A scrupulous writer, in every sentence that he writes, will ask himself at least four questions, thus:
- What am I trying to say?
- What words will express it?
- What image or idiom will make it clearer?
- Is this image fresh enough to have an effect?
A paper should not just be a sequence of formulae or logical steps.
It should also be organized and motivated in such a way that the reader is always aware what the near-term and long-term objectives of any portion of the paper are, how the current arguments are advancing towards these goals, how crucial they are to those goals, and why the claimed results at each step are at least plausible (or if they are surprising, to indicate exactly why and how they are surprising).
Informal, heuristic, or motivational reasoning is therefore very welcome, but should be clearly indicated as such to distinguish it from formal, rigorous reasoning (for instance, these portions of the paper can be placed in remarks or footnotes).
At the start of each section, it is often a good idea to give a brief paragraph describing the purpose of that section. For instance, if a section is devoted to proving a key milestone in the paper, the milestone can be stated near the start of the section, next to a discussion as to why this milestone is important and perhaps a brief sketch as to how one is going to prove it in this section.
Before presenting your most general result, it can help to first discuss a less technical special case or “toy” result first to give some flavour of the significance of the main result, and also on the strategy of proof. This can be worthwhile even if this toy result was already known in the literature. For instance, it often happens that the key to generalising the proof of the toy result to the more general result was to re-interpret an existing proof of the former in a way that generalised to the latter, and discussing this re-interpretation near the beginning of the paper can be enormously clarifying to readers.
See also “Be considerate to your audience“.