How to write math well?

Let's learn about writing good mathematical texts.

For some people it could be especially interesting to answer about writing texts on Math Overflow, though I personally feel like I've already mastered a certain level in writing online answers while being hopelessly behind the curve in writing papers. So,

What is your advice in writing good mathematical texts, online or offline?

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I'm testing the new soft-question tag. Feel free to comment on that or delete it, or perhaps pick it up. –  Ilya Nikokoshev Oct 19 '09 at 20:14

One trick that my advisor, Ronnie Lee, advocated was to use a descriptive term before using the symbolic name for the object. Thus write, "the function $f$, the element $x$, the group $G$, or the subgroup $H$. Most importantly, don't expect that your reader has internalized the notation that you are using. If you introduced a symbol $\Theta_{i,j,k}(x,y,z)$ on page 2 and you don't use it again until page 5, then remind them that the subscripts of the cocycle $\Theta$ indicate one thing while the arguments $x,y,z$ indicate another.

Another trick that is suggested by literature --- and can be deadly in technical writing --- is to try and find synonyms for the objects in question. A group might be a group for a while, or later it may be giving an action. In the latter case, the set of symmetries $G$ that act on the space $X$ is given by $\ldots$. Context is important.

Vary cadence. Long sentences that contain many ideas should have shorter declarative sentences interspersed. Read your papers out loud. Do they sound repetitive?

My last piece of advice is one I have been wanting to say for a long time. Don't write your results up. Write your results down. You figure out what I mean by that.

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OK, Scott, it's two months later and I can't figure out what your last paragraph means. I give up! Can you put me out of my misery? –  Tom Leinster Dec 28 '09 at 0:49
Tom, There is a tendency among us to always write things in the most elegant manner. Writing things "up" often makes the reader have t rework the elegance in a plebeian way. When you write things down, you are thinking of your audience as students who are trying to learn the material. As I said, I don;t know if I always succeed at this. Often I am constrained by coauthors, and often I constrain coauthors. Despite appearances, we spend a lot of time editing ourselves and each other. –  Scott Carter Dec 28 '09 at 16:19
"have t" should be "have to." "don;t" should be "don't" This edit box spills into the box on the right. –  Scott Carter Dec 28 '09 at 16:21
Aha! Thanks. Good thoughts. –  Tom Leinster Dec 29 '09 at 0:34
@Scott: So "Don't write your results up. Write your results down" is an example of writing things up? –  Sune Jakobsen Feb 3 '11 at 8:51

I can't say much about textbooks from the perspective of the producer, so I'll say something from the perspective of a consumer.

I like books that are good at what Terry Tao referred to as "information hiding".

My favorite textbooks somehow manage to give me all the excruciating details when I need them but only the intuitive picture when I don't.

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As of now, Terry Tao wins in a landslide :) –  Ilya Nikokoshev Oct 23 '09 at 18:15

P.Halmos in How to write mathematics (easy to be found via your favorite www search engine) gives some nice advice. Or try Serre's exposition Writing mathematics.

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Ferran didn't mention that the Serre exposition linked above is a rather amusing video of a lecture on how to write math badly. :-) –  Alison Miller Oct 23 '09 at 22:41
Thanks for the pointer! Of course, after your comment, I just had to watch Serre. Fun :) –  Ilya Nikokoshev Nov 16 '09 at 21:43

Halmos's article contains a lot of good advice at the tactical level, the writing equivalent of "face your audience". This is stuff you have to get straight: if your notation is crazy then your potential readership is already zero. Fortunately this is largely a matter of acquiring good habits.

Next, you have to constantly keep your reader in mind. Whenever you are faced with a choice, ask "What would be best for my reader(s)?" It can be useful to assign someone you know well to this role, because it makes it easier to stay consistent. Remember that it is much more important to be clear than it is to be complete. Be prepared to do a lot of rewriting.

If you follow these suggestions, you will with practice become a competent writer. To become a good writer is much more difficult. Writing good mathematics is no easier, and no harder, than writing good prose on other topics. Read a lot, find good examples that you like and think carefully about why they work.

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I don't feel experienced enough in mathematics writing to give advice. But I like Terry Tao's advice on writing mathematics papers; he also provides links to some other people's advice.

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There's a set of notes from a class on mathematical writing that was run by donald knuth that you can find in various locales, here's one link. Its not up to date with respect to online writing (being 20 years old), but it should still have some good gems!

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Those notes are amazing ! I spent a delightful evening reading them, and I plan to go back as often as necessary to digest the wisdom... The other references there (tex.loria.fr/typographie) seem worthwhile, especially for French would-be typographers. –  Simon Pepin Lehalleur Aug 9 '10 at 22:11

Currently I am reading the book "A primer of Mathematical writing" by Steven Krantz. I have found it extremely useful. It is helping me with my thesis and 2 papers. The book covers everything from grammar, writing papers, CV, grants, job applications, books, book reviews, referee report, expositions also technical aspects like bibliography, index, appendix and even time management.

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You may also wish to take a look at the references on math (and more broadly, science) writing listed here.

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Nice list, thanks! –  Ilya Nikokoshev Dec 28 '09 at 0:05

Halmos has already been mentioned, but I'd like to record the following: It may be helpful to employ spiral-writing (a-la Halmos) but at the paragraph level, rather than the chapter or section level as he suggests. So, write a paragraph, then the next one, looking back to see if the previous can be clarified in light of the one written. I suggest this because the paragraph is supposed to be the natural unit of composition, and if C.S. Peirce was right, clarity is utility. Spiral writing at this level may help not only exposit mathematics but also do mathematics. Each subsequent thought we attempt requires a better utility from what has been previously written...and the only way to guarantee such increased utility is to have attempted this subsequent thought and then carefully had a look at exactly what was needed in prior paragraphs to develop that thought.

I guess that in summary I have only two new things to add here. 1. Halmos's spiral writing is philosophically sound from the standpoint of early pragmatism, and 2. Maybe it can be effectively employed at the paragraph level instead of the section/chapter level.

Of course, Halmos suggests writing a chapter at a sitting as a way to fight inertia...so my second suggestion in the previous paragraph may not be actually better, only personal.

Anyhow, I was going to record this thought in my own personal notes, but thought maybe someone else may have a use for it as well.

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