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I have noticed that among mathematicians there is a great diversity in style of expression. Some of these are exceptionally neat and elegant. What I mean by that is that proof, or argument takes few rows of text, is very concise and uses extraordinary well-defined vocabulary. I assume to acquire this those mathematicians have some systematic strategy that they are aware of when they acquire knowledge. What would be that strategy ? Furthermore living mathematicians develop even their own style which is not simply reproduction of the material found in books. These structures are so beautiful. Could somebody explain in what way could this be acquired ?

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Teaching, realizing typical obstacles people have in absorbing material. Adapting, developing a framework to work-around those obstacles, giving people things to "hold on to" when they're learning confusing new devices. That's perhaps the best device I can think of. –  Ryan Budney May 29 '11 at 23:40
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Read and write a lot of mathematics. Get someone to comment on what you write. If you want to work on exposition, pick your favorite result and write a clear proof. –  Yuval Filmus May 29 '11 at 23:54

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This is an expansion of my comment above.

The first step to literacy is reading. There are stylistic conventions for written mathematics, and they are not covered in any course, though there are some prescriptive books around. As you mention, there is more than one style, though in any given branch of mathematics, most of the papers probably form a homogeneous group.

The second step is writing. Writing by itself is immensely helpful, but it's even better if a "dialect expert" comments and corrects your output. This is especially important for non-native speakers of the target language (usually English). As for the comments, don't take them too seriously. There's a difference between conventions and style, and even regarding the former, sometimes it's good to be innovative.

Finally, one way to find enough material to write about (unless you're a phenomenal researcher even at this early stage) is to write expositions of proofs (or subjects) you like. There is often more than one way to explain a proof, and you can practice both your understanding of the material, your explanatory skills, and your literacy. You can complement it with lecturing on the material.

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> Thanks. Could you name some of those prescriptive books that you were thinking of ? I am also interested to understand more sharply difference between proposition, lemma, corollary and theorem. I have heard some who disagree if something is theorem or not. How formally to be sure ? If there is no absolute answer, then at least I would like to know how to challenge that something should be called a theorem. –  rivocantus May 30 '11 at 8:42
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@rivocantus: have a look at D. Goss's Some Hints on Mathematical Style (based on Hints by Serre); they also contains a longer list of references. There's a classic by Steenrod, Halmos, Schiffer, Dieudonné, How to write mathematics, worth looking at. Finally, Milne's Tips for Authors have some good points that seem obvious afterwards. As always with such things: Take away what you like, ignore what you don't like. But it's usually better to ignore something deliberately than not knowing about it :) –  t.b. May 30 '11 at 11:51
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Another point might be: writing with a particular audience in mind; trying it out on that audience; then adapting your writing based on the feedback. –  GEdgar May 31 '11 at 18:45

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