I hesitate to ask this question. However I read many advices from math.stackexchange, and I couldn't find anything similar.

A good time always goes too fast! Two years are fled. In the third year of PHD, my major is general topology and I'm facing with graduation from PHD. I do enjoy research, however the pressure to publish makes me be agitated and not quite, for I haven't publish any paper. I find, sometimes, doing research and to publish are contradictory.

Here is my question: How to write a good mathematical paper? Could anybody give me some suggestions?

Thanks ahead.

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    $\begingroup$ It appears that you have done research and produced some results. Writing a paper should be easy once you have something to write about. You should write done your results and proofs as clearly as possible. Try not to get bogged down by details, and consult your advisor to determine what details an expert should be able fill in themselves. Check your spelling and grammar. Talk to advisor about the known journals of your area and those likely to accept your paper. The important thing is to consult your advisor, but since you have results, I think the hardest part is already done. $\endgroup$ – William Aug 11 '12 at 1:38
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    $\begingroup$ Don't do the classic textbook approach of having your main statement and the preceeding lemmas be a total secret until the reader get to the specific page. The most negative comment I recieved on the only thing I have ever written close to a mathematical paper was that I started out too "heavy". I was told it was better to have an abstract (which should only be a sentence or two) and then a relatively short section explaining basic ideas in a way that wouldn't be considered a wall of text or an overload of definitions and constructions. $\endgroup$ – Arthur Aug 11 '12 at 1:48
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    $\begingroup$ You have to have a result first. $\endgroup$ – user2468 Aug 11 '12 at 3:44
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    $\begingroup$ @Paul Unfortunately I cannot edit it, although it does work for me. Here is the direct link: terrytao.wordpress.com/advice-on-writing-papers $\endgroup$ – Sniper Clown Aug 11 '12 at 6:58
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    $\begingroup$ Having just refereed my first paper, I'll try to say a few of meaningful things. (1) Don't obfuscate with formally correct notation where a general idea -- simply expressible in English with perhaps a few mathematical symbols -- will suffice. (2) Be consistent with notations/conventions. (3) If your proof involves a long, tedious, technical component, break it up into segments and explain what it is you are attempting to do in each segment. (4) Remember that while after two years of intense study everything seems natural and clear, it might not be for someone seeing it for the first time. $\endgroup$ – user642796 Aug 11 '12 at 7:07

As someone who is currently working on my first mathematical paper, I've found this guide from MIT to be very helpful. It covers both writing a clear and precise paper in general as well as the specific challenges presented by a mathematical paper. It's also fun to read! For example, the author likes to illustrate common mistakes within the text. One of my favorites is:

Don’t string adjectives together, especially if they are really nouns. Many high quality pure mathematics original research journal article sentences illustrate this problem.

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    $\begingroup$ Thanks for your answer and your paper. The paper seems very helpful for me. I'v downloaded it:) $\endgroup$ – Paul Aug 11 '12 at 1:58
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    $\begingroup$ +1 for the intelligent sentence. Made me laugh this morning! $\endgroup$ – Gottfried Helms Aug 12 '12 at 8:34
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    $\begingroup$ it is funny)))) $\endgroup$ – Seyhmus Güngören Aug 23 '12 at 8:54

There are some notes on Mathematical Writing from a course taught by Knuth. They are quite extensive; I've only read the first few pages and those were already quite helpful to me, but there are also notes from guest lectures by various people, e.g. Wilf and Halmos.


You say you haven't published a paper yet. Then perhaps this would be useful to you: How to Write Your First Paper by Steven G. Krantz (Notices of the AMS, December 2007, pp. 1507-1511).


I would like to add the above mentioned Paul Halmos work on How to Write Mathematics.

In too short (taken from the MAA):

  1. Say something. To have something to say is by far the most important ingredient of good exposition.
  2. Speak to someone. Ask yourself who it is that you want to reach.
  3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.
  4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.
  5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on. (Annotation: 1,2,1,2,3,1,2,3,4,...)
  6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.
  7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.
  8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.
  9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
  10. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

I would like to add

  1. "If you work eight hours to save five minutes of the reader's time, you have saved over 80 man-hours for each 1000 readers, and your name will be deservedly blessed down the corridors of many mathematics buildings." p. 134

and emphasize that at least the former holds for every scientific writing.


http://padic.mathstat.uottawa.ca/~mnevins/latex/sample.pdf, here is a good short and simple guide. other is the following: www2.kenyon.edu/Depts/Math/Aydin/.../Report.doc, which goes to the main aspects of the body of a math report.

Hope to be useful. Greetings.


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