# Drama in Numbers

## Putting a passion for mathematics on stage

As the curtain rises, an illuminated mathematical expression dominates the scene. "Do you see that theorem?" the narrator asks. "In 1637, Pierre de Fermat . . . wrote it down in the margin of a book. Then he added this tantalizing note." A spotlight suddenly reveals a bearded, bewigged, flashily dressed Fermat, who promptly sings,

"I have discovered a truly marvelous proof, a truly marvelous proof of this, which this margin is not large enough to contain."

Next, a quick succession of vignettes portraying centuries of immense mathematical frustration spawned by Fermat's never-recorded proof–known as Fermat's last theorem–unfolds across the checkerboard floor of a sparsely furnished stage.

The musical number that closes this prologue introduces the play's hero–Daniel Keane, a modestly dressed, mildly bewildered, Princeton mathematician who is claiming to have proved Fermat's last theorem. Pursued by a gaggle of reporters, Keane fumbles to explain what he has done. One reporter inquires, "What is a proof, and who cares?"

"Fermat's Last Tango" is billed as a musical fantasy inspired by real-life Princeton mathematician Andrew Wiles and his encounters with Fermat's last theorem (SN: 11/5/94, p. 295). It's one of several mathematics-rich stage productions of the past few years. Although these plays, with their overtly mathematical themes and number-enthralled characters, have especially captivated mathematicians, they have also attracted remarkably diverse and enthusiastic audiences.

David Auburn's "Proof," which hinges on the disputed authorship of a mathematical work, won the 2001 Pulitzer Prize for drama and is still running on Broadway. Tom Stoppard's 1993 play "Arcadia," which brings fractal geometry and chaos theory into a 19th-century setting, continues to thrive in a variety of venues.

"Each of these plays gave me considerable pleasure, albeit in very different ways," says Robert Osserman of the Mathematical Sciences Research Institute in Berkeley, Calif. He has interviewed the playwrights of "Proof" and "Arcadia" as part of public programs including excerpts from the stage productions.

These three plays depict the pursuit of mathematics as a painful joy–an intense endeavor that can unveil an alluring beauty in ideal objects or bare mathematical symbols. In "Fermat's Last Tango," Keane lauds the power and purity of mathematics. The beauty of numbers is everywhere, he tunefully proclaims.

At the same time, the pursuit of mathematics can humble or even crush a practitioner who fails to measure up to the field's exacting demands. The plays bring to mind people's powerful needs for recognition and connection with others doing similar work. The scripts explore the counterpoint between pure logic and the emotional complexities of everyday life, and they elucidate the meaning of *proof* in different settings.

**Beautiful proof**

Auburn's play "Proof," first produced in 2000, centers on the younger daughter of a brilliant mathematician. The father, Robert, had become mentally unstable in his later years. Emotionally drained after years of taking care of him and neglecting her own education, 25-year-old Catherine must face her father's death, deal with her manipulative, estranged sister, and cope with the amorous attentions of a former student of her father.

The plot centers on the authorship of a potentially outstanding mathematical proof in number theory, which was found among notebooks filled with Robert's less-than-lucid scribbles. At first glance, the play appears to be both a mystery and a romantic comedy. On a deeper level, it raises questions about proof in human relationships as well as in math and about the stereotype that links youth and creativity.

"Auburn's script is well-crafted, fast moving, and marked by sparkling dialog," Donald J. Albers of the Mathematical Association of America remarked in a review of the play. "The mathematicians portrayed in 'Proof' come off as delightfully human and rather attractive people with whom you would probably enjoy having dinner."

Auburn himself has stated that he did not set out to write a play about mathematicians. He was interested in exploring the question of whether mental illness, as well as talent, can be inherited. And he was attracted to the idea of sisters fighting over an item of ambiguous significance found after their parents had died. The mathematical connections came later.

Auburn ended up immersing himself in works depicting the mathematical mind. He read popular books about mathematicians such as Paul Erds, Srinivasa Ramanujan, and John Forbes Nash, whose biography became the basis of the 2001 film *A Beautiful Mind*. The example of Wiles working in his attic for 7 years to prove Fermat's last theorem gave Auburn a sense of the romance of mathematical work.

Members of the mathematics department at New York University offered Auburn advice and visited rehearsals. Auburn also sent copies of the play to other mathematicians, including Jean E. Taylor of Rutgers University. "I was surprised to find myself quite captivated by it," Taylor says, noting also that she was jarred by a minor mathematical error in the preliminary text.

In "Proof," Catherine's spiritual mentor is the 19th-century mathematician Sophie Germain, who sent her own highly original mathematical results to Carl Friedrich Gauss under a man's name because women then had no credibility in mathematics. Her work centered on certain numbers that are now known as Germain primes. Taylor pointed out that the example of a Germain prime given in the preliminary text was missing the term "+ 1."

"When I first went to see 'Proof' and that moment came up in the play, I was happy to hear the 'plus one' clearly spoken," Taylor says.

When the play opened just over 2 years ago, the Courant Institute in New York City hosted a daylong symposium that addressed some of the issues raised in the play, including the role of women in mathematics.

In Taylor's view, the play accurately portrays some of the barriers that women still face as professional mathematicians. A man, for example, wouldn't be expected to forego his education to take care of an ailing parent. His claim to have authored a proof would also be considered more credible. "Try to imagine the play with the sex of the characters [Catherine and her father's student] reversed," Taylor says. "Imagine that and you see that everything changes."

In "Proof," Auburn found a witty, engrossing way to explore the notion of proof in several different senses–in the idea of a mathematical proof with its particular iron-clad inevitability, the notion of establishing the authorship of an intellectual work, and the daily proof that people seek to reassure themselves of the stability of reality and of their personal relationships.

**Arcadian chaos**

Like "Proof," the play "Arcadia" features a very clever young woman who has remarkable mathematical insights yet faces the skepticism of well-trained scholars who are less original in their thinking.

Thomasina Coverly is a mathematically precocious teenager of the early 19th century. Noting the irregular or branched nature of natural forms such as mountains and trees, she sets out to invent by trial and error the first mathematical framework to portray such structures.

Later, she playfully writes in her notebook, "I, Thomasina Coverly, have found a truly wonderful method whereby all the forms of nature must give up their numerical secrets and draw themselves through number alone." Then she slyly adds, "This margin being too mean for my purpose, the reader must look elsewhere for the New Geometry of Irregular Forms discovered by Thomasina Coverly."

The play nimbly switches back and forth between the early 19th century and modern scenes that take place in the same house. One of the contemporary characters is mathematician and biologist Valentine Coverly, who discovers Thomasina's notebook and marvels at her insights. As it happens, Valentine himself is modeling population changes among grouse on his family's estate. Valentine tries to explain iteration, algorithm, chaos, and other mathematical terms to a best-selling author and garden historian who's visiting.

Stoppard's play delves into the unsettling experience of facing new ideas, the interplay of hypothesis and evidence, and the role of human character in discovery. Yet the conversation remains sprightly and amusing, and the often-befuddled characters are engaging.

"I consider ['Arcadia'] one of the outstanding plays of the last couple of decades, with or without the math," Osserman says. "Stoppard is one of our greatest playwrights, and 'Arcadia' is one of his best–if not the best–of his plays. . . . Apart from that, the amount of serious mathematics he fits in–in a totally natural way–is just amazing. And unlike most plays that touch on mathematics, he really gets it right."

"Arcadia" serves as an antidote to the common impression that mathematics putters along in inscrutable increments and hasn't changed much since Euclid's time. The play also brings mathematics to, as Valentine puts it, ". . . the ordinary-sized stuff which is our lives, the things people write poetry about–clouds–daffodils–waterfalls–and what happens in a cup of coffee when the cream goes in."

Mathematician Robert L. Devaney of Boston University originally learned about "Arcadia" from his son, an actor in New York City. When the play was scheduled to be performed at Boston's Huntington Theater, Devaney became an informal advisor. "I spoke several times to the director as he was putting the play together and eventually he had me talking to the cast about the mathematical ideas in the play, showing them fractals, and so on," he says. "I became the 'chaos consultant' for the production."

"Since my original involvement, I have helped out a number of school productions by getting together the math, humanities, and science teachers and students to understand the different aspects of the play," Devaney adds. "I can't imagine a better way to get liberal arts students involved in some contemporary mathematics."

Devaney has developed an informative Web site describing and animating some of the mathematical ideas lurking in Stoppard's play (*see* http://math.bu.edu/DYSYS/arcadia/).

"Arcadia" was not Stoppard's first venture into mathematical terrain. His 1972 play "Jumpers" features a professor of moral philosophy who tangles tangentially with Zeno's paradox, the nature of infinite series, and other mathematical questions while his wife struggles with a corpse in the bedroom.

"It is vintage Stoppard, enormously clever, very funny in parts, but I don't think it can compare to 'Arcadia,'" Osserman says.

**Song-and-dance numbers**

"Fermat's Last Tango" is a rare foray of mathematics-inclined playwrights into the arena of musicals. Written by Joshua Rosenblum and Joanne Sidney Lessner, "Fermat's Last Tango" revels in the drama and passion associated with the centuries-long quest to elucidate Fermat's tantalizing marginal hint.

"As a musical fantasy on the subject of wrestling with the mysterious process of discovering a proof, 'Fermat's Last Tango' is in a class by itself," Osserman says.

The musical focuses on the traumatic period between the 1993 discovery of a flaw in the proof Wiles originally presented of Fermat's last theorem and Wiles' circumventing of that deficiency a year later. Keane, the character standing in for Wiles, must confront the possibility of failure. Riddled with doubt, taunted by a mean-spirited Fermat, and haunted by a ghostly chorus of Pythagoras, Euclid, Isaac Newton, and Gauss, Keane returns to his singular, possibly ill-fated pursuit. He retreats to the attic, again leaving his wife as the "math widow" she had been for so many years before.

"This drama is so powerful because it describes the clash between frail humanity on the one hand and intellectual destiny on the other," says mathematician Arthur Jaffe of Harvard University. "And it all rings true."

The musical is also cheerful, clever, and appealing. The extraordinarily inventive lyrics of its wide-ranging score are laced with numerous mathematical and historical references. Indeed, the play's mathematical vocabulary is surprisingly sophisticated.

Imagine a song in which the phrase "Taniyama-Shimura conjecture" is heard not just once but several times.

"Of all popular portrayals of mathematics in the media, I believe that only this play contains real mathematical content," Jaffe contends. "The authors had real insight.

The characters think about mathematics just the way a real mathematician would."

"Fermat's Last Tango" was originally performed by the York Theater Company in New York City in December 2000. Fascinated by the play and encouraged by Wiles, Jaffe went to considerable trouble to have the production videotaped on behalf of the Clay Mathematics Institute in Cambridge, Mass., where he worked until recently. The institute continues to sell the tape (see http://www.claymath.org/Publications/Fermats_Last_Tango/). A London production of the play is now is the works.

Mathematics on stage is a scarce commodity. Only rarely do playwrights and composers delve into the seemingly forbidding world of mathematicians and their abstruse concerns. The recent spate of successful productions may be just the start of a beautiful marriage between mathematics and the theater.

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Donald J. Albers

Mathematical Association of America

1529 18th Street, N.W.

Washington, DC 20036

Web site: [Go to]

Robert L. Devaney

Department of Mathematics

Boston University

Boston, MA 02215

Web site: [Go to].

Arthur Jaffe

Department of Physics

Harvard University

17 Oxford Street

Cambridge, MA 02138

Web site: [Go to]

Robert Osserman

Mathematical Sciences Research Institute

1000 Centennial Drive

Berkeley, CA 94720-5070

Web site: [Go to]

2000. Math and science on stage and screen. Science Friday (Oct. 20). See [Go to].

1999. Behind the scenes–mathematics in Arcadia: Tom Stoppard in conversation with Robert Osserman. Mathematical Sciences Research Institute (Feb. 19). See [Go to] and [Go to].

Abbott, S.D. 1999. Turning theorems into plays. Math Horizons 7(September):5-11.

Albers, D. 2000. A beautiful Proof. MAA Focus (August/September):8.

Alexanderson, G.L. 2002. Osserman interviews David Auburn, author of Proof. MAA Focus (February):6-7. Available at [Go to].

Bayer, D. 2000. Proof: A theater review. Notices of the American Mathematical Society 47(October):1082-1084. Available at [Go to].

Corbett, G. 2000. New York story: The Courant Institute meets Broadway. SIAM News 33(December):1-2.

______. 1999. Math program plays to packed house. SIAM News 32(April):1,3. Available at [Go to].

Jackson, A. 1995. Love and the second law of thermodynamics: Tom Stoppard's Arcadia. Notices of the American Mathematical Society 42(November):1284-1287. Available at [Go to].

Kirby, R. 2002. Conversations about mathematics. Notices of the American Mathematical Society 49(March):333-335. Available at [Go to].

Lustig, H., and K. Shepherd-Barr. 2002. Science as theater. American Scientist 90(November-December):550-555. See [Go to].

Osserman, R. 2001. Fermat's Last Tango: A theater review. Notices of the American Mathematical Society 48(December):1330-1332. Available at [Go to].

Peterson, I. 2002. Song-and-dance Fermat. Science News Online (May 11). Available at [Go to].

______. 1997. Math in Arcadia. Science News Online (Feb. 2). Available at [Go to].

______. 1994. Fermat's famous theorem: Proved at last? Science News 146(Nov. 5):295.

Robinson, S. 2002. Proof: An admirable approximation of mathematical culture. SIAM News 36(January-February):1. Available at [Go to].

Rockmore, D. 2000. Uncertainty is certain in mathematics and life. Chronicle of Higher Education (June 23):B9.

Saul, M. 2001. The mathematician's Proof. Notices of the American Mathematical Society 48(June/July):596-597.

Spencer, J. 2001. Geek chic. Notices of the American Mathematical Society 48(February):165. Available at [Go to].

For information about "Proof: A Symposium," held Oct. 16, 2000, at the Courant Institute, New York City, and audio files of the meeting proceedings, see [Go to].

Robert L. Devaney's Web pages devoted to "Chaos, Fractals, and Arcadia," can be found at [Go to].

For a list of plays with a mathematical theme, see mathematician Alex Kasman's "Mathematical Fiction" Web page at [Go to].