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    • CommentAuthorYemon Choi
    • CommentTimeDec 12th 2010

    See here. Although I wasn't one of those who voted to close, my initial reaction was that the question was too "subjective and argumentative", or more suited for blog discussion. Since I see that the question has attracted some impassioned defence and a vote to re-open, I thought a meta thread should be opened for further debate/discussion/clarification.

    • CommentAuthorWillieWong
    • CommentTimeDec 12th 2010

    Ditto. I don't think that question is meaningful. If you ask ten mathematicians working on ten different problems, they will probably give you ten different answers to this question. For a list of what people find intricate and beautiful, why not just read the "What is..." section of the AMS Notices?

    • CommentAuthorYemon Choi
    • CommentTimeDec 12th 2010

    In fairness, the question is about objects rather than topics. But even then this seems the sort of question which will attract some over-eager answers. (Which also seems to have happened to one of Gowers' recent questions, but that's another debate...)

    I was one of the people that voted to close. Beyond being subjective and argumentative, my problem with the thread is I don't see any upside to any possible answers. At best an answer hypes a field or and object of study in some field. As you can see from current responses, not only is this exactly what's happening but it's happening in an entirely uninformative and uninteresting way. Wikipedia is a more useful source of descriptions of mathematical objects.
    I think the question is more interesting than the answers it has attracted so far. I wish people had focused more on the "Are these sensible statements?" part than rushed to post about what they think is cool. If it does get reopened, I think reformulating the question might help.
    • CommentAuthordeane.yang
    • CommentTimeDec 12th 2010
    For me, any question that provokes thoughtful and interesting answers is worth keeping; I don't like a lot of the rigid definitions of MathOverflow that I guess are in its FAQ and enforced by many. I think this question had potential value, but the answers posted so far fall far short of the original example, $E_8$. I was just about to post an answer expressing the view that the exceptional Lie groups epitomize the combined qualities of "intricate" and "beautiful" better than anything else I can think of, but the question was closed. Given the quality of the answers so far, I have no objection to leaving it closed.
    I'm not sure that this question can be considered "subjective and argumentative" just because it include the words "arguably" and "perhaps", as Pete L. Clark observes, for after all these words are in a quotation from a paper in *Scientific American*, and reflect by no mean the opinion of the OP.
    Pietro, I think you might be misunderstanding the point of Pete's post. The subjective and argumentative part of the post is the main question: "What are some other candidates for the most intricate structure and for the most beautiful structure in all of mathematics?"

    I'm not at all happy with the gestapo references in the comment thread.

    In fact, I'm not at all happy with the entire comment thread. I feel like I should (copy & paste it to here,) delete it, and email everyone involved to say "You should have started a thread on meta, these comments were inappropriate."


    Let me confirm that Ryan Budney's interpretation is the one I had intended. I'm sorry for any misunderstanding.

    Indeed the main question that Ryan has quoted above is as subjective a mathematical question as I can imagine: surely no one thinks that a question about mathematical beauty has a single, definitive answer?

    As others have pointed out, there may be a MO-appropriate question in here with regard to the Scientific American paper itself. I myself would be interested to know what purely mathematical claims this paper makes and to what extent they can be justified. This is still not quite on-topic for MO, but it's getting there...

    Added a minute later: while I think the merit and the appropriateness of the question is up for debate, I don't feel the same way about the answers that have been given so far. So far as the answers are concerned the question may as well have been "Tell me some of your favorite mathematical objects". Thus I invite people who think the question is a good one for MO to suggest changes that would elicit more useful responses.


    Personally, I thought that Tim Gowers's question was equally argumentative and equally unlikely to give any interesting answers, but there wasn't even any debate about its appropriateness, and it keeps getting up-votes. So, while the present question definitely seems argumentative to me and not very likely to generate good responses, in a precedent-based system it would have to stay open. Or is there a fundamental difference between the two questions that I am completely missing?

    Let me add some general idea to the words said in the original Richard's topic.
    I think that a priori each questioned should remain opened, and only some obviously unappropriate questions (spam or homeworks, for example) should be closed.
    the main aim of MO is being helpful to professional mathematicians, and if at least some of them find the question helpful,interesting, meaningful etc - why should we think about closing it? My answer is: if it is not only helpful, but bad in some sense - say, concerns private life of real people, or is offensive and so on. This question definitely is not of that kind. Advices to post it on the blog instead are strange: MO has much wider audience, then any blog. And it makes sense to have all good questions in the same place.

    But Fedor, "being helpful to professional mathematicians" is not the main aim of MO.

    MathOverflow is for questions of interest to research mathematicians that admit definitive answers. These are generally going to be technical questions, but those that aren't should be formulated carefully. The software is not designed for hosting discussions, and whether or not such discussions are fun, or helpful or interesting, they belong elsewhere.

    I don't think it's a very good question, beauty being in the eye of the beholder. Its theme seems ill-suited to this site. So if it were still open, I would vote to close.
    • CommentAuthorWillieWong
    • CommentTimeDec 12th 2010

    @Alex: it has been argued many times before that MO closing of question is not precedence-based. Different community standards at different times, different people being awake with voting powers, and other reasons conceivably contribute to different receptions of the question. To borrow a phrase from Walt Whitman:

    MO contradicts itself. MO is large. MO contains multitudes.

    If you want MO to be completely consistent in these voting matters with regards to precedence, you'd probably need to have all 3K+ users take a course in jurisprudence...

    @Scott: I would support that. I don't quite like the name calling.

    @All: I largely agree with Pete on why I think the question is not a good one. But in view of Deane and Thierry's comments, if the question was rephrased as a narrower question about exceptional lie groups, or one with a longer, more objective list of criteria, I may be convinced to support it being re-opened or posted anew.

    • CommentAuthorYemon Choi
    • CommentTimeDec 12th 2010 edited

    It seems that the question is on the verge of being re-opened. I still think that the question is likely to attract ill-thought out examples (many who think "maths is cool!" have heard of "E8" or "The Monster", but I suspect comparatively few have been introduced to $\beta N$ or the quasinilpotent DT operator). Note that the question asks about intricate objects, not profound ones.


    The current answers come off as pseudo-profound hogwash.

    • CommentAuthorYemon Choi
    • CommentTimeDec 12th 2010

    @Harry: I wouldn't say they are hogwash, but at least one of them seems to willfully ignore the usual meanings of the word "intricate".

    • CommentAuthorHJRW
    • CommentTimeDec 13th 2010


    For me, there's an important difference between the question under discussion and Tim Gowers'. The phenomenon that Tim wants examples of is technical, even if it's not entirely well defined. It is possible to produce evidence that a certain proof 'requires a fundamentally new way of thinking'. But what kind of evidence are you going to produce that a given mathematical object is beautiful? From this point of view, Tim's question was, in an important sense, less argumentative than this question. (I agree that Tim's question didn't elicit, to my eyes at least, many good answers.)


    Henry, I am actually quite intrigued because I suspect that I am missing something that at least 50 other people see. Is there any sensible way of distinguishing between fundamentally new ways of thinking and "merely", say, vast generalisations from a special case? I simply don't see how to make the main words in the question precise in any meaningful way. No idea comes out of nowhere and usually, even the most innovative idea must have been suggested by some hint, a hunch, a heuristic, a vague parallel. I see neither how to confidently assert that a way of thinking is "fundamentally new", nor the use in such a distinction.

    If anything, then it seems easier to me to confidently say "this mathematical object is more intricate than that object", at least in some cases. E.g. I guess, noone will dispute that any sporadic simple group is a more intricate objects than a finite cyclic group.

    • CommentAuthorfedja
    • CommentTimeDec 13th 2010
    In response to the last paragraph in the last comment, I've heard that some young and extremely talented mathematician (he is very well-known now) came to visit Bernstein and talked to him for over two hours on how beautiful and intricate the recent (as of then) discoveries in the theory of analytic functions were. Bernstein answered with just one phrase: "You know, as for myself, the function $\sin z$ is still so full of mysteries!".

    This is not to say that there is no objective criteria for "beauty" or "intricacy", just to say that our field of sight is usually so limited that we are hardly any better than those 7 blind man who commented on what an elephant is like. Now imagine the same 7 blind men to start a (civilized and logical) discussion of whether the elephant is beautiful or not.

    As to Gower's question, yes, there is a "sensible way of distinguishing between fundamentally new ways of thinking and "merely", say, vast generalisations from a special case?". One of such steps was done somewhere between Thales and Euclid, when the concept of mathematical proof was born. I don't know the name of the person who was the first to think of this possibility to use pure logic to deduce valid non-trivial conclusions about the real world but he definitely didn't generalize from a partial case. Of course, some such "new ways of thinking" took more than one person and more than one year to arise. Also, not every example one can give here will look as majestic as the birth of mathematics. Still, the distinction is there though I would hate to construct a Turing machine that takes the idea description as an input and produces "new way" or "generalization" as an output. But of course, every time this happens, it is a major event that reverberates through centuries (from Galois to Banach) and the answer here is in front of everyone's eyes. The only problem is that, as pater Brown said in one of Chesterton's stories, "the thing you look for may be not too small, but too big to notice".
    I leave the question of whether or not the question should be reopened to more experienced members of MO, but as a physicist I find the motivation from the Sci. Am. article on Lisi's E8 theory unfortunate since most professional physicists don't think the model makes much sense, and to the degree it does make sense, it is not consistent with what we know about the structure of the Standard Model.
    • CommentAuthorgilkalai
    • CommentTimeDec 13th 2010
    It may be the case that one of the things we will learn (or some of us will learn) from Gowers's question is that the notion of "fundamentally new way of thinking" is problematic. This does not mean that the question is not a good MO question, on the contrary. One of the successful outcomes in studying a question is to realize difficulties that the question raises.
    • CommentAuthorHJRW
    • CommentTimeDec 13th 2010


    I think fedja answered your question beautifully. For my own part, I can't imagine what evidence you would cite to suggest that one mathematical object is more or less beautiful or intricate than another. On the other hand, I can imagine what evidence you would cite to argue that a certain proof required a new way of thinking.

    No idea comes out of nowhere and usually, even the most innovative idea must have been suggested by some hint, a hunch, a heuristic, a vague parallel. I see neither how to confidently assert that a way of thinking is "fundamentally new", nor the use in such a distinction.

    With respect, this is just positivist silliness. Working mathematicians talk about new ideas all the time. For instance, I feel fairly confident in asserting that Hamilton had the idea of using Ricci flow to prove the Poincare Conjecture. If I were told that Hamilton got the idea from someone else, then I'd revise my assertion about its attribution, but it wouldn't change the fact that it was a new idea.


    I am in complete agreement with Jeff Harvey. In fact, I find remarkable that Scientific American would have devoted an article to this work.

    • CommentAuthordeane.yang
    • CommentTimeDec 13th 2010
    I'm still with Gil on this. I think the assumption that I disagree with is that one can tell in advance whether a question has good well-defined answers to it or not. It certainly isn't obvious whether anyone can make a convincing argument that a certain object is more beautiful and intricate than anything else, and I concede that it seems unlikely that anyone can. But I'm all for letting people try anyway. If after enough time and enough tries, we are all unimpressed, then we can downvote the answers and close the question.

    This question and the other one about object whose study amounts to a subdiscipline seem to suffer from similar problems. Some of them have already been identified, but here's one that annoys me: there are a lot of mathematical objects these days which have some definition like "the space of all widgets" and it's extremely unclear to me where the line is dividing "the study of the space of widgets" and "the study of widgets."


    I would like to lend my moral support to people who voted to close. I think MO needs a "closing population". Of course, one can argue that one or two soft questions a day does no harm, and one great answer by Terry Tao outweights 20 mediocre ones by Joe Maths. But without the closing population, the site would be overrun by those questions, since they are much more popular, understood and appreciated than the technical questions, which are the main purpose of MO.

    (No surprise why these questions are more popular: within 30 minutes of this question being reopened, 3 answers have appeared, none of them have any justification, and frankly even I can come up with answers like that.)

    I think that ultimately, mathematicians thrive on answering technical questions within their expertise. Talking with one or two prominent people in my field about why they are not on MO, certainly one of the reasons for their lack of interest is the (their impression) lack of quality questions they would like to answer. Soft questions add to that impression.

    Finally, one more reason why I have a lot of sympathy for the people who voted to close on questions like this one. They chose the difficult, unpopular choice, sometimes against the opinions of more senior colleagues. Quite often, their votes are met with abusive languages, I have seen them compared to "Spanish inquisition", "moral police", "Gestapo", and being accused of "against the advancement of mathematics", just from a couple of recent threads. They are the ones who have to come to meta and spend a great deal of energy to defend their vote, and I found their arguments professional and carefully constructed.

    Yes I agree, and I think MO is suffering to some (small) extent from not having enough members willing to stick their neck out and make unpopular choices, especially when a "big name" poster is involved, like this case.
    • CommentAuthorgilkalai
    • CommentTimeDec 13th 2010 edited
    I agree with Ryan that this makes a small difference. So if we are in disagreement about the sign of the difference we still have to remember that this effects MO only to a small extent. The effect on MO's individual users is even smaller since users can filter out questions they do not like. On the other hand I do not understand some other statements by Ryan. I do not see what is the point in making unpopular choices regarding closing. This is precisely the place to be tolerant to views of others and certainly to the popular opinion, and not to be overlly ideologic. Also I do not see in what sense people are "sticking their necks".

    Certainly strong language is inappropriate and offensive language should not be permitted. I propose zero tolerance to offensive expressions and suggest to have offensive comments deleted.

    Some arguments by some closing advocates do not take into account the small stakes. Statements asserting that if MO will move in this direction or in that direction will mean "the end for MO" are apparently effective but unjustified (I worry even more when I hear such statements from politicians.)

    There are a few places where happy-closing tendencies make some difference that MO is suffering from. One is questions in applied mathematics. Those very rarely have "unique answers" and the wording of the faq can be read as discouraging applied math. (And the practices of some of our closing members confirm this.) Also I dont think it is a good idea to quickly close a question (say as too elementary) when it is not in your area of research.

    Apart from the sign of the derivative, I agree with what Gil's just written. I know that I use strong language (meaning, that I express myself strongly, not - I hope - that I use offensive language) here on meta but I hope that I don't come across that way on MO. And whilst I try to express my point of view forcefully (as I hope to convince others of it, otherwise why would I bother?), I do try not to go over into hyperbole. When I talk of things like "the end of MO", I mean (and I try to remember to say this) that MO would become in practice useless for me and so I would leave. I try to justify such statements, and so not sound like an opinionated politician. But sometimes I find myself making the same point over and over again, and sometimes I forget to put in the technical details in the repetitions.

    (I'm strongly tempted to "write up" my thoughts so that I can just put a link to them each time.)

    On to Deane's last post:

    I think the assumption that I disagree with is that one can tell in advance whether a question has good well-defined answers to it or not.

    which - to me - completely scuppers the hypothesis that "good answers make good questions"!

    I would like to elaborate on a point that Gil made: besides being disproportionately under-voted (I have in fact run the numbers on relative votes per question on arxiv tags), applied questions also seem to be closed with greater alacrity. We would do well to collectively take more care in closing questions in areas that are clearly under-represented both on MO and in the expertise of its participants.
    • CommentAuthorHailong Dao
    • CommentTimeDec 13th 2010 edited

    Gill, Steve: the point about applied questions is certainly valid, but I think it belongs to another thread. I was writing about big-list questions which are often popular. Personally, I do not get involved with technical questions in other areas.

    And of course we need a "reopening population" too, for check and balance.

    • CommentAuthorYemon Choi
    • CommentTimeDec 13th 2010 edited

    I think Steve's point (or his elaboration of one of Gil's points) is a good one, and I personally have much more time for applied maths questions on MO, even if I can't contribute to answering them, than for questions such as the one under discussion.

    As will be seen from some of my comments to answers on that question, one reason I am pessimistic about the quality (rather than the shininess) of the answers is because people seem to be ignoring the word intricate that was used in the main question. Perhaps I am too hung up on what words mean - that is, like, so outmoded, man - but to me "intricate" does not mean the same as "profound and fundamental", or "capable of generating complexity" - it means something like "having fine or delicate detail, or complex internal structure".

    So, to use an example I've mentioned upthread and which someone else mentioned in answer to the original post, the semi-topological compact semigroup obtained by taking the Stone-Cech compactification of the natural numbers and equipping it with the induced "Arens-type" multiplication, would qualify as "intricate", and according to authors such as Hindmann and Strauss would be considered beautiful. The natural numbers themselves might be beautiful, but I think the case remains to be argued that they are as intricate. As for claiming the empty set is intricate, that just baffles me.

    • CommentAuthorYemon Choi
    • CommentTimeDec 13th 2010 edited

    By the way, my thanks to those who have come on this thread to defend the original question or argue for its re-opening, even if I don't agree with you. (Have there been five here? it seems like fewer.)

    • CommentAuthorEmerton
    • CommentTimeDec 13th 2010

    Dear fedja,

    Thank you your eloquent posting above.

    Best wishes,


    • CommentAuthordeane.yang
    • CommentTimeDec 13th 2010
    Andrew Stacey, I didn't understand your comment about my comment! Please explain. (I apologize if this is due to my own senility)
    • CommentAuthorsean tilson
    • CommentTimeDec 13th 2010 edited
    It is very important to have a "closing population" for precisely the reasons mentioned by Hailong! As a student, I appreciate thework that you guys do to keep the site clean and fun. It seems that no one has suggested that the questions be moved to a more appropriate forum such as Math.SE. It seems to me that there is only one reason to ask a question such as the above on MO rather than MSE: the audience. When I post a question there rather than here it is because I think that is a waste peoples time, to trivial etc, and that I think there is somewhere there that can answer it. I think these questions are more argumentative than people want for MO, and there is no reason to keep them here (other than the votes) when there are venues like MSE. Initially, there won't necessarily be the same quality of answers, but if people know that the questions have migrated, than wouldn't they add MSE to their to do list?

    I am curious of peoples thought on this. I initially hoped that MSE would be a place where certain questions inappropriate for MO could live, but I have found it to be more enjoyable than that.

    I am certainly interested in the answers to the above questions, but I can understand why they are not appropriate for MO.
    • CommentAuthorgilkalai
    • CommentTimeDec 14th 2010
    I am happy with the surprising emerging agreement with Andrew. Of course, in all our disussions, we only just scratched the important question if good answers make good questions. So I am not sure we have yet good answers to this (good) question. Be the answer as it may be I think the only people who can support the idea that questions can be judged entirely on their own merit are mathematicians. There are plenty of places where a question (any question) is just a ceremony to enable the answerer to say whatever he or she wishes.

    There are various ways to try to approach this problem. By analogies perhaps (e.g., do good students make a good teacher?). I really think that in real life mathematics, good answers make good questions (and also good answerers make a good question). Even if you do not believe it as a fundamental law you may still agree that in cases of uncertainty you can infer from the quality of answers to the quality of questions. You may also agree that a good question need to take into account the potential answerers. I am willing to agree that there are some questions that are so hopeless that no good answers will help them.

    Of course, an MO item consists of a question + a few answers + edits+ comments and we are interested in the merit of such combined enterprises more than the quality of their parts.