# 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. However, among these approximations, my co-author wants to state (and label) important definitions and results (lemmas, equations, etc). He then wants to, later in the paper when we are doing the corresponding part carefully and rigorously, refer back to these (say, "by equation (2.4)", or "by Lemma 2.2"). Moreover, he is very against redundancies, so does not like things being stated twice precisely (including in the outline) -- once precisely in the text and once approximately in the outline is fine.

To me, this seems insane. (Of course, I did not use such a phrase when speaking with him!) When I read a paper, I never 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.

However, I'm a pretty junior author -- 2nd year of PhD -- while he is a first year postdoc. That doesn't mean that I haven't read a reasonable number of papers (and in fact my lack of experience and knowledge means that I am less able to understand poorly written papers); moreover, he has said that he feels writing papers well isn't his best attribute.

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 precisely important, even key, results/definitions that will be referred back to in the main body of the paper when giving proofs?

Just as some extra comments... I'm not here to try to get people to tell me that I'm right and my co-author is wrong and/or being silly! I know that sometimes some people come to Stack Exchange for such comments (see, particular, Workplace/Interpersonal Skills SEs!). I should have perhaps made the following clear: if everything my co-author does is standard in the field, and I'm in the wrong, I definitely want to know that and will accept it! -- I'm here to learn :-) please have no qualms about hearing criticism! (assume that it's constructive, of course)

• I would think your co-author follows best practices; in a scholarly article redundancy is to be avoided; once an equation is given and you need it again you refer to it, you don't repeat it. – Carlo Beenakker Aug 13 at 17:30
• Yes, you don't repeat formal statements or definitions. You can mention a technical term and provide a brief gloss, to be followed later by a full definition, and then refer to the definition. But you don't repeat statements or equations. – Arturo Magidin Aug 13 at 19:47
• I hope you view this as an opportunity both to learn how to write and to cowrite. It also seems that it is an opportunity to learn how to navigate difficulties to a compromise. Of course show respect and deference to experience, but also try ways of expressing your ideas. E.g. "Hey Joe! I tried it two ways, one where the definitions are here, and one where they are over here. I think option 2 reads better. What do you think?" Gerhard "'Joe' Is A Conjectural Name" Paseman, 2018.08.13. – Gerhard Paseman Aug 13 at 20:41
• Thank you for the suggestions. And yes, Gerhard, I hope it will be an opportunity not only to learn how to write maths better, but also to collaborate better! (Going straight in the deep end: he's Israeli and, while I didn't know this about Israelis, he describes the Israeli way of discussing as "I come to the table with my idea and you leave with my idea", haha!) :-) -- in the end, he has a raft of publications, and he collaborated with top people, and I'm very grateful for all his advice! – Sam T Aug 13 at 21:34
• What Carlo Beenakker and Arturo Magidin say is common practice, but occasionally I will see, especially in a long and complicated paper, the main theorem stated twice, once in the introduction and once in the main body of the paper (the latter is for ease of reference). However, almost always the theorem will be given the same theorem number. If you state exactly the same theorem twice with two different theorem numbers, I think that many readers will be confused. – Timothy Chow Aug 13 at 22:52

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.

• I find this especially annoying when the references are to other papers, in particular when a proof of an interesting/impressive-sounding result in the paper one is reading seems to consist of "basic observation, Theorem 3.14 in [23], Lemma 2.5 in [17], the rest is routine". I'd be much happier if people said "basic observation, now recall that subalgebras of Type I C*-algebras are always nuclear, see e.g. Theorem 3.14 in [23]", and so on. – Yemon Choi Aug 14 at 0:03
• @YemonChoi :) ... I do really believe that some of this is a stylistic fetish that unfortunately resonates with the psychology of many math people... but/and which is self-destructive. – paul garrett Aug 14 at 0:08
• At least Bourbaki labels things like "Proposition." Have you seen Aschbacher's Finite Group Theory? As I recall, statements are just labelled by numbers, and then proofs are of the form: This follows from (3.2), (4.8) and (5.12). – Kimball Aug 14 at 1:03
• @Kimball, this is why I don't believe in finite group theory. :) – paul garrett Aug 14 at 1:07
• So this is pretty close to my feelings on the matter. I don't like it when papers say "which follows by (3)", and (3) is some fairly innocuous result, but you really want to know what's being applied; "which follows by the standard bound on the binomial coefficients (3)" adds a few more words, but is extremely beneficial, I feel – Sam T Aug 14 at 9:10

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.

• Same here. Readability is more important than avoiding redundancy. – Nik Weaver Aug 13 at 23:01
• I think Paul Garrett answer is in support of redundancy also, but it's somehow more compelling if there are multiple answers arguing in favor of redundancy. – Kimball Aug 14 at 1:06

(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.

• I particularly like the suggestion of calling it "Preliminary Results and Outline of Proof". As a pretty junior member of the maths fraternity, sometimes it can be difficult for me to know how experts read papers. I know when I read papers, I'd fine the style he suggests pretty difficult (and indeed do when I read papers with this style). Doesn't mean that in 5yrs time, when I'm more familiar with reading papers (an art in itself), I won't be advocating his method! -- I appreciate your comments, thanks. – Sam T Aug 13 at 21:37

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

• 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.

• 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.

• Thank you for your clear, well referenced answer. Could you expand on your second piece of advice (on continuously contradicting)? I don't really understand what you mean... – Sam T Aug 14 at 11:54
• I've edited it a little bit. Do you feel clearer? And more importantly, do you think it's helpful? – Ooker Aug 14 at 16:27
• Yes and yes :) -- thanks, +1 – Sam T Aug 14 at 17:08

(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.

• Yeah, so it sounds like your approach is more similar to mine -- but that, likely being far more experienced than I am, are more aware of other people's writing styles. Thank you for your comments. – Sam T Aug 13 at 21:41
• Aha! Darij, you are evidently a much more self-disciplined mathematician than I am! :) But, yes, tastes differ. – paul garrett Aug 14 at 0:12
• @paulgarrett: On the downside, I get fewer things written up... – darij grinberg Aug 14 at 0:29
• @darijgrinberg, also, I do think that there is an insidious "perfection the enemy of the good" at play, amplified by our not truly knowing at one point in time what the criteria (for perfection) will be in the future... "But, hey, if you enjoy it, and they pay you, do it!" :) – paul garrett Aug 16 at 1:36

(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) 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).

• I understand. Without meaning to be harsh in response, I don't fully see that this answers the question though? Maybe it's my misunderstanding of your answer. I'm particularly interested in stating precisely results in amongst more informal discussion, and for the reader to then understand which bits she/he needs to understand, compared with bits that will be expanded on. Would you be able to expand your answer to comment on this more explicitly? Thanks – Sam T Aug 13 at 21:40
• I appreciate a concise answer asserting "mathematics...values extreme...concision" – Words Like Jared Aug 13 at 21:42