Paul Zorn, St. Olaf College

Retiring MAA Presidential Address, Joint Mathematics Meetings, San Diego, January 11, 2013

Abstract: Mathematicians don’t just do mathematics. We communicate our subject too, by speaking, writing, teaching, illustrating, editing, explaining, and professing it, for expert and non-expert audiences alike. Words, pictures, equations, and other media may be well or poorly suited to the special purposes of mathematical exposition, and mathematical exposition may be good, bad, or indifferent, depending largely on its audience. But, as I will argue with examples, mathematical exposition is at its best real and valuable mathematics—and no less challenging or deserving of professional reward than other forms of mathematical activity. Mathematics is a big tent, and its vitality and growth depends on contributions from many directions.

**Part I: Introduction**

I’m here to talk, as the title suggests, about mathematical communication in various forms. Mathematics itself takes many forms, and a lot of it is hard, as the doll Barbie once said, though in another context. Communicating mathematics is hard, too. Doing it effectively, I’ll argue, requires our best efforts, not just mathematical but also linguistic, artistic, visual, pedagogical, sociological, psychological—pick your favorite academic adjective. Doing it well, as I’ll also argue, deserves not just our respect and admiration, but also recognition in the academic currency of tenure, promotion, and professional advancement.

That heavier stuff comes later. Let’s talk a little first about mathematics and art and literature, and their symbiotic relationship in service of communication, broadly understood.

As evidence that such marriages can work, I adduce Juan Sánchez Cotán, a Spanish painter born around 1560. Cotán had a successful art career around Toledo until his mid-40s, when he shuttered his workshop and took holy orders in a Carthusian monastery. Thereafter, he painted mainly religious scenes.

Cotán is now famous mainly for his still lifes. I find this first one (*Still Life with Game Fowl, Vegetables, and Fruits*, 1602) quite beautiful.

It’s also arrestingly strange. Notice the eye-popping colors and the flash-photo contrast: bright birds and produce against a pitch-black background. How did Cotán achieve these effects, or even imagine them, 300 years before the light bulb? Notice, too, Cotán’s eye, and his care, for geometry and structure. The apples are aligned in classic sphere-packing style.

(As you well know, Johannes Kepler—a German contemporary of Cotán—conjectured in 1611 that the standard packing is indeed the tightest possible. This obvious-seeming result was finally proved—almost 400 years later—by Thomas Hales.)

The lemons form a triangle, the little birds are lined up in parallel, and the bunch of celery looks like a family of curves emanating from a point. This would all be dullly pedantic, like a standard story problem in a derivative calculus text, were it not for the gorgeous textures and high detail. You can almost smell the lemons, and perhaps a hint of decaying meat.

This second picture (*Quince, Cabbage, Melon, and Cucumber*, 1602) is more austere, but to my eye even better.

Better still than the image you see here is the *real* painting, just up the road in the San Diego Museum of Art, where you can see it up close and personal. Here the mathematical elements are even clearer: the spherical pear and cabbage, the cut-spherical melon, and the cone-shaped cucumber, all arranged in a curve that looks parabolic. Or is it exponential? And notice the contrast between the curvaceous natural forms and the sharp geometry of the shadows and the rectangular window opening.

Is all this math stuff really *in* these pictures? Did Cotán really have it in mind? I think he did, but he kept it—literally—in perspective. This is a painting, not a math drill, and we see this in the hyper-fine detail of each cabbage leaf, the overripe and dented pear, and the textures of melon and cucumber. Maybe the artist is saying, though not in so many words, that both the universal forms of geometry and the particulars of real pears and cabbages and melons are worth caring about, and paying attention to.

The Israeli artist Ori Gersht has certainly been paying attention. Here is a still from a 2006 video work, called “Pomegranate.”

Gersht keeps a lot of Cotán’s ingredients but adds a few of his own, including the eponymous fruit and one other notable addition—a high-velocity rifle bullet.

Perhaps this says something about the ephemerality of ideal forms, or about man’s inhumanity to produce.

Mathematical communication is a big subject, and an important one. As Professor Suzanne Weekes reminded us in her invited address at this meeting on bringing mathematics and industry together, un-communicated mathematics is useless mathematics.

I’ll have little or nothing to say about many interesting aspects of this big subject. Here are a few of those omissions:

- Mathematical art, or artistic mathematics, with forbidden symmetries, as in this cover picture, by Frank Farris;
- mathematical blogging, and crowd-sourced mathematics, as practiced at the Huffington Post by Frank Morgan;
- how to become a mathematics writer, as discussed by Dana Mackenzie;
- characteristic habits of thought and expression employed by mathematicians, and how they can sometime lead to misunderstandings, as discussed by Gerald Folland;
- the importance of a good “lede” in mathematical writing, as explained by Barry Cipra.

The good news is that you can hear more about most of these things, straight from the horses’ mouths, at the MAA Invited Paper Session on Writing, Talking, and Sharing Mathematics, also at these Meetings. [For a report on the session, see the article by Evelyn Lamb.]

Nor will I have much to say about mathematical communication and the role of mathematics in popular culture. But here’s one book (*Mathematics in Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media*, 2012) with plenty of information and speculation on that interesting subject.

One of the editors, Jessica Sklar, was slated to speak in the Invited Paper Session just mentioned, but is unfortunately ill and unable to attend these meetings. I’m especially sorry about Jessica’s absence, both for her sake and because of the tendency of mathematics and mathematicians to appear in popular culture either as impressive but unapproachable geniuses, or as obsessive, half-cracked weirdos.

Part II: Mathematics in Literature

Part III: What Is Mathematics?

Part IV: Mathematical Language: Learning from Barbie

Part V: Rule Books and Tour Guides: Two Live Questions

Part VI: Lessons from History

Part VII: Valuing Communication