Writing and Reckoning: Sign Systems and Argument in Verbal and Mathematical Communication

  Carolyn Collette,

English

Giuliana Davidoff,

Mathematics

Index | Syllabus  |  Language of Mathematics  |  Fermat's Last Theorem

In the spring semester of 1997-98, English 210, in conjunction with the Math-Across-the-Curriculum program at Mount Holyoke College, incorporated a unit on the similarities and differences between mathematical "language" and literary language. English 210 is an advanced composition course based on the book Finding Common Ground: A Guide to Personal, Professional and Public Discourse (Collette and Johnson, New York: Addison Wesley, 1997). Through close analysis of language, style, and format in a wide variety of literary and non-literary texts, the course helps students develop a range of voices, styles and structures to express their ideas in various disciplines and public discussions. Students in this course tend not to be English majors; the course enrolls students across the disciplines at Mount Holyoke. The portion of the course devoted to the discourses of mathematics comprised three class sessions and a unit of readings. Two professors, Carolyn Collette of the English Department, and Giuliana Davidoff of the Mathematics Department, designed the unit to focus on the question of whether mathematical signs operate differently in the mind from verbal signs. We were interested in the language of mathematics, and in exploring with students the points of intersection between the two sign systems. We began the unit with a class discussion comparing a poem and a proof. For this purpose we selected an English language translation of Wislawa Szymborska's poem "Pi" in View with a Grain of Sand: Selected Poems (trans. Baranczak and Cavanaugh, New York: Harcourt Brace, 1995) and a proof of a theorem that two line segments drawn through a triangle contain the same number of points. Initially the discussion focused on the free play of signs in the numerous possible readings of the poem as compared to the precision and direction of the proof; however, as the discussion proceeded, students began to express their sense of structure of the proof to a written argument. Once they made that connection, they began to think in terms of "holes" in the argument, and began to recognize the "hidden" similarities between the proof and the patterns of verbal signs that comprise the poetic text, specifically similarities of inherent ambiguity in the sign patterns of mathematics. This appeared in a discussion of whether the theorem implied that the line segments in question had to be parallel, and whether parallelism or non-parallelism made a difference. By the end of class they recognized the inevitable free play of the mind working with both sign systems.


On the second day of the unit the class watched a NOVA video, "The Proof, " which tells the story of how Andrew Wiles "solved" Fermat's last theorem. In this class we were interested in how mathematicians convey information to one another and to the public, and in how they used signs to represent their thinking; we were also interested in how the film-makers addressed the issue of making a complex topic accessible to viewers, who, even granting the self-selecting nature of the probable audience, were likely not familiar with the specific problem or the strategies of twentieth-century mathematics to solve it. Students were asked to view the film with the question of how mathematicians and how the video communicate with the ultimate aim of writing a paper on the subject. In the discussion that followed the video, students noted immediately that the video depended on verbal analogies, metaphors, and the visual representation of metaphor as a major structuring device-- a dark room, a bridge, a doughnut, all appeared as visual correlatives to the language of mathematics. Students were also quick to point out the universality of the "language" of mathematics, the common discourse of the field that allowed the video-makers to splice together fragments of different interviews into a seamless presentation of the idea of modular forms and elliptical curves. They noted, too, the difficulty the various mathematicians interviewed had in explaining what they were thinking. The papers that came out of this assignment challenged the students to find the language to talk about both the language of cinema and the language of mathematics.

On the third day of the unit, Giuliana Davidoff came to class to discuss the reading for that day, a set of readings including f a Scientific American article on the same topic as the video ("Fermat's Last Stand," Nov. , 1997, 68-73), written by some of the mathematicians who appeared in the previous class video, a New York Times Book Review of the book The Number Sense: How the Mind Creates Mathematics, by Stanislaus Dehaene (Oxford, 199 ? ), and a February 10, 1998 "Science Times" article, "Useful Invention or Absolute Truth: What is Math?" (NY Times). The class began with students were asked to respond to their experience of reading the Scientific American article. When they said that they had gotten "lost" in the explanation in both the movie and the article, Giuliana Davidoff proceeded to explain the mathematics of the article in a manner that was so clear and direct that the class immediately recognized the dependence of higher mathematics on verbal symbols and, quite interestingly, the use of substitutions in the process of proving that elliptical curves are modular forms. That pattern of substitutions was the subtextual foundation of the thinking in both the video and the article; making it explicit allowed students both to understand the large issues involved in solving Fermat's last theorem, and to recognize in mathematics a form of a principle we had been working with all semester: the fundamental usefulness of analogy in communication in all kinds of writing.

The unit ended but it didn't conclude. In the first place, it ended differently from where I thought it would, in that it was much more complex and that it opened many new ways about thinking about the "languages" of math and English than I had imagined when we first formulated the idea of metaphor as a property of verbal signs and not of the mathematical sign system. Clearly the role of analogy and substitution in both systems is a complicated and intriguing area. We were able to raise student awareness of this fact. We were also able to interest students in reading on current thinking about whether or how the human brain is hard-wired for math as it seems to be for language; this leads to thinking about all the kinds of mathematics that humans can create, and the question of whether there can be as many kinds of mathematical systems as there are languages. This topic lead to consideration of current interest in defining mathematics not so much as reflecting eternal Platonic verities and cosmic harmonies, as a construction of cultures that, in many ways, explains and derives from natural patterns in the external world, as well as physiological patterns in the human brain.

Because the class was an advanced writing course elected by students many of whom are already strong writers and thinkers, the discussions we had were wide-ranging, open and allusive. The discussion of mathematics was in part shaped by student interest and experience (only two or three of the 18 students in the class were currently taking math). The discussion was also shaped in part because of the orientation of the course whose primary goal was to help students c communicate effectively in language and forms that sounded natural and felt comfortable. The course paid a good deal of attention to structure of writing and the importance of reader's imagination. In this respect some basic principles of semiotics underlay all our discussions; as a result a great deal of attention was directed to analogy and metaphor as crucial ways of engaging the reader's mind. In that way students were oriented to think about mathematics as one kind of language, and to think about how it communicates a way of describing the world.

Carolyn Collette, Department of English
Mount Holyoke College

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