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'?
Best Answer
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.