In summary, I see MO as a place where research-level mathematicians come when they are in research mode. So that's completely different to when we visit blogs and the like.

Harry Gindi:

I think that the more important point here is the one that Andy originally made about it being a "fishing expedition". It seemed like the question was posted on a whim, just so the user could do something and maybe spark a little discussion. I am against this sort of posting in general. I'm not against all big-list questions (some of them have been quite good), but the good ones were asked by people who put in time, effort, and some thought.

These are probably closest to how I feel. I want to try and help answer specific questions (what I think used to be called SMART targets in UK educational jargon), and personally would like to use MO as a research tool not a learning aid.

As for the question "why not just ignore these questions?", because I have interests outside my nominal speciality, I don't want to set up a large number of filters. (I used to filter out big-list questions, but then found there were one or two "useful resources" or "recommended journals" type questions that I actually wanted to see the answers to.) Then I find the repeated bumping of these questions mildly irksome.

Perhaps a discrete, er, discreet inquiry in one or two well-chosen locations …

in the future, there may be an excellent mechanism for diverting people "there" from MO. If we migrate to StackExchange 2.0 (which we hope too), then we will likely be able to "migrate" questions instead of "closing" them. This mechanism already exists amongst the "trilogy" of Stack Overflow, Server Fault, and Super User, and I'm not sure that it will apply to all the new Stack Exchange 2.0 sites about to appear.

However, I do not see why such a question will drive away even a research mathematician who do not like this question, and I dont see any evidence for that.

I don't think that I've ever argued that such questions will drive away research mathematicians. I've tried to argue that it will drive away me since that is the only data point that I can be sure of. I've also tried to explain why it would do so so that others can decide whether I'm just being unreasonable or I might be indicative of a wider group.

For once, I think that Jon's said something sensible!

If you say enough things with no regard for the topic at hand, you will probably eventually say something relevant.

Then again, Andrew, remember that you only "think" that he's said something sensible. You may be reading too much in to it.

For once, I think that Jon's said something sensible! If we think of MathOverflow as the be-all and end-all of maths on the internet and the only place that we (as mathematicians) can interact, then it will break under the pressure. If we treat it as a part of a larger game then it has a chance to work and to be something really useful.

I don't think that the answer is by partitioning MO (using tags or whatever) as the boundaries are too fluid. Rather, we need other sites that can be used in conjunction with MO. Then everyone can pick and choose the bits that they like according to taste. Big one-stop shops are ugly and frustrating. It may be necessary for me to buy my groceries in a massive supermarket, but it doesn't mean that I enjoy the experience at all. Browsing the shops in a mall, or a quaint English village (with a pub, of course) is a much more pleasant experience.

The argument that that doesn't exist yet doesn't wash. I'm busy building my area of this vast conglomeration, Anton and Scott (and ...) are busy building theirs. If you like what you see, great! But if you think something is lacking then build it yourself! It's a lot of fun and there's plenty of space around.

I've learnt things from reading the answers to these "soft questions", just like the others have. I've even answered a few. However, it's not an efficient way for me to learn things, or an efficient way to communicate my "wisdom" (such as it is). So if that becomes the predominant type of question at MO then sheer economics of time will mean that I'll not come here. There are so many little bits of lore in all sorts of places that missing out on the few buried deep in the junk at MO will not discombobulate me overmuch. Losing the ability to short-circuit the journey from "topological spaces in which singleton sets are G_delta subsets" to "countable pseudocharacter" would be a far greater loss.

In summary, I see MO as a place where research-level mathematicians come when they are in research mode. So that's completely different to when we visit blogs and the like. I suspect that many are coming here when they are in "goofing off" mode and I don't like that because it's distracting. Anyone with kids will have seen this: when all are playing nicely and quietly then it's great, but once one starts acting up then it doesn't take long before they're all at it.

So you're saying that diagonalization is a logical fallacy?

At any rate, observing discrepancies between espoused values and enacted values affords invaluable information toward the design and maintenance of social-technical architectures. When people persistently behave in ways that depart from their touted principles, it may be an indication that (1) there is a problem with their actions, (2) there is a problem with their principles, (3) all of the above.

I can think of a way that this question could have been put which I would have been happy with:

I'm learning some algebra, beyond the normal undergraduate syllabus, and am getting a bit bewildered by all the different types of structures that there are. In other subjects where I've encountered this, such as topology, I've found it useful to have a list of examples showing the subtle differences - often these are called counterexamples. In topology and analysis, there are the classic "Counterexamples in X" books but there doesn't seem to be one in Algebra. So I'm compiling my own. I've made a start at link but, as I'm sure will be appreciated, it's hard for a beginner to find counterexamples for everything. So I'd like to ask if anyone has a favourite counterexample in algebra.

Community wiki rules, of course. And I'll add the answers to my list (though anyone is welcome to add them themselves since it is a wiki) and I hope that it will be a useful resource for others.

Reasons why I'd be happier with that:

1. The person asking explains why they want the list to be made and how it will help them in their current work.
2. The person asking has already done some work towards answering the question themselves.
3. The person asking has thought about how best to use the answers once they've been given.
4. The project is bigger than the question.

Anyway, Andrew has said a number of times that his opinions have changed since he first came here.

It's almost identical to a thread "What are your favorite theorems in algebra?", which is more clearly off-topic.

Transferring a question to your favorite site and closing it here may be aggravating to other people, as well as ineffective.

For example, I am not a member of nLab, I am not familiar with its overall purpose, editing software, or editing culture, and I don't want at this very moment to commit to participation there (I followed the nLab link before reaching this decision). Thus while it will take mere moments for me to contribute to answers at MO, I will most certainly not do it over at the other site. The argument of the type "I spent a huge effort just copying the answers and adjusting, why can't people just do it themselves?" just invites a response: "Why should they do something that you like, and they may disagree with?" Additionally closing the question on MO is unmistakenly dictatorial and this seems like a truly Bad Idea.

MO exists because of good will of its participants and we have to ask ourselves whether alienating people for an illusory and subjective purpose of "maintaining purity" is a worthwhile trade-off (as John Stillwell's remarks indicate, the issue of what's best is far from simple). Andy Putman's suggestion that big-list questions from inexperienced users should not be encouraged is worth listening to: can this be codified somehow? I know that FAQ explicitly mentions it, but as regular flare-ups on meta demonstrate, this, evidently, is not robust enough.

Where I agree with Andrew Stacey is in that maintaining two synchronized copies of the answers is a tedious task. But highlighting the fact that the migration process is tedious, even for an experienced user of both systems like he, only reinforces my impression that things should best be left alone for a while. Any kind of editorial work is tedious, and I can see an advantage in eventually creating an authoritative version of the answers, which is rubricated, supported by citations, cross-referenced, and edited for style and uniformity (just like a real book!). It is short-circuiting the process that I find both short-sighted and ineffectual.

(Not that I've never asked or enjoyed a soft question. But as the site grows I find myself becoming more and more of a "strict constructionist" with respect to the FAQ.)

I knew when I posted that, someone was going to ask about quantum field theory (perhaps in jest)?

Questions like this are just fishing expeditions.

QFT

Quantum field theory??

@Dylan: the closure system is the community in action. I am not a moderator, nor affiliated with any in any way. I've just been around a long time (comparatively speaking) and so have learnt what does and does not work on MO. In order to encourage what does work, then I often vote to close on stuff that I know (from experience) won't work. I'm sometimes more forceful on questions like this one where it's close to the borderline because I know that the obviously bad questions will just get ignored. Ultimately it's selfish - I want people spending their time looking at questions that are directly useful to the questioner, not on other questions. In the spirit of "blatant self-aggrandizement", I'd say that my latest question is a good example of MO working as it should: there was something that I didn't know and that would have been very difficult for me to track down by myself; but to someone who did know then it took only seconds to point me in the right direction. Lots of time saved for me, almost no time lost for KP Hart.

As for how "big questions" and "soft questions" are meant to be used, that's simple: they aren't. They're tolerated to some degree, but if it's felt that there are too many then they get stamped on. Sometimes people post here saying "I think there are too many" and then others will probably go through and vote to close a few; sometimes, everyone just gets fed up with them.

In your case, I'd remind you that I did/do see value in the question! If it could work, I'd be happy with a system whereby someone posted a question "I've started a list of counterexamples in semiquasihemidemi-ring theory; if you have a favourite counterexample, please add it to the list at <link>" which by design would not have any answers itself but which would direct everyone to the more useful place. However, I'm sure that there are flaws aplenty with that which haven't occurred to me right now.

And so to the nLab itself (by the way, I didn't ignore what Gil said; I just didn't respond to it). You're right. It is just for the category theorists. Clearly pages on Banach spaces, DF spaces, Frechet spaces, Hilbert spaces, barrelled spaces, bornological spaces, the closed graph theorem, complete topological vector spaces have no place there whatsoever. And most certainly there is no place for a nice diagram of properties of locally convex topological vector spaces.

Silliness aside, yes I would have moved "counterexamples in analysis" to the nLab. Even faster than counterexamples in algebra! I certainly don't describe myself as a category theorist - I'm a differential topologist if I'm anything. I invite you to take a closer look and learn what the nLab is really all about since from what you wrote above, you don't know yet. (That isn't your fault, I admit we don't do a lot of PR, but then we're too busy actually working there!)

@Campaigner: normally I just silently ignore any post by someone who won't even deign to tell me their name, but that was just so funny! Thanks for lightening the tone.

The overall mechanism of the site results in "broad" and "easy" questions getting lots of upvotes, with "narrow" or "technical" questions getting fewer, solely as a result of the relative readerships! As such, there is a natural tendency for pretty much all of the questions that "we don't like" to receive lots of upvotes --- because these questions tend to be relatively comprehensible to a wide audience. If we want to promote (as we do!) the use of MathOverflow for answering narrow and technical questions, there will always be some tension between voting totals and community moderation.

Moreover, we're strongly committed to maintaining MathOverflow at the "research-oriented" end of mathematics discussion on the internet. Our target audience is at one extreme of potential audience for "mathematics on the internet". If we have an entirely democratic policy, it seems there's just no way to control the slide towards mediocrity. That said, we've always wanted meta to provide the sort of democracy we don't really believe in via the "voting" system, so make your case! :-)

Questions like this are just fishing expeditions.

QFT

There aren't so many algebraists on the nLab

All of you algebraists hear that? We'd love you to contribute. There is a noticeable dearth of algebraic geometers...

MO is more suitable place to ask this question compared to nLab simply because in nLab the question will attract fewer answers and fewer readers.

I would like to say that MO is a more suitable place to ask this question than the nLab, but the nLab is a more suitable place to answer this question.

If I didn't think that this question had any merit whatsoever then I would have just voted to close and ignored it for the rest of time. I've actually spent a fair amount of energy on this question, maybe more than anyone else today!, because I think that the answers to this question are worth putting somewhere better than MO. So actually, I'm not hostile to this question at all and such an accusation shows that I haven't made my case properly.

I don't remember exactly where I saw this, but somewhere on SO is a Venn diagram with wikis, forums, and blogs as the main sets and SO as the intersection. I think that's a good diagram, but I think that I take from it something that the person who drew it didn't intend. Rather than saying that SO combines the best of wikis, forums, and blogs, I see it rather as SO simply combines them. But in so doing, it has to sacrifice some of the strengths of each. This particular question is a clear case where the strengths of a wiki would be an enormous benefit. Has anyone looked at that page that I created? I just copied over the answers; since then, Urs Schreiber has organised it by area and added loads of hyperlinks. So now someone who sees a particular counterexample can click on the relevant links and remind themselves of what the definitions are! (Admittedly, some links are awaiting filling, but then there aren't so many algebraists on the nLab.)

http://ncatlab.org/nlab/show/counterexamples+in+algebra

1. One doesn't want to sort such counterexamples by popularity! One wants a decent index, cross-referenced, and all that jazz. At the very least, a wiki if not a proper database (though I'm not sure how to set such up usefully).
2. It won't get added to after an initial surge, and later additions won't get checked against earlier additions. So it will remain incomplete, but anyone wanting to do it properly will get told 'This is on MO'.
3. As a question, it's not focussed: it's not easy to think of what would make an answer that could be accepted!
4. As a question, it's too broad.

Original question: http://mathoverflow.net/questions/29006/counterexamples-in-algebra