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I think that the history of this question about connections is unfortunate. I would like to discuss (respectfully, without throwing blame around) whether/how a better outcome might have been achieved.

  1. OP asked, as their first interaction with MO, a clear, well-posed question about connections in differential geometry.
  2. I think that if you looked in the right graduate-level textbook you could find an answer, but also that you could easily look in several relevant graduate-level textbooks without finding an answer.
  3. In the light of 2., I think that the question could have been asked at MSE, but that it is also perfectly acceptable at MO. (The level seems comparable to A and B, which are currently on the front page at MSE, for example.)
  4. The question has 4 upvotes, and there is an accepted answer with 5 upvotes, which seems to me to be clear and helpful.
  5. There were also two votes to close, with one comment suggesting (along with a typographical correction) that the question was "not research level" and therefore better suited to MSE.
  6. I don't agree with that comment, and I do not think that it is consistent with the guidelines to be considerate and welcoming to new users. However, I would accept that this is a matter of judgement.
  7. OP was very offended by the comment about MSE. They stated (in a now deleted comment) that they are a tenured academic giving graduate level lectures on differential geometry, and that it was rude to suggest that their question should be migrated to a "site for helping undergraduates to cheat". They reiterated this view of MSE in another comment that has also been deleted (apparently by a moderator).
  8. I think that most of us would agree that OP's opinion of MSE is inaccurate, and disparaging to many MSE users who we respect (including many people who are active on both MO and MSE). However, it is not hard to imagine how someone might form that opinion; ideally it would be corrected by exposure to more information.
  9. While OP's feeling of offense is clearly enhanced by their low opinion of MSE, I think that it is natural to be somewhat upset at the suggestion that a clear, well-posed, graduate level question needs to be migrated. If people feel a need to make such a suggestion, I would hope that they could do so respectfully and with more explanation.
  10. So OP got an answer to their question, but it seems that they did not enjoy the experience, and are unlikely to return. I find that result disappointing.
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    $\begingroup$ +1 - I was not at all happy with the first comment on that post, but at the time was too busy to do more than upvote the question and upvote the OP's comment, even if I did feel that the "shade" being cast on MSE was unfortunate $\endgroup$
    – Yemon Choi
    Dec 3, 2021 at 11:39
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    $\begingroup$ Just to point out one minor detail here, the account is unregistered. So we don't quite know if the person sitting behind the keyboard was here before, or will be here again. Having said that, I think there is a long-term consensus that graduate level questions, if well-posed, definitely have a place on MO. They also have a place on MSE, but general dictum of SE is "do not migrate on-topic questions". I don't know if making that suggestion was necessarily a bad idea, but it definitely triggered something in this case, which is indeed unfortunate. $\endgroup$
    – Asaf Karagila Mod
    Dec 3, 2021 at 12:20
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    $\begingroup$ @AlexM.: "I would like to discuss (respectfully, without throwing blame around) whether/how a better outcome might have been achieved." - This seems like a question to me. $\endgroup$ Dec 3, 2021 at 16:10
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    $\begingroup$ @AlexM One issue with your approach is that if one person thinks a question is very poor, they can make the question-asker feel attacked and not want to come back and ask another question (likely a better one after getting more experience with the site), even if everyone else on the site thinks the question is reasonable. $\endgroup$
    – Will Sawin
    Dec 3, 2021 at 19:47
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    $\begingroup$ @AlexM. "who is not able to prove an elementary fact about connections" - that would be me, then, since I never learned diff geom properly. Would you be content with my withdrawal from MO, or would you like me to ask Newcastle to annul my PhD? $\endgroup$
    – Yemon Choi
    Dec 3, 2021 at 22:47
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    $\begingroup$ @AlexM.: This wasn't a request for a pf of an elementary fact about connections, but whether there was a conceptual description of something defined by a big mess in local coordinates. When I first learned about the ordinary d operator, the local coordinate defn was totally unenlightening, and it was only much later in my education (after reading many books, and well after getting my PhD) that I learned the coordinate-free formula. I personally have never seen the formula for the extension of a connection to higher differentials given in the answer. Should I give back my PhD and resign? $\endgroup$ Dec 3, 2021 at 23:02
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    $\begingroup$ @AlexM.: This is hardly elementary, and strikes me as entirely appropriate for MO. $\endgroup$ Dec 3, 2021 at 23:03
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    $\begingroup$ @AlexM.: I'm happy for you. I got a PhD in geometry/topology from a reasonably fancy place, and it never came up. Probably people working in differential geometry were taught it, but not everyone. $\endgroup$ Dec 3, 2021 at 23:06
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    $\begingroup$ P.S. I think drawing any conclusions from someone writing $\nabla$ instead of $\Delta$ is utterly absurd - I make this mistake around half of the time, and I have written/am writing several papers on connections. $\endgroup$
    – dhy
    Dec 4, 2021 at 0:00
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    $\begingroup$ This is a typical example where a small initial misunderstanding gives rise to a somehow bitter discussion. Instead of focusing on "who offended whom", I think that the best thing in these cases is trying to keep calm when things have gotten hectic and give people the benefit of the doubt. We should also recall that new users are probably not familiar with the site, so a certain dose of tolerance is always appropriate. $\endgroup$ Dec 4, 2021 at 6:37
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    $\begingroup$ @AlexM I want to say something about "I refuse to believe that ... is an appropriate user of MO." While the question-asker has clarified that she is faculty, just from the original question, I think by far the most likely guess is it was asked by a graduate student. Who reads orange books, spends a long time filling in the details in the arguments there, and wonders if they can be done a better way? All these are more or less famously done by graduate students. $\endgroup$
    – Will Sawin
    Dec 5, 2021 at 14:15
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    $\begingroup$ @AlexM But graduate students also learn math very rapidly! So even if you are certain that no one who doesn't know X and Y should be on MO, if a grad student asks a question displaying a lack of knowledge, in a few months or a year they might know X and Y very well, and contribute positively to MO. But if you shoo them away they will never get the chance. $\endgroup$
    – Will Sawin
    Dec 5, 2021 at 14:17
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    $\begingroup$ @Emil: There is a difference between saying that MSE has a problem with cheating and what was said in that deleted comment. Nobody here pretends that there are no problems on MSE with regards to cheating, but to claim that it's the only thing that is going on that site is offensive to the many of us who are taking part on both communities. You're being very reductive in your comment. $\endgroup$
    – Asaf Karagila Mod
    Dec 7, 2021 at 10:07
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    $\begingroup$ @Emil: well, "devoted to" is different from "sometimes, unfortunately involved with". To make an example, in Italy we had several problems with corruption, but still it would be unjust to say "Italian people are devoted to corruption", as there are many honest people. At any rate, I do not think that "chastisement" is the term here, it was rather a clarification. $\endgroup$ Dec 7, 2021 at 10:19
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    $\begingroup$ @AlexM. I personally assess MO questions not according to how elementary the question seems to me (especially if I'm an expert, because then the risk is particularly high that I will rate the question as more elementary than it really is), but to ask whether I could imagine a colleague of mine who works in an entirely different area of mathematics coming to my office and asking me the question. In part, MO is an attempt to virtually replicate the latter situation. If I put it this way, do you agree that one of your colleagues might ask you such an elementary question about connections? $\endgroup$ Dec 13, 2021 at 18:17

3 Answers 3

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I just want to comment, I think at this point, fighting and unpleasantness about whether questions are appropriate is a much bigger problem for the site than the presence of less-advanced questions on the site. So, I think one way for this unpleasantness to be avoided is for people to just be a little more openminded about what’s an appropriate question for this site. The site will not collapse because you didn’t migrate a borderline question; it well could if we make life unpleasant for all new users.

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    $\begingroup$ I have always advocated for the threshold to be: "is this question something that some particular strong second-year graduate student might not be able to easily sort out?" I still think using this threshold and granting the benefit of doubt is generally a good idea. $\endgroup$ Dec 8, 2021 at 13:51
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    $\begingroup$ Graduate programs are not uniform: they vary, also in a substantial way, from school to school and from country to country. So the only possible discriminant is common sense. And yes, also common sense can be subjective. Thus the golden rule should be be respectful, be nice. $\endgroup$ Dec 9, 2021 at 9:40
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I want to distinguish two different things that were said in the first comment: that the question was not research level and that the question should be migrated to math stackexchange.

It is not really possible to know in advance what opinion the question-asker has of math stackexchange. But pretty much everyone at a graduate level and above is offended when their question is criticized as not research-level, and frequently very offended. This is one of the biggest things that leads people to have a negative experience on MO and leave the site. Thus it is best to avoid saying this except in really extreme cases.

If someone is hurt or offended by an initial comment, and makes this clear, it's better to acknowledge and apologize for this in follow-up comments. If A says something rude to B and B responds by saying something rude to A, then C shouldn't show up and correct only B's rudeness, even in a perfectly polite way. Better to apologize on behalf of A and then correct B's rudeness, or say something nice and give people time to cool down. (One user, Somatic Custard, had good intent here but probably could have phrased it better.)

One way to prevent situations like this is to think about the opinions of others on the site before criticizing. If you think a question is bad but it is reasonably well upvoted, or similar questions have been upvoted in the past, then you disagree with other MO users on what is an acceptable question. If the question gets significantly more upvotes than a critical comment, then you are likely in the minority. Disagreement is fine, and you could be right, but you shouldn't express this disagreement by attacking the person who wrote the question. Imagine walking into a room and having people start yelling at you to leave, even though most people in the room think you should be there.

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    $\begingroup$ Maybe a rough rule of thumb is: it's okay to say "this question is not appropriate for MO" (or the same sentiment but more bluntly) to a question that is obviously at undergraduate level; but not to a question that is at graduate level. $\endgroup$ Dec 3, 2021 at 19:52
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    $\begingroup$ If migration to MSE is the right option, I would avoid completely the expression "not appropriate for MO" (that might appear exclusionary, especially to new users) and instead say "possibly more appropriate for MSE". $\endgroup$ Dec 3, 2021 at 22:47
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    $\begingroup$ I was confused by the strength of the questioner's response, which indicated they seemed to feel told that they were not capable of, or did not perform, research in mathematics, rather than commenting on just one of their questions (whether or not the comment was founded). I am a tenured professor of mathematics, but that doesn't mean that every mathematics question I ask is research level. For example, in retrospect I find Traces and projectors a low-level question that didn't belong here; if someone had told me so, then I wouldn't have minded. $\endgroup$
    – LSpice
    Dec 6, 2021 at 2:03
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    $\begingroup$ That is to say, I believe that any comments—which perhaps have been made here in MMO, but I believe did not appear on MO—to the effect that a user doesn't belong on MO are very much untoward; but I think that one can suggest that a particular question doesn't belong on MO without saying anything about the user, and much of the upset that followed seemed to come from conflating these two sentiments. (Again, I pass no judgement on whether the question was appropriate; I am not a geometer.) $\endgroup$
    – LSpice
    Dec 6, 2021 at 2:08
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    $\begingroup$ This is a thorny issue, at the root of many of the discussions about "the unpleasantness of MO" that we had over the years. Many people react to criticisms about their questions as it was criticism about their person (I was insulted). But discussing if this is a social problem or not is outside the scope of MO. Our priority is discussing Mathematics in a calm environment, so, from a practical point of view, it is better (if possible) to avoid expressions like "does not belong", "it is not appropriate" and so on, and use equivalent formulations less likely to hurt people susceptibility. $\endgroup$ Dec 6, 2021 at 6:42
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    $\begingroup$ @FrancescoPolizzi Yes, given what a high percentage of people react in this way, blaming people for doing so is not really productive compared to adapting our language to account for it. $\endgroup$
    – Will Sawin
    Dec 6, 2021 at 15:07
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I note that frequently users complain about unexplained downvotes and anonymous downvotes. Discouraging comments from users who indeed make a comment of this nature ("this is off-topic") could be, in this respect, counterproductive.

The comment says "Anyway, since it is not of research level, this question is more appropriate". OP called this "rude" and "I was the one who was insulted". I strongly disagree that this comment was rude, nor any kind of insult; still I believe it was peremptory and hope the meaning of "rude" will not be too extrapolated. From the fact that this comment wasn't deleted by moderators I understand that the moderators also didn't believe it qualifies as rude. On the other hand, I found the tone of the OP in the comments excessive (and also peremptory, notably about the scope of MSE), and I'm a bit worried (also as a regular user of MSE) that these comments got so many upvotes.

By the way, and as an analogy, as a referee one often has to recommend rejection of a paper in research journals; as an author one often has to undergo such rejections: one should not view this as exchange of insults, even if the negative assessment turns out to be unfair. Of course here standards are different, but it's the same principle: one has to accept criticism. And here, more than in a journal process, one has room to reply to criticism.

P.S. I also tend to disagree with that initial comment, in the sense that the post better suits MO than MSE: Milnor's book being really advanced level maths, and connections in vector bundles is far from the "common background". But I think its tone is acceptable. On the other hand, I find the comment by the same user (in one comment in this meta thread) "I refuse to believe that a person who does not understand (…), and who is not able to prove an elementary fact (…), is an appropriate user of MO" very excessive. This extrapolates from one question to a judgement about the user. None of us should be blamed for asking a possibly naïve question.

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    $\begingroup$ I suspect what's going on with the upvotes is that people are voting up that comment to express disagreement with the previous comment, as it is not possible to downvote a comment. I didn't upvote that comment, but I sometimes upvote one commenter because I think another commenter is being unreasonable and want to show some support. I don't think there are necessarily 8 MSE-haters. $\endgroup$
    – Will Sawin
    Dec 8, 2021 at 15:05

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