Experts advice on writing mathematics papers

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Easy reading is damn hard writing!

The late number theorist David Goss in his very brief article Some Hints on Mathematical Style quotes famous mathematician J.P.Serre commenting on the original version of his article:

It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathematicians have a good ear; some not (and some prefer the slangy expressions such as “iff”). That’s life.

Indeed, that is how I’ve learnt the grammar of this language; by reading from others and trail and error of writing. However, more recently I had to co-author a 50 plus page paper, and over the course of writing it, I was increasingly concerned about readability of the final outcome. Or as Hawthorne said Easy reading is damn hard writing. So, I decided to search on the web to see what the experts have to say on the matter. Goss’s paper, despite its brevity, has very good tips on writing maths well with style; I like the analogy of writing papers with transmitting of ideas:

All basic definitions should be given if they are not a standard part of the literature. It is perhaps best to err on the side of making life easier on the reader by including a bit too much as opposed to too little (Rule 1). Some redundancy should be built into the paper so that one or two misprints cannot destroy the understandability. The analogy is with “error-correcting codes” which allow transmission of information through noisy and defective channels.

Goss gives a set of minimal rules. Most rules there must be observed in any good math writing, and maybe one or two seems subjective to me. As a great companion to Goss’s paper you should also watch famous Serre’s talk How to write mathematics badly. It is rather fun to watch! You get to watch lots of funny mistakes, including mispellingsA self-referential word. The correct spelling is, of course, misspelling. of different kinds: e.g. “Principle bundles have moral fibres.”

If you are unsure about certain typographical issue when writing maths, such as spacing, breaks, hyphens, etc. then you can consult the book Mathematics into Type by E. Swanson. (It is freely accessible.)

There are also some books and articles that are more rigid on how you should write your mathematics. However, I think although some of these rules can improve the quality of piece of writing dramatically, especially for the reader, we must not forget that mathematics is a highly social activity and at the end of day any mathematics paper reflects the idiosyncratic habits of thinking of its author, and these could make an otherwise monotone stylish. Speaking of stylish papers that come to my mind are: I agree with Alain Connes that

My main point is that mathematicians are so “singular”, (and behave like fermions as opposed to the physicists who behave like bosons) that making general statements about them often produces something obviously wrong or devoid of any content.

Finally, I cannot finish this point without mentioning that during my chasing of various interesting links folks shared here and there, I came across a book by Carl E Linderholm from 1972. Among other things, Mathematics made difficult is a humorous and entertaining satire of stereotypes of French mathematicians, the project of formulating all of mathematics on category-theoretic foundations, and some other pedestrian jokes. Here is a quote that stuck in my mind from it:

The simpler the things a man gets difficulty out of, the better his mathmanship.

More references on writing with style

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