The question is whether we should reopen it (link here). My point of view is that we should because

a) It is asked by an intelligent and highly respected user.

b) It hasn't generated a single ridiculous or flaming response.

In general, I would prefer to close "opinion-based" questions only if they are either directly put in a way that provokes a conflict, like "So, in the Russia-Ukraine conflict, which side are you on?" (the problem with that way to pose it is obvious even on a web forum that is devoted to the discussion of the conflict itself), or when the discussion escalates into a real and uncivilized clash of opinions (a civilized clash of opinions is what the intelligent conversation is always about: if you agree on everything, what's the point of talking?). Neither one has really happened in this particular case. Moreover, I do not even think that the question is in a "high risk" group for that to ever happen (though, I may mistake here) and some answers and comments were quite interesting.

I have no desire to start the usual tug of war game, so I am not casting a reopen vote on the main site at the moment,but I would really love to know what other people think :-)

Edit: It's been reopened.

Edit: It's been reclosed.

  • I wasn't keen on the question, but I refrained from voting to close, mainly because of your observation (b). I think I am still ambivalent and will wait to see if a consensus decision emerges :) – Yemon Choi Nov 30 '14 at 19:11
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    I am aware of (a), and I mean no disrespect to the author when I say this, but questions like this or "Who was the greatest mathematician?" can be entertaining in private (with enough alcohol). But I am less sure of its place in a public forum like this. – Donu Arapura Nov 30 '14 at 21:07
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    Another issue with the question is that the extent to which the resolution of a particular problem advances mathematics often depends also largely on how the resolution looks like (does it give further insights, or is it only a long technical chain of arguments that somehow does the job?) -- and that is usually not known before one has actually solved the conjecture / problem in question. – Stefan Kohl Dec 1 '14 at 0:07
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    I'm not sure I find (a) particularly compelling, and it makes me uneasy that the same question asked by user-xyzuv would very likely get closed without a murmur as irremediably subjective and opinion-based (what makes it suddenly appropriate when asked by someone with high rep?). It's a great question for a blog, obviously. As for (b): knock on wood. I'm sorry to say that with the exception of Paul Siegel's answer, the ones that have so far appeared don't look like they were composed with much care. :-( – Todd Trimble Dec 1 '14 at 3:53
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    @ToddTrimble: I just saw you welcome Igor Kriz to MO. Clearly having someone like him is a good thing for MO. Wouldn't you agree that that gives a compelling argument for (a): there are a priori reasons (that we can all agree on) for welcoming Igor Kriz rather than user-abcde. Re the question: I too am ambivalent about how useful it is, but I am sympathetic to fedja's reasoning. – Lucia Dec 1 '14 at 4:56
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    @Lucia Of course, but I don't see a strong connection with fedja's (a). Kriz gave a very insightful and thoughtful answer, and based on his IRL reputation we can expect more of the same and very much wish to encourage that. Joseph O'Rourke is a very valuable member of our community, to say the least, but in my view the MO-appropriateness of his post is questionable and I don't want to encourage the asking of that sort of question, by him or anyone else. What his high rep and intelligence do is make it uncomfortable or at least harder to say this openly. – Todd Trimble Dec 1 '14 at 5:37
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    @ToddTrimble: You make a perfectly valid point of course, although I still like the idea of cutting more slack to people (posting under their real name) based on their IRL reputation. – Lucia Dec 1 '14 at 5:49
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    @Lucia I do understand that point of view; thanks. Another interesting test case came up here: mathoverflow.net/questions/38639/thinking-and-explaining A part of the problem I'm having with the current question is that it's a crystal-ball question, asking us to imagine (and defend a pov on) a scenario we necessarily have a dim grasp of (cf. Stefan Kohl's comment), whereas Thurston's question was encouraging answers based on deep inner experience. – Todd Trimble Dec 1 '14 at 6:00
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    Perhaps as a compromise, let us keep the question open as long as (b) holds? – user9072 Dec 1 '14 at 18:44
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    I cast the final vote to reopen. However, I regard that as tentative. I might reverse my opinion if we do not get insightful answers in the next couple of days. – Neil Strickland Dec 1 '14 at 19:06

I apologize to the community for provoking this controversy. I believe I have learned from the experience, and will avoid similar questions in the future.

Please let the question remained closed.

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    Something ventured, something gained. At last count, no serious casualties from this venture. I am looking forward to your next question. – The Masked Avenger Dec 2 '14 at 4:04

I was considering an answer to the original question when a new question popped up. Rather than post an additional question, let me analyze and change Joseph's question, and then make a brief response here. If others approve, I will then post/edit a version of the question and answer on the forum.

If I go through what I know of the history of mathematics, most of the things that I consider advances arise from changes in its foundations, specifically in terms of the language used. Using as milestones Euclid's elements, Cartesian coordinates, Frege and followers attempts at formalism, many advances resulted from our being able to acquire perspectives that were phrased using new modes of expression, and communicate these perspectives to others. Showing that trisecting the angle was classically impossible, that hyperbolic and spherical geometry existed to refute the parallel postulate, and that certain theories were incomplete from being recursively enumerable instead of recursive, all these are small steps compared to the large ones of being able to find the language and ask the questions in that language. Similarly, I would consider question-asking moments (Pacioli before probability, are certain constructions possible, Hilbert's list) to be at least as great an advance as the answers to these questions.

So I would ask the question: what would herald the next major advance in mathematics, be it a development of a certain theory, solving a set of conjectures, or laying the foundation for future work? And I would answer: fulfilling the program of reverse mathematics so that it is understood what can be done with the current tools and systems of reasoning, and then trying to apply that to other systems of reasoning.

More can be said. However, if this is enough to edit and reopen the question (which isn't quite for MathOverflow but I could see it being shoehorned in), I am willing to do the edit and the flag, and then post an extension of the answer when it is reopened. I am also willing to do a brand new question.

Gerhard "Ask Me About System Design" Paseman, 2014.12.02

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