I'll be the first to admit that this is a risky question to try to get away with on math overflow, but I'm going to give it a shot anyway.

Roughly speaking, the question is this: **Is it good to try to explain the main ideas of a complicated proof before giving the details?**

One the one hand, the naive answer might be "Yes of course, why wouldn't you want it explained in clear language before you have to go through the fiddly computation or tricky argument". But this seems not to actually hold up to much scrutiny... because well...It's actually relatively rare to do so. Papers are full of tricky, fiddly, deep and complicated proofs in which the authors don't try to outline the argument in more natural language. Should they? Why don't they?

- Is it just bad writing?
- Or is it an example of over-adherence to formal style that harms the richness and intelligibility of the literature? (this retired MO classic comes to mind: What are some examples of colorful language in serious mathematics papers? )
- Is it because to do it, we'd have to make the mathematics less precise and we just hate doing this?
- Or is it because: (and this might be the real reason I think this question is interesting):
**when done only quite well but not brilliantly, it can actually make it harder to understand the proof:**Somehow you might expect to gain clarity by reading the more top-level description of the proof, but actually it is mired in the way the math*seems*or*feels*on a more intuitive level to the author, which just might not click with you (cf. Thinking and Explaining ). - Another blander aspect is: It takes up space. I sometimes get into a period of a few weeks where I do it to a few of the proofs in a paper and then later I cut them all out again because the paper has gotten long.

Similar to, but not the same as: When to postpone a proof?

yes: it is really good to some kind of high level description of the proof before you get into it. But, as you hint at, it is critical that a high level description should be expressed in terms that make sense to a general reader who is knowledgeable in the field of the paper. I have seen terrible high level descriptions that sort of conjure up an image of how the author sees the proof (something like "the proof is like making toffee, and then adding nuts at the end"), which convey nothing to the reader who is not in the author's head. $\endgroup$ – Anthony Quas May 25 '18 at 0:5911more comments