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My co-authors and I sometimes have different views on the organization of our papers (in mathematics). While some of them insist on writing down all the details, I try to keep emphasis on the main lines of thought and avoid diverging from them whenever possible.

This difference of opinions comes up in several aspects: whether to discuss all the results (positive and negative) that we obtained in the project or select only the most important/interesting; how much details should the proofs contain; etc.

My two questions are:

  1. if there is a way to make a paper contain all the details and at the same time have a clear structure not obfuscated by these details? Are there any organizational tricks that help to find some balance?

  2. reading and writing research papers, what do you value more its comprehensiveness or clarity?

I would like to emphasize that this question is only about what should appear in the final draft after all directions of research have been explored and all proofs have been completely checked and re-checked.

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    First, you should ask a single focused question. Second, you do need to select and prioritize what you want in your paper. It can't be everything for everyone. Think about your main target audience.
    – Kimball
    Aug 9, 2021 at 23:03
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    This is a hard question to answer. An experienced writer will omit some details because he/she knows that the community or target audience is well verse in certain concepts or should 'get it' or make the necessary jump in a derivation without issues. However, a newbie may write down every little detail or information that is considered too basic, which is likely to annoy an experienced reader. To balance this tradeoff, you may want to leave the details in the appendix of your paper. Aug 10, 2021 at 0:08

4 Answers 4

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I would say that clarity should be the king in the sense that if you want the paper to be read beyond the introduction and built upon, the best thing you can do is to present the main idea (or ideas) in the simplest nontrivial setting where you can spell everything out with all due details without making it a technical nightmare.

The standard trap people fall into is trying to present the strongest and the most general result they can prove. Nine times out of ten it turns out that the direction of the generalization they chose is not the one the other people get interested in later and one has to spend hours or weeks undoing all the sophisticated build-up and reducing the proof to its core from where everything else has to be rebuilt in a completely different way. So clearly presenting this core from the beginning would be very beneficial, IMHO. The word "clearly" means "with all relevant details and with minimal number of outside references"+"in as easy to comprehend way as possible".

By tradition, a mathematical paper has to contain some new result (preferably a solution of some reasonably old unsolved problem) just to prove that the techniques you promote are worth something but, for majority of articles, the results themselves are of much less interest than the new twists in the proofs. Your "central theorem" will, probably, be either forgotten or greatly improved in a few years but some little lemma may easily get an independent life and the simpler that lemma is, the more chances it gets (provided, of course, that it is above a certain threshold in terms of novelty and ingenuity).

The main "organizational trick" is to try to write about one thing a time. There are cases when you can and should set up "explosive fireworks of ideas" but they are rare, few people are capable of that and even fewer know how to do it in a way that illuminates things rather than blinds and confuses the reader. So, the (in)famous KISS principle is a good rule of thumb on most occasions. That one and the classical "divide and conquer" approach can work miracles as far as the clarity of communication is concerned. I heard once that a bad lecturer is always thinking of whether he forgot to include something into the course and a good one is worried if he put in something unnecessary. The same applies to mathematical writing if you want your papers to be read rather than just quoted in long reference lists under the category "and Vasya also was there and did something that nobody has ever really looked at but it would be impolite not to mention him".

Proofs should be complete and fully spelled out. It is often said that two omitted trivialities in a row can create an impenetrable barrier to the reader. Prove a weaker or a less general result than you can, if you feel that the argument becomes too entangled otherwise, but prove it in full and spend some time thinking of how to present it in the most logical and easiest to understand way. One hour spared by the writer often results in a few days wasted by the reader (which should also be multiplied by the number of readers).

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There is no hard-and-fast rule, because good communication is as much an art as a science. However, in my view, the majority of mathematics papers are written with too little detail. It is possible to give too much detail, but in practice I almost never see it. So I would encourage you and everyone else to err on the side of more detail, rather than less.

If you are worried about details distracting from the main thrust of the work, the best way to mitigate this is to organize the paper very clearly, with lots of signposting. Divide long arguments into lemmas whenever it makes sense to do so. This allows the reader to skip the details they don't want to read.

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    I don't know about the majority. By number, there are a lot of papers of minor results of little interest to anyone, but that's not what your reading. I think some people might say the majority of papers shouldn't even be published in the first place.
    – Kimball
    Aug 10, 2021 at 1:27
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Appendices, footnotes/endnotes and supplementary materials are your friends

I am in your camp here, insofar as I prefer to write in a way that "flows" well by sticking to the main lines of thought and avoiding diverging into side issues that distract the reader. The first three rules of good writing are: clarity, clarity, clarity. Consequently, almost every mathematics paper I write shifts substantial amounts of material out of the main body of the paper. I like to put proofs of theorems in appendices (unless the proof method is important to understand the material in the body), side aspects of the material and minor caveats into footnotes, and coding and other materials into supplementary materials. This allows me to write in a way where the main body of the paper flows naturally and gives the reader a clear and intuitive discussion of the research problem at issue.

  • Don't be afraid to give heuristic assistance/discussion: At the extreme end of brevity, some mathematics papers and notes are essentially theorem-proof-theorem-proof, etc., and it can be difficult for the reader to see how the authors figured out to use the methods under use. In my view, it is useful to break this up by giving at least some heuristic discussion to assist the reader to understand why a particular theorem or method of proof was chosen, and what motivated it. It is okay to speak heuristically here, and a small amount of additional detail on motivations, methods, etc., can make a world of difference in understanding the topic.

  • You can often put proofs in an appendix: In some cases a mathematical proof can highlight important aspects of the subject under discussion (and in this case you might want it in the body of the paper), but often the proof is there just to ensure that the theorem is proved. Moving proofs to an appendix allows you to just assert your theorems and then discuss their significance, and interested readers can look up the proof if needed.

  • Footnotes/endnotes can be used to prevent interruption of arguments: Often you want to have a main line of discussion where you speak in general terms, in a way that does not tax the attention of the reader. In some cases there may be some minor technical caveats or other minor points you want to make, but you are afraid that they will interrupt the flow of argument. In such cases, footnotes provide a useful way of dealing with a minor caveat without interrupting the flow of argument for the reader.

  • Supplementary materials are good for, well, supplementary materials: Some mathematics papers are augmented by computer simulations, reproducible coding, etc., and in such cases these should generally be made available in supplementary materials. It is rarely useful to include large amounts of computer code in a paper (unless part of your task is to explain how to implement something in computational software) so it is usually best to move this out of the body.

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  • Also when the coworkers want something in you can always say:'yes! let us put it in the appendix'
    – lalala
    Aug 10, 2021 at 12:51
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    This advice may be good for papers in applied math. But in pure math, the difficulty/novelty of the proofs is the main thing your paper is judged on. Appendices are useful for long-but-routine proofs that won't impress anyone, but a paper won't get into a decent journal if a significant fraction of the proofs are in the appendix. People will see the paper as trivial. Again, I'm speaking only about the publishing culture in pure math here.
    – anon
    Aug 10, 2021 at 14:41
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    I have never seen an appendix used to provide proofs in a pure math paper. Aug 10, 2021 at 14:47
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    @AlexanderWoo I have, but very rarely and only for proofs which can be summarised as "follow the proof of [similar lemma] from [previous paper] making the necessary minor changes". This can be good if you want the paper to focus on the new parts of the proof while still being self-contained. Aug 10, 2021 at 16:29
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    @anon: With respect, I feel like I covered that case specifically in the answer; I have said that if the proof itself is important then it should be in the body.
    – Ben
    Aug 10, 2021 at 21:57
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Realistically, none of your readers will read past the introduction. Write a long introduction with your main lines of thought. For a long paper, people frequently now even write an introduction to the introduction, so that the first page or two says what the paper is about, then the next four or five papers lays out the main ideas of the paper, and then the remainder contains all the details.

Particularly technical lemmas whose proofs are long but not enlightening can be proved in sections at the end of the paper. In particular, you do not need to always give the proof of a statement before you use it, though if you defer a proof you should be careful to make it clear that you do not have a circular proof.

As to what level of detail to include, my suggestion is to pick some specific person as your intended reader for the paper, and include all the details that will be necessary for them. Do not pick the leading expert in your field or one of your close collaborators. Frequently it is best to pick a graduate student, and it's okay if you're not more specific than X's hypothetical PhD student who is just starting research.

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