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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.
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?
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'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?
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.
@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.
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.
@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".
Alex,
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.
Alex,
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.
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.
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"!
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.
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.
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.)
Dear fedja,
Thank you your eloquent posting above.
Best wishes,
Matthew
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