A similar guide, aimed at people writing research papers, is “How to Write Mathematics” by Paul Halmos (1970) [1].
They both start from a similar assumption:
Lee: “When you write a paper in a math class, your goal will be to communicate mathematical reasoning and ideas clearly to another person. The writing done in a math class is very similar to the writing done for other classes. You are probably already used to writing papers in other subjects like psychology, history, and literature. You can follow many of the same guidelines in a mathematics paper as you would in a paper written about these other subjects.”
Halmos: “The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea.”
[1] https://www.mathematik.uni-marburg.de/~agricola/material/hal...
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.
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.
Do you have TL;DR for the difference between aimed at undergraduates vs research papers in terms of writing?
I think the reason is that his writing was like a clear 1-on-1 tutorial session, with many of the "writing mathematics" practices described in the article here. It had a conversational style as if it was someone trying to explain but in written words. I recall phrases like "now look at the expression we have here, what does it tell us?" Or, "what follows is a somewhat long derivation, but you will find the effort of working through it pays off".
Most other textbooks read like stilted reference manuals by comparison, with "exercises left to the reader".
My professor didn’t use Griffiths for the exercises, so I actually sold my copy back to the bookstore and bought a used copy of Shankar’s book: it is certainly drier, but I think it’s much more clear and precise.
This would be insane to do in code, why is it normal in mathematics?
Note: I had this example in my head while writing https://www.traffic-simulation.de/info/IDMsstar.png
For programming, the same constraint does not apply.
For most people, with their limited writing skills, understanding and precision have a tradeoff, and there is no perfect paper.
All in all, imprecise probably was the wrong term. The statements are precise for the people working in this specific "bubble" of research, because they might know the implicit assumptions made and in the worst case, the only one who knows the implicit assumptions is the author.
