While my question topic is that of mathematical writing of papers, which is a broad subject, the particular question is specific.
I am writing a paper, in which we have a section called "Outline of Proof". (It's Section 2.)
The outline is fairly informal, and we omit some technical details, making approximations.
Among these approximations, should we state (and label) important definitions and results (lemmas, equations, etc), with the intent of, later in the paper, referencing thes?
This raises the point of redundencies: some people don't like things being stated twice precisely (including in the outline), so wouldn't want anything explained/stated (even in the outline) re-explained/stated.
This seems ill-advised to me. When I read a paper, I rarely carefully read the outline:
I just read it, and try to get an overview (or 'outline') of the proof;
if there are parts that I don't really understand, I don't get hung up on them, trusting that with the more rigorous explanation later I'll be able to make sense of what the authors are saying.
So my question is this:
(a) is it standard to read an outline of a proof carefully?
(b) is it standard (or at least not discouraged) to state in the outline precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?
Best Answer
I am aware that my personal opinion is somewhat of an outlier, given the style of "minimalism" for many decades now in mathematical writing. But, by this time, I am unabashedly annoyed by internal references that require me to flip back to have _any_idea_ of what's being referred-to. The most hilarious case of this is Bourbaki's textbooks, which strive mightily to avoid naming anything, nor referring to anything by traditional names, nor even telling what the actual content is, but will give without telling you that it's ... oh, say, the intermediate value theorem.
Sure, Bourbaki's texts are an extreme case, but they've made me sensitive to the issue, since (long ago) they were the best source for several things.
Many of my colleagues have said that it's an irrelevant comparison, but I do prefer to think of mathematical papers as things to be read through, like novels. So overt necessities (or commands) to flip back seem perverse to me. Rather, I like "recall, from section (5.2), that blah=blah. Also, theorem (4.5) says that blah."
It costs relatively little in terms of space, and (to my mind) adds hugely to the readability.
I do think that it may be worthwhile to separate fetishism about minimalism from other considerations...
Oop: to answer the original questions, ... I do take seriously any offered outlines, although, yes, also, I do try to skim through as rapidly as possible to see what's going on. Probably in part because I've seen quite a few things, it is not hard for me to sort the "usual" from the "novel", and this doesn't slow me down. When I was younger/less-experienced, I would have been very happy to have outlines and such, and to have repeated/reminder definitions/statements throughout any longish essay.
Apart from a possibly outlier claim that we should require human-readable documents (as opposed to computer-verifiable), there is a still further possible-outlier claim that we should require documents readable by not-only world-class experts. The caricature of this requirement is that an attempt at communication is ... poor... if the message can only be understood by a recipient who already knows the message. In this guise, it sounds silly, but, in fact, many mathematics papers come very close to this. Makes things top-heavy and un-sustainable.