by Neal Koblitz
The title of your recent posting "Mathematics' need for narratives" resonated with me. I am a research mathematician, and when I write a paper, usually jointly with my collaborator Alfred Menezes, we decide whether it deserves a place on our website (anotherlook.ca) by asking ourselves if it truly "tells a story." In our case the "story" is typically an analysis of mathematical proofs of security of computer protocols that reveals a dark underside --- an overlooked flaw in the proof, a misleading interpretation of the result, reliance on a model that is woefully inadequate for the intended application, a hypothesis to the theorem that is so strong as to render the argument essentially circular. In our critiques of the paradigm of "provable security" (which in my opinion is an oxymoron), we depict scenarios in which the promised mathematical guarantees lose their validity, if possible with humorous references to popular culture and current events. One of our papers, jointly written with Ann Hibner Koblitz, raises some technical issues involving the math while presenting a historical narrative that draws on research in the social construction of science and technology; the subtitle of this paper is "the serpentine course of a paradigm shift." In this paper we introduce the term "narrative inversion" to refer to narratives of mathematical certainty behind which one finds a reality that is full of doubt and contingency.
However, when I read the introduction and perused the table of contents of Circles Disturbed... by Barry Mazur et al, I saw that the contributors' definition of mathematics and of the narratives that guide mathematicians' thinking is narrow and insular. To them, mathematics means pure theory. They seem uninterested in the stories that arise from applied branches of mathematics or from misguided and self-serving attempts to apply mathematics to social and economic questions. If anyone needs to be convinced that the mathematical enterprise encompasses a lot more than pure theory --- and can provide many dramatic narratives about topics other than theorem-proving --- it should suffice to point out that the largest employer of math PhDs in the world is the U.S. National Security Agency.
Moreover, in Circles Disturbed... mathematics is identified with the Eurocentric tradition starting in ancient Greece. This is not the version of the history of mathematics that I present to my students when I teach a course every year on the subject. Rather, I expect my students to know about the ancient Chinese and Indian traditions, as well as the criticisms of Eurocentric history of mathematics that have been made by Martin Bernal in the controversial book Black Athena (1987) and by ethnomathematics researchers such as Marcia Ascher.
Here are a few of my favorite examples of mathematical narratives in the broad sense in which I would define the term:
(1) Cathy O'Neil (a former PhD student of Barry Mazur, ironically) has a forthcoming book with the clever title Weapons of Math Destruction. She was a "quant" during the days of the economic meltdown, and writes entertainingly about the misuse of mathematical models on Wall Street and elsewhere. Her blog address is mathbabe.org.
In 2000 David Li, a quant with a PhD from the University of Waterloo, developed a mathematical formula that purportedly predicted the likelihood that a set of companies would successively default on their debts. His model was based on an optimistic narrative of economic progress under capitalism --- a narrative that ignored the stories of repeated boom/bust that should have been clear to anyone who studies history --- and was based on data of the most recent years of relative prosperity. His formula was used on Wall Street to give a mathematical justification for sharply increasing investments in newly-created exotic financial instruments such as mortgage-backed collateralized debt obligations. The failure of Li's mathematical model was a factor in the 2008 economic collapse.
(2) John Ewing (president of Math for America and former executive director of the American Math Society) wrote an article in the AMS Notices titled "Mathematical Intimidation: Driven by the Data" exposing the scam of value-added modeling, a much-hyped pseudo-mathematical approach to evaluating teachers.
(3) The late William Thurston (winner of a Fields Medal, often called the mathematical world's Nobel Prize) led a valiant but ultimately unsuccessful battle to get the AMS to do something about mathematicians' excessive reliance on military and NSA funding. Thurston was part of a long tradition of mathematicians who rejected the dominant narrative in the profession, which maintained that the profession was at the service of whomever had power and money --- wealthy patrons in medieval and early modern times, the U.S. Department of Defense in our day. (As the prominent computer scientist Phil Rogaway put it, most researchers have never seen a funding source they didn't like.) Other famous mathematicians of the last century who advocated an alternative humanistic vision included the British pacifist G. H. Hardy (who lauded his field number theory as "gentle and clean" because in his day it was pursued only for aesthetic reasons and had no applications) and Norbert Weiner (see S. J. Heims' 1982 joint biography John von Neumann and Norbert Weiner: From Mathematics to the Technologies of Life and Death, which contrasts the outlook of Weiner with that of the militarist von Neumann, another leading mathematician of the same time period).
(4) In the 1980s Serge Lang led a successful fight to keep right-wing political scientist Samuel Huntington from being elected to the U.S. National Academy of Science. (I described this battle in an article in the Mathematical Intelligencer titled "A Tale of Three Equations: Or the Emperor Has No Clothes.") A small example of Huntington's misuse of quantitative methodology was that he favorably cited a statistical study purporting to show that South Africa (this was during the apartheid period) had a population with a high "satisfaction index." How could Huntington (who, according to wikipedia, was a "valued adviser" to the apartheid regime, advising them to increase the repressive power of the state) seriously expect anyone to believe that? He, like most capitalist economists and political scientists, was using a narrative of economic progress that attached huge importance to numbers such as gross domestic product per capita, number of telephones per capita (which was one of the ingredients in the "satisfaction index"), and so on --- while ignoring the fundamental issue of how those resources were distributed. In the case of South Africa, it was certainly true that the whites owned a lot of telephones.
The introduction to Circles Disturbed... says that mathematicians are "delighted" to see the ways that mathematicians have been portrayed in recent works of literature and popular culture. Undoubtedly many are, but I have a less sanguine view of the direction of popular imagery of science and mathematics. I believe there is far more superstition, ignorance, and anti-scientific bias among Americans now than there was when I was growing up a half-century ago. If one compares the portrayals of mathematicians in recent popular works with those of earlier decades, the picture is not one of constant upward progress. Compare, for example, the 1980 movie It's My Turn (in which Jill Clayburgh plays a mathematician) or the 1988 movie Stand and Deliver (about the high school math teacher Jaime Escalante) --- the latter movie was cited by several of my students as what inspired them to want to become math teachers --- with the more recent movie A Beautiful Mind (2001) and the play Proof (2000, film in 2005), in which all the mathematicians are schizophrenic. The view of the mathematical profession that emerges from the latter portrayals is pretty dismal. Both works suggest that there is an inevitable connection between mathematical creativity and mental illness. In A Beautiful Mind some of the scenes of John Nash's schizophrenia (e.g., when he almost drowns their baby) are horrific, and the depiction of nerd behavior in the Princeton math department also reinforces some of the worst stereotypes about mathematicians. From the standpoint of mathematicians who want to improve the popular image of the profession and convince young people that it's a rewarding profession to go into, these portrayals are not helpful.
Or take the 1999 novel Cryptonomicon by Neil Stephenson. I am embarrassed to say that this book was highly recommended to me by a mathematical colleague whom I much respected, and so I read it. The book's portrayal of the great mathematician Alan Turing is full of juvenile and homophobic humor, with the author inventing an imaginary affair with a gay Nazi during World War II (as if to try to justify the later persecution of Turing by the British government that led to his death). That the novel is full of racism (against Asians and New Guineans) and misogyny did not prevent it from becoming a best-seller among mathematicians and computer scientists.
The mathematical world -- like the world outside -- is full of struggles and full of narratives, but judging by its introduction and table of contents, the book Circles Disturbed... doesn't come close to telling the whole story.