a masterpiece. A brilliant mathematical career suspended by a paranoid schizophrenia manifested in a tendency to see the hand of the government in everything. Messages encoded in newspapers and the stars and television. All phenomena devolving back to a world in which every single act and action must be part of some grand Nash-centric universe. All of the world part of a grand design and pattern whose revelation became a turbulent-minded obsession.

But this is, in essence, a mathematician's worst nightmare: pattern seeking taken to its infinite limit; mathematical skill and talent run amok. Surely, if we were forced to sum up in a single word the guiding principle of mathematics and mathematical research, it would be the principle of pattern. Numbers as the distillation of the pattern common among equinumerous collections of objects; geometry as the spatial patterns of the Platonic proxies of the real objects around us; logic as the pattern of argument and reason.

And the patterns don't stop there. We then connect those first-order patterns with further patterns: Any number ending in 0 is divisible by 10; any number whose digits add to a multiple of nine is divisible by nine; the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the hypotenuse. Patterns upon patterns upon patterns, without end.

NASH'S NOBEL-WINNING WORK was the distillation and axiomatization of human commerce. It began with a brief, jewel-like work: a multifaceted, sparkling seven-page paper titled "The Bargaining Problem." This was a fitting title for a man whose life seems itself a real Faustian bargain, in which flashes of brilliance and clarity were traded for long periods of depression and confusion, creativity lying fallow, waiting for a day in which the ravages of disease might be plowed under so that productivity might spring anew.

"The Bargaining Problem" marked Nash's foray into the subject of game theory, but his lasting achievement followed soon with the publication of "Equilibrium Points in n-Person Games." Here Nash achieved a startlingly broad extension of the utility and applicability of game theory for economics, moving it out of the impossibly idealized and simple model of two-person zero-sum games, in which one actor's loss is the other's gain, to the highly nuanced and real-life scenarios of equilibrium through compromise, a result of many players sharing and hiding information, forming coalitions and cartels -- in short, acting as people do.

Nash laid waste to Adam Smith's Invisible Hand, that unseen force guiding any competitive market to natural equilibriums of price and value. He instead made possible an analytic theory of a world of economics in which personal interest and gain were fundamental forces, a world in which any individual's actions were of worth and mattered, a world without a divine cosmic scheme. Nash's work made irrelevant an omniscient and omnipotent tyrant that, later, while in the thrall of his illness, he found impossible to deny.

Nash's game-theoretic work places the real world of human interaction in the confines of the ideal and Platonic, and his achievements in geometry were of the same flavor. The physical world is a world modeled not by the perfect lines, angles, and circles of Euclidean geometry, but one in which Riemannian geometry holds sway, a description of shape and distance, of spatial (rather than emotional) relationships, that seems to lie beyond the possibilities of rigid Euclidean description. Riemannian geometry is the mathematics of Einstein's and Hawking's space-time, a geometry capable of describing a curved universe, black holes, and knots of stringlike tendrils of energy.

On the surface, Euclid's and Riemann's worlds would appear to be completely different. The classic example of a Riemannian geometry is the surface of a sphere. In this setting, even our familiar triangles acquire puzzling possibilities. Its gentle, constant curvature entails a land where a triangle's angles add up to an amount greater than the Euclidean, or "flat," 180-degree paradigm. Nevertheless, the sphere can be seen in the mind's eye and even modeled by hand, providing a realization of this exotic two-dimensional world (on the surface of a sphere, two numbers -- latitude and longitude -- suffice to give a precise location) within a Euclidean three-dimensional world.

But what of Riemannian spaces of higher dimensions and of more elaborate and complicated curvatures, whose twists and turns would seem to defy any such mundane coordination? These spaces are beyond imagination, defined only as solutions to families of polynomial equations, just as a sphere can be defined as the locus of points at a unit distance from some ideal center. It was with a shock to many mathematicians and scientists that in the early 1950s, in his paper "The Imbedding Problem for Riemannian Manifolds," Nash showed that, in fact, many of these Riemannian worlds (more precisely, Riemannian "manifolds" of sufficient smoothness) could actually be described in a Euclidean setting, provided that enough dimensions are used, showing that under certain conditions, the real and Platonic worlds can coexist.

These major intellectual achievements stand like a synecdoche for a mind bent on integrating real and imagined worlds, or a life bent on finding order in the messiness of real relationships, and even, identity.

Walking the tightrope between the Platonic and the worldly is the hallmark of great applied mathematics. Great ideas can be like thunderbolts, brilliant flashes of illumination that explode from a tumultuous and sometimes dark cloud of thoughts born of a serendipitous collision of nature and nurture. Revealing by a powerful light that facet of the real world deserving of the distillation into theorem and proof and, in so doing, unearthingthe essence of a phenomenon. Nash's model of human behavior in his theory of noncooperative games, his breakthrough achievement in geometry, his work on equations that describe turbulent fluid flow -- each of these was such a thunderbolt.

Ultimately, the creation of a beautiful mathematical model is about making choices -- what to omit and what to include, what to ignore and what to magnify -- and in this way, it is like any work of art. It was Edna St. Vincent Millay who said that "Euclid alone has looked on beauty bare." Perhaps it would take someone who was a little bit of a mathematician to turn the embarrassment of riches that is the truth of John Nash's remarkable life into a beautiful movie.

*Daniel Rockmore is a professor of mathematics and computer
science at Dartmouth College.*