May I wonder why this question is not welcomed:

accidental-unplanned-breakthroughs-in-mathematics

see the votes-down. And unfriendly responses.

but many others are even more broad or more opinion-based are welcomed in MO?

your-favorite-surprising-connections-in-mathematics

what-are-some-slogans-that-express-mathematical-tricks?

what-are-some-mathematical-sculptures?

see the votes-up.

Could the reason behind is an effect of a broken window theory or some sort of biased/discrimination against the post person? (Do people know the communities suppression may hurt a person esteem?) Is there some way to improve the question?

To me, this quesiont accidental-unplanned-breakthroughs-in-mathematics is more serious than the other threes. It is meant to be a good starting point to see how many commonly easily granted-taken math studies have some nontrivial twists behind. It is a serious question.

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    You got very good answers to this question in the comments on the original post. If you don't understand or agree with those answers, then I doubt that anyone can say anything that will satisfy you. – Steven Landsburg Dec 27 '13 at 5:45
  • I can't understand your original question since the example doesn't seem to be an example. – Noah Snyder Dec 27 '13 at 23:41
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    Dear Idear, I disagree that your question (even with improved presentation) is "more serious" than the three questions you listed, which are actually good MO questions. – Gil Kalai Dec 29 '13 at 11:23

Here are some (somewhat disconnected) thoughts.

The main sticking point expressed in the objections is over the word 'breakthrough' (which is indeed difficult to characterize). But the Calabi conjecture example, and Sam Hopkins' answer, suggests a slightly different sort of question which might (might) gain more traction. "What are some examples of theorems which were eventually proven after initial attempts to prove them wrong?"

It's probably not a great question posed this way, but the situation I have in mind is familiar in research: we have a statement whose truth we are unsure about, but we have our instincts about which way it goes, and this initiates a kind of proofs-and-refutations dialectic where our initial hunches and approaches morph into their opposite, e.g., efforts to prove something is false eventually yield insights into the fact that the statement is actually true. Or something like that. Anyway, I think some such question might have appeal, and would highlight one of those "nontrivial twists" (as you put it) that helps make mathematics interesting -- and it might also be free of the vagueness or opinion-based aspects that people are objecting to.

About those other three questions: keep in mind that MO 'culture' is always evolving, and it's not necessarily the case that a question that was popular in 2010 would be popular (or even remain open) in 2014. I can see an objection that the first of those is opinion-based (what does 'surprising' mean, etc.). The second strikes me as less opinion-based somehow -- but I was personally disappointed by the quality of many of the answers. The third was not at all opinion-based or too broad: it was quite concrete, and unobjectionable IMO.

Comparing the merits of these sorts of soft questions (yours included) is largely a matter of taste; I have just given you my own personal reactions. It would probably be best not to try to argue matters of taste (e.g., which is "more serious" than the others). It would also be best not to press the idea that there is some sort of bias/discrimination against you personally; it would be impossible to prove, and such discussion would be headed in an ugly and fruitless direction. It is however true that there is a built-in looking askance and being more strict with 'soft' questions, since most users don't want MO to turn into a repository of subjective opinions, gossip, etc. (this speaks more to the broken windows metaphor).

Here is one other possible reason why your question might have got down voted:

Soft questions are not considered like other questions, they are kind of special gray area in the scope that are treated differently by people from technical questions. Many users think they should be infrequent. So there are higher expectations from a soft question and also an author of a soft question. In particular many don't like new users asking soft questions for various reasons that I won't get into here.

If you are a new user who have not asked/answered enough (which is subjective) good technical questions and you ask a soft question which is not exceptionally good (e.g. not a good motivation why you are asking the question) then it is likely that your question will get down voted and closed.

ps: I am simply explaining a possible reason, not expressing agreement/disagreement.

pps: For improving your questions you might want to check the advice about asking subjective questions in Good Subjective, Bad Subjective although not all of them might be applicable in this case.

I think that this (the MO question) is potentially a good question, or, in other words, it can be made into a good question. It should be made, a little more focused and then I don't think that it will represent a huge list of answers. Then it can be quite a useful and educating list. (Also the proof-theory tag does not seem to belong there. Update: it was removed)

The Calabi-Yau example was not so convincing (and so are the examples from physics) and it could be better to have a small list of good motivating examples.

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    I agree. And yes, having the proof-theory tag is definitely a red flag; removing it from the question would already be an improvement. – Andrés E. Caicedo Dec 27 '13 at 7:25

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