Mathematical Writing: Proof Outlines/Overview in a Paper 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?

 A: There are at least two cognitive processes happen during the information transmission:  


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*Illusion of transparency or curse of knowledge: the author always overestimates how the readers understand their work. In an experiment about guessing a song by listening to how it's tapped, the tappers guess that 50% of time the listeners will guess right, while in fact the number is 3%.  

*Cognitive load: the readers read the paper to find solutions for their problems, not the author's problem. What is worse is that the desire to get them solved will cloud their minds and make what is written black and white in the paper be unavoidably exotic, misunderstood or, to an extent, distorted. Not getting what you are expecting is so frustrating, like you are in a rush and your car suddenly stops working. A related phenomenon is information overload.    


Of course it is impossible to write a smooth research paper, and the readers are expected to overcome it anyway, and I may exaggerate a bit, but overall it's still better to keep them in mind when writing anything. It's important to have a concise writing, but when the work is long, complicate and abstruse like in a math paper, only relying on conciseness may make it become too dense, and will backfire the intention.  
My advice are:  


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*Make a concrete analogy. A concrete analogy will intertwine to the text and allow room for the readers to project their background into it.  

*Make the ideas constantly contradict each other. "Contradiction" here doesn't mean as a logical contradiction, but more about "a surprising, but still logical step of development". It introduces why the topic is important, and is the source of excitation, enlightenment, and satisfaction. Being able to solve contradictions is the reason why the ideas survive and are worth the attention.  

*Notice where the flow emerges and dissipates. This will help overcome the jargon barrier without having to oversimplify them. Imagine the article is like a heatmap, and each jargon/theorem/proof is a heat source, then the writer's job is to locate them not too hot (too dense) or too cold (too uninformative).

*Viewing the topic in as much perspective as possible. Each unit of the article (phrase, sentence, paragraph, section) should be thought as an unique, different perspective, so the readers can see the topic in fresh perspectives. This makes a profound, advanced topic more playful, imaginative and transformative. The simplest trick is to start a new sentence with a different subject than the preceding one. If the readers are about to leave and we have only one more sentence to keep their attention, what should it be?    


Though not about math paper, research about myth debunking and web usability (widely known for the F-shaped pattern) will provide experimental insights about what conciseness really means. For complicated and long writings, I also have an article for this, you can check it out: Making concrete analogies and big pictures.  
A: (a) There is no "standard" way to read a paper.
(b) It is certainly common to state your main theorem precisely in the introduction or outline.  For example, Theorem 0.5 in the Introduction of Wiles's paper on Fermat's Last Theorem states Fermat's Last Theorem precisely.  I cannot imagine what else he could have done—not mention Fermat's Last Theorem in the Introduction?  mention it but state it imprecisely and defer a precise statement until later in the paper?  Now, many conjectures and theorems have complicated statements.  It may be more natural to give an informal statement in the outline and defer a formal statement until later.  But if you do state something precisely in the outline, then there's no reason not to cite it later.  Even if the reader skipped or skimmed the precise statement on a first reading, the precise statement is still there in the paper, and can easily be referred back to.  Unlike Snapchat photos or Mission Impossible directives, the outline does not self-destruct after you move on to the rest of the paper.
A: (b) This is something some authors do and others don't. I lean heavily against this (I believe outlines should be fully skippable). If you end up doing this, make sure that it remains clear what is being used (e.g., label definitions and refer to them by label rather than referring to the whole outline section) and that the setting used in the outline section is not less general than that used in the rest of the paper (so, no restricting to characteristic $0$ "for simplicity").
(a) Not sure. But if the references are explicit and concrete, it shouldn't be necessary: Readers can just read the specific definitions they need.
A: *

*(a) I (and many others) read proof outlines carefully in order to understand the key ideas and the flow of a rigorous proof, even given the details are missing.  Think of such an outline as a overview description of a trip:  "I take a taxi to San Francisco Airport, then fly to Boston, then take the "T" (subway) to Harvard Square, then walk to my colleague's office."  Every step is clear, and the reader interested in details will have natural questions:  "which airline?", "which flight?", "which T lines and stations?", "which building is the colleague's?", ...  If any key step in the outline were omitted (nothing about riding the "T", for instance), the reader would be puzzled.  These will be provided by a more-detailed description.  So too with a proof.  

*(b) I don't think there is a universal approach to this.  However mathematics, broadly speaking, is a field that values extreme precision and concision, with rejection of redundancy more than any other discipline.  (That's just one reason some of us are drawn to the field.)  


I think you will be better guided by worrying instead about correctness, clarity, persuasiveness, and if possible beauty/elegance (and hence memorability).
A: 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.
A: I don't think anyone has yet argued in favor of redundancy, so let me try it. I don't mean to  just repeat things but rather to give useful reminders. If a concept was defined in Section 2 and its first use outside that outline is in Section 7, there's a good chance that the reader (even one who paid attention to the outline) will have forgotten it by then. 
Technically, it's enough to give a reminder of the form "By Definition 2.10" at the point where it's needed; readers with imperfect memory should then go back and reread Definition 2.10 (and probably some of the surrounding material to recapture the context). If they find it annoying to have to go back like that, well,it's their own fault for forgetting the definition. 
I'd prefer a bit more kindness to the reader (especially when I'm the reader). A reminder of the form "Recall that [concept] was defined (in Definition 2.10) to isolate the [such-and-such structure] that played a key role in Jane Doe's proof of the Gauss conjecture" might jog the reader's memory enough to save some work and some annoyance. 
A: Having that outline is very useful for editors, to see quickly whom to send the paper for reviewing. 
Then for referees, to see if they want to review or not the paper, and also to have a quick idea on whether they intend to make a positive/negative report. 
And finally, for referees again, in case they make a positive report, it's so much easier to copy a bit from that outline part when making your report.
Passed this, and getting to regular users of the paper, no rule of course. Just write your paper for yourself, I mean make it pleasant to read for yourself, guess that's the best rule when writing something.
A: (a) In my mind, the point of a proof outline is to convey the flow of the argument, when this argument may be obscured by technical details when the proof is actually being presented, or when it requires a lot of steps and there is danger of the reader "missing the forest for the trees". If the author(s) feel the need to include such an outline, especially in a separate section labeled as such, then that signals to me that I better pay attention to it. This is different from where one tacks on a paragraph at the end of an introductory or preparatory section saying something like "the idea of the proof is..." which signals that one is just trying to indicate the main point. 
(b) It really depends; it is common to have a section with all the basic definitions, some preparatory minor general results, and perhaps embed them into a description of how the proof will go. On the other hand, one may have just an outline of the proof with the broad strokes to orient the reader in what follows. The main point is to try to make the reading flow and help the reader go through it. If, as you read this section, it feels like you are being buffeted from one side to the other by unexpected definitions/technical results/equations, then they shouldn't be there. If they seem natural (if they help illuminate the outline), then they should stay. But if you do include preliminary definitions and results, then the section should not be titled "Outline of the Proof"; perhaps you should call it "Preliminary results and outline of the proof" instead.
