Writing tips for students

In a room with three mathematicians, there are five different opinions on mathematical writing. Here I will try to collect not really my opinions on the subject, but some resources that I find useful.

The target audience are (my) students. Everywhere I have been teaching, it seems that students are expected to pick up writing by themselves (if at all), and the lack of training becomes clear once they start writing a thesis.

Resources on writing in general

Mathematical writing is writing. To improve your writing you should read, preferably non-mathematical literature. George Orwell wrote a nice essay, called Politics and the English language, containing many valuable lessons about writing. Two of his rules are very useful:

If it is possible to cut a word out, always cut it out;
Never use the passive where you can use the active.

Another book that helps me a lot is the following:

W. Strunk Jr. and E.B. White, The Elements of Style, Pearson, Boston, 2000.

Resources on mathematical writing

Halmos wrote a famous essay on mathematical writing:

P.R. Halmos, How to write mathematics, L'Enseignement mathématique 16 (1970) 123-152.

Here is perhaps his most important advice: plan ahead, specially regarding notation. Halmos suggests that we make a table of notation before starting to write. To do so, we need to anticipate all the symbols and letters we will need. I think it is useful to go into detail here, making choices for indices, names of variables, etc., before you start writing.

In my experience, this alone saves a lot of time. It is likely you are not going to be able to anticipate everything you need: at some point you will realize that a certain letter was used for something and now is needed for something else. But, if you don't write the table before, it will be much worse.

There are also lecture notes from a course by Knuth on mathematical writing. These even present several different opinions on many topics.

One advice by Knuth is also very useful, if not always possible to follow if you are in a hurry (more for psychological reasons than because of real necessity): write by hand first. In my experience, this ends up saving a lot of effort later, because it forces one to think more slowly and carefully. Even if you don't follow this advice to the letter, it is useful to write a rough draft or outline by hand, and if there is a complicated proof, writing it by hand first is a good idea.

Douglas West maintains a very useful website with writing tips for mathematicians; the tips for non-native English speakers are specially useful.

Learn TeX and LaTeX properly

You are likely going to typeset your work using LaTeX (or, perhaps, TeX). Then learn to use it properly!

Invest time and read Knuth's TeXbook. It is well worth it. Not only will you learn about TeX, you will be reading what can only be considered a major achievement: a user's manual for a piece of technical software that is pleasant to read!
If you are writing a math paper, use the AMS document classes. They look much better than pure LaTeX. Read the user's manual.
There are tools like chktex that can check your TeX file and show you warnings.
Learn what a non-breakable space is and use it!

Read good math

Perhaps most mathematical papers (and even books) are poorly written, and that is not only a matter of taste. In order to improve your writing, you should have access to examples of good writing. I highly recommend anything that Halmos wrote, but one of his books is particularly good (and accessible):

P.R. Halmos, Finite-dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1993.

Things to avoid

Here is a (growing) list of things to avoid.

The majestic plural

For some reason, one is not allowed to write in the first person in a mathematical (or scientific) paper. Whatever the reason, it is just not done. To work around this issue, many people resort to using the third person instead, referring to themselves as "we". So it is that "we show", "we do", "we prove a new result", "we decided", etc.

This use of "we" to refer to yourself is called the majestic plural. The pope for instance uses it when speaking as the head of the Church ("our beloved predecessor" and so on), but not in daily life. This should hint at how ridiculous it is to use it in a mathematical paper! (And I have done it, mea culpa, mea maxima culpa.)

Luckily, it is possible to avoid the majestic plural completely. Here are some ways of doing it:

Use "we" as "you (the reader) and me (the writer)". For instance, it is fine to say "we will see in Theorem X", or "we have that", or "let us show", etc. Every time you write "we", see if it is the same as "you and me".
Use the imperative. Instead of saying "we multiply both sides by 10 obtaining...", say "multiply both sides by 10 to get..." etc.
Last resort: use the passive. Sometimes, but not very often, you need to say what you have done. This is usually the case in the introduction of a paper and specially in the abstract. For instance, "We prove a theorem etc.". If you are writing with co-authors, then "we" refers to the authors, and is not an instance of the majestic plural. If you are writing alone, then the situation is trickier. Instead of using the majestic plural, I much prefer to use the passive voice ("In this paper, it is shown that every sequence etc.").

When writing a research proposal, I find the use of the first person acceptable. You are talking about what you plan to do, after all, and it is weird to refer to yourself all the time as the "primary investigator" or the "researcher".

The passive voice

In some languages (like Portuguese) there are good ways to use the passive voice. Not so in English: as Orwell says, it is best to not use the passive at all if possible (but sometimes it is not possible; see 3 above).

Texts that overuse the passive tend to sound like "a theorem is shown", "a result is obtained", "a program is written". Nobody knows who does all these things.

References are parenthetical

"In [2], it is shown that...". This kind of construction is very common, and in my opinion quite terrible. It forces the reader to go to the bibliography and check what [2] is, and it is not much more work to write the better-sounding sentence "Hilbert [2] shows that...".

Ideally, references should be parenthetical, that is, if they are removed the text should remain coherent. In many old journals, references were footnotes, so it was actually impossible to write something like "in [2]" as above.