Beauty in numbers
Mathematics historian Judith Victor Grabiner, SB’60, teaches math to the liberal arts masses.
Mathematics historian Judith Victor Grabiner, SB’60, took UChicago’s Core to heart. Now a Pitzer College professor, in high school she found math to be the only subject “taught with any intellectual rigor and integrity” and chose the University of Chicago because of its strong math department. There Grabiner discovered that problems in the humanities were “just as challenging and just as interesting and just as important as problems in the sciences,” she says. Humanities 3, taught by Herman Sinaiko, AB’47, PhD’61, was the turning point. She realized “that history, philosophy, and literature were going to be part of my intellectual approach to the world after that.”
Grabiner wedded the humanities to her mathematical literacy with a PhD in the history of science from Harvard, where she read Galileo’s and Newton’s original writings for the first time. After a few years as a Harvard instructor, she and husband Sandy, a mathematician at the Massachusetts Institute of Technology, headed to California looking for jobs. Grabiner bounced between teaching and working on her first book—The Origin of Cauchy’s Rigorous Calculus (MIT Press, 1981)—before spending 14 years at California State University, Dominguez Hills, teaching history.
Now in Pitzer’s math department, Grabiner teaches courses such as Mathematics, Philosophy, and the “Real World” and Mathematics in Many Cultures to humanities students looking to satisfy their general education requirement. Her writing covers similar ground: the importance of historical context when teaching mathematics, the way proof writing changed over time, and the diverse traditions that led to modern mathematics.
In January Grabiner won the Mathematical Association of America’s (MAA) prestigious Beckenbach Book Prize for A Historian Looks Back: The Calculus as Algebra and Selected Writings (MAA, 2010). She is the only four-time winner of the MAA’s Lester R. Ford Award for best article in American Mathematical Monthly. After finishing her current project on math’s place in the liberal arts, Grabiner plans to research the history of optimization in mathematics and philosophy. In an interview with the Magazine, edited and adapted below, she talks about her life, teaching, and research.
There’s a guy who taught music at the University of Chicago, back when I was a student, called Leonard Meyer [PhD’54]. And he wrote a book called Emotion and Meaning in Music (University of Chicago Press, 1956). I was flipping through it one day, and this just jumped out on the page at me: that musical beauty is a violation of your expectations in a way that afterward seems natural. That’s what I think mathematical beauty is too. I’d like to see liberal arts students get that and have that experience.
Almost every culture that we know anything about has had some kind of mathematics, and the mathematics that they have is developed to solve problems of that particular culture. There are cultures that have mathematically complicated kinship systems and geometric representations of how those kinship systems work. There are cultures that have experimented with all sorts of patterns and symmetries to do art. And then the historic cultures like the Chinese, the Indian, the Islamic, that have very sophisticated mathematical systems. One of the subjects that’s been discovered in many different cultures is elementary combinatorics. Say in medieval India, you’ve got a line of Sanskrit poetry that has five syllables in it, and every syllable can be either heavy or light, and a good poem that’s got five syllable lines uses every possible combination of heavy and light—well, how many are there?
Fear and loathing in math class
My students don’t take mathematics as part of their major. They want to find a math course that interests them, and they don’t think there’s going to be one. I joke that I teach courses whose prerequisite is hatred and fear of mathematics. That’s an indictment of how we do K–12 mathematics, you know. It’s taught as a collection of recipes. There’s no “why was this discovered and why would you ever need it?”And if there’s any subject in which the answers to “why” are very well known, it’s mathematics.
Not your mother’s calculus
Aristotle said to understand motion is to understand nature, because nature is that which changes and the eternal principle of change itself. OK, Plato’s mathematics is the study of that which does not change, right? Two and two is always four, the circle’s always round, etc. Mathematics is a model for what the search for absolute truth could be, because although the objects in mathematics aren’t as good as Plato’s form of good, they are unchanging, and so you can have unchanging truth about them. So the idea that you could have a mathematics of change, at least in the ancient world, well, that doesn’t seem like a reasonable thing to do. But that’s what calculus is.
In political life, a little quantitative understanding can help you speak intelligently about public policy. You have to figure out whether the average is an arithmetic mean, where the outlier cases pull the mean very much, or whether it’s a median, half above and half below, or whether it’s the most-often-achieved value. Statistical abstractions are very handy, and they’re nice shorthand for a large amount of data. But what’s real, ultimately, are the individuals, and you really have to look at how things are distributed. Francis Galton, a famous 19th-century British statistician, is supposed to have said that, on the average, Switzerland is flat.
Math, philosophy, and the “real world”
At the end of my class, every student has to do a report on how mathematics is used in an area of interest to that student. I just had a student report on mathematics and music in my class last semester, and it’s news for a lot of people that there is mathematics in music. When he was done, a guy got up to talk about ski jumping and the way mathematical physics describes the curved course. Then I had a student do a logical analysis of arguments using symbolic logic and arguments about free will. For probably the first time in my students’ lives, the student is an expert in the class on that particular piece of mathematics. And let me say one more thing about teaching math for a liberal arts course. Every math course these students have ever taken is a prerequisite to something else. You can’t do Algebra 2 if you didn’t get Algebra 1. But in a liberal arts course, you’re not going anywhere. So you can try to build a sense of mastery. Because the exciting thing about mathematics is when you get it.