“Vividness” in Mathematics

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.

Comments

  1. #1 MNb
    February 15, 2013

    “declare that the second passage is more vivid than the first”
    It’s your full right to declare what you want, but I had no trouble reading the first quote and couldn’t make it through the other, trying twice (the second time after I read your declaration). Matter of taste, I suppose.

    “nearly all math textbooks leave out the vividness altogether”
    I don’t know about the USA, but in the Netherlands vividness is mandatory. Being a little conservative I would argue that this demand sometimes goes at the expense of the level of the content.

  2. #2 AlanF
    Manchester, NH
    February 15, 2013

    I enjoyed your comments about vividness in math teaching. I’ve long had more or less the same thoughts.

    My favorite math textbook was one on differential equations (I no longer remember the name off the top of my head) that included historical notes about the mathematicians who made significant contributions to the field. The history made the basic material far more interesting than a dry presentation. The author also included stuff about *why* the material was important.

    Writing this called back my memory of Professor Arthur Mattuck of MIT, who taught my differential equations course. He was very entertaining and managed to make the course fun — something not often seen in that course.

  3. #3 Petri J
    Europe
    February 15, 2013

    During my study times, I heard that very many books are written to impress, not to inform. It hasn´t changed…

  4. #4 dean
    February 15, 2013

    But doesn’t a good deal of this come from the way we’ve taught to write during our training (passive voice, third person)? After years of learning to rewrite material to pare it down during graduate school, dissertations, and publications, it shouldn’t be surprising that the ability to switch presentation style is rare.

  5. #5 Eric Lund
    February 15, 2013

    Oddly enough, the second description of students arriving on campus is more vivid because it is more precise than the first. The first passage speaks of generic cars full of generic stuff, and the author tells us that the parents are socioeconomic peers. The second tells us what kind of cars are involved (station wagons are cars with a larger carrying capacity that were common for several decades in the US; in the last two decades they have been mostly replaced by SUVs), what kind of stuff they are carrying, and that they are carrying more stuff than can actually fit in them. The second passage also shows, not tells, us that the parents have similar socioeconomic backgrounds.

    The analogy to mathematical language isn’t perfect, because ordinary language often isn’t precise enough for mathematics. But for students who are learning about mathematics, both kinds are needed: the technical language for giving a precise definition, and the more vivid language for explaining why the student should care about that definition. Mathematicians are often not the best judges of what should be in a textbook because they are a self-selecting group: this stuff always was easy for them, and they may have forgotten that it isn’t easy for everybody.

  6. #6 Sean T
    February 15, 2013

    Eric,

    I think the real problem might not be so much that mathematics came easy to the textbook writers. Maybe it didn’t; maybe the writer had to struggle to learn his subject before finally mastering it. The real issue is that math textbook writers are so familiar with the mathematical notations and formal definitions that they don’t realize that students, especially students just beginning to study mathematics, are not. The textbook author looks at a formula that formally defines a concept and thinks “gee, that seems pretty clear to me,” whereas the student gets little to no understanding of the concept from that formula.

    A good example of this might be the definition of a limit. A mathematician can look at the formal definition, with all the epsilons and deltas and understand what that definition says. A beginning calculus student gets next to nothing from that formal definition.

    This can be extremely detrimental in an attempt to educate students. The student exposed only to the formal definition of limit, for example, will probably never actually understand the concept. Since the concept of limit is absolutely fundamental in calculus, that student likely will never be able to learn calculus properly.

  7. #7 konrad
    February 15, 2013

    I agree with the general point you are making, but I’m with MNb when it comes to the specific example. I found the first passage well written and very readable, but the second was unbearably tedious to the point that I couldn’t bring myself to read it all the way through. Ditto on the second attempt.

    When it comes to text books, I have also seen ones that make the mistake of wading through endless tedious examples before getting to the point. This is a real danger – one needs to strike a balance.

  8. #8 eric
    February 15, 2013

    I, too, am with MNb on preferring the first passage about returning college students. However, I much preferred the second passage on actual math.
    There are occasions where excessive detail can kill a ‘story’ just as dead as not enough. Think John Galt’s speech or Umberto Eco’s three-page descirption of a single door in Name of the Rose. Writers need to convey their point and help their readers undertand it; if detail helps do that, add detail. If detail gets in the way of your audience understanding the point, take it out.
    All IMO, of course.

  9. #9 Another Matt
    February 15, 2013

    For the last 10 years I’ve been teaching courses in computer music (and done the odd music theory tutoring here and there). One thing I’ve found with pedagogy is that students who have little experience with math tend to be much more adept at getting to the abstract concept by generalizing from “vivid” examples rather than extrapolating implications from a precise definition at the outset. The precision can always become explicit later, but the trick is to be consistent with all terminology in the informal presentation so that the precision is already built in and the student does not have to “unlearn” anything.

    This makes some intuitive sense from a cognitive point of view — I think we have a lot more practice at induction and analogy building than formal deductive reasoning. This may be one of the reasons that “doing math” often feels like a process of empirical investigation and discovery.

  10. #10 Jason Rosenhouse
    February 15, 2013

    I actually like the first passage too, even though the second passage is more “vivid.” Show don’t tell is a good maxim, but it often leads to bloated writing.

  11. #11 Oliver
    February 16, 2013

    Good point, Jason. What most writers forget is that it is possible to show and not tell without stuffing tedious details into the prose. It reminded me of Mark Twain’s Cooper’s Prose Style actually. Quite a humorous look at the use of detailed descriptions in writing, and it could probably apply to the second “vivid” passage above.

  12. #12 joemac53
    February 16, 2013

    I taught high school math and science for 35 years. I hated almost all the textbooks I had to use. We used to start the year with “How to read the text” lessons. I forbade students to look at the algebra that the physics books used to solve sample problems. I informed students that they would usually find answers to multiple part questions in reverse order of their being asked in the problem.
    I used one precalculus book for many years because it was so small the kids did not mind taking it home and back to school each day.
    I did the explanations as many ways as I could think of. I did not expect the books to do that for me. I could be very vivid at times!

  13. #13 Blaine
    February 18, 2013

    Perhaps its because most mathematicians are in the main, cognitively deficient except when it comes to mathematics. The older, perjorative term for this was idiot savant.

  14. #14 Sean T
    February 19, 2013

    Blaine,

    I really hope you intended that to be tounge in cheek. Generalizations such as that are rarely true and can be hurtful to those who have been so mischaracterized. Jason, for one, is certainly not cognitively deficient outside of mathematics, as is evident from his blog posts (which I dont’ always agree with, but which certainly are good evidence against the idea that he’s cognitiviely deficient). On the contrary, in my limited experience with dealing with mathematicians, I find that most are fairly conversant on many topics outside of mathematics, and are generally well-rounded individuals. There may be some that are as you describe, but that seems to be the exception rather than the rule.

    I honestly believe that the real reason that math texts are written as they are is that they ARE comprehensible to mathematicians who write them. Mathematical formalisms have developed because the precision of ordinary language is insufficient for mathematical use. Writing textbooks in such formalism increses the precision of the texts, and that seems to be the priority of the textbook authors.

  15. #15 Blaine
    February 19, 2013

    Thanks, I am a mathematician

  16. #16 Blaine
    February 19, 2013

    Of course I mean it tongue in cheek along the lines of C.P. Snow’s two cultures. I certainly would not include Jason in that comment. But, there does seem to be a divide between quant types ( of which I am one ) and liberal art types. The level of specialization required to be able to not only consume and apply modern math, but also produce it is quite high…the same goes for high end literature. Those skills are often not found among mathematicians with notable exceptions which only reinforces the observation. I often noted distinct personality traits that also make it more difficult to explain mathematics to others. My set theory professor, Thomas Jech at Penn State, would tell us how stupid Americans were before he called us up to the front of the class to finish off a proof he started…not a good bedside manner.

  17. #17 Blaine
    February 19, 2013

    A good many times I have been present at gatherings of people who, by the standards of the traditional culture, are thought highly educated and who have with considerable gusto been expressing their incredulity at the illiteracy of scientists. Once or twice I have been provoked and have asked the company how many of them could describe the Second Law of Thermodynamics. The response was cold: it was also negative. Yet I was asking something which is the scientific equivalent of: Have you read a work of Shakespeare’s?[4]

    I now believe that if I had asked an even simpler question — such as, What do you mean by mass, or acceleration, which is the scientific equivalent of saying, Can you read? — not more than one in ten of the highly educated would have felt that I was speaking the same language. So the great edifice of modern physics goes up, and the majority of the cleverest people in the western world have about as much insight into it as their neolithic ancestors would have had.[4]
    C.P. Snow

    I suspect the reverse observation would often apply as well.

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  20. #20 Tom
    Brisbane, Australia
    February 27, 2013

    Your premise (that more explanation and vivider imagery is a good thing) seems almost impossible to argue with, so I’m slightly disturbed that I’m not entirely sure I agree. I’ve always enjoyed reading math textbooks in my spare time, and while some of my favourites have been full of vivid and passionate prose (Halmos’ Naive Set Theory, Lanczos’ Variational Principles of Mechanics) there’s something to be said for a concise text which requires the reader to fill in the imagery and motivation (I’m thinking here of Rudin’s POMA and Grimmet and Stirzaker’s Probability and Random Processes – two beautiful texts which reward patient study). If nothing else, compacter texts are much easier to skim through later if your memory’s as bad as mine.

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