Thank you your eloquent posting above.

Best wishes,

Matthew

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.

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

(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"!

(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.

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.

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.

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.)

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.

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.

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.

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."