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Regarding What is the most compelling reason to believe Church’s thesis?, right now I am not convinced it should be closed. This is exactly the kind of philosophical question I would like to see answered by mathematicians. Maybe the reasons to avoid such questions outweigh the reasons to allow them, but what are they?
I agree that philosophical questions can be extremely interesting, and I like having discussions about philosophical topics, but I think they are generally very poorly suited to Math Overflow.
The framework that Math Overflow runs on is designed to prevent discussions. The answers to a question will get shuffled around as they are voted on, so any discussion is forced into the comments. The comments are limited to 600 characters and basically don't allow any formatting. This was specifically done in order to keep comments to the point. This isn't because discussions are bad; it's to make the framework really good for focused questions. Math Overflow is extremely good at handling "questions that have answers" (see this other post of mine), and that actually makes it bad for other things, like discussions.
So I've been thinking a lot about what MO is good for and what it's bad for. When it's bad for X, instead of trying to do X anyway, I think we should team up with another site that is good for X. For example, we really don't want to have homework questions on MO, so when a homework question comes up, we close it and direct the asker to Ask Dr. Math, NRICH, or Art of Problem Solving. Another example (recently hashed out here on meta.MO) is that it's probably bad to have people effectively asking other people to write expository articles on MO, so we will close these questions and direct the asker to Wikipedia or nLab.
The case of discussion topics is trickier, because MO is clearly poorly suited for discussions, but I don't really know where to send people. Blogs are really the appropriate place for discussions, but not everybody has access to a popular math blog. One point that Scott brought up when we were chatting earlier is that many math blogs have request threads where you can ask the owner to blog about a particular topic. But that isn't really a very satisfying solution.
When it's bad for X, instead of trying to do X anyway, I think we should team up with another site that is good for X.
Sounds great, and I agree with your analysis that MO is bad for discussions as a symptom of being great for Q&A.
It seems like the reason to avoid philosophical discussions is because they're discussions, not because they're philosophical. If so, then the task is not to prevent philosophical Q&A itself from happening, but to prevent it from getting too discussiony. And that can be as much the fault of the answers as the question.
For example, I think a succinct question like "What is the most compelling reason for..." can be succinctly answered in the form "One compelling reason is..." without necessarily turning into a discussion. If it does, I'd say that's problem with those perpetuating the discussion, not the question itself.
By contrast, a question like "<here is my opinion on Church's thesis>...<here are some counterarguments>... What do you think?" is more poignantly discussiony in that it lacks a direct question, and probably consists mainly of statements that are just begging to be analyzed and refuted.
Might I propose "too discussiony" as an official reason for closing posts?
Forums are the best place for discussions. Alongside meta.mathoverflow.net you could have another site, say forum.mathoverflow.net (or, if you feel like being silly, overflow.mathoverflow.net since it would be for the overflow of mathoverflow) for discussions.
One big lesson we (mathematicians) should learn from the OSS community is that the best way to implement stuff is that each piece should do one thing, and do it well. The rest is just making sure that they link well together. So redirecting people to wikipedia or the nlab is best for expository articles whilst to a forum for discussions, and to a forum for discussions.
You'd absolutely have to have LaTeX support, even if just inline images (though MathML would be best) and OpenID sign-in would be nice too. bbpress has both, I think, but vanilla doesn't have OpenID yet.
Welcome to the world of feature creep!
Before we go ahead and start a discussion forum, I'd like to make sure there isn't already a good one that we could just point to. Googling math discussion forum brings up a lot of them, but they seem mostly plagued by homework questions. If it turns out that math discussion forums are generally crappy, is there some reason to believe we can make a better one?
if f(n) is the length (in characters) of the longest minimal-length proof of a theorem statable in at most n characters, ...
This looks like a mathy answer, I'd be happy to see such a conversation on MO. But I think the problem is, you cannot know in advance if a "philosophical question" is "well-defined logic problem" or "an invitation to talk about popular science".
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