# How to write a good mathematical paper?

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?

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Thanks William for your careful comment. The terrible thing is that I have not good results to write up to now. –  Paul Aug 11 '12 at 1:45
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. –  Arthur Aug 11 '12 at 1:48
You have to have a result first. –  user2468 Aug 11 '12 at 3:44
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. –  Arthur Fischer 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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+1 for the intelligent sentence. Made me laugh this morning! –  Gottfried Helms Aug 12 '12 at 8:34
it is funny)))) –  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.

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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).

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I have several pet peeves when I'm reading a paper, online article or blog which involves mathematics.

So if you are not constrained by space, and writing to the general public (and not for an expert community) try to avoid these:

• Starting from an already simplified form: Even if it's tempting to remove a term from the both sides of the equation right away, I would like to see the formula or law used with all the substituted values in full glory and unchanged, and do the simplification in the next step.

• Doing too many symbol folding in one step: when the author solves an equation, substitutes the resulting expression to another, solves it too, integrates it and factors it in a single step and shows the result only, and even not saying anything about how they get that result, and is even dare to write that it's left as an exercise for the reader...

• Omitting the arguments of functions: using only $f$ to refer the value of $f(x)$. By brain decodes these argumentless variables as an arbitrary constant I can substitute any number to, then I wonder why they get that weird expression when they differentiate it using $x$. Differential equations and physics derivations involving speed and acceleration are guilty of this all the time...

Mathematicians should not strive to confuse their audiences.