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I think that having a list of counterexamples in algebra is a Good Idea. I don't think that MO is the appropriate place to host it. Here's my reasons:
Original question: http://mathoverflow.net/questions/29006/counterexamples-in-algebra
Having just finished grading a load of papers, I felt inspired to act on my own recommendations:
(PS If you don't have anywhere better to put your "counterexamples in AG", why not try the nLab?)
Arguing "But question X wasn't closed" is a Bad Idea with me. My reaction is usually to go back and vote for the others to be closed as well! After all, I've clearly decided right now that I don't like the current question (as an MO question, let's be clear on that) so saying that certain others are like it is more likely to make me think that those others aren't good MO questions than to make me revise my current opinion.
Exactly how will the answers to this question be useful to students? Compared to, say, a wiki page such as this one. Let's be clear on this: I am definitely not saying that this question has no point to its existence whatsoever. I am saying that it doesn't make sense for it to be on MO when there's a better place for it to be. Often when I vote to close some discussion question (like the "should π be 2π?" one) then I get the plaintive wail, "Where else should I ask this?". Most times, I just shrug and say, "Don't know, I just know it shouldn't be here.". Here, at last, I can actually say where it should be!
(Addressing Gil's point). Having copied over most of the answers, I can attest to the fact that it would be much, much better if people added their counterexamples directly to the nLab page. It would be a pain to click on each question, view the source, cut-and-paste, modify for different syntax, and repeat. In fact, I just rewrote each question but whilst that was okay for about 20 answers, it would rapidly get annoying for more. Far, far easier if everyone adds their answer to the right place first off - no more work for them, no additional work for anyone else.
@Akhil Mathew: I voted to close all of those too. Just because some of us are inconsistent doesn't mean that we now have to leave open every big-list question. Imagine getting arrested for something and telling the judge "but one of my friends did it one time and the cops let him go, so come on..."
I'm going to ignore most of what Gil's just said (my blood pressure won't take it) and focus in on one very specific aspect:
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.)
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...
Questions like this are just fishing expeditions.
QFT
@Dylan, and others who wonder why we close questions.
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! :-)
@Charles: many apologies! I should have checked the page history. Urs told me (via the nForum) that he'd edited the page and I assumed that that was the next edit after mine. Mea culpa, and welcome to the nLab!
@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.
Please do! If you're not sure about the nlab culture, just come to the forum and ask!
Questions like this are just fishing expeditions.
QFT
Quantum field theory??
Quoted for truth.
I knew when I posted that, someone was going to ask about quantum field theory (perhaps in jest)?
I think what Andrew was trying to get at is that soft questions are missing the point of MO, which is (to borrow a buzzword from Michael Nielsen) to redirect expert attention. The fact that people learn a lot from soft questions doesn't invalidate the fact that they are somewhat contrary to the basic purpose of the site; if anything, the fact that soft questions are so popular means they take up space on the front page and make it harder for experts to read the questions they can answer. Of course, there are experts like Terence Tao who excel at and enjoy answering soft questions, but they are rare exceptions!
(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 am in agreement with Gil Kalai both in that "such a question can be used as a resource for doing it on another platform" and that "It is also not reasonable to vote to close a question simply because one feels that there is a better platform on earth for this question". Let me state up front that when I saw this particular question, my first reaction was: "Oh, no, not again!". While I recognize its potential merit, I am not particularly biased in its favor. Andrew's desire to find the best forum for the question is praiseworthy and his effort in setting up a hosting page is acknowledged. However, there is an important issue of unintended consequences to consider.
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.
I would have voted to close because as Andy said, this question seemed like a fishing expedition. It was very much like a "I'm not particularly interested in anything in particular, so everybody write down a bunch of crap I can read!"
It's almost identical to a thread "What are your favorite theorems in algebra?", which is more clearly off-topic.
Ive been a bit busy to formulate a coherent contribution to tis thread, but for the record my first impulse/inclination is closest to Andrew Stacey's than to others. I am not instinctively sympathetic to the reasoning that "now MO is more popular and has reached a broader audience, it should cater to them" - going back to Andrew's own oft-repeated point that MO can do one thing very well (bringing a specific question to the attention of people for whom it is easily answerable) which is not done well elsewhere, and it is something of use to research-active mathematicians. I don't really want it to be some kind of "portal/gateway for the mathematical community", although that may not be what people are suggesting it should be - forgive me for any misreadings caused by haste.
In the light of what has been said above, it seems that the motto ensconced in the “welcome” box may be logically ambiguous. “A place for mathematicians to ask and answer questions” leaves open the interpretations (1) that people other than research mathematicians are welcome to ask questions of mathematicians and (2) that people other than research mathematicians are welcome to answer questions asked by mathematicians. Perhaps a stricter grammatical construction would serve to close out these misinterpretations.
@Dylan: Tu quoque
Anyway, Andrew has said a number of times that his opinions have changed since he first came here.
For the record, it's really Anton's oft repeated point that MO should do one thing and do it well. It took me a while to come round to it, as witnessed by questions like "Theorems for nothing" (which I've just voted to close, by the way, since I would not ask it now), but I'm now fully convinced that he's right. As I changed my mind on this issue, I am perhaps more vocal than others who were in agreement with Anton all along. It's worth all of us remembering that these issues have been discussed before, so those to whom they are fresh should be aware that the others of us have done so, and those of us who have discussed it before should be aware that the newcomers haven't!
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:
Et tu, tu quoque?
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.
-1 Jon Awbrey. I never know what the heck you're talking about.
@ QFT (Quantum Fishing Trip?) — Not to worry, someday there will be an app for that.
I am a research mathematician who read the answers to the question under discussion and learnt something from some of them, so I am happy that it was asked. As John Stillwell already remarked, one shouldn't presume that these kinds of questions are disliked by all participants. As with many things, some like them, some don't.
It seems to me that much of the continuing dissonance at MO Math U is really a problem of software design, in particular, the fact that we are using a 1-page attention buffer, in effect, a 1-room schoolhouse model. Try to imagine your average math department scheduling all of its diverse and sundry graduate seminars in the same room at the same time. How could anything but gnashing of teeth and grinding of gears be expected to result?
@Andy: you're worse than a politician! Agreeing with the last person who spoke!
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.
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.
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.