Exercises in (Mathematical) Style: Stories of Binomial Coefficients

The story should start with Raymon Quéneau and a little book called Exercices de Style. Quéneau was a French novelist and poet, the co-founder of a group called Oulipo (which stands for “Ouvroir de littérature potentielle”, i.e., workshop on potential literature). The several writers in Oulipo seem to have been fascinated by certain literary games, particularly combinatorial ones. They wrote, for example, a book that (potentially) contained \(10^{14}\) sonnets: you had 10 choices for each of the 14 lines that make up a sonnet. The claim, of course, is that all of those sonnets made sense — but how could one check, given the huge number of potential texts? Another example of Oulipoian antics is George Perec’s famous La Disparition, an entire novel written without using the letter e.

Exercices de Style fits the mold: it tells the same story 99 times in a vast array of different styles. The story is banal: a strange-looking young man riding a Paris bus gets angry with another rider, finds a seat, and later is seen again having a conversation about a button on his overcoat. Quéneau starts by telling it straight, but then the games begin. The second entry is called “double entry” — “towards the middle of the day and at noon…” Some of the variations are funny, some are poetic, some are just plain weird. It may well be Quéneau’s most popular book, and it is almost certainly his most accessible. It certainly is a powerful demonstration of the power of language. Quéneau once said that

… the finished product may possibly act as a kind of rust-remover to literature to help to rid it of some of its scabs. If I have been able to contribute a little to this, then I am very proud, especially if I have done it without boring the reader too much.

Boring the reader is without a doubt the major risk: it is the same story each of the 99 times, after all. I find that I have to take it in small doses. There is an English translation as well, which must have been an interesting exercise in itself.

Something about the idea seems to appeal to mathematicians. There is, for example, Rationnel mon \(\mathbb{Q}\), written by Ludmila Duchêne and Agnès Leblanc (but the names are almost certainly pseudonyms) and published by Hermann in 2010. This book collects 65 proofs that the square root of 2 is irrational, in different styles. There are two axes of variation in this case: the proofs are written in various ways, but there are also many different proofs. In other words, they vary in the style of mathematical writing and in the style of mathematics.

So we come to John McCleary’s book, whose subtitle, Stories of Binomial Coefficients, tells us a little more about what he is up to. McCleary doesn’t take the big risk: these are 99 different stories on a common theme, rather than the same story told 99 times. But once again we find variations in both writing style and mathematical style. There are poems, a section in the style of Bourbaki (complete with the “narrow turning” symbol they made popular), one presented as a sheet of scrap paper (complete with coffee spills), one “mathematical idol” program. Some stories are combinatorial, some probabilistic, some have an algebraic flavor. There is even a final story that takes place on a bus, perhaps the same bus where Quéneau’s original narrative was born.

At the back of the book, McCleary gives us notes on what he was trying to do in each of the sections, including references when appropriate. Among other things, he hopes that this will help us think further about mathematical writing and the question of style.

In summary: this is a delightful experiment, both amusing and thought-provoking. There is a lot of good mathematics in the book, as well as some mathematical jokes. And I suspect our prose style could well use some rust-remover. I think most mathematicians will enjoy it.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.