## New York Story: The Courant Institute Meets Broadway

**December 21, 2000**

CATHERINE (Mary-Louise Parker): HE's dead; I don't need any PROTEGES around. HAL (Ben Shenkman): There will be others. . . . You think I'm the only one? People are already working over his stuff. Someone's gonna read those notebooks. . . . CATHERINE: He's my father, I'll do it. . . . HAL: You don't have the math. It's all just squiggles on a page. You wouldn't know the good stuff from the junk. . . . CATHERINE: I know mathematics. HAL: It would be pretty high-order. It would take a professional to recognize it. Proof, Act I, Scene 1 (Photo courtesy of Joan Marcus)

For several months this year, New York theater goers had their choice among a surprising array of plays with mathematical themes. Outstanding among them was *Proof*, by the young playwright David Auburn, which moved to Broadway in October after a widely acclaimed, extended Off-Broadway run at the Manhattan Theatre Club.

The theater hasn't been unique among the arts in providing an unusual new home for mathematical ideas. *A Beautiful Mind*, Sylvia Nasar's 1998 biography of John Forbes Nash, Jr., won a National Book Critics Circle Award, and Nasar recently sold the movie rights---Good Will Hunting and *Pi* don't seem to mark the end of filmmakers' fascination with mathematics. And the triangle at the heart of a serious and compelling new novel---Rebecca Goldstein's *Properties of Ligh*t---is united by intense collaboration on the mathematics underlying one of the fundamental problems in modern physics.

With all this creative activity unfolding, the Courant Institute of Mathematical Sciences, New York University, in collaboration with the Manhattan Theatre Club and with the support of the Alfred P. Sloan Foundation, put together a one-day symposium on issues and ideas raised by *Proof*, just as the play was going into previews on Broadway. Held October 16 at the NYU law school, the symposium featured three panel discussions:

I. What's a Proof and What's it Worth? Chaired by Peter Sarnak, the panelists (four mathematical scientists---Arthur Jaffe, Dusa McDuff, Michael Rabin, and Jack Schwartz---and two philosophers---Kit Fine and Thomas Nagel) discussed the nature of mathematical proof in relation to the play.

II. Women and Proof. Five female mathematicians, chaired by Margaret Wright, discussed the position of women in the mathematical sciences, touching in particular on issues faced by Catherine, Proof's main character.

III. Images of Proof in Performance and Prose. Ben Shenkman, the actor who portrayed a young mathematician in *Proof *(and who had also starred in *Pi*) and playwright David Auburn joined Rebecca Goldstein and Sylvia Nasar in a discussion chaired by Michael Janeway, a professor of journalism at Columbia University.

**The Proof**CLAIRE: What IS it?

HAL: Oh, uh, it's a result. A proof.

I mean it looks like a proof. I mean it is a proof, a very long proof, I haven't read it all of course, or checked it, I don't even know if I COULD check it, but if it IS a proof of what I think it's a proof of, it's . . . a very . . . IMPORTANT . . . proof.

CLAIRE: What does it prove?

HAL: It looks like it proves a theorem . . . a mathematical theorem about prime numbers, something mathematicians have been trying to prove since . . . since there were mathematicians, basically. Most people thought it couldn't be done.

The title of the play is "Proof," not "The Proof," but as the preceding lines (Act I, Scene 4) show, a specific proof is central to the action. The play's element of mystery, it has been found among the papers of Robert, one of the central triangle of characters, father of Catherine (and Claire) and former thesis adviser of Hal. Before years of insanity, Robert had been, in Hal's words, "the BEST. . . . He revolutionized the field twice before he was 22." A few years before his death (the play opens the day before his funeral), he had emerged from his madness.

A lot is happening here, and the symposium did an impressive job of capturing it, with ideas from the play sparking connections with current mathematical research and, given the choice of panelists, connecting with other ideas and characters, both real and fictional.

Computer scientist Michael Rabin of Harvard University, a member of Panel I, drew connections between "the new field of computer science and the ancient field of mathematics," citing some implications for mathematical proof. Countering the view of a mathematical proof as "a paragon of certainty," he introduced the audience to the concept of randomization in computations. With cryptography and e-commerce as his examples, he made a convincing case that real lives are affected by developments in the, to many in the audience, oxymoronic area of probabilistic proof. But it was with another development in computer science that Rabin found the deepest meaning for the play.

*Panelist Michael Rabin introduced the audience to the concepts of probabilistic and zero-knowledge proof. *

As portrayed in *Proof*, mathematics is a very solitary activity, he pointed out. Catherine had a proof, and she was very generous with it. Making use of techniques of interactive proof, and specifically zero-knowledge proof, Rabin explained, she could have proved to the satisfaction of a research journal that she had a proof, without revealing anything of its nature or details.

What Catherine needs to prove is at the heart of the play, entering into the love, trust, and betrayal that make the characters' lives so moving for the audience. In writing *Proof*, Auburn explained at the symposium, he was "trying to draw an analogy with the kinds of proof we demand from people in a relationship"; although worried about using a rigorous scientific concept "promiscuously," he "couldn't resist doing it."

**Stereotypes Unchecked**The lone genius in the garret is but one of the stereotypes about mathematics woven into

*Proof*. (It's a stereotype that doesn't fit the person who comes to mind as the basis for the character of Robert: John Nash. Nash "was no Ramanujan, "Sylvia Nasar pointed out in the final panel discussion. Highly trained, "Nash had a sense of what the true problems were," Nasar said; for him, mathematics was a competition, and he set out to beat the other young mathematicians in solving those problems. Auburn, although aware of the story of Nash, said that he hadn't consciously based the character on him.)

*Sylvia Nasar, a member of the third panel at the Courant Proofsymposium. She is the author of A Beautiful Mind, a biography of JohnForbes Nash, Jr., with whom the Proof character Robert has much incommon. In awarding her the 2000 Communications Award for the book,the Joint Policy Board for Mathematics commended her for giving "the general public a glimpse into the world of mathematical research and an understanding of its impact on society." It is thanks to plays like Proof and Arcadia, and movies like Good Will Hunting, Nasar said at the symposium, that the public has come to perceive mathematics as glamourous [!], related to music and art, rather than as "boring number crunching."*

The perception that great work in mathematics is done early in a career is another of the stereotypes to appear in the play. Hal laments his unspectacular output: "At a certain point you readjust your expectations. . . . I'm 28 . . . on the downhill slope." Robert himself at one point tells Catherine and Hal that "it does get harder. It's a stereotype that happens to be true, unfortunately for me---unfortunately for you, for all of us." Members of Panel II strongly disputed this stereotype, pointing out that many mathematicians achieve their greatest results at 40 or beyond.

David Auburn set out to write a play not about mathematics, but about the fear of inheriting mental illness, in a character the same age as a parent when the illness struck. Coming to mathematics "through the back door," as he explained in the third panel discussion, he saw it turn out to be completely apt, especially when combined with the idea that mathematicians do great work only when young.

Rebecca Goldstein's new novel and earlier *Mind-Body Problem* both depict scientists who, like Auburn's Robert, have been unable to continue their work. She pointed to a "spooky similarity" between the *Proof *triangle and that of her own aging, mentally diminished physicist, the daughter who lives with and cares for him (and has inherited much from him), and his younger (male) prot�g�. As they discovered during the discussion, both Auburn and Goldstein had been deeply affected by *A Mathematician's Apology*, by Hardy's despair to be writing not mathematics but "about mathematics," lacking "the freshness of mind, the energy, or the patience to carry on effectively with my proper job."

Mental instability and mathematics, Cantor, G�del, Nash . . . it's an irresistible theme, Auburn said, one that jumps out at a nonspecialist. At least one member of the audience wasn't comfortable with the theme: Robert Osserman (who set the standard for events like the Courant symposium with last year's Berkeley discussion of Tom Stoppard's *Arcadia*), quoted a *New York Times* review of *Proof* to the effect that Catherine "has witnessed firsthand the jumble that mathematics can make of a working brain." There's "something subversive here," Osserman said, and it turns up as well in *Pi*, in which the older mathematician has had to stop doing mathematics, and in *Uncle Petros & Goldbach's Conjecture*.

**Women in Mathematics**HAL: Really original work---it's all young guys.

CATHERINE: Young guys. . . . .

HAL: There are some women.

CATHERINE: Who?

HAL: There's a woman at Stanford, I can't remember her name.

CATHERINE: Sophie Germain?

HAL: Yeah? I've probably seen her at meetings, I just don't think I've met her.

Catherine sets Hal straight, relating the story of Sophie Germain's correspon-dence with Gauss, in which she used the name Antoine-August LeBlanc.

HAL: Did he ever find out who she was? Gauss.

CATHERINE: Yeah. Later a mutual friend told him the brilliant young man was a woman.

He wrote to her: "A taste for the mysteries of numbers is excessively rare, but when a person of the sex which must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in penetrating the most obscure parts of them, then without a doubt she must have the noblest courage, quite extraordinary talents and superior genius."

I memorized it. . . .

Stereotypes about women in mathematics are implicated in the situation of *Proof'*s Catherine, and, as made clear by every one of the Panel II speakers, have much to do with the careers of women in mathematics today. Margaret Wright opened the panel's discussion with some facts about women in mathematics: In 1981, the American Mathematical Society's Gibbs Lecture was given for the first time by a woman (that woman, Cathleen Morawetz, was a member of the panel); 4.5% of the members of the two mathematics sections of the U.S. National Academy of Sciences are women, and 50% (four) of those women were also present at the symposium. In 1999, in Group I U.S. mathematics departments (the top 48 as ranked by the National Research Council in 1996), women made up 6.4% of tenured faculty at public institutions and 4.6% at private institutions. Despite the small numbers, the situation is improving, Wright said, but "the problem is not completely solved."

Mary Pugh, an assistant professor of mathematics at the University of Pennsylvania, added another layer to Wright's statistics on the numbers of women at various steps in the mathematical hierarchy: The numbers of women decrease steadily as the levels ascend. Pugh cited the huge numbers of candidates for positions of all types, from assistant professor to department chair, journal editor, or conference speaker. Study after study reveals unconscious biases, she said; it's hard for an institution to take a chance on the nonstandard option of appointing a woman.

Whatever faculty it is that's highly prized is the one women are said to lack, panelist Dusa McDuff (also a member of Panel I) pointed out, quoting Rousseau and Kant; women have been characterized as having deficiencies both for creativity and for reasoning. "These old ideas resonate," she said.

Jean Taylor considered *Proof* with some gender reversals: What if the person caring for Robert had been a son instead of a daughter? First, she suggested, it's unlikely that a son would be taking care of an elderly parent. Second, the son of a famous mathematician would be readily accepted as a promising mathematician, with no elaborate proof required.

For panel member Karen Uhlenbeck, Catherine was a plausible role model; the style of her interactions "felt a lot like when I was an undergraduate." Uhlenbeck focused on two themes in the play: the way the burden for the care of elderly parents tends to fall on women, and the idea that women can't be brilliant. Uhlenbeck, one of the NAS members referred to by Wright in her introduction and also a former MacArthur fellow (and a recipient of this year's National Medal of Science, announced as this issue of *SIAM News* went to press), is an obvious counterexample to the latter point, yet she spoke with complete sincerity about obstacles facing women in mathematics.

Cathleen Morawetz, whose career (at Courant) has been marked by success and abundant recognition, recounted the tortuous path by which she reached that happy state; low points in the early days include rejection from actuarial school on the grounds that she would be likely to go off and have children and, once she had settled on aeronautical engineering, with a dream of studying with von K�rm�n at Caltech, a closed door there too: Caltech did not admit women undergraduates until 1970.

Uhlenbeck feels that she and Morawetz have led "pioneering lives," and her sense of discouragement arises from a "grim view" of the opportunities for women in the profession today. She concluded with the suggestion that the panel, or perhaps a slightly expanded version of it, meet again "to try to figure out how things can be fixed."

One thing about Panel II: When the speakers had finished, members of the audience were lined up at the microphones to comment and ask questions. The organizers of the *Proof* symposium, having seized on an upbeat local event to draw the attention of wide new audiences to mathematics, had got people thinking and offering opinions on the discipline-its fascination, its achievements, and its failings.---*GRC*

**For Further Information**

*Proof*is now playing, with the original cast, at the Walter Kerr Theater. At least two mathematicians have reviewed the play: Don Albers, in the

*MAA Focus*(August/September 2000), and Dave Bayer, in the

*Notices of the AMS*(October 2000). (Michael Frayn's

*Copenhagen*, among the season's plays on scientific/mathematical themes, is also on Broadway; it won this year's Tony Award for Best Play.)

The Courant symposium marked the beginning of a three-year collaboration between the Sloan Foundation and the Manhattan Theatre Club. Doron Weber of Sloan (in attendance at the Courant event with former SIAM president Hirsh Cohen, also of Sloan), explained that the program was established to support new works that present the lives of scientists and mathematicians in compelling ways. (See www.sloan.org.)

Sylvia Nasar's *A Beautiful Mind* was reviewed in *SIAM News* in June 1999. Also available in the *SIAM Ne*ws archives are an article on the Berkeley/Tom Stoppard event and Philip Davis's review of *Uncle Petros*.

*Properties of Light*, by Rebecca Goldstein, was published this year by Houghton Mifflin.