Is usually terrible. Possibly because writing is just plain hard, harder in your second language (and scientists tend not to be writing in their mother language), and possibly because a shadowy conspiracy of lecturers aims to create demand for remunerated means of education delivery, such as lectures, where they explain well in person what they obfuscated in books, as in don’t be Big Talk,
I am especially interested in good mathematical style, but academic style in general could be good to know.
J.S. Milne, tips for authors.
If you write clearly, then your readers may understand your mathematics and conclude that it isn’t profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities.
Cosma Shalizi, Practical peer review:
- The quality of peer review is generally abysmal.
- Peer reviewers are better readers of your work than almost anyone else.
Rob Hyndman’s Writing seminar also has some bullet points on weathering reviews.
Van Savage & Pamela Yeh, Cormac McCarthy’s tips on how to write a great science paper:
- Use minimalism to achieve clarity. While you are writing, ask yourself: is it possible to preserve my original message without that punctuation mark, that word, that sentence, that paragraph or that section? Remove extra words or commas whenever you can.
- Decide on your paper’s theme and two or three points you want every reader to remember. This theme and these points form the single thread that runs through your piece. The words, sentences, paragraphs and sections are the needlework that holds it together. If something isn’t needed to help the reader to understand the main theme, omit it.
- Limit each paragraph to a single message. A single sentence can be a paragraph. Each paragraph should explore that message by first asking a question and then progressing to an idea, and sometimes to an answer. It’s also perfectly fine to raise questions in a paragraph and leave them unanswered.
E.T. Jaynes channelled though Tom Leinster:
If you differentiate a function \(f(x)\) without first having stated that it is differentiable, you are accused of lack of rigor. If you note that your function \(f(x)\) has some special property natural to the application, you are accused of lack of generality. In other words, every statement you make will receive the discourteous interpretation.…
fanatical insistence on one particular form of precision and generality can be carried so far that it defeats its own purpose; 20th century mathematics often degenerates into an idle adversary game instead of a communication process.
The fanatic is not trying to understand your substantive message at all, but only trying to find fault with your style of presentation. He will strive to read nonsense into what you are saying, if he can possibly find any way of doing so. In self-defense, writers are obliged to concentrate their attention on every tiny, irrelevant, nit-picking detail of how things are said rather than on what is said. The length grows; the content shrinks.
Mathematical communication would be much more efficient and pleasant if we adopted a different attitude. For one who makes the courteous interpretation of what others write, the fact that \(x\) is introduced as a variable already implies that there is some set \(X\) of possible values. Why should it be necessary to repeat that incantation every time a variable is introduced, thus using up two symbols where one would do? (Indeed, the range of values is usually indicated more clearly at the point where it matters, by adding conditions such as \((0<x<10 \lt x \lt 1)\) after an equation.)
For a courteous reader, the fact that a writer differentiates f(x)f(x) twice already implies that he considers it twice differentiable; why should he be required to say everything twice?
Shaun Lehmann, the vagueness problem in academic writing:
[It is..] likely that your writing was suffering from ‘vagueness’ – a constant problem in English. English-speaking readers (especially in an academic context) will only do a very small amount of work to figure out what you mean before they respond with confusion. […] A useful technique is to learn to read your work through the eyes of a kind of caricature of the low-context communication mode. You need to imagine a reader who is highly intelligent and logical, but who has no common sense and will fail to interpret any multiple meaning in the way you had intended.
I call my version of this the Commander Data Meditation […]
There is also some stuff there about “high and low context languages”:
In a high context language you can take a lot for granted and don’t have to explain yourself. You may also see cultural communication styles like this referred to as listener/reader responsible. As it happens, some of the most common first languages of students writing in English are derived from high-context environments: Chinese, Korean, Japanese, Indonesian, Thai, Arabic, and to some extent Spanish and French.
On the other hand, a low-context culture relies much more so on the content of the message. Low context languages developed in situations where people living next to each other were different – such as in trading ports and countries that have been repeatedly colonised – such as England was for thousands of years.
I’m not convinced by that part.
Knuth, Donald E., Tracy L. Larrabee, and Paul M. Roberts. 1989. Mathematical Writing. [Washington, D.C.]: Mathematical Association of America. http://carmenere.ucsd.edu/drichert/reference_docs/Mathematical_Writing-Knuth.pdf.
Lamport, Leslie. 1995. “How to Write a Proof.” The American Mathematical Monthly 102 (7): 600–608. https://doi.org/10.2307/2974556.
———. 2012. “How to Write a 21st Century Proof.” Journal of Fixed Point Theory and Applications 11 (1): 43–63. https://doi.org/10.1007/s11784-012-0071-6.
Savage, Van, and Pamela Yeh. 2019. “Novelist Cormac McCarthy’s Tips on How to Write a Great Science Paper.” Nature, September. https://doi.org/10.1038/d41586-019-02918-5.