A similar guide, aimed at people writing research papers, is “How to Write Mathematics” by Paul Halmos (1970) .
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.”
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.
I really appreciate when papers get to say at some point "for the sake of clarity, let drop this or that notation" that are not that relevant to the key ideas. Usually it's in the good quality papers.
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.
Are mathematicians ever frustrated that they don't understand what the variables mean? Or are you able to look at the above equation and infer the meaning of the variables based on experience?
I understand that there are symbols such as "delta" which basically always mean the same thing. But would you have been able to tell what "b" meant without someone telling you?
Yes, if the presenter does not define them, or if the variables don't have a "usual" definition within the field.
> But would you have been able to tell what "b" meant without someone telling you?
Mathematicians, being humans, are not perfect by any means. They will sometimes fail to define things properly. But they are not stupid - they do have a cultural norm of defining everything, that is obeyed most of the time. Conversely, readers of mathematics are not perfect either. They will sometimes skip over the definitional part of a paper/book and jump straight to the results, and wonder why they can't understand.
In praxis one should be aware of the following points:
- In programming, the computer will complain if an undefined variable is used. In mathematics, this is sadly missing. (The next best things are other proofreaders, i.e. other mathematicians.)
- Variable names aren’t just picked at random (or as a, b, c, …), but nearly always follow sensible patterns. (Natural numbers are n, m, k, l, …; vectors are v, w, u, …; indices are i, j, k, l, …; radius is r; …) Different authors may use different conventions, but they still allow mathematicians to kind of understand what the variable means just from looking at its name.
- Every area of mathematics has certain keywords which the reader has to be aware of. Again, some authors may use (slightly) different conventions, but there are typically only few conventions out there, and they often don’t differ much. (Example: The space of homomorphisms/linear maps between two vector spaces V and W is commonly denoted by Hom(V, W), hom(V,W), ℒ(V, W) or Lin(V, W).) One can oftentimes tell what a keyword means just from it’s name, its signature, and its usage. Keywords also often consist of more than one letter or are typeset in a special way to distinguish them from regular variables.)
> Are mathematicians ever frustrated that they don't understand what the variables mean?
So to answer the question: if a mathematician doesn’t understand what a variable or a notation means, then one of the following has happend:
- The variable was already introduced beforehand, but the reader forgot about it. (This is the most common scenario.)
- The variable is explained in the upcoming line. (Also very common. The reader will—of course—only notice this after going through the previous part of the text multiple times in seach of just this explanation.)
- It is a standard notation that the reader is not familiar with. (Often happens if the reader is missing the background knowledge assumed by the author, or if the author uses some outdated notation (e.g. because they have been dead for over 50 years).)
- The author made a simple mistake while writing. (Typo; forgot to change a variable name after shuffeling things around).
- The author actually forgot to define the variable: a mistake that is hopefully catched by their peers.
- The explanatory text was left out for time reasons (giving a talk, writing some rough/informal lecture notes, quickly scribbling down homework in the morning).
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.