There are three rules for writing the novel. Unfortunately, no one knows what they are.(W. Somerset Maugham)

Everyone has to develop their own writing style, based on their own strengths and weaknesses, on the subject matter, on the target audience, and sometimes on the target medium. As such, it is virtually impossible to prescribe rigid rules for writing that encompass all conceivable situations and styles.

Nevertheless, I do have some general advice on these topics:

- Writing a paper
- Use the introduction to “sell” the key points of your paper; the results should be described accurately. One should also invest some effort in both organising and motivating the paper, and in particular in selecting good notation and giving appropriate amounts of detail. But one should not over-optimise the paper.
- It also assists readability if you factor the paper into smaller pieces, for instance by making plenty of lemmas.
- To reduce the time needed to write and organise a paper, I recommend writing a rapid prototype first.
- For first time authors especially, it is important to try to write professionally, and in one’s own voice. One should take advantage of the English language, and not just rely purely on mathematical symbols.
- The ratio between results and effort in one’s paper should be at a local maximum.

- Submitting a paper

I should point out, of course, that my own writing style is not perfect, and I myself don’t always adhere to the above rules, often to my own detriment. If some of these suggestions seem too unsuitable for your particular paper, use common sense.

Dual to the art of *writing* a paper well, is the art of *reading* a paper well. Here is some commentary of mine on this topic:

- On “compilation errors” in mathematical reading, and how to resolve them.
- On the use of implicit mathematical notational conventions to provide contextual clues when reading.
- On key “jumps in difficulty” in a mathematical argument, and how finding and understanding them is often key to understanding the argument as a whole.
- On “local” and “global” errors in mathematical papers, and how to detect them.

Some further advice on mathematical exposition:

- Michèle Audin’s “Conseils aux auteurs de textes mathématiques“
- Henry Cohn’s “Advice for amateur mathematicians on writing and publishing papers“.
- Oded Goldreich’s “How to write a paper“.
- David Goss’ “Some hints on mathematical style“
- Timothy Gowers on “writing examples first!” (see also this followup post)
- Paul Halmos’ “How to write mathematics” (the book also contains similar pieces by Dieudonné, Schiffer, and Steenrod); the article can be found here.
- “Mathematical Writing” – notes from a lecture course by Don Knuth, Tracy Larrabee, and Paul Roberts.
- Dick Lipton on an analogy between paper writing and city planning.
- Ashley Reiter’s “Writing a research paper in mathematics“
- Jean-Pierre Serre’s “How to write mathematics badly“

## 114 comments

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13 March, 2015 at 9:01 am

portonI am writing a research monograph. What’s about this experiment: In my book write instead of ?

13 March, 2015 at 9:04 am

portonNote that I don’t use $\frac{}{}$ construct to express division in my book (I use no division at all, as far as I remember), so it is free to be used for an other meaning.

18 March, 2015 at 6:37 am

portonWhat also about a set definition laying on its side? . This may greatly reduce formula width (and this is a great problem for me).

18 April, 2015 at 6:56 am

portonI am an amateur mathematician without official scientific degrees. I am writing a breakthrough research monograph in abstract mathematics. When I finish it, what is better: to publish it traditionally or to put its LaTeX files into GitHub.com under a free copyleft license so that everyone could be able to edit my work. That it needs editing, is quite probable because this is a very new field of research and the book may require changes to make it better and more general.

The main issue here is that after the decision there is no way back: If I publish it traditionally I may lose copyright and be not able to distribute my LaTeX files for free, and reversely if I put it online with a free license, this may be an obstacle for publishing it.

I ask you the advice, what to choose?

I am also afraid, that if I don’t publish it traditionally, math community may refuse to cite my work (and thus the world is not worth to receive my discoveries). This is even despite that publishing under copyleft is better for hunting errors, as in GiHub and similar free Git hosters there are error reporting zillas.

Please help me to make the correct choice.

30 May, 2015 at 10:12 pm

Stephen King to share writing tips in new short story collection | Ismael Olson[…] I also came across with this article as well: https://terrytao.wordpress.com/advice-on-writing-papers/ […]

14 June, 2015 at 11:53 pm

observerHi Terry,

Some time ago you were an advocate of publishing no further than arxiv. However, with their comments it is more of the same. Is there a time stamping technique you can recommend so that we can move on and publish at our websites?

13 September, 2015 at 10:25 am

Mathematic Reading | futileinfo[…] Terry Tao’s https://terrytao.wordpress.com/advice-on-writing-papers/ […]

23 October, 2015 at 4:20 am

SalinasDear Tao Terence,

Just for my curiousity, 1+2+3+4+5+6 …. = -1/12 is it true ????? It is used to resolve the “Casimir effect” ! (see demonstration :

In french but easy to understand:

https://sciencetonnante.wordpress.com/2013/05/27/1234567-112/

Do you work on this topic or it is not a serious topic ???

Thanks

Miguel SALINAS

23 October, 2015 at 9:12 am

SalinasOk I found the answer here:

https://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

thanks to being awesome

26 December, 2015 at 11:42 am

portonIf a lattice has no least element, then the difference a\b is undefined for every a and b in this lattice. So, is it worth to explicitly say about the lattice “with least element” when mentioning a\b?

Omitting it would decrease the length of theorem conditions. But without this condition conditions would be warrantenly false.

Example: “for a distributive lattice with least element there exists no more than one difference a\b of elements a and b” vs “for a distributive lattice there exists no more than one difference a\b of elements a and b”

I suspect that omitting existence of least element would hinder exposition, because in this case instead of saying “the least element” I need to say in the proof a little more subtle “an element of the set of least elements” (this set is always of one or zero elements).

What is better for: a. research monograph; b. textbook?