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quid wrote: " Also, I am too not clear in which sense there was such an impressive success for MO."
I'm not too clear on that myself. The best successes on MO, in my opinion, are where someone has a question that they really want to know the answer to because they need it to advance their research, and someone comes in and provides just the insight or solution they need, with a consequent happy ending that winds up in a publication. When MO succeeds in providing an environment for that, it's very deeply gratifying for all concerned, and we can all justly celebrate the very existence of MO.
Here it's something rather different. It looks more like a case where someone has asked something rather casually, and several answers appear which, yes, are very good, but probably much deeper and more intense than the casual questioner bargained for. The number of participants who are really able to understand and make substantial use of the answers is probably pretty small, certainly much smaller than the upvoting would reflect, and very likely the original questioner is excluded from that number.
The answers by Minhyong and Marty particularly are, to a casual reader, beautifully and eloquently written -- a showcase for eloquently written MO answers as it were -- but how they really benefit the masses is not terrifically clear. You could think of them as akin to colloquium talks that are absolutely wonderful for experts but quickly get into things way over the heads of the general audience. You could try to say feel-good words here, e.g., they provide education for a general audience. Well, perhaps. Perhaps to the same extent that colloquia given by experts on deep and beautiful but difficult mathematics are lastingly educational for a general audience[1]. Or, to put it more cynically, they might provide good sources of material to bone up on before a cocktail party, so that you can look smart if the subject of the Mochizuki's work comes up.
Anyway, it's not so unreasonable to me to inquire more deeply why people are upholding this as among the great success stories of MO. (Notice I didn't even touch upon Vesselin Dimitrov's answers, which are different in character and perhaps even more intense, and which have raised real concerns among some MO participants as discussed before.)
[1] There is the obvious counter that unlike most colloquium talks, these answers are written down. Fair enough. It's still debatable the extent to which casual readers are still going to wrap their heads around these answers, and retain them firmly in memory.
Let us have an informal guiding principle that when we have a question which is not offensive, not a big list, and not very soft, then once it is clearly supported by ten (say) 3K+ participants we let it stay open.
What are some more examples of such questions, beyond the ABC question? I've lost track of the overall context, and I'm having trouble thinking of controversial questions satisfying these constaints. I'm probably overlooking a bunch of examples (I haven't really tried looking), and it would be useful to see what the consequences of such a principle would be.
My interpretation of the proposed guideline is that if an inoffensive, non-soft, non-big-list question is closed and re-opened twice (and thus has accumulated ten 3000+ reputation supporters in the re-opening), then it should not be closed again.
I was curious about the statistics, so I just look quickly at all the discussions in the "Is this question acceptable" category with at least ten comments in them. Of course, some of the questions under discussion have since been deleted (so I can't see them), I may have missed some due to sloppiness, and some controversial questions may not have led to discussions in this category or may have had fewer comments. I also looked at the top 20 closed questions by number of votes. Based on this, I'd guess that the ABC question is the only non-soft, non-big-list question that has ever been closed three times, and it ended up being re-opened a third time.
So I don't think the proposed guideline would ever have changed anything. On the other hand, that also makes it seem harmless. Certainly it's no more arbitrary than the current system (where 15 supporters can re-open for a third time, while 14 accomplish nothing).
Regarding big-lists: here it is really importangt to me to keep two things separated: a) Should a list be started. b) When should it be ended. (It happens that eventually even OPs want their own lists closed; so these are really two separate things.)
The 'ten-votes rule' should (if at all) apply to a) only, with a reasonably but not overly long period of validity.
Regarding close-wars in general, they are indeed very very rare but they have occurred a few times. The modus operandi is that after three or four close & reopen, the moderators step in to lock the question in the state they judge most appropriate on a case-by-case basis. We never really reached the point where that kind of intervention was necessary, but we have always had this understanding in anticipation because of precedents on SO. I can only recall less than a handful of questions that went back & forth twice.
@Alexander Chervov: Yes, I did write that. Also, my last name is Trimble.
You'll notice that I said "best successes", and that this was my opinion. My opinion is that these are exemplars of good cases where the people involved really care about both the question and answers. I did not say MO should accommodate only research-oriented questions for purposes of publication.
I repeat, however, that I see problems with the abc question, and I am uncomfortable with one or two of the answers. I won't repeat my reasons.
@Alexander Chervov: the term "exciting" is a bit vague. Personally, I do not need or even want that much "excitement" here. Browsing the daily summary of arXiv (in some subject categories) is also not something that typically creates "excitement" for me. Still I do it basically everyday (or at least for every day) with varying level of detail (depending on the time I have available), as more often then not there is something interesting and sometimes something very relevant for me.
Regarding the main mission: in my opinion this is summarized very well in the first sentence of the FAQs.
MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books.
It does not only say 'writing articles' (ie, direct relevance to active research) but also reading ariticles and graduate level books; so of course somehow general interest mathematical questions are also welcome, and even a very important part of the content. (Yet there is no mention of question one comes across when writing ones CV for instance! Sometimes, they can still be alright, but definitely they are already outside the primary goal.)
But at least to me, as for Todd Trimble it seems, it is important that the OP really and in a reasonable sense cares about the question. What I find frustrating is to answer a question to then find out somehow the person asking it did not really care in the first place (be it on MO or some student I teach or another context) or subjectively cares but it is completely unreasonable. If this should happen on MO too often, then I will stop dedicating some of my time to answer questions here.
Let me give you an example of a recent MO question that I found a good MO question, while it seems not for writing an article (it in principle could be, but it seems not in the present case):
There is a clear mathematical question OP found interesting enough to dedicate some time to look (before asking) for an answer. This was unsuccessfull so one asks on MO. IMO this is a very good way to use MO.
@Alexander Chervov:
ad 1. yes, but that it is hardly written anywhere and as such interesting, was checked beforehand by OP (because he cared). It can happen that people do not check and do not care and get lucky sometimes with their question, like the proverbial blind chicken; still I prefer a site where the default is that people are careful and competent when they ask to increase the ration of 'good' over 'bad' questions.
ad 2. I think one of the more "exciting" times on MO was "Department Closure time"; I almost left just because of this question and some of the reactions to it. ABC was also "exciting"; I did never consider to leave because of this, but took a bit of an almost complete MO-vacation after it (also though not only) to recover from that excitement (you can check my rep history to see the latter). Also some other of the excitement around here to me is as appealing as excitement over the releases of certain new cell-phones or what not .
ad 3. Well, there is something to your point of view, and this helps a bit to overcome occassional frustration. But then, the site is designed to answer some specific person's question; only the questioner can accept an answer and as soon as it is accepted it is the first answer in any view (oldest, newest, votes, regardless you always get that answer first as 'the answer' to the question; so I prefer the person at that button having some interest and competence on the thing being asked).
No problem at all, Alexander.
But I'd like to respond to the sentiment "It seems many people here think -- the main mission is to help publications and all other quests should be closed (strength might be different) but the idea is like this." From what I can see, it's really more complicated than that. For example, one of the most upvoted questions recently is a career advice question (the one with "loneliness" or "lonely" in the title). While the MO appropriateness of this has been queried (being not a mathematical question, hence outside of anyone's expertise around here), no votes to close have been cast, and indeed this (and many other questions of this type) are largely tolerated.
Very different in character is the recent smiley-face question posed by Joseph O'Rourke. Virtually without exception, the questions he poses are well-liked and appreciated (count me a fan as well!). Now, I don't know what his research is precisely, but I would guess that many of his questions are driven by sheer curiosity and not necessarily by publication needs -- those I guess could be a happy by-product. I've never heard a breath of complaint about such questions, even from those who maintain the strictest standards of MO-appropriateness.
You might ask yourself: what do these questions have in common?
It's not that either question was put with publication needs in mind -- and that's not the benchmark, even for the very strict among us. It's more the thoughtfulness and seriousness of intent ('serious' here doesn't mean lacking in humor or morose or anything limited like that). The point is that both Flora and Joseph evidently put a lot of thought into their questions; their questions are expressed with clarity and honesty. In other words, these are questions by people who really want to know; they have thought matters through, considered their questions well, and put care into their formulations. To a very large degree, that's what we want! And that's what almost everyone accepts.
I'll join the chorus and say that I too find MO less interesting these days, and would welcome more questions coming from a background of non-expertise. Gerry Myerson once said words to the effect, "outside of our own areas of expertise, we are all graduate students". It's too bad so many people feel that they have to have hat in hand (asking here beforehand about appropriateness, or assuming by default that they should go to MSE first, etc.) before bringing to MO a question asked at the level of a competent graduate student -- realizing here that "competent graduate student" is a somewhat fuzzy concept. As long as such a one has "done one's homework", makes a good faith effort to think about the question a bit before posting (googling, Wikipedia, etc.), and asks a question with care and seriousness in the sense above, that is, really wanting to know, it ought to be be welcomed here.