Mathematics in the Stage Play "Proof"

"I think there is some connection between extremely prodigious mathematical ability and craziness. I don't think that math drives people crazy, but those with edgy or slightly irrational personalities are drawn to it." (math.cofc.edu) These words were spoken by the playwright of "Proof", David Auburn, and basically deal with one of the main themes in the play. The main characters in the play are mathematicians and deal with issues of personal relationships, authorship, and even mental sanity. The themes of the play are accentuated by the presence of mathematics in that everything can be mathematically characterized and interpreted in various ways. The play was made into a critically acclaimed film in 2005 with the same title. Gwyneth Paltrow took the title role of Catherine with Anthony Hopkins, Hope Davis, and Jake Gyllenhaal co-starring. Paltrow played the same role in the play in London with British director John Madden, who also directed the film and David Auburn adapting his own screenplay. Auburn was born in the setting of the play, Chicago, Illinois, and attended the University of Chicago where he eventually received a degree in English literature. His play won the 2001 Pulitzer Prize for drama as well as a 2001 Tony Award.

The play is about Catherine, the daughter of a famous mathematician Robert who is recently deceased. Robert made major contributions to various mathematical fields before the age of twenty-eight but was beset with mental illness for the rest of his life. Catherine previously dropped out of the mathematics department at Northwestern to take care of her ill father and saw his slow descent into madness. Catherine always seems to have a fear instilled into her that she is exactly like her father. Hal was a student of Robert's at the University of Chicago and now teaches there. Hal wants to look through Robert's tons of notebooks for one lucid notebook among hundreds written as he was deteriorating so that he can publish it with well intentions. When he and Catherine have a relationship, Catherine gives Hal a key to a notebook that contains an extraordinary proof which proves something mathematicians have been working on for ages. Hal and Catherine's sister Claire who flew in for the funeral are not sure if she or her father actually wrote the proof because they have similar handwriting. Eventually, Hal brings the proof to some mathematicians in order to make sure that the proof checks out and when it does Catherine is not sure she wants the credit because she will have to dip back into her former life. But eventually when her sister's intentions of saving her from illness are revealed, Catherine chooses to go through the proof line by line and continue on the path that was meant for her. In the film, some additional scenes were added to make sure that the audience is not sure of the proof's origin even after the conclusion.

The title of the play obviously has multiple meanings; of course the literal proof that is discovered and the proof of which person is its true author. The basis of the mathematical proof that the characters in the play discuss is based on one of the well known theorems in history; Andrew Wiles' proof of Fermat's Last Theorem. (math.cofc.edu) Fermat was a mathematician in the 1600s who wrote a note in the margins of his book that he proved a certain theorem. This proposition was concerning an equation that is similar to the Pythagorean Theorem. This theorem by Pythagoras states that x² + y² = z². The whole number solutions to this are 3² + 4² = 5². Fermat also considered the cubed version of this to be x³ + y³ = z³. Fermat's question was could you find solutions to the cubed equation? For this he claimed that there were none and that for the family of equations: xª + yª = zª where a is larger than 2 it is impossible to find a solution. The basics of a mathematical proof are that there is a line of reasoning that consists of many steps and the more rigorous it is, no one can ever prove it wrong. This proof and many others are extremely abstract in that they do not actually prove anything that comes up in the world that we live in. Andrew Wiles tried his entire life to prove Fermat's Last Theorem and he became famous in the mathematical world when in 1993 he announced that he had finally achieved this. But eventually an error in his calculation severed his dream until another mathematician from Cambridge helped him to repair it. (pbs.org) In all of the mathematical work that was left by Fermat, only one of those was a proof. Basically, the proof says that the area of a right triangle cannot be a square. This means that a rational triangle cannot be a rational square and in symbols, there do not exist integers with x, y, and z with x² + y² = z² with the same conditions that were stated before. (mcs) This is a less abstract way of explaining Fermat's Last Theorem and applies it to some aspects of geometry.

In the play, Hal describes some different areas of mathematics that Robert had made great contributions to during his lucid period. Those areas were the game theory, algebraic geometry, and rational behavior. The game theory is basically an interdisciplinary approach to the study of human behavior. This involves areas of some different disciplines; mathematics, economics, and certain areas of social and behavioral sciences. Game theory was founded by the great mathematician John von Neumann and he co-authored a very important book titled The Theory of Games and Economic Behavior. There are various issues and almost infinite questions in game theory that deal with its rationality. Some of those questions are what does it mean to choose strategies rationally when outcomes depend on the strategies of others and when information is complete? Also, in games that allow mutual gain or loss, is it rational to cooperate to realize the mutual gain or to avoid the mutual loss or is it rational to act aggressively in seeking individual gain regardless of mutual gain or loss? (drexel.edu) A rather famous example of game theory is that of the prisoner's dilemma. In the game two prisoners have been partners in a crime and have been brought in by the police. Each person is then put in a separate holding cell and offered an opportunity to confess. Each outcome is given a number and the higher the number the better. If no person confesses, they split the proceeds of the crime and this is represented by 5 units of utility for each person. If one person confesses and the other does not, one testifies against the other and goes free and receives 10 units of utility while the other receives nothing. And finally, if both people confess, they are still charged with the crime but with lesser sentences, this is noted with each person receiving 1 point of utility. This situation is famous because it involves many different and important situations. (ucla.edu) Algebraic geometry is a subject with roots in analytic geometry. At a very basic level this subject is concerned with the geometry of the solutions of a system of polynomial equations. This involves various techniques from geometry, number theory, and analysis. The results from algebraic geometry can be applied to problems from some different disciplines such as what Andrew Wiles used to prove Fermat's Last Theorem. (math.tamu.edu) Examples of algebraic geometry deal with analyzing shapes and the spaces inside of those shapes using certain mathematical theories. Rational behavior deals with economics and individuals maximizing some objective function under the constraints that they face. "The concept of rational behavior has - in addition to making the analysis of individual behavior a good deal more tractable than a less structured assumption would permit - two interpretations. First, it allows to derive optimal economic behavior in a normative sense. Second, models of rational behavior can be used to explain and predict actual (i.e., observed) economic behavior." (uni-mannheim.de)

"The great thing about math is that it's a kind of scientific activity that's still done in a solitary way. Most science is now with big teams on big projects. In math, someone could have done something major working alone in the attic." (math.cofc.edu) This is another quote from David Auburn that shows us the life of some mathematicians like Robert, Catherine, and Hal in his play. They all worked on their proofs and theorems alone and when they thought that maybe they had something, only then did they run them by another scholar. Andrew Wiles admitted some of his tendencies while working to prove Fermat's Last Theorem; "When I got stuck and I didn't know what to do next, I would go out for a walk. I'd often walk down by the lake. Walking has a very good effect in that you're in this state of relaxation, but at the same time you're allowing the sub-conscious to work on you. And often if you have one particular thing buzzing in your mind then you don't need anything to write with or any desk. I'd always have a pencil and paper ready and, if I really had an idea, I'd sit down at a bench and I'd start scribbling away." (pbs.org) During the play Hal brings up to Catherine that when he was at a conference in Toronto, many of the mathematicians took amphetamines in order to peak their creative potential. This is a rumored occurrence and one of the very large rumors involves famous Hungarian mathematician Paul Erdos. It was rumored that he took amphetamines most of his career and once said that when he stared at a piece of paper with the drugs, his mind was filled with ideas but without them, it was just a piece of paper. (wikipedia) It is obvious that Auburn researched into the life of mathematics because it is such a rarely documented area that remains apart from much of mainstream society which has very different goals in life.

Both the play and the film are remarkably fascinating to me because of the many questions that the works bring up. And one of the important messages that arise is that many of these messages do not have a clear solution. The dialog of the play is truly academic language, and I also believe that the tone of the University of Chicago campus is portrayed exactly right especially in the film. In the film, David Auburn added two scenes that were not in the play in order for the audience to still not be completely sure who the true author of the proof was. I believe that this was to show that in certain cases, it is almost impossible to know for sure who the true author is.

David Auburn, the author of the play and screenwriter of the film is not a mathematician. This allows the people who read the play to identify with the characters even if they are not fluent in mathematical language. This also shows that Auburn was forced to go elsewhere for research into the mathematical portions of the dialog. The proof that the characters in the play are concerned with is in relation to the proof of Fermat's Last Theorem by Andrew Wiles. It is also possible that the character of Robert is based on the now famous John Nash, who fought against schizophrenia during his mathematical career and was the basis for the 2001 film A Beautiful Mind. And the text of the play deals with fascinating authorship issues that are present in almost every academic profession.

WORKS CITED

-Kasman, Alex. "Proof". Mathematical Fiction. College of Charleston. December 3^rd, 2006. < http://math.cofc.edu/kasman/MATHFICT/mfview.php?callnumber=mf139> (math.cofc.edu)

-"The Proof". Nova Online. PBS. December 3^rd, 2006. http://www.pbs.org/wgbh/nova/proof/ (pbs.org)

-"Paul Erdos". Wikipedia. December 3^rd, 2006.

< http://en.wikipedia.org/wiki/Paul_Erdos> (wikipedia)

-McCain, Roger A. "The Basics". Game Theory: An Introductory Sketch. Drexel University. December 3^rd, 2006. < http://William-king.www.drexel.edu/top/eco/game/intro.html> (drexel.edu)

-Stiller, Peter F. "What is Algebraic Geometry". Texas A&M University. December 3^rd, 2006. < http://www.math.tamu.edu/~Peter.Stiller/agpage.html> (math.tamu.edu)

-Arapura, Donu. "Introducton to Algebraic Geometry". September 27^th, 2005. Purdue University. December 3^rd, 2006. < http://www.math.purdue.edu/~dvb/algeom.html> (math.purdue.edu)

-Winter, Joachim. "Rational Behavior". June 17^th, 1999. December 3^rd, 2006.

< www.sfb504.uni-mannheim.de/glossary/rational.htm> (uni-mannheim.de)

-Levine, David K. "What is Game Theory". Economic and Game Theory. UCLA.

December 8th, 2006. < http://levine.sscnet.ucla.edu/general/whatis.htm> (ucla.edu)

-O'Conner, J J. Robertson E F. "Fermat's Last Theorem". The MacTutor History of

Mathematics Archive. February 1996. MacTutor. December 8^th, 2006.

< www-history.mcs.st-andrews.ac.uk/HistTopics/Fermat's_last_theorem.html> (mcs)

-Auburn, David. Proof. New York: Faber and Faber, Inc. 2001.

-Proof. Dir. John Madden. Perf. Gwyneth Paltrow, Anthony Hopkins, Jake Gyllenhaal.

Miramax, 2005.