Posted March
27, 2002:
Beautiful opinions
about Mind: Exploiting
a Beautiful Mind

By DANIEL ROCKMORE '84

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 clichÈ 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, unearthing the 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.