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Background. I am someone at the very very beginning stages of my career as a mathematician, and am, for example, almost ready to (re)submit my first paper. I have already read a lot of essays on how to write well (i.e. http://terrytao.wordpress.com/advice-on-writing-papers/ is golden), but I still sometimes have doubts on what the best course of action is in certain situations. So that's the reason of this question. If it gets closed, because the only reasonable answer is 'it depends', I completely understand. But it would be great if you could bear with me for a moment.

Problem. To me, an article is (or at least feels like) a written monologue. But, on the other hand, the purpose of a paper is to explain new ideas to your reader, which generally is much more easy in a dialogue. Because in a dialogue you can interact with someone, ask questions to make sure he or she understands and re-explain things in a different way if he/she so desires. So if I am writing mathematics, I try to remind the reader where we are in the argument, why we are there, where we are going next, what the main ideas are behind all the calculations, etc. Let me give you an example: when I read something mathematical that Timothy Gowers wrote, it is immediately clear to me that his goal is to make me understand his ideas. If I remember correctly, his article on his proof of Szemeredi's Theorem by Fourier-analytic methods is over 100 pages long, but his writing style is very leisurely and he first goes through the case of arithmetic progressions of length 4, to make some important ideas clear, to show what else is needed to complete the proof, etc. To me, this is a delight. So I'd like to model my writing style after his. But, unfortunately, I'm not Timothy Gowers. So I'm not sure if I can get away with writing like he does, or if being that leisurely and involved with the reader actually makes me look unprofessional. And thus my question is:

When in doubt, should I acknowledge the fact that I am a human being, writing and explaining concepts to another human being? Or should I try not to break 'the fourth wall'?

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Having seen attempts by people-not-named-Tim-Gowers at informal writing, I have to say that it is more difficult than it looks. Formal and professional style very naturally forces you to be precise and exact. An informal conversational style needs at least the same amount of effort in making things precise, choosing the right things to explain, etc., while its appearance might make it look like a write up of a conversation. –  Arend Bayer Apr 15 '12 at 1:21
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Whatever you do, resist the urge to show off. –  Paul Apr 15 '12 at 1:44
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Nowadays you can do both: use your personal style for your website/blog to describe things conversationally, and have the professional article vetted by appropriate readers before submitting to a journal. Even mathematicians outside your field are willing to comment on style, especially if you reciprocate. Gerhard "Ask Me About System Design" Paseman, 2012.04.14 –  Gerhard Paseman Apr 15 '12 at 2:48
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Having a quick look at your preprint I noticed that the introduction is missing. Introduce the reader to the problem, explain how it relates to previous results in the literature and why it is interesting, state your main result clearly, etc. Most importantly, write some introduction! (Look how it's done in other papers and discuss it with your advisor) –  Robert Haslhofer Apr 15 '12 at 8:15
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@SpiceTheBird: shouldn't your comment read "One thing that we like to do..." –  Federico Poloni Apr 15 '12 at 8:59
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2 Answers 2

I think in writing professionally, you have a character or persona that you are maintaining within the conversation. I agree there is some shifting among the imperative: "Consider [...]," the plural: "We consider [...]", and the personal: "When I first considered [...]" Such shifts of point of focus and points of view, if executed correctly, can add greater clarity. Also, you can step out of character once in a while. My belief is that it is important for the reader to know what motivated an idea and when a course of development did not lead anywhere.

Even when you are maintaining the professional persona, be true to yourself. If for example, you don't like an excess of notation, then indicate how you will cut the notation to a manageable size. If you use figures to think through the ideas, then draw them. Still always define terms. Look for meaningful words in English that serve as metaphors for the mathematical ideas. Let the reader know what the theorems are, what the connections are among the theorems and the lemmas, when the proof is complete, and why such and such is not a counter-example.

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I can't help but reiterate here Serre's great maxim: "Precise yet informal". Striving for this goal in mathematical writing will avoid your looking unprofessional, but will make you look more human at the same time. It seems that mathematics that does this best is very professional (I can't help but think of Milnor's style, here).

Regarding writing that appears purely to guide the reader between theorems: as far as the guidance ALSO is precise yet informal, there should be no foul. It is when these comments are not really true...but close to true...that they should be omitted or left to the talk. Too many "sort-ofs" should be avoided in the body of a mathematical paper. Serre's maxim seems a good guidepost for this, too.


Comments to supplement original answer:

Also, as long as mathematical papers are valued for being interesting it seems that there will always be a bit of 'ought' along with the 'is'. Otherwise, one could just rattle along formally proving theorems without regard to how they will be received. This is evidence of the human part of mathematics that stands in opposition to the dispassionate precision of formal proof. Elegance is far from formal perfection, and is perhaps closer to human utility or aesthetic sense. Somehow, Serre's maxim expresses something important about mathematical life and work.

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