Is it standard for a math research paper to include less detail in proofs than a typical textbook would?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

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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.

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_{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.}

Answer 5

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.

Answer 6

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.

Answer 7

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.