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Firstly, I want to thank the people who have commented on or reviewed my attempted proofs over the years. I'm not a student, so that kind of feedback is invaluable.

I've gotten a little bit better at math, and am starting to answer proof-verfication questions. Although I'm only answering easy ones and I'm not doing it consistently.

On a few of the questions I've considered answering and maybe once or twice after I've given an answer, I've seen comments saying that a proof-verification question should focus on one narrow issue within an attempted proof.

This is different from the formula that I follow for my own proof-verfication questions, which basically consists of a) asking if the proof as a whole works and b) identifying the parts of the proof attempt that I think are weakest, least elegant, or most suspect. Pointing out weaknesses in an otherwise well-thought-out proof attempt is similar to but not exactly the same thing as focusing on those weaknesses only.

I also can't find a meta reference for this, so I'm asking this question in the hopes of getting some guidelines for proof-verification questions. I'd also like to ask that we sharply distinguish questions that are acceptable (i.e. answering them doesn't constitute an EoQS violation) and questions that are good, in such a way that the class of acceptable-but-not-good questions is broad.

My interest in distinguishing the two buckets is partially selfish, since the questions that are easy enough for me to answer typically have one or two problems that the asker needs to address, like insufficient context or bad use of MathJax. I still think it's a good idea to distinguish them though.

Further reading:

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    $\begingroup$ A proof can be used as context for an on-topic question, but lacking any such on-topic question it is off topic, i.e. "is my proof corrrect?" is off-topic. If you peruse older threads you will find that many users think we should abolish the various verification tags. Answers to such often are dupes, or so extremely localized that they are of little use to others. $\endgroup$ Mar 26, 2023 at 19:25
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    $\begingroup$ My heuristic is, essentially, to ask "Is this question asking me to do the work that a TA or instructor would do when grading this problem as part of an assignment?" If the answer is "Yes", then the question is probably off-topic. Hence presenting a proof-attempt with the question "Is this correct?" is off-topic. On the other hand, if it is the kind of question you might ask an instructor during office hours, it is more likely to be on-topic, e.g. "Here is my proof. I think that [this step] is likely incorrect. Can you help me to fix it?" $\endgroup$
    – Xander Henderson Mod
    Mar 26, 2023 at 20:22
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    $\begingroup$ I think it's fine to post proof-verification questions with your formula (e.g. write most of the proof as context and make your question on the part that's troubling you), but know that some people are reluctant to answer proof-verification questions in general. They are long and bothersome to check, which is why you see comments saying proof-verification questions should focus on one narrow issue within an attempted proof. If someone's skilled enough to write a long and interesting proof, then they're certainly skilled enough to verify it for themselves. $\endgroup$ Mar 27, 2023 at 3:41
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    $\begingroup$ @XanderHenderson What is the problem with such questions, if I may ask. Many people do not have access to teaching assistants, since education can be really expensive, and if people are doing it for free, why not? $\endgroup$
    – Nico
    Mar 28, 2023 at 22:28
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    $\begingroup$ @Nico The goal of the SE network is to produce a repository of questions and answers which are useful to a wide range of people, not just the person who asks a question. This site is not designed to grade homework, and would rapidly become less and less useful if everyone posted their work and asked for someone to review it. The site works best when each question is asked once. $\endgroup$
    – Xander Henderson Mod
    Mar 28, 2023 at 22:32
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    $\begingroup$ Someone who is looking for feedback on their work would be advised to find another resource. Math SE chat is a possibility, as are other sites on the internet, e.g. reddit or quora. $\endgroup$
    – Xander Henderson Mod
    Mar 28, 2023 at 22:33
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    $\begingroup$ @Accelerator While your last line is technically true, someone who knows the math well enough to create a proof and check it may still miss an error, possibly even an obvious one. That's why the peer-review process doesn't start at the time of submission to a journal; generally one will have colleagues look at it first. -- It seems to me that, if that's not what MSE is for, then a site for that would be really useful to those who aren't affiliated with an academic institution. $\endgroup$ Mar 31, 2023 at 7:55
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    $\begingroup$ Euler, Gauss, all the great published proofs that were faulty or had errors. Certainly they were capable of finding their mistakes, but sometimes the human brain accidentally overlooks mistakes; hence Rubber Duck debugging and the like. $\endgroup$ Mar 31, 2023 at 8:03
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    $\begingroup$ @EricSnyder I agree that having a place on the internet where people can get their problems checked might be useful. I would imagine that Quora or reddit fit the bill. Math SE's chat can also work. And there is probably a discord server out there that gets the job done. $\endgroup$
    – Xander Henderson Mod
    Mar 31, 2023 at 21:48
  • $\begingroup$ @XanderHenderson Hmm, possibly true. I don't know if r/math would take that sort of thing or not. I wonder if it's too crazy to suggest it as a separate SE site. $\endgroup$ Apr 2, 2023 at 7:19
  • $\begingroup$ @EricSnyder The idea for a proof verification site has come up a number of times. FWIW I think it's a good idea. There's also precedent in the SE network (Stack Overflow vs Code Review). $\endgroup$ Apr 2, 2023 at 17:30
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    $\begingroup$ @GregNisbet Note that a proof verification site is likely to get overrun by people asking about the correctness of their disproof of Cantor’s theorem or their proof of RH… $\endgroup$ Apr 2, 2023 at 17:42
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    $\begingroup$ @Shinrin-Yoku Or, more voluminously, overrun by hundreds to thousands of dupes asking to verify proofs of common exercises that reoccur every semester (which has already begun here). $\endgroup$ Apr 2, 2023 at 18:09

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Stance on proof-verification

I think the stance of proof-verification questions for some time is that on the positive side :

  • They contain attempted proofs and therefore effort from the poster. Hence, they differ from a "do my work for me" question, and many users consider such questions worthy of an answer for that reason.

  • Answers to such questions consist of looking at proof writing, organization of proof, notation etc. with more scrutiny : something that's not typically present in other tags. Learning how to present one's work is an equally important part of proofs, and visitors learn this by observing others' proofs and seeing the lacunae in their work and presentation pointed out.

However, on the negative side :

  • If not tagged properly, such questions can get in the way of an authoritative question-answer pair. Duplicate closure is a problem because it requires people to say "xyz answer in the dupe candidate is the perfect version of OP's proof that they need critiqued" and people are not very confident with that assertion or with the belief that OP will be helped by duplicate closure, particularly if their maturity differs vastly from the maturity of the writing in the answer of the dupe candidate.

  • When the proof to be verified is perfectly fine, one of three things occurs, all of which aren't universally desirable. The first (and probably the safest) is that comments to the effect "this is fine!" etc. are left which, from the archiving point of view, is slightly worse off than a question-answer pair. The second is that people answer (maybe as a community-wiki) "yes , this is fine" etc. which tends to attract downvotes. The third is that people give alternate proofs for the same fact instead of critiquing the proof, which isn't good if visitors came specifically for the proof critique.

  • Finally, if the OP is miles off their proof-attempt (note : I'm talking about basic logic and coherent structure being missing in the proof, or about theorems used which completely lack context etc.) or if the proof is utterly huge, then the question is probably too broad. Also, users have a tendency of not pointing out specifically incorrect steps, but instead presenting their proof and saying "is this right?", which also leads to very broad questions.


On improvement in writing and attitude

Personally, I'm not comfortable with the removal of the tags itself, that seems too strong as an action. However, I am comfortable with demanding more of such questions. Here's what I would suggest helps people write better proof verification questions :

  • It is difficult for OP's to find mistakes in their work : that's natural to expect. Therefore, in the event that the OP believes that none of their reasoning is shaky, they should mention details that allows experts in presentation and notation to take over and provide quality answers. For example, do they feel that their notation, in parts, is cumbersome? Do they think that a couple of expressions are long and winded and can be simplified? Then those parts should be pointed out specifically. To put it in a nutshell, there should be context that aids the answering of "if the proof is correct, then what exactly can be improved in xyz parts of the proof vis-a-vis presentation, notation and organization of the proof as a whole?".

  • Another important part of proof-verification is : where you write your proof also matters. For example, if your aim is to exercise the use of a particular theorem because the question appeared in a section of a book which has that theorem as part of its contents, then the proof should contain a line to the effect "xyz theorem is applicable because its hypothesis abc and def are satisfied by equations ... and ...". Another example is in academic papers, where a proof need not be written in much detail but parts of it can be skipped or cited. That context helps users with expertise deliver more quality to the OP's doorstep.

  • Users have to continue to look up similar problems and provide evidence for why their proof is unique : otherwise, duplicate closure should be in order. Another "prove that $\frac 1n \to 0$" question should not stick because there is a huge chance that it has been reproduced authoritatively in a q-a pair. Users wishing to clarify their proof are clarifying more than just the mathematics, it's also about the structure , organization and notation, so if they haven't seen similar questions or proofs and tried to copy the structure of those proofs, they aren't doing a service to themselves or to the site.

Here's how I think attitudes apropos proof-verification questions need to change :

  • For questioners, there has to be a desire to not just know if some proof is correct, but in fact, how it can be the best version of itself, and how they can improve their own writing skills. Questions phrased as "here's my work, is this right?" don't show that ambition. Questioners wish for quality from answerers, but that starts with demanding improvement from themselves.

  • Answerers have to work harder to use their expertise, and not just work for the points. They should go through the OP's context, the notations, the organization of the proof, the presence of complicated expressions which can be simplified, and the likes. They have to present quality. This is personal, but I don't think we should entertain answers containing alternate proofs, particularly if those answerers don't even attempt to contrast the OP's proof with their own in a fine manner. Basically, answerers have to feel that their time is valuable and dedicate themselves completely and properly to excellent questions : that's how quality is delivered.

  • For curators, I believe they have to take a slightly harder stance than the present on questions that merely contain work and "Is this okay?". Alternate forms of context, or a finer analysis of possible lacunae in the proof from OP itself should be demanded. If this is not possible then closure should be in order, although people should feel free to help out in the comments to improve the question.


On "good", "acceptable" etc.

I wrote , earlier, three bullet points for OP's of proof-verification questions : mention the finer points in your proof that you think require more scrutiny, mention the source of your problem and what your proof is going to be written for (exam, paper, article etc.) and contrast with the existing corpus of proofs. Here is some more delineation :

  • A large proof's verification should almost always be closed as broad.

  • Any non-broad question that does at least two of the above satisfactorily and , if possibly a duplicate, does a satisfactory job of contrasting the attempt with the proof in the dupe candidate, counts as a good question in my eyes.

  • Any question that doesn't do any of the above and admits an authoritative question-answer pair is not acceptable (with emphasis : improving such questions generally takes away more time than finding the next good question). Any question that doesn't do any of the above but is not a duplicate (example : work + "am I right/is this fine?") is not acceptable but can be improved with OP made to point out problematic parts.

  • Any question that goes a little further than "am I right/ is this fine?" by partially satisfying one of the requirements above is , in my opinion, acceptable. For example, consider "work + "am I right/is this fine" + "we were asked to use Taylor's theorem by our instructor/ this can be found as Exercise 7.12 in Chapter 7 of Rudin". A good question would go further : "I tried to use Taylor's theorem rigorously and I would like to know if I got the hypothesis check and usage correct" or "I have seen xyz tool in Chapter 7 of Rudin and would like to know if its usage can be simplified".


Conclusion

Our aim, conditioned on the premise that we are not entirely removing the proof-verification tag, is to maximize the benefit that questioners and future visitors receive from the website as a whole, and not merely those questions. Keeping this in mind, the entire community has to work together to raise the quality of such questions and the organization of the rest of the site conditioned on such questions being present. If, by way of experiences or evidence, such actions are found to not scale or are simply untenable, only then would I consider the removal of the tag as an option.

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    $\begingroup$ "Hence, they differ from a 'do my work for me' question, and many users consider such questions worthy of an answer for that reason." Sure... instead of "do my homework," they are "grade my homework." :/ $\endgroup$
    – Xander Henderson Mod
    Apr 6, 2023 at 13:46
  • $\begingroup$ @XanderHenderson But "grade my homework" helps the OP learn, much more than a spoon-fed answer. It helps future visitors learn because they see OP's mistakes and don't want to repeat them. These arguments have been hashed out before and I know the counterarguments (my stance is that SV-questions need to be of higher quality, as you've already seen). I believe that intangibles (such as "how many people are helped/hindered by the existence of the SV tag") will not let us have a constructive conversation on the SV tag. Conditioned on that, raise the quality is my call. $\endgroup$ Apr 7, 2023 at 19:01
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    $\begingroup$ It might help the original asker to learn, but that is not what this site is about. This site is about building a repository of questions and answers. Most "grade my homework" questions are too local, in that the only person who is likely to be helped by the question is the original asker---even other askers with exactly the same mistake or concern are unlikely to ever find the question. "Grade my homework" questions are not a good fit for this format. For those kinds of questions, there are other resources: SE chat, reddit, quora, your TA, etc. $\endgroup$
    – Xander Henderson Mod
    Apr 7, 2023 at 20:06
  • $\begingroup$ @XanderHenderson If the community were strongly supportive about the idea of building a repository, then why has the SV/PV tag been hanging around for so long? I am making these suggestions conditioned on the premise that the tag is going nowhere. If you strongly believe that the tag can be removed, then call a vote (like the homework tag) and get it done. If that doesn't work, then arguments against its removal won't involve repository building for sure, in which case, not everybody is here to build that repository. $\endgroup$ Apr 8, 2023 at 6:00
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    $\begingroup$ You seem to be reading something that I did not write. I said "grade my homework" style questions are not okay. It is very possible to write a solution-verification question which is not in that style. math.meta.stackexchange.com/questions/35592/… $\endgroup$
    – Xander Henderson Mod
    Apr 8, 2023 at 12:43
  • $\begingroup$ @XanderHenderson I agree that I was reading too much into what you wrote, and see your point. That is what I was indicating in the section on improvement tips and attitude changes. Having said that, asking an SV question requires an OP to attempt a proof of a question. A comparison which I've seen made on this site is that an SV question helps the OP learn better than a "do my work for me". That's not my opinion but is considered a "positive", which is why I put this point there. $\endgroup$ Apr 8, 2023 at 13:57

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