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Context: As a researcher you need to read many research papers by others. In the case of mathematics, a paper consists of:

  1. Results (theorems, lemma, propositions)
  2. Proofs that consist of the technique or method by which the researcher obtaining their result.

Clearly, results can be used without learning the proof. However, understanding a proof may help you develop future results (but not necessarily always).

Ideally, it is best to learn the proof and techniques used in detail in order to fully understand the reality.

Question:

Do researchers (graduate students, professors, research fellows) in mathematics have a detailed understanding of the proofs/techniques outlined in contemporary journal papers? What is the convention in math?

Please note that my issue is not whether the proof is correct or not, I want to know what most academics do in the most of the cases when a result is published in a peer-reviewed journal paper. Do they read and understand the proof or they just skip the proof and remember the result? (provided that the proof technique is not out of the box)?

Example: I was reading Simon Singh's book on Fermat's Last Theorem, he wrote that only half dozen people understood the proof of Andrew Wiles. (The number maybe slightly erratic since I read the book long ago.) But the modularity conjecture was proved by him at the same time, I guess this is quite important. This made me think how much actually people understand contemporary work? How much it is actually important? Mathematics is a very very technical subject now a days.

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    It depends on what kind of math you do, what you need the math for, whether the result is interesting, etc. My primary job is to prove things, and very often, I need to understand how someone else proved things so I can copy their techniques and prove more things. So, yeah, I read and try to understand contemporary proofs in detail, when those proofs are useful for the things I need to prove. (Also, when was math ever not a technical subject?) – artificial_moonlet Jan 21 at 16:03
  • As a thought experiment, suppose that you take the premier textbook in some field of math and memorize the statement of every theorem in the book. Do you suppose that you know that domain? That you can use it effectively? You can almost certainly trust that the author(s) got it right, of course. – Buffy Jan 21 at 18:09
  • @Buffy Text-Books and Journal-Papers are 2 different things, you build your basic reading books in detail, doing exercise, journal paper is not same, research paper might not be compared with books, you don't learn basics from research paper. – Jim Jan 21 at 18:13
  • Why do you think learning is different? Ok, then. Take 50 papers in some field and memorize the statements of every theorem stated. Do you think you are now conversant in that field? – Buffy Jan 21 at 18:27
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    You seem to be resisting something obvious to most mathematicians. Mathematics isn't about facts. It is about discovery and the establishment of some truths. A process, not a thing. But a paper that is interesting to me, and worth a serious examination, may not be to you, so a skim will do. Or even ignoring it. – Buffy Jan 21 at 18:39
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The answer, of course, is that it varies with each reader and with the specific needs of each reader. For many people, a skim is sufficient for most papers. The question is "does this seem reasonable" and if so, there may be little need to go into the details.

This is especially true about proofs. If an overview of the proof suggests that the techniques are standard for the field, and the results don't seem to clash with what the reader knows, then it is unlikely that the reader will spend the time on every detail.

The exceptions, of course, are many. Students want to learn new proof techniques that they are relatively unfamiliar with. Surprising results require a close look. Long standing problems, when finally proven, also require a close look.

And, in many cases, the way something is proven is more important than the result itself. If a skim of a proof suggests that there is something new here, then experienced mathematicians will want to examine it in detail as well as look for gaps and errors.

But if you are a student, or otherwise feel the need for an answer to this question, then I'd suggest that you err on the side of completeness. Keep your skeptic hat on until you are satisfied.

  • Please note that my issue is not whether the proof is correct or not, I want to know what most academic do in the most of the cases when a result is published in a journal pepper? Do they read and understand the proof or they just skip the proof and remember the result (provided that the proof technique is not out of the box)? – Jim Jan 21 at 14:54
  • @Jim But rarely is one interested in the results without also getting a feeling for how trustworthy these results are. If your question is "how often does a mathematician read the results without caring whether they are right or wrong", my assumption is that the answer is "basically never". – xLeitix Jan 21 at 15:00
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    Not for a serious mathematician. Maybe it is a bit flip but: Theorems are boring. Proofs are interesting. The most interesting part of my doctoral work was one of the proofs, which broke new ground. The statements were fun, but that one proof is what gave it value. Don't assume reviewers never make mistakes. See this recent question, for example: academia.stackexchange.com/q/143215/75368 – Buffy Jan 21 at 15:10
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    @Jim In most cases most academics just skim the paper and skip large parts of it. However asking for the average is highly misleading. Perhaps 90% of the papers I'll read, I'll never use. And in general those are the papers I skim. In theory I could move the average to reading all papers in detail by simply not looking at those 90% at all and still do the same mathematics as before. It's just that the best way to gauge if a paper is worth reading in detail and has results or techniques that can be used, is skimming it first. – mlk Jan 21 at 15:24
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    Maybe to add to my last comment. Things are also source dependent. As a student, papers generally get recommended to you. Assuming the recommender knows what they are doing, those will be relevant and worth a detailed reading. As an active researcher trying to stay at the "state of the art" (whatever that may be), most of the papers you read are whatever just has been published, which as said often have a promising abstract but then turn out to be less relevant after a few pages. – mlk Jan 21 at 15:33
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It depends. If I'm just browsing MathSciNet or the arXiv to see what's out there, I'll read a large number of abstracts but few papers. If I find a paper that looks like it might be related to my research area (in a broad sense) then I'll likely read the introduction but nothing more. If I see a paper that looks like it might make use of techniques that could be useful in my research then I'll read the introduction and skim the proofs. If a paper contains a theorem whose proof I need to modify or extend for a paper that I'm currently working on then I'll likely read the proof very carefully.

To put the above in context, I'm a professor, and thus have to teach, write papers as part of my research program, serve on administrative committees and meet with students. This takes up a lot of time! And any time I spend reading a paper is time that I am not necessarily spending on any of the aforementioned responsibilities. Therefore, if I think that a paper might be valuable for my own research or that of a student, I'm much more likely to spend time with the paper and perhaps read through the actual proofs than I am a paper that isn't really related to my research at all. (Even if the latter paper is more 'important'.)

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From personal experience, I read a few papers very carefully and understood the proofs in full detail. For most papers a rough look at the results was sufficient. It depends mainly on the goal of reading the paper.

The ones I read very carefully were usually the ones where I wanted to apply the technique to a similar setting for my own research. In order to do that it is not sufficient to know the result in the paper is true, I needed to understand the nitty-gritty detail of the proof to see where I could just use what is there and where I needed to adapt something to fit to the new setting.

For most of the various other situations where I looked at some paper just reading the big theorems and maybe skimming the proofs was usually enough.

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Like anyone else, mathematicians by and large read papers selfishly, i.e. to the level needed to advance their own thinking, and no more.

So if the result matches my intuition, I may or may not even read past the initial statement in the introduction.

If it seems to open up intriguing vistas, or is a bit surprising, I will read enough to understand the method of proof, to "visualize the scaffolding" so as to be able to consider whether this scaffolding could be broadened in some way.

If it's surprising, unexpected, or I find it suspicious, I will read deeper into some areas of the proof, basically to figure out where my intuition needs correction.

I will read technical parts of the proof in detail if and only if I feel those technical elements are important to the above aims (e.g. they introduce a methodology that I would like to apply, mutatis mutandis, more so than the actual result, or I can't figure out why my intuition is misleading without digging into technical details); if I feel a sense of responsibility (reviewing a manuscript pre publication, or the paper is surprising and published in a journal where I'm not sure I can trust the review process); or -- rarely but sometimes -- the exposition is so elegant, I don't want to put it down.

(All of this written in the present tense; however, I'm actually no longer active in pure math. However, it describes what I did, what I continue to do in my new area, and what I believe many other math academics continue to do.)

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In my opinion, your question contains a false assumption. You say that it is not about correctness, and that one can safely use the results of peer-reviewed papers, but reality shows that still enough papers are published that contain at least in details some flaws, need extra assumptions or similar stuff, and if this happens you are also in the boat! Hence, if I want to use a result and am not sure that it is correct (may it that I know that close colleagues who I trust have verified it or I know that the relevant experts really have read it) then I check it myself.

  • Mate, you got it wrong, I am not saying peer reviewed papers are flawless, but that is not my concern in this post, I want to know how much professional mathematician read the paper. – Jim Jan 22 at 18:34
  • I think you missed my point ;) As far as I understand you, you proclamate that the central point to read a peer-reviewed paper carefully is to understand techniques for broadening the own understanding or for adaptation. But I say that verification is also an important point (which is ignored too often from colleagues in my opinion). And there is no general measure for this effect, if I work in a rapidly-developing field, there are more new result which cannot be that intensively checked by the community, hence there I´d have more often the need to carefully read the stuff myselfs. – Sebastian Bechtel Jan 22 at 18:44
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    Agreed. If I don’t personally know that the proof is correct, I dare not use the result as a step in one of my proofs. – WGroleau Jan 22 at 19:47
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    @Jim: you’re asking about how much professional mathematicians read papers, and, implicitly, about their motivations for that choice. So an answer saying that correctness-checking is an important motivator for (at least some) mathematicians seems completely on-topic. – PLL Jan 23 at 11:07
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Usually, I read all the paper I cite when I write a paper. However, the word "read" does not have the same meaning in function of the kind of citation. The more the reading of a paper can help to enlarge the vision of a field and improve the understanding, the more carefully I will read it.

One can also not read all the paper he cites in detail. The hierarchy (to my mind) is as follows.

  1. We cite a paper in the introduction in order to talk about more or less closely related works. In this case, one can only read the abstract or the main results of the paper.
  2. We cite a paper in which a quite similar work was one, but using different assumptions and we want to compare them. In this case of course we have to read in detail the results and the examples of applications. And also the idea of proof, which can help to understand more deeply the differences between the other assumption.
  3. We used a result in an other paper P during a proof. The contribution of this paper is not decisive, in the sense that for example, if the result of the paper P is wrong, the proof can be saved without more assumption. Just that the use of the paper P made the proof shorter. In this case, we can read the ideas of the proof to be convinced that it works.
  4. We used a result in an other paper P during a proof. The contribution of this paper is important; for example, if paper P is correct, then one can prove a result under the assumption that a random variable has a finite moment of order two, but without this paper, one need moments of order higher than two. In this case it would be wise to read the proof into details.
  5. We used a result in an other paper P during a proof. The contribution of this paper is crucial, in the sense that if paper P is wrong, then all the paper that we are writing is wrong. Then one should read into details the proofs and also look at the references therein to be sure that everything works well.

Overall, we always have to read with criticism, in the sense that we should not take anything for granted. I mean even a paper written by big names or accepted in a good journal, whatever it means.

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