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I learned to do math proofs in college. But recently I have begun studying more advanced math books and I've noticed some mathematicians frequently make assumptions that I don't. When I write proofs, I tend to state every detail possible. But mathematicians often omit details or repetition. I used to think this was annoying but having written more proofs I realize such omissions create concision and may aid in understanding as a result.

Having said this, I do not feel I have a good grasp on what makes for good mathematical writing. Does anyone have any good resources on this topic?

Remark 1: I have extensively studied general writing and verbal communication and very few of the principles used in explaining things (eg use of analogies or metaphors) work well for mathematics -- at least at higher levels. So math writing is clearly a distinct skill.

Remark 2: A very trivial, simple example of such concision is that when we refer to $\mathbb{R}^{n}$, we mean a set of numbers for $n=1$ but a set of ordered pairs for $n>1$. Even I wouldn't state this, but for younger learners, I'm sure that distinction isn't known initially. I certainly didn't and had to learn it.

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  • $\begingroup$ Are you saying that $\mathbb{R}^n$ isn't a set for all $n$? $\endgroup$
    – Robin Goodfellow
    Commented Dec 21, 2014 at 4:09
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    $\begingroup$ @StanShunpike: $\mathbb{R}$ is the set of all real numbers, while $\mathbb{R}^n$ is the set of all ordered $n$-tuples of real numbers. Both are sets, but the elements of the latter aren't real numbers. $\endgroup$
    – Daniel Hast
    Commented Dec 21, 2014 at 4:20
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    $\begingroup$ Easy joke: "First step, write poorly." $\endgroup$
    – Linear
    Commented Dec 21, 2014 at 4:35
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    $\begingroup$ You will find some related posts on math.SE. For example, Book about technical and academic writing and posts which are shown there among linked questions (and perhaps among related questions). $\endgroup$
    – Martin
    Commented Dec 21, 2014 at 9:51
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    $\begingroup$ Hi again. This is one of the main reasons I've assigned plenty of proofs and graded them myself in my upper-division and graduate courses. The best way to learn to write is to do so, and plenty of feedback is necessary. $\endgroup$
    – Ted Shifrin
    Commented Dec 21, 2014 at 22:29

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I'd suggest Halmos' "How to Write Mathematics"

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My favorite little tidbit on this topic is due to Jean Pierre Serre: One should aim to be "precise, yet informal". Here, "informal" means not using symbols and notation, i.e. not using "formalism".

Regarding the steps suppressed in mathematical writing, this is very sensitive to the audience one is addressing. Routine arguments that can be checked by a reader of appropriate background do not need to be elaborated on (provided you are absolutely sure they are correct). It is generally a good practice keep a draft with all the messy details, and abridge them in a final draft, in a way sensitive to one's intended audience.

I should elaborate a bit more on which details should be hidden. Generally, people read papers on topics they are interested in, either because they have nearby expertise or the topic has some bearing on their own work. Most of us immediately go looking for what is new in a paper, and appreciate finding it as soon as we can. If you don't know your audience and just write like an automaton, giving all details, it looks like you are a novice and have no idea what is routine and what is new. Worse yet, this sort of approach that avoids nuance makes your paper look more error-prone...because it looks like you are treating everything with the same amount of care. As a rule, one should omit or abridge those things that are sufficiently routine (after checking them to death), include those things that are delicate (i.e. potential pitfalls that should be treated with care...most experts are aware of these things in their areas...knowing these gives your paper credence with these experts) and last, but not least, highlight the new ideas so they are clearly visible. A paper that does these things looks more like a mathematician wrote it...indeed it would be very difficult for an amateur to effectively write such a paper convincingly!

Finally, regarding Serre, his talk "How to Write Mathematics Badly" is fun.

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This is not the type of resource you request, but if you seek examples of excellent expositions geared toward the general public:

The Best Writing on Mathematics series (Princeton link).

Now five ten years: 2010, ..., 2019. All edited by Mircea Pitici.

         

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This is a classic reference: "Handbook of writing for the mathematical sciences" by N. Higham. See here

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Mathematical writing is very different from other kinds of writing, including other kinds of technical writing. Aside from low-level textbooks, the audience is generally assumed to be other competent mathematicians, with their expertise in the particular subfield depending on how technical the work is. It's awful form, and it will infuriate your readers, if you try to hold their hand and assume that they need helpful analogies and metaphors (which invariably just cloud the issue more) to deal with the concepts. Just present the data straightforwardly and without narration, giving enough details to be precise but not too much to be pedantic. (Note in particular that a math paper or book intended for other mathematicians is not a problem set to be graded; it's actually preferable to omit trivial details that a competent reader should be able to fill in themselves.) The line between the two is hard to define but easy to spot in practice. Think of it as something like a legal document: The audience is other lawyers, who are assumed to be competent enough to handle the material if it's presented directly, and who expect and want the terse style and precise vocabulary that presents the argument without beating them over the head with it.

The best-written math textbook I've read is Bott and Tu's Differential Forms in Algebraic Topology. Try copying the style there.

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Franco Vivaldi has a useful book on mathematical writing available from Springer: Mathematical Writing (2014).

I would also say in response to Remark 1 that analogies and metaphors remain valuable in mathematical writing, as they can aid understanding, but they must be used with care. For instance, if a metaphor contradicts an established mathematical definition, then the definition wins.

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I strongly recommend Steven G. Krantz's A Primer of Mathematical Writing. I consult it regularly when I'm not doing recreational writing.

If you are writing something for fun or for your own benefit alone, do whatever you think will help you think best.

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  • $\begingroup$ I went to my local university mathematics library and this looked like the best book on writing from the 6 or 7 I looked at. Thanks for the suggestion. $\endgroup$
    – Stan Shunpike
    Commented Dec 22, 2014 at 20:57
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The first example I know of that comes to mind when I read your question is "Clean Writing in Mathematics" found in Appendix 4 of Calculus: A Liberal Art by W. M. Priestley.

In general, writing mathematics is just like any other skill: The more you practice and the more you see others do it, the better you'll get. I find I take bits of pieces of styles from mathematicians that I read and admire most, thus creating my own unique style.

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