I am slowly working my way through the anthology Circles Disturbed: The Interplay of Mathematics and Narrative, edited by Apostolos Doxiadis and Barry Mazur. The book includes an excellent essay by mathematician Timothy Gowers titled, “Vividness in Mathematics and Narrative.” It makes a point that has often bothered me about mathematical discourse.
Gowers opens with two passages meant to describe the beginning of an academic year. Here’s the first:
It is September again, and the campus, which has been very quiet for the last couple of months, is suddenly full of cars bringing students back after their vacation. The parents are well-to-do; the cars are expensive and packed with things that the students will need during the term, though not all these items are strictly necessary. There is a general, if unspoken, sense among the parents that they are all from the same sector of society, with similar attitudes and similar ways of life. This makes them feel comfortable, and perhaps even a little smug. At first their sons and daughters are a bit shy, which causes some of them to be quiet and others to be overexcited. But this will wear off very soon: the transition will be forgotten and the term will have properly begun. The weather is typical for the time of year, with just a hint of the change that will take place over the next three months.
The second passage is this:
The station wagons arrived at noon, a long shining line that coursed through the west campus. In single file they eased around the orange I-beam sculpture and moved toward the dormitories. The roofs of the station wagons were loaded down with carefully secured suitcases full of light and heavy clothing; with boxes of blankets, boots and shoes, stationery and books, sheets, pillows, quilts; with rolled-up rugs and sleeping bags; with bicycles, skis, rucksacks, English and Western saddles, inflated rafts. As cars slowed to a crawl and stopped, students sprang out and raced to the rear doors to begin removing the objects inside; the stereo sets, radios, personal computers; small refrigerators and table ranges; the cartons of phonograph records and cassettes; the hairdryers and styling irons; the tennis rackets, soccer balls, hockey and lacrosse sticks, bows and arrows; the controlled substances, the birth control pills and devices; the junk food still in shopping bags — onion-and-garlic chips, nacho thins, peanut creme patties, Waffelos and Kabooms, fruit chews and toffee popcorn; the Dum-Dum pops, the Mystic mints.
I’ve witnessed this spectacle every September for twenty-one years. It is a brilliant event, invariably. The students greet each other with comic cries and gestures of sodden collapse. Their summer has been bloated with criminal pleasures as always. The parents stand sun-dazed near their automobiles, seeing images of themselves in every direction. The conscientious suntans. The well-made faces and wry looks. They feel a sense of renewal, of communal recognition. The women crisp and alert, in diet trim, knowing people’s names. Their husbands content to measure out the time, distant but ungrudging, accomplished in parenthood, something about them suggesting massive insurance coverage. The assembly of station wagons, as much as anything they might do in the course of the year, more than formal liturgies or laws, tells the parents they are a collection of the like-minded and the spiritually akin, a people, a nation.
I don’t think I need a precise definition of “vividness” to declare that the second passage is more vivid than the first. The second is the opening passage of Don DeLillo’s novel White Noise. The first was something Gowers made up just for the purpose of comparison. The curious thing, as Gowers explains, is that a similar phenomenon happens in mathematics. There are vivid and non-vivid ways of presenting math. He uses the example of defining what a “group” is. The non-vivid way is simply to state the precise, technical definition: “A group G is a set S with a binary operation, such that the binary operation is associative, there is an identity element with respect to the operation, and every element in S has an inverse with respect to the operation.” That’s a perfectly correct definition, but it lacks vividness.
Contrast that with this description:
From early childhood, we are all familiar with the idea of symmetry. On looking in a mirror, we note with amusement that when we move our right arm, our reflection appears to move its left arm — an effect that depends on the fact that human bodies look approximately the same if they are reflected in a vertical plane that separates them down the middle. …
The mathematician’s take on symmetry is slightly different. To a mathematician, symmetry is not so much a static property of an object but rather something you can do to an object. For example, take an equilateral triangle. The layperson might say that it is quite a symmetrical shape: it is symmetrical about three lines of reflection, and has rotational symmetry as well. A mathematician would say that the equilateral triangle has symmetries rather than is symmetrical. These symmetries are the three possible reflections, the two possible rotations (clockwise through 120 degrees), and the seemingly pointless “identity transformation,” which consists in doing nothing at all. Thus, to a mathematician, a symmetry of a shape means something you can do to that shape that leaves it looking the same afterward as it did before….
A simple observation that turns out to have ramifications throughout all of modern mathematics is that if you do two symmetries, one after another, the result is a third symmetry. …
Gowers goes on for a while like this, but I hope I have presented enough to make the point. The first presents the essential details in a concise way, but it is very flat and gives the reader no inkling of why anyone should care about groups. The second gradually builds up the idea of a group by relating it to simpler and more natural notions. It is far more vivid.
Both sorts of prose have their place. At some point you need to present the sharp, precise definitions of the abstract objects you are studying. The trouble is (and this is me talking now, not Gowers) that nearly all math textbooks leave out the vividness altogether. It is as if their authors are determined to make the subject as boring as possible. They are so eager to get down to the business of stating definitions and proving theorems that they never place themselves in the position of people learning the subject for the first time.
Worse yet, the social pressures among mathematicians favor that sort of presentation. It has happened to me several times that I have heard colleagues praising the most horribly opaque textbooks precisely because they were so concise and efficient. “No wasted words!” they would say, enviously. As far as I can tell, such books have nothing to do with good pedagogy, and everything to do with showing off the bits of esoterica mastered by their authors. Woe betide the author who presents two consecutive sentences of exposition!
In an undergraduate math class of mine the professor was writing the textbook for the class as we went along. It was an appalling book. Mostly just a textbook on how not to write a textbook. Excessive use of jargon and notation? Check! Little or no context for anything that was being done? Check! Applications that were as abstract and esoteric as the thing being applied? Check! It was horrifying that such a thing could be published.
More horrifying than that? The book later won an award for mathematical exposition.
And then we wonder why people hate math.