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## The Not-So-Mad Mathematician

*Written by Susan Goldstine, Associate Professor of Mathematics*

Georg Cantor was the first mathematician to develop a theory of the infinite. Then he went insane.

The desire to link these events is hard to resist. In fact, there is no direct connection between the theory and the mental instability. Cantor did make regular visits to sanatoria for mental breakdowns toward the end of his life. But his condition was chronic depression, and his first severe bout came 10 years after his most important ork on infinite cardinalities.

Moreover, there is nothing in the study of comparing infinities that would drive a person to madness. As with much in mathematics, measuring sizes of infinity is largely a matter of careful bookkeeping, albeit with staggering results.

Nonetheless, I often tell my students (and then dutifully debunk) the story of Cantor’s insanity. I’m not alone in this, and I think it points to the love-hate relationship that mathematicians have with the notion of the insane mathematical genius. On the one hand, we live in a culture in which mathematical talent is seen as freakish, in which “Oh, I was never any good at math” is a badge of normality. The last thing we need is the image that people who do math are crazy. On the other hand, the tinge of madness lends a certain dark glamour to a field that is often perceived from the outside as dull and lifeless.

Perhaps this is why the two most popular movies about mathematicians in the past decade, “A Beautiful Mind” (2001) and “Proof” (2005), each focus on a mathematician struggling with debilitating mental illness. And the most famous Hollywood depiction of the mathematical community in the previous decade, “Good Will Hunting” (1997), is the story of a troubled young math genius and his psychological rehabilitation. With the mythology of mathematical madness in mind, I decided to give these films a closer look.

“A Beautiful Mind,” an adaptation of the book by Sylvia Nasar, tells the life story of John Nash (played by Russell Crowe), a real-life Nobel-prize winning mathematician afflicted with schizophrenia. I have resisted watching it, even though I felt compelled to do so, because many people view my profession through the lens of this film. I was reluctant because I know enough about John Nash and his work to be irked by Hollywood’s inevitable warping of facts. And to be honest— Russell Crowe as a Princeton mathematician? Please. This article gave me the impetus to finally sit down and watch the movie.

In the film, mathematics and insanity are inextricably linked. Nash’s paranoid schizophrenia ultimately lead him to compulsively search for patterns, and “A Beautiful Mind” blurs the boundary between his delusional quest for order and his proper mathematical work. There is a scene early in the film in which Nash looks at screens full of numbers in a National Security Agency facility and patterns emerge from the noise; clumps of numbers light up and dance off of the screen as he scans them. The episode is later revealed to be a paranoid delusion, but the image of patterns magically coalescing is one of the best representations I’ve seen of what mathematical insight feels like.

Frustratingly, the scenes depicting Nash’s genuine mathematical breakthroughs seem anemic by comparison. “A Beautiful Mind” works very hard to show the inner workings of the mathematical community and the schizophrenic mind, neither of which is an easy task. It does both aptly, but the end result strikes me more as an admirable expository feat than as a great movie.

By contrast, I admire “Good Will Hunting” as a film. It stars Matt Damon and Ben Affleck, who wrote the screenplay back when no one had heard of them, and features Robin Williams playing his stock benevolent healer role. Will Hunting, Damon’s character, is a juvenile delinquent and a self-taught mathematical genius.

This is a very different film from “A Beautiful Mind,” but there are intriguing overlaps. Early in “Good Will Hunting,” while trying to solve a mathematical puzzle he has spotted while cleaning floors in the MIT mathematics department, Will stares into a mirror and scrawls equations across his reflection with a marker. Early in “A Beautiful Mind,” Nash stands in front of his dorm room window jotting down formulas and diagrams with a wax pencil and stares through them. In each case, the camera shows us the mathematics floating in front of the man’s gaze, a strange confl uence of the tangible and the intangible.

Despite this common ground, mathematics and mental health have a very different relationship in “Good Will Hunting.” Will’s delinquency and instability stem from a traumatic childhood, not from some internal defect linked to his talent. Gerard Lambeau, a prominent MIT professor, discovers his gift and rescues Will from a jail term on the condition that he study math with Lambeau and undergo therapy with Sean McGuire, Williams’ character. Instead of mathematics being intertwined with mental illness, it offers escape from it.

But mathematics is just a stepping stone. In the arc of the plot, Will comes to terms with his mathematical brilliance and then casts it off to move on with his life. I find “Good Will Hunting” much more satisfying as a coming-of-age story than as a study of the mathematical life.

For me, it is “Proof,” the film adaptation of the David Auburn play, which strikes the perfect balance between storytelling and capturing the world of mathematics. It also has the most complex and satisfying relationship between insanity and mathematical insight.

At the center of “Proof” is Caroline, played by Gwyneth Paltrow, who is struggling with two potentially inherited conditions: mathematical talent and mental illness. She is the daughter of Robert Llewellyn, played by Anthony Hopkins, a famous mathematician who, like John Nash, suffers from schizophrenic delusion interrupted by a miraculous remission. As the story unfolds, mathematical discovery is both a conduit to mental instability and a passage back to sanity.

On reflection, part of the reason “Proof” speaks to me more that the other two films is the presence of a third mathematician, Hal, a former student of the senior Llewellyn played by Jake Gyllenhaal. Hal is neither famous nor insane. He is an average math professor, a little self-centered but essentially a decent guy who tries his best to navigate the maelstrom of Caroline’s family.

When it comes down to it, most mathematicians have more in common with Hal than with Caroline or Will or John Nash. And at the end of the day, the romance of the solitary mad genius can’t hold a candle to the richness of shared human experience.