In Praise of the Notation Section
As a junior mathematician, I feel strange writing this sort of post. Usually, I opine on things that it’s perfectly OK for someone at my level to write about: what it feels like to do math of any sort, specific tools a researcher needs when s/he’s just starting out, and so forth. Today, though, I’m going to be a little bolder than usual and argue that many papers and books today are missing something very useful: a full glossary of notation.
By that, I don’t mean just an index of symbols, although those can be helpful too, and I don’t mean a section entitled “Introduction” or “Preparatory Lemmas” (and, tangentially, did you know that the proper plural of “lemma” is “lemmata”?) with notation sprinkled throughout. I mean a well-defined point in the text, the primary function of which is to act as a ready reference for every piece of notation you use. If \(\alpha_t=(1+H)/(t+H)\), then this notation section would remind me of that, as well as of what H was.
There are several benefits of this. Quite beyond ease of reading and standardization throughout a paper, which can be an issue in and of itself – I have worked with tenured professors, one MacArthur Genius in particular, who had trouble keeping track of all the notation in a twenty-page paper, and a friend of my mother’s recently saw her thirteen-year-old daughter reduced to tears by a poorly-defined symbol in a geometry textbook – a notation section can clarify a paper or a book in a unique way. This is especially true if the section is written in a style that explains the ideas behind each definition: “We define operator X to be this, and function Y to be that, and by applying the one to the other we get a result X(Y), which we denote Z.” Reading this, I know not only to watch out for X, Y, and Z, but also something of the strategy with which the paper will combine them, and, if nothing else related to Y gets its own symbol, I would know that nothing else that will be done with X and Y is quite as important as calculating Z. Defining your notation elegantly isn’t quite a proof sketch, but in some cases it can almost lead to one.
As to what goes in this section, the question is not as contextual as you might imagine. We can reasonably imagine that most mathematicians know what \(\sum\) is. I would suggest one very simple rule: if you made it up for the paper, and if you use it in more than one place, it should go in the notation section. This way, the section can also work as a way of helping readers to pick out what’s most important. Without a notation section, I might gloss over function Y the first time I see it, not knowing that it’s going to come up again later. Similarly, I might spend ages trying to work out exactly how a particularly complex piece of algebra is carried out, only to find that it’s just a minor bit of rigor orthogonal to the main thrust of the proof. By having a notation section of any kind, papers can help readers look ahead; by having a well-filtered notation section, papers can help the reader prioritize what they see when they do.
I should mention here that I do not believe a bullet-point list is the right way to format a section like this. As I said, a large part of the purpose of a notation section is to be a microcosm of the paper, and that means it has to flow like the paper. On the topic of stylistic decisions, I don’t have much of an opinion on whether or not papers reiterate the definitions of particular pieces of notation when they actually first use them, as long as those definitions are consistent throughout. This, by the way, is another benefit of a fixed notation section: it fixes the notation in the rest of the paper. I remember reading one paper, written by three authors, who considered two different and unequal cost vectors. Different sections, depending on who wrote them, referred to either one vector or the other as \(c\), and the other as either \(c_0\) or \(\tilde{c}\).
Many proofs require no special notation beyond a few variable names. Many more are perfectly comprehensible even when defining notation as it’s used. Many, however, in all fields, are not, and so I would ask my readers, as a favor to theirs, if they would consider starting their papers with notation.