As I’m teaching a course called “Scientific Communication” to a few Honours students this semester, I’ve been reading or re-reading many of the standard articles on “How to Write Mathematics” and doing quite a bit of online searching on the topic.
After reading many of these articles, a bunch of blog posts and several sets of slides, the main conclusion that I’ve reached is that Halmos’s classic article “How to Write Mathematics”, written nearly 40 years ago, is an absolute gem. It’s still totally relevant to today’s mathematicians and moreover, it is beautifully written itself. I actually went down to our library and got out the book in which it appears, but there seem to be plenty of copies on the web.
The basic rule is simple: focus on trying to convey your ideas to the reader as clearly and simply as possible. The aim is not to impress the reader with your vocabulary or erudition or by labelling difficult logical jumps or complex calculations as “obvious”. Instead the object is to use simple language, clear and consistent notation, and plenty of examples and figures to convey the ideas to your intended audience.
Of course, looking at the still-too-low average standard of mathematical papers, this is obviously more easily said than done!