A Guide to Writing Mathematics [pdf]

> Are mathematicians ever frustrated that they don't understand what the variables mean?

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?

Probably not.

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.

Variable names in mathematics have no intrinsic meaning (as they have, for example, in physics). In a mathematical text, every occuring variable must be properly defined. This is most commonly done before they are used, with formulations like “let x be …” or “x := …”, or immediately after they have just been used in a formula, with something like “where x denotes …”. Failing to do so is just as much of a mistake in mathematics as it is in programming. (In an homework assignment or exam, doing so will lose you points.)

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.)

Good mathematical writers will oftentimes go out of their way to explain their notation at the beginning of their text, just to be sure.

> 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).

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.