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In my understanding the introduction of a paper is a very important section because it not only gives a concise summary of the results, but it is also the only section most people will read at all. Therefore I often ask myself how to arrange it in such a way that, for example,

  • the results are easy to grasp,
  • the results are clearly motivated,
  • the reader gets convinced of the significance of the results,
  • the reader gets interested to read the rest of the paper.

Now there are surely several ways to do this, but in order to keep it focused, for this question I would only like to know the following:

Do you prefer to give a summary of the results, including short remarks about how the objects you have studied are defined, and then later give some context and motivation for these results, or do you prefer it vice versa? Or do you prefer to give statement, context and motivation of a result and then move on to the next result? And what are your main reasons for doing so? In particular, how are your goals of an introduction, for example those mentioned above, achieved by these decisions? Is there an arrangement which is more common?

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    $\begingroup$ Just a comment on your fourth point, on how to arrange the introduction so that the reader gets interested to read the rest of the paper: I always considered it more important not to get the reader interested, but to let them know whether they personally should be interested (as this depends on the individual reader). And furthermore - since this question had a downvote when I saw it - I think this is very relevant to research mathematicians! $\endgroup$ – Alex Amenta Nov 10 '16 at 16:30
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    $\begingroup$ Part of the difficulty is that there are really two kinds of introduction: one where you advertise your results to the experts (providing a high-level overview and motivation and possibly connections to existing literature), and one where you actually introduce them for non-expert readers. Bad things tend to come out when these two are mixed together. $\endgroup$ – darij grinberg Nov 10 '16 at 16:33
  • $\begingroup$ @AlexAmenta: I agree. I should have said that the 4th bullet only refers to those readers who study the particular topic. Thanks a lot for your remark that this question is very relevant. The downvote made me doubt it, even as if no question about mathematical writing was allowed here. $\endgroup$ – HeinrichD Nov 10 '16 at 17:16
  • $\begingroup$ @darijgrinberg: absolutely, I didn't think of that! It's always worth reminding yourself who your audience is. HeinrichD: no problem, I'd like to see some more experienced mathematicians share their thoughts on this question myself. It's true that most people only read the introduction, so we should make an effort to get it right $\endgroup$ – Alex Amenta Nov 11 '16 at 0:20
  • $\begingroup$ One method that I often use (introduced to me by Máté Wierdl) is to write simplified statements of the main results in the introduction, followed by more technical versions later. This serves to motivate for non-experts, and also gives a good flavour for experts. $\endgroup$ – Anthony Quas Nov 11 '16 at 7:53
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To some extent this depends on the nature of your result, but the principle I generally follow is this:

State your result as soon as possible, subject to the constraint that when readers arrive at the statement of your result, they will have some idea of what question you are answering and why.

For example, if you're proving a conjecture that is easy to state, or is likely to be known already to anyone who is interested in your paper, then you can pretty much launch right in and say that your paper proves so-and-so's conjecture, and state the conjecture.

If you're proving a conjecture that isn't so well known or easy to state, then I'd start by saying that you're proving so-and-so's conjecture. Then give the minimum necessary background to understand the statement of the conjecture and why it was made. After that you can give further context and motivation if it seems appropriate.

If you're not proving someone's conjecture but are proving something that bears a simple relationship to an existing result (e.g., it's the converse of some known theorem, or a generalization), then again I would start by saying that you're (for example) generalizing so-and-so's result. Then state the other result, and state your result. Further context and motivation can come afterwards if appropriate.

If your result does not fall neatly into one of these categories but is more of a general theoretical advance, then it may be trickier to decide how much background to give before stating your result. I usually lean towards giving as little background as possible, because it's easy to make the mistake of giving too much background, and losing the interest of the reader. As a first draft, try stating your result without any context, and then ask yourself if your typical reader would react by thinking, "Why on earth would anyone prove such a result?" If so, then back up and give just enough context to pre-empt such a question.

One general comment is that if your result is difficult to state precisely because the precise statement is very complicated, then see if you can come up with a simplified version, even if you have to be slightly vague or have to weaken your result to do so. This will allow you to state your result sooner, in accordance with the highlighted principle above. The more precise version of your theorem can come later, after the reader has decided that it's of sufficient interest.

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