Exposition is not underrated

It seems to be the conventional wisdom (for example, some of the comments here) that exposition is undervalued in our profession. I disagree. To cast things in economic terms, let’s take “valued” to mean one of two things: increased salary (cash) or increased recognition by peers (ego). First, I think it is unquestionably the case that a reputation as a good expositor is more likely to lead to invitations to conferences, to give colloquiua, and to give invited addresses, all of which also affect one’s career in a positive way. Second, a well written paper is more likely to be accepted by a higher ranked journal, and is also more likely to be cited by others – factors which also have a direct impact on one’s career (the effect here may be marginal, but is, I think, non-trivial.) Third, I think that certain forms of expository writing — such as graduate texts (think “the Arithmetic of Elliptic Curves”) — are widely known and (deservedly) widely praised. So what is the complaint?

I think the key point here is to distinguish between several flavours of expository writing. The first concerns articles which might once have appeared as short articles in the Monthly. Here a highlight of the form is something like Elkies on Pythagorean triples and Hilbert’s Theorem 90. This is the amuse-bouche of the exposition world.
Second is the account of a known result whose proof is not readily available in the literature, perhaps something like A proof that Euler missed. Third is an attempt to come to terms with some body of work by either filling in details, giving plenty of examples, or offering a slightly different perspective; let’s say Vakil’s algebraic geometry notes. Finally, there is the survey/overview style paper which seeks to convey a vision of the field and its connections to mathematics, pehaps something like Mazur’s paper “The theme of p-adic Variation” (a title that is both poetic and yet almost a pun).

The majority of expository writing falls in the third class. It usually takes the form of notes for a graduate class that someone posts on their webpage. The “level” of mathematics is usually that of a graduate class, or an advanced undergraduate class. Let me freely admit that it is wonderful to have such sources freely available, and that they can be useful. They play an important educational role. But how much of a contribution do they make to the advancement of mathematics? I think the level is relevant here. An exposition of Dirichlet’s theorem on arithmetic progression is essentially worthless — it is a topic covered well in an endless numbers of textbooks. And let me pass on without mentioning (apophasis alert) any article concerning the discrete geometry of Chicken McNuggets. Then, as the difficulty of the subject matter becomes higher, and the number of available resources become scarcer, the utility of such notes are increased. However, there’s a catch. The most inspiring, fundamental, insightful, and useful expository pieces can only possibly be written by a very few people. This is due to two obvious restrictions: few people can write well, and few people have interesting and deep things to say. Take, for example, the topic of recent progress in the Langlands programme. It’s perfectly possible for many people to give an anodyne talk to a broad audience on the latest developments. With some effort, a smaller number of people can also present some intuition for some of the core ideas. But anyone qualified to give a detailed exposition of the latest modularity lifting theorems is more interested in proving new theorems themselves.

By all means encourage good exposition, by all means cherish it when the masters commit their intuition to paper, by all means enjoy the wealth of expository notes available on the web, by all means encourage (through the reviewing process) authors to write clearer papers and describe their intuition, by all means use NSF money to fund instructional workshops. But don’t, as Cathy O’Neil suggested (update: I heard this suggestion from Cathy in person, but it was pointed out to me that she says something similar here), pay good mathematicians to spend six months learning topic X in order to produce a purely expository treatment of some important piece of mathematics; either they won’t be up for the task, they will have better things to do, or they would have done it naturally out of their own accord and inclination.

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6 Responses to Exposition is not underrated

  1. I essentially agree with these ideas. One thing I’d like to point out is that I really like the way the Bourbaki Seminar basically tries to convince mathematicians to write survey expositions of some works that they understand very well, and (most of the time) that is specifically not theirs. This can be a fair amount of work for the writer, but this approach has had outstanding successes.

    I’ll add a fun little-known fact concerning that note of Elikies on Theorem 90 and Pythagorean triples: this proof was already published (at least) in 1970 in the American Math. Monthly in a paper of Olga Taussky on sums of squares (Monthly, vol. 77, No. 8, 1970, pages 805 to 830 (see page 807.) That paper is also quite a nice example of survey writing…

    • There are some excellent Bourbaki seminar articles. Part of the reason is that they have good success in convincing experts to write the articles, and the best such articles contain insights that could only come from an expert. There are not many people who could have written Travaux De Shimura.

  2. Drew Armstrong says:

    Comment here:

    • I don’t think this issue is really about finding a balance between research and education. It’s about individuals wanting to be rewarded professionally (tenure, salary, etc.) by writing expository articles which neither make a contribution to the corpus of mathematics nor (outside exceptional cases) to the education of the next generation of mathematicians. The language of the debate also seems to ignore the fact that the web is flooded with more expository material than would even have been conceivable thirty years ago, from Wikipedia to MathOverflow to math blogs to online course notes (this is, mostly, a good thing). Many mathematicians have a desire to explain their ideas, and it’s not clear to me that anything profitable can be gained by skewing the incentives. (If you want to argue that teaching should be taken more seriously when it comes to hiring/tenure decisions, that is another debate.) I find myself in agreement with the comment you quote from Dan L: “I also do not believe that your hypothetical “great storytellers of mathematics” really exist, in the sense that I do not believe that a person can achieve that level of mastery over a field without being a successful researcher.”

  3. Georges Elencwajg says:

    Very thoughtful article, going against a tired cliché. One might also mention historical articles like Dieudonné’s “The historical development of algebraic geometry ” or Schappacher-Schoof’s “Beppo Levi and the arithmetic of elliptic curves” where the authors explain in particular Levi’s elementary proof that an elliptic curve over Q cannot have a rational 16-torsion point. (I won’t digress on the stylistic point that one doesn’t need a Greek word like “apophasis” when a plain, cosy Latin word like “preterition” works just as well.)

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