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I am a fourth-year undergraduate student in Mathematics. Recently, I read a mathematics paper. It seems that the proof in the paper is not as convincing as what one might find in a textbook. For instance, the authors skip some calculations and arguments in the proof. I feel that the detail in the textbook is better. In addition, the instructor in my undergraduate year always checked the completeness and the detail of the proof in exam and homework.

Is this a standard practice in writing mathematics papers? What is the advantage (if any) of skipping some calculation and argument?

I plan to apply to a graduate school in the future. If “the skill of minimizing the proof in math papers” is important, then how do I learn this habit and unlearn the old habits of my undergraduate years?

TRiG
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fourth year
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    Terry Tao has a blog post on this general subject, which he calls the post-rigorous stage of the practice of mathematics. – Peter Taylor Sep 04 '17 at 07:32
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    Here's a related story : http://curiosamathematica.tumblr.com/post/122398968526/obvious Note: It might be a joke. – Eric Duminil Sep 04 '17 at 08:18
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    @EricDuminil Thank you. It is interesting story. – fourth year Sep 04 '17 at 09:12
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    Things that is commonly omitted are details in proof by induction. Base cases are often trivial and not spelled out, and the assumption and induction step can many times be clear from the context. Simplifications that Mathematica can perform by using FullSimplify can also be omitted, as the reader can easily verify these steps themselves. – Per Alexandersson Sep 04 '17 at 11:08
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    Note that many textbooks are rather brazen about proofs as well (most calculus books, set theory in discrete math books, ...), only you perhaps didn't notice at the time. – Kimball Sep 04 '17 at 13:04
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    In addition to what others have noted, omitting routine details may actually improve readability (for readers familiar with the topic). It's easy to get lost in the details of a long and complex proof, and skipping the less essential details makes the overall structure more clear. However, this helps only if done well. – Sasho Nikolov Sep 04 '17 at 16:43
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    Well, at least research papers don't leave the proof as an exercise :) – Vim Sep 05 '17 at 00:15
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    @Vim: actually, many do. Mostly for easy stuff, or things that can be done by rehashing arguments already in the paper. In any case, it is fairly common. – Martin Argerami Sep 05 '17 at 01:42
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    Because textbooks and papers have different purposes. Textbooks want to teach, while papers want to impress. – polfosol Sep 05 '17 at 04:52
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    @polfosol : I hope that your claim is not true. – fourth year Sep 06 '17 at 01:51

7 Answers7

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Yes, it's normal. Homeworks and exams are written to prove that the writer has certain skills; papers are written to prove something new. The reader's skills are not under question, so a different style of writing is appropriate. Also, journals used to have stricter page limits than they do now, so there was quite some pressure to be terse. Conversely, somebody who has a hundred exam scripts to mark needs all the details to be spelled out because they don't have more than a few minutes to give to each script.

It is conventional to omit "routine" calculations that the reader should be able to do themself. For example, one might just assert that a certain function reaches its maximum at x=2p/(1-pq) and assume that the reader is capable of setting the derivative to zero and solving. The reader will typically trust the writer (and the peer-reviewers!) to have done the calculation correctly.

In my view, some authors take this too far and omit calculations which can take hours or days to reconstruct, which is a royal pain when trying to adapt or extend the result. Over time, as you read more research papers, you'll learn what is an appropriate level of detail: the big hints come when you start to co-author papers with your advisor.

David Richerby
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    One note: in exam scripts the details are frequently needed not because of time constraints in the grading, but to understand where an error in a calculation comes from. – Massimo Ortolano Sep 04 '17 at 07:44
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    @David: If one can not fill the gap in the paper for days/months, is it appropriate to ask the author about the detail? – fourth year Sep 04 '17 at 09:07
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    @fourthyear Yes. Certainly don't let it go for months! If there's somebody you could ask locally (a colleague or advisor), it's probably a good idea to ask them, first. – David Richerby Sep 04 '17 at 09:10
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    "The reader will typically trust the writer (and the peer-reviewers!) to have done the calculation correctly." One could argue that if someone actually already did the calculation, then it's probably only a medium sized effort to store/attach it somewhere (as a footnote maybe). So maybe some readers could fill in details, so other readers do not have to do them again. Would require a more Wikipedia style approach to journals though. – NoDataDumpNoContribution Sep 04 '17 at 13:07
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    @Trilarion It would make more sense for the authors to just include it in the first place. – David Richerby Sep 04 '17 at 13:08
  • Omitting easy calculations gets weird when a major component of your results requires 30+ pages of very easy if not trivial (to a research mathematician) calculations in sequence. Some problems are best, possibly only, solvable by savagely beating them with the algebra stick. Too many omissions, or a step omitted too many times, can suddenly transform "reasonable omission of something easy/well-known" into "I can't even figure out what's going on anymore." But include everything and you might be chided for including too much easy stuff and making the paper seem bloated. – zibadawa timmy Sep 04 '17 at 16:43
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    @DavidRicherby - I would say it is appropriate to ask the author if one cannot fill in the gap and one is an expert in the area. But, in saying this, I also want to say that many people underestimate their expertise. Any advanced graduate student, or anyone having ideas on open problems in the area, should be considered an expert! – Alexander Woo Sep 04 '17 at 18:49
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    @zibadawatimmy - Surely if you have 30 pages of calculations, there is an organization and a story to them: pages 1-5 are meant to establish some important intermediate goal, and page 3 helps in this particular way towards that goal, and the 2nd equation on page 3 fits in for this particular reason. If you can do that, then you can highlight the major steps. If you can't, then no one will be able to follow your 30 pages anyway! – Alexander Woo Sep 04 '17 at 18:53
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Yes, it is common. It saves time and space for the reader.

Keep in mind that when you're doing math (and computer science) you need to pick, from the wide continuum of possible abstractions, the right level for the intended reader/student/recipient. It's among the most important skills for a writer or teacher. For any level of reader, there are things that are "obvious" that would be tedious for the reader if written out fully.

An example: We know that 3x + 5x = 8x. Why? Technically it's because 3x + 5x = x∙3 + x∙5 [commutative property of multiplication] = x(3 + 5) [distributive property of multiplication over addition] = x(8) [addition of natural numbers] = 8x [commutative property of multiplication]. Now, to the extent that "combining like terms" is a relation with which you've worked so much that 3x + 5x = 8x seems obvious, then we could have skipped those atomic-sized steps from fundamental axioms.

So too, the expected audiences for those papers you're reading probably find all the skipped steps "obvious" and something they can fill in mentally (or at least approximate or sanity-check on the fly) as they read it; and hence it would be a waste of space and most readers' time to fill them in. You can get to this point by reading more of the papers at that same level (and as you level-up, keeping pencil & paper next to you, working slowly, and filling in the missing details as you read). Hopefully by working through a master's and PhD program and specializing deeply in one particular area, one can get to the point of reading those papers just like you would read an algebra or calculus book right now. Of course, you'll simultaneously need to maintain the skill of filling in the extra details any time you're serving as the teacher and trying to explain things to lower-level students.

Daniel R. Collins
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  • Thank you very much for your answer. How do we estimate the knowledge of the reader? – fourth year Sep 04 '17 at 02:34
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    @fourthyear: Assuming a standard undergraduate education in your field is usually a good place to start. – aeismail Sep 04 '17 at 02:38
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    "something they can just fill in mentally as they read it" seems exaggerated. It's something they can fill in, but not necessarily "mentally as they read". – Michael Hardy Sep 04 '17 at 03:44
  • @MichaelHardy: I added a parenthetical note there to try and dial down the claim. If you can offer better language for that spot, then I'd appreciate seeing it. – Daniel R. Collins Sep 04 '17 at 05:23
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    Depending on how the distributive property is expressed, you might not need the commutative property ;-) – Massimo Ortolano Sep 04 '17 at 05:37
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    Your example is useful for explanation but I think it's overly simple. It's not unusual for somewhat lengthy, complex calculations to be omitted. This certainly doesn't "save time" for anybody who's trying to extend or adapt the result, as they first have to spend something between half an hour and an afternoon reconstructing the reasoning. I think there is a tendency to omit "boring calculations", regardless of whether or not one feels the reader could easily reconstruct them. – David Richerby Sep 04 '17 at 07:07
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    Wouldn't it be cool if there would be a plus button on papers on almost anything, so that everyone else from different backgrounds could just zoom in with the details until it's satisfiable for a large range of different knowledge levels? Of course this would have to be generated somewhat automatically to be manageable by the creators. – NoDataDumpNoContribution Sep 04 '17 at 13:03
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    @Trilarion: I feel like that was always held out as the promise of hypertext, but was just not profitable for the level of effort. – Daniel R. Collins Sep 04 '17 at 16:16
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    @DavidRicherby I think the example here is intentionally simple, for explanatory purposes. A more realistic example might be the definite integral ∫ exp(-x^2) dx = sqrt(π). It requires a nontrivial insight to realize how the circle constant comes out of the integral. It is common enough that some textbooks devote a page or two to the proof the first time it crops up, but rare enough to imagine an advanced undergrad or an early grad student finding it and spending an hour (or more) rediscovering that the indefinite integral can't be solved. – rob Sep 06 '17 at 01:48
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Without knowing the details of the paper you read, I'd guess that details were omitted from the proof because the authors considered them so elementary that the reader would readily fill in the specifics. That doesn't mean they'd expect the reader to do it all in their head effortlessly as they read, but only that they'd expect the reader to be able to do it without help from the authors. Such omissions would be inappropriate in an undergraduate textbook because those specifics would be the very thing that they are trying to explain to the reader.

Michael Hardy
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Others have already provided good (and bad) reasons to write concise proofs, but since you are planning to publish yourself, I will share my own approach when writing papers.

As an author, you have to satisfy very different types of readers, from undergraduate students to highly experienced researchers that work exactly in your field of research. But even for a single person, different levels of abstractions are necessary, because I (and surely many others) read papers top-down:

When reading a paper for the first time, I skip the proofs completely. Too many lengthy proofs and I have problems to get the overall picture. Then I skim over the proofs, looking for the main ideas of them. This step would be really annoying if there are too many steps. I only look at the proofs in detail if they are interesting to support my own research, I review the paper or something is suspicious¹. In this final step, I am happy about every detail that saves me time and effort.

How to address the different needs?

My usual approach for publishing proofs is as follows.

  1. I use pen and paper to construct the proof. This results in a huge pile of unreadable garbage, but at some point, I am confident enough that my proof works.

  2. With the ideas still fresh in mind, I write down the complete proof in a publishable form, i.e. in LaTeX², including every conversion that (in my opinion) is necessary for an undergraduate to directly understand every step.

  3. With the fully proof at hand, it can now be condensed. For example, by presenting only very high-level steps in the main part of the paper (maybe only the final result and a textual description of the proof ideas) and a moderately condensed version in the appendix that leaves out all steps that seem trivial.

  4. The full proof should be submitted as supplementary material if permitted by the journal (see below for an alternative).

Of course, the second step comes with extra effort compared to going directly from the pile of unreadable garbage to the most condensed form. However, it pays off in the long run:

  • By writing down every single step in a clearly readable form, a lot of errors are directly recognized. Otherwise, it wastes the time of your supervisor, your reviewers and (if the paper even gets published with the error) other researchers, not to mention the shame and effort when the error is eventually detected.

  • Even if you clearly understand your handwritten notes and your publication now, this won't be the case after a year (if you can even find them). So you have to waste time to redo your work.

  • If more detailed proofs are published as supplementary material, it will eventually save time for everyone.

  • Last but not least, it improves the credibility of your work. Even if only a very minor portion of your readers will actually benefit from or even read your detailed version, they trust you more if they see that you have a complete proof as supplementary material³.

What if I can not submit supplementary material?

In my field of research, only a minor part of the journals and conferences allow submitting supplementary material. An alternative is to submit supplementary material to e.g. arXiv.org. With good timing, you can even mutually cite the original paper and the supplementary material. You should not use a personal website because the probability is high that it will not be accessible for a long time.

Unfortunately, this is very difficult in a double blind review process. It would be much better if the submission of supplementary material is widely available at every journal and every conference.

¹ The "That has to be wrong!" effect. You might think that leaving out proof steps will help you as an author in this case. To the contrary, it increases the incentive to prove you wrong.

² If you dislike writing long and complex formulas with LaTeX: My wife is very happy with LyX.

³ No excuse for hiding a wrong proof by using an excessive amount of formulas. That will be detected eventually.

koalo
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  • +1. This answer is awesome and deserves a lot of upvotes. Do you have any example of journals that accept/encourage supplementary material, especially proof details. – Taladris Sep 05 '17 at 00:09
  • @koalo: Thank you for your answer. I that that your suggestion is very useful. – fourth year Sep 05 '17 at 03:28
  • @koalo: For submitting the supplementary material, how do we avoid the conflict with the journal where we publish our main paper or the issue with self plagiarism? – fourth year Sep 05 '17 at 03:30
  • There's no timing issue between ArXiv and a journal paper. The ArXiv version can be edited at any time. – David Richerby Sep 05 '17 at 10:43
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    "It would be much better if the submission of supplementary material is widely available at every journal" It would be much better if journals didn't regard proof as merely "supplementary". – David Richerby Sep 05 '17 at 10:43
  • @Taladris One example I know are the Proceedings of Machine Learning Research – koalo Sep 05 '17 at 11:55
  • @fourth year: Of course you have to make it very clear that something is supplementary and not a publication on its own. Since arXiv is not a journal or conference proceeding itself, I do not think someone will criticize the novelty of your main paper or see it as self-plagiarism. – koalo Sep 05 '17 at 12:11
  • @David Richerby: I am nitpicking, but you can not edit a submission, but only upload a new version with a new submission date and the old version is still available. I agree that it is not a big issue, but you have to be aware of that. – koalo Sep 05 '17 at 12:20
  • @koalo Which is functionally no different from what I said. Let's not waste time. – David Richerby Sep 05 '17 at 12:21
  • @David Richerby: It very much depends on the field of research. If the whole point of your paper is to prove something, then it certainly does not make sense to not include this proof. But having pages full of highly detailed proofs on undergraduate level in a paper where the proof is not the main part does not make sense in my opinion. – koalo Sep 05 '17 at 12:28
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    If it's not proven, it's not true. You wouldn't include scientific claims based on experimental results that don't appear in the paper, so why would you include mathematical claims based on mathematics that doesn't appear in the paper? – David Richerby Sep 05 '17 at 12:35
  • Let us continue this discussion in chat. – koalo Sep 05 '17 at 14:09
  • @DavidRicherby, most scientific papers don't include their experimental results in the sense of all of the raw data. You have to take it on trust that the authors correctly calculated the summary results which do appear. – Peter Taylor Sep 06 '17 at 08:02
  • @PeterTaylor Yes but they do include those summary statistics. As I stated in the linked chat, a few small (non-systematic) mistakes in the experimental results won't affect the aggregate statistics. A few small mistakes in a mathematical proof can completely invalidate the whole thing. Mathematical proof is not the same thing as experimental data. – David Richerby Sep 06 '17 at 08:11
  • @DavidRicherby, I agree that minor errors in the raw data won't affect the statistics (that's what statistics is all about, after all!), but fundamental errors in the calculation of the statistics cannot be detected without access to the raw data. On the other hand, if some details of a proof are skipped as obvious to the target audience, they can still check the proof by filling in the details themselves. The vast majority of published proofs cannot be passed directly to a theorem prover: the standard of acceptable proof at a research level is that it convinces your peers. – Peter Taylor Sep 06 '17 at 08:28
  • @PeterTaylor Absolutely. I'm commenting on the tendency of non-mathematical journals to publish articles with essentially all the proof provided only as "supplementary material" (my experience in this regard is with probabilistic models of evolution). – David Richerby Sep 06 '17 at 08:56
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When you are sufficiently familiar with a specific topic, you will also know which parts of a proof are key steps and which are not (either just tedious algebraic manipulation or case checking or some standard argument or...). You hence know which parts should be included and which parts can be left out. Even when you are talking about undergraduate courses, your instructor definitely takes some prior mathematical facts for granted, such as 2 = 1+1 and (1+2)+3 = 1+(2+3). So when you talk about completeness of a proof you are actually saying that it contains all the steps that have not been taken for granted. Similarly in a paper, the authors will omit all the steps that can be taken for granted, often because anyone in that field can easily fill it in. It also avoids making the reader tired by presenting as concisely as possible the information the reader actually wants.

user21820
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    As I commented on another answer, the omitted calculations are often much more complicated than primary school arithmetic. – David Richerby Sep 04 '17 at 07:10
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    @DavidRicherby: That's right, but how is that relevant to my answer? – user21820 Sep 04 '17 at 11:00
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    It's relevant because the examples you give are exactly primary school arithmetic! – David Richerby Sep 04 '17 at 11:15
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    @DavidRicherby: I don't think you read my answer very carefully. I gave those examples in a sentence beginning with "Even when you are talking about undergraduate courses, your instructor definitely takes some prior mathematical facts for granted", and I said later that "Similarly in a paper, the authors will omit all the steps that can be taken for granted". So I do not see how your comment is relevant. – user21820 Sep 04 '17 at 11:17
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    The concept of "key step" becomes vague to the point of no longer making sense in advanced mathematics. What can be "taken for granted" is a function of the writing, the reader and any amount of mental state on the reader's part. And "anyone in that field" isn't very well-defined either; does a graduate student learning the field count? – darij grinberg Sep 04 '17 at 18:58
  • @darijgrinberg: Of course "anyone in that field" isn't well-defined. This is a social issue, not a mathematical one. Some papers are written to serve as expository material, and so might have graduate students among their target audience. Some other papers are written to publish solutions to open problems, and have other experts among their target audience. I thought it should have been obvious to anyone reading my answer that "sufficiently familiarity with a topic" necessarily includes familiarity with the terminology and conventions in that field. – user21820 Sep 05 '17 at 07:18
  • @darijgrinberg: Papers that don't follow the norms take longer to be understood by the international mathematical community, if at all. My answer is aimed at those who wish to write papers to convey their mathematical ideas to others, and is not intended to be an accurate description of all mathematical papers written in history. – user21820 Sep 05 '17 at 07:23
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I feel (but may be completely off the mark here), that to some authors its also a matter of pride. By spelling out details that are not “deep”, but may be difficult to reconstruct nonetheless, they’ll give the impression that they struggled with these details themselves at some point. I have the impression that this hurts some authors’ egos, and hence they do not include the details, opting instead to write very difficult-to-read articles.

TRiG
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user2520938
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    It's not that simple. On at least two occasions I was told by a referee/editor to shorten my paper significantly because way less detail was needed. And it was true that many experts didn't need that much detail, but also that many graduate students would do. In the end, the level of detail is an arbitrary assumption on the expertise of the would-be reader. – Martin Argerami Sep 05 '17 at 01:49
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It is standard practice in math. The advantage of skipping trivial details like "this matrix is invertible" without computing the determinant or a easy induction is readability. However, it is also quite frequent to skip larger details that undoubtedly annoy most readers except the most experienced experts. That's great for those three experts and unhelpful for everyone else including students and even experienced mathematicians in other fields. By reading more papers you'll be able to distinguish between the two kinds of omission.

When you write your own papers, you should include those harder details. Err on the side of overexplaining. The worst that you'll get told by a reviewer or editor is that you'll have to shorten a bit, which is far better than having too terse a paper.