[Air-l] Exploiting a Beautiful Mind
john.white at wku.edu
john.white at wku.edu
Tue Jan 22 08:56:59 PST 2002
This article from The Chronicle of Higher Education
(http://chronicle.com) was forwarded to you from: john.white at wku.edu
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The following message was enclosed:
For the technology and hollywood thread.
We often think of it as gadgets, but technology should change
the way we interact with our environment...not just the way we
make coffee.
:)
--JW
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From the issue dated January 25, 2002
Exploiting a Beautiful Mind
By DANIEL ROCKMORE
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 builda 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.
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Copyright 2002 by The Chronicle of Higher Education
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