As a math major at UCD, I never saw Lee's essay, so I'm guessing it's from the past few years.

I did read Halmos's, though. It was helpful for me because I was starting from a very programming centric frame of mind and I was surprised at how "conversational" math writing was. Reading it helped me start to learn how to express ideas precisely and clearly without a strict code-like structure.

Thanks for linking this! I like Halmos‘ clarifications on the editorial “we“. I don’t know if this is because I‘m living in a non English speaking country but at our university students often get told to use ”we” instead of ”I” in their papers. I always found this weird. If you‘re the only author you can‘t just refer to yourself as ”we” — just to avoid the use of ”I”. It sounds wrong — especially if the reader knows that you’re the only author — and eventually leads to absurd constructs such as the example in the essay: ”We thank our wife for her help”.

I understand the problem. I used to manage an English academic writing program at a university in Japan, and what guidance to give to students about pronoun usage was a frequent topic of discussion among the teachers.

One problem was that the students had learned a moderately informal version of English in which first-person pronouns are common. Also, they were young and used to writing and speaking about themselves. That led to what some teachers perceived as excessive use of “I” for the research papers the students were being taught to write.

Another issue was that the teachers themselves all had academic backgrounds, most with doctorates, and, we discovered through our discussions, pronoun usage varies a lot by field. Curious, I once looked through journals in a variety of fields—sociology, nursing, physics, gender studies, literature—and found that in some fields the authors never seemed to refer to themselves by “I” or “we” while in others it was common.

The use of “we” in mathematical writing, especially proofs, may be a special case. The “we” in a sentence like “If we assume that M is a compact metric space, then we can prove that ...” doesn’t really refer to the author or authors; it seems to have a more abstract referent.

Paul Halmos, by the way, was an excellent teacher as well as writer of mathematics. I was fortunate to take several classes from him when I was an undergraduate at the University of California, Santa Barbara, in the 1970s. Though I ended up not going into mathematics, I still have a very fond memories of learning with him.

I'm going to read through the guide linked in the post, but I don't have time (Or really the desire) to read through the guide aimed at people writing research papers.

Do you have TL;DR for the difference between aimed at undergraduates vs research papers in terms of writing?