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I've just written my first mathematical research paper. It proves some new results, which while not ground-breaking are (according to an expert in the field) at least somewhat interesting and surprising. At the moment however, I spend more of the paper developing the background material (giving standard definitions and constructions, proving standard lemmas) than proving the main theorems.

Is this a problem? The way I see it, there are several arguments for and against:

For:

  • The background material is "standard" in the sense that anyone who works on this class of problems would know the definitions or results in some form. However, this is at most a few hundred people, while if I include the background material my paper should be comprehensible to an advanced undergraduate.

  • Some of the background results are part of the folklore of the field, and I've never been able to find a proof of them in literature. While they are believable and not hard to prove, I feel someone should bother doing it. More selfishly, this is one more reason for people to cite my paper.

  • I don't know of any one reference which states all the background material I need, so if I don't include it my readers have to chase down multiple sources and I have to use conflicting notations.

Against:

  • It may be annoying to an expert in the field, although they could skip much of it and mainly refer to the background section for notation.

  • It makes the paper longer, although even with the background the paper is not long (13 pages).

  • From what I've heard, it is generally considered bad practice to restate definitions and constructions states elsewhere and to reprove theorems available in literature. In part this is because it gives the impression that I haven't read the literature. This is exacerbated by the fact that I only cite ~5 previous works, mostly for further reading or alternative presentations of some of the background.

I'd like advice on this from someone with experience writing such papers.

  • 1
    I wonder if this might be better suited for MathOverflow. – JeffE Jun 6 '13 at 0:19
  • 3
    There are many reasonable answers below, I just want to stress a few principles: clearly separate the background from the new stuff; clearly and plainly indicate that what is not new is not new, and why you include it; especially for proofs of well-known lemmas, only include them if you cannot find a reference after trying at least twice to find one from MR or ZB databases (but in any case, include the precise needed statements). Inside these guidelines, you should be ok whatever choice you make. – Benoît Kloeckner Jun 7 '13 at 19:01
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I'd recommend putting in as much explanation, context, and background as you reasonably can. In principle you could certainly put in too much, but in practice I don't think I've ever seen a mathematics paper I thought actually did have too much. If you structure your writing so experts can easily skim things they already know, then I don't see why extra explanation should trouble anyone. Tradition requires being concise, but that tradition was based on the economics of publishing. Space in printed journals used to be a scarce resource, and if you spent it on unnecessary exposition you were keeping someone else from publishing at all. Publishing still isn't free (and readers have only so much patience, as well), so you shouldn't include enormous amounts of unnecessary background, but traditional writing styles should be adjusted to account for the greater availability of space.

It's possible that a referee or editor will object that they don't like your style. You may be able to resist making any changes (it depends on how exciting your paper is and how high your own status is within the community), and if you do make changes you can still keep a lot of explanation.

If necessary, it's worth having two versions as Paul Garrett suggests. If you do, then you should make it very clear which is which and what the differences are, to avoid serious confusion if someone refers to the paper without realizing they need to specify which version.

The background material is "standard" in the sense that anyone who works on this class of problems would know the definitions or results in some form. However, this is at most a few hundred people, while if I include the background material my paper should be comprehensible to an advanced undergraduate.

One reason to be generous with explanation is that it's easy to overestimate how much people know. Aiming at advanced undergrads is a good way to make sure grad students really can read the paper. If you aim at experts, you may write a paper only your own collaborators can comfortably read.

Some of the background results are part of the folklore of the field, and I've never been able to find a proof of them in literature. While they are believable and not hard to prove, I feel someone should bother doing it.

This is a great reason to include a proof, although it's important to consult with experts to confirm that there really isn't an accessible proof in the literature.

It makes the paper longer, although even with the background the paper is not long (13 pages).

You should make sure it doesn't look like padding, say with just a few pages of original content among the 13 pages.

From what I've heard, it is generally considered bad practice to restate definitions and constructions states elsewhere and to reprove theorems available in literature. In part this is because it gives the impression that I haven't read the literature. This is exacerbated by the fact that I only cite ~5 previous works, mostly for further reading or alternative presentations of some of the background.

This does seem like a worry, and five references is not very many, even for a 13-page paper. You might not strictly need any more, but I'd recommend tracking down additional references for background. As a general rule it's best to cite generously, giving plenty of credit to other authors and offering many resources to readers.

Regarding giving the impression that you haven't read the literature, it's important to be clear about what's background from the literature (with a citation), what's folklore, and what's your own contribution. If you reprove something you say is known but don't give a citation for, it can look like you were too lazy to track it down.

  • Additionally, some journals may allow publishing supplemental material online, which may be a better place to put more background. References themselves are often a good way to give more background for the reader, I mean... if it's supposed to be well known in the literature then the OP should be able to cite it right? – ttbek Sep 20 '17 at 19:51
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As you rightly perceive, there are conflicting desiderata for "formal papers". The main tradition for refereed-journal publications is to assume one is writing for the experts, not for anyone who'd need much background or context. Yes, this makes reading such papers needlessly difficult for non-experts. Yes, the necessary background is likely scattered among several prior papers, and some occurs "nowhere", in the sense of being apocryphal... lost in the mists of time? :) Nevertheless, the highest-status refereed journals would probably want more-discursive broader explanations edited out... and the referees and editors might interpret your inclusion of such things as "amateurish" ("It's just not done..."), to your detriment.

But the expectations or standards of refereed-journals is certainly not the only criterion, and is manifestly antithetical to the obviously desirable outcome of wider dissemination of ideas, collection and organization of otherwise-chaotic literature, and so on. Some people will tell you that somehow it's not ok to "mix" "research" and "exposition"... but this is silly, even if traditional. But, then, given the traditional predilection of refereed-journals, if you want a more discursive version of your write-up, you may have to reconcile yourself to having two versions, one for referees, one for exposition/wider-dissemination.

The for-referees version should probably not cite the discursive version, but, instead, do the quasi-unhelpful bit of citing the disparate standard sources. The discursive version could cite the for-referees version, as well as the standard sources, and still include your own updated presentation, bringing apocrypha to the light of day, and so on.

But, I fear no single refereed-journal-publishable version could meet the nicer-exposition requirements you (completely reasonably) would like to impose.

  • How would the version for wider dissemination be disseminated? Through the arXiv? – Alex Becker Jun 5 '13 at 22:07
  • Sure, arXiv is one reasonable venue for wider dissemination of a more discursive version, if you don't have a personal web-page that's well-known and that'll be stable. – paul garrett Jun 5 '13 at 22:17
  • to your detriment — And the journal's. – JeffE Jun 6 '13 at 0:20
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If there are proofs in the literature for some of your background materials, it is better you do not prove them in your paper unless your proofs are much simpler or they use a method which will be used again in your other proofs (I have seen both instances in papers written by well known mathematicians). However if you cannot find a proof for some of the easy (or well known) results in the literature, it is a great help for your readers to add your proofs. About introducing your notation and/or definitions, you can be even more generous to your readers. Although there is a downside for including lots of backgrounds, not including some of them can delay the referee process of your paper. Do not hesitate to cite more references if they help to find some of the background materials. And finally, I suggest you devote some paragraphs to explain the motivation of your work.

Good luck with your paper!

2

It's not entirely clear whether the background material is just background, or is needed as scaffolding for the results you prove. Also, if certain lemmas are "standard" why are you proving them ? surely there's some text or other paper that proves these lemmas, and which you can cite ?

In general, in a paper you provide the scaffolding needed to prove your results. It's not necessary to provide other atmospherics unless its part of a larger discussion of related work.

  • Most of it is necessary for the theorems I prove. Regarding the results being standard, as I stated in the first point for including the background: The background material is "standard" in the sense that anyone who works on this class of problems would know the definitions or results in some form. However, this is at most a few hundred people, while if I include the background material my paper should be comprehensible to an advanced undergraduate. – Alex Becker Jun 5 '13 at 21:35
  • And as I said, some of the lemmas are widely-known but I've yet to find anyone who knows where to find a proof. – Alex Becker Jun 5 '13 at 21:49
  • 4
    My personal preference would in this case to state the lemmas without proof (stating that they're "folklore", and then add proofs in an appendix "for completeness". As for the background material, @paul makes the correct point that unfortunately or otherwise, you're not expected to cater to an "advanced undergraduate" audience in a formal paper. – Suresh Jun 5 '13 at 22:15
  • I've considered adding an appendix. Also, not including any background material is the difference between an advanced undergrad being able to understand the paper, and only someone who has worked on irrational billiards being able to understand it. – Alex Becker Jun 5 '13 at 22:28
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One factor to add is that in many mathematics papers, there are many different definitions and notations for a subject. Some of these approaches may make your results seem more or less natural. Even someone who is familiar with the subject may become confused by the way in which you present things.

One good reason for a background section is so that you can "set the scene" using the presentation that you think is best. Sometimes just setting things up in a coherent way is more valuable than the new results themselves.

0

Warning: I have experience with "mathematical" papers in the Computer Science community. In our field, papers are supposed to be (to some extent) self-contained; moreover, it's more the writer's responsibility to make stuff clear, than the reader's responsibility to study the stuff (this generally helps to get more readers and make your research have impact; also, readers don't have time to read all papers they'd want to, so be gentle to them).

I'd optimize background for skippability. In fact, optimize all sections for that, but especially such background.

For concreteness, I'll give an instance of what is reasonable/can be recommended in our field.

\section{Background}
In this section, to make this paper self-contained and to fix notation,
we summarize
the theory of representable functors % don't be as vague as
% "background on X"
which we'll use later in the rest of the paper/in Sec. YYY. % Help readers figure out
% whether they actually need this background,
% if it's only needed for part of the paper.

\subsection{Standard definitions}
In this subsection, we summarize background which is available in the
literature, though spread across different references~\citep{1, 2, 3}.
% Don't order the material necessarily by reference, but by how they
% are best presented.

% ...

\subsection{Folklore theorems}
In this subsection, we present some basic results. Although they appear to be
folklore/almost obvious/believable % what you prefer
for experts, we found no proof in the literature, so
we include the proofs for completeness.

You can include something that is part of the literature; to show you know the literature, you just need to also add a citation — doing otherwise can be bad practice (if it's very clear the result is not yours), and might be taken for unethical by somebody who misunderstand what you claim.

But you should decide whether to re-include the proofs though — can you lift the statement directly, adapting the notation? Or is the setting different enough that you need to adapt the proof? Or is it important that a reader knows the proof to get your paper, because you reuse similar proof ideas?

However, things in mathematics might be different; advice still should have similar consequences, but conventions are different. Let me say that the habits of mathematicians are quite frustrating to e.g. computer scientists. I've seen at least one respected computer scientist define some standard references (in mathematical style) "unreadable" or "write-only".

protected by Alexandros Oct 3 '18 at 18:16

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