Writing About
Mathematics Writing About
Mathematics This is my first attempt to write about writing. There is a good SIAM book on this subject by Nick Higham, and this article adds only a few suggestions. Some of the suggestions are almost tricks, and I hope they will help. Others go deeper because writing is very hard work---if you seriously want to be read and understood. Maybe that is the point of this article.
In all these suggestions, I assume that the mathematical ideas have been thought through---there is something to say. But the saying is distinct from the thinking. To get the equations and proofs correct is absolutely not the final step. You have to move beyond the research, where the goal is insight. In writing, the goals are less abstract and more human:
Some writing is only for the record. I am not speaking about that style at all. This article is about communication to a reader, not storage in a journal. If you want the attention of readers, you have to give them your attention. What do they already know, and how does the new part fit in? This comes naturally with speech, so I often say the words aloud. That tends to keep the sentences simple.
The enemy of good writing is not lack of space. It is lack of time and energy and patience. Sometimes the ideas flow easily into words, sometimes they get stuck and won't move. I try to write when I feel psychologically strong (these words are written in the early morning; I am impressed by anyone who can keep writing after dinner). Above all, remember the reader.
Creating a string of equations is easy, and no good. It's much better to express your idea twice, and in two ways-- first in words and then in the equation.
Innocent bad example:
The angle between two subspaces is inf cos-1|<u,v>|/ ||u|| ||v||.
Better to have said it in words too:
The angle between two subspaces is the smallest angle between vectors in the subspaces: inf
....
Tell the reader what the damn equation means! That person has other things to do. Given half a chance, he or she will stop reading your article. You must grab the reader's attention and hold on. If the point is not clearly made, and decoding the notation requires an effort, phooey on the rest (unless you are proving Fermat's Last Theorem).
The notation maps ideas into symbols. If the notation is good, the reader won't keep searching back to the start of the paper. Just add the words the reader needs:
Poor: Analogous to (3.1.5), if we take the union of the bases in (3.1.12), N=1,2,...,
we arrive at (3.1.14).
Better: We can include the following simple proof of (3.1.14).
Best: Just say it: We can quickly show that T is bounded.
Another useful device has been hiding in these paragraphs (and in this sentence). I think it is called anthropomorphism. An inanimate and abstract concept is assigned human properties. It hides in a paragraph. It gets stuck and won't move. This is nonsense, of course. An idea can't do such things, it just sits there. I can't stop doing this....
Humanization is a simple way to write more actively. It definitely adds life to the paper.
The first words are a signal to the reader, green or red. Work on those important words. Read them aloud, their rhythm is important. A mixture of long and short sentences will help. You absolutely must find an interesting way to start.
With a specific example, I can try to show in detail how the pieces might fit. The nearest example is the beginning of this article. May I ask you to look back at the first paragraph? We can analyze it and improve it.
The first sentence has nine words and no commas. It introduces the subject: "to write about writing." In a sense it also introduces the author: "This is my first attempt." The first-person adjective my is informal and personal (not always appropriate). The important point is that you can not only read those nine words, you can hear them. Living speech is extra powerful--- it is impossible to keep up such strength in writing.
In that opening paragraph, notice the word suggestions. The repetition of that word connects everything. A few suggestions are separated into some of the suggestions and others. It is like multiresolution, moving from the broad topic (writing) to the specific contents (suggestions) and then to the details.
The last sentence of the paragraph is suitably short, but too weak. A purist would complain about the vague reference to that. Better to repeat the key idea: "Maybe serious work is the point of this article." Now I have to think again about that gentle word maybe.
At this rate we won't finish! This isn't a poem, it is just a little article about writing. Every word counts, but the first words count the most. A typical mathematics paper will change gear, as it goes from introduction to exposition. For this main part, where the research is described, I add only one new thought:
A list of more than four items is stretching it. I will end with with two very small suggestions, which you can safely ignore
"The components yi, i = 1,2,...,n, are all positive." I bet that i = 1,2,...,n is obvious from the context. Stating it just interferes with the reader's thinking.
I don't think that L2 and a Besov space should be defined in the same paper. If L2 is not already an old friend, there is no point whatsoever to introducing Bp,ks. I believe that definitions of L2 should be forbidden in research papers.
I can't end with grumbles. Maybe unhappiness is good for novelists, but mathematics is basically optimistic. We are writing for friends, and they really want to understand. Talk to them as you write!
Gilbert Strang, a professor of
mathematics at MIT, is the author of six textbooks, most recently Wavelets and Filter
Banks. He has also written (and spoken) about wavelets for SIAM; his 1989 paper in
SIAM Review is often held up as a model of expository writing. Currently he is SIAM
Vice-President for Education.