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A few months back someone asked about the difference between questions appropriate for mathoverflow.net and those appropriate for math.stackexchange.com. And someone answered, saying something along the lines that those appropriate for this site are at a more advanced level than those appropriate for that other site. That seems to me like a definition by non-essentials. Some questions on math.stackexchange.com are on some fairly sophisticated topics. But I think they might be inappropriate for this site simply because their answers are applications of known methods and would not be a part of anything suitable for publication, unless it's in something like a textbook.

It seems to me that what qualifies something as a "research" question may be that novel ideas are involved in answering them, that might be worth publishing for the first time in the kind of journal whose purpose is to publish novel ideas.

There are obvious reasons why such things occur more often in subjects requiring much prerequisite learning before one can even understand the question, than in things like elementary-school arithmetic. But it seems to me that defining "research" purely by the "level" in that sense is a definition by non-essentials.

(And of course, one often posts in ignorance of whether a question is a well known long-solved problem or something whose solution would be a publishable novelty. That's another complication, but not quite the point. (E.g. this one: Testing contrasts in statistics: Is this provably a hard problem, or not? I certainly didn't know if this had been done before.))

So what would be a better answer than merely the "level"?

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    $\begingroup$ In my opinion, in practise, the answer is the level. (Only that on math.SE there is no upper-bound, so in principle there are no questions that are too high-level there, it might just be that one gets better reception on MO, while on MO there is a lower-bound.) Also MO is not for research question, but research-level question, which addmittedly is a vague term, but the FAQs explained it. I'd say advanced graduate-level (5-6+ studies at university) is roughly speaking the appropriateness-level for MO. $\endgroup$
    – user9072
    Commented Jun 30, 2013 at 19:48
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    $\begingroup$ I'd agree with quid, and add that it's comparatively rare that answers to "MO-appropriate questions" actually involve novel ideas. Many such questions arise when a mathematician needs to investigate something outside his/her comfort zone, and feels sure that some expert out there would be able to supply the knowledge he/she is missing. $\endgroup$
    – Todd Trimble
    Commented Jul 14, 2013 at 0:25
  • $\begingroup$ I thought the point is not at all that the answer should involve novel research, but that the question should arise in research. When MO is called "the best math department in the world" I thought the idea is a researcher might run up against a question not exactly in their field, but needed for their research, and not answerable by normal references, and go ask someone whose field it is. That latter expert might not find the question hard -- only the answer should not be obvious to everyone equipped to understand and use it. $\endgroup$
    – Colin McLarty
    Commented Dec 9, 2014 at 19:40

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Over at tea we recently hashed out the following brief summary:

MathOverflow is a questions and answers site for research mathematicians. We accept mathematical questions at the Ph.D. student level and higher. Questions of all levels are accepted at our sister site Math.SE.

(Personally I'd change "research mathematicians" to "professional mathematicians," since I think that's clearer. Most people don't know what "research" means.)

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    $\begingroup$ This is good, though I also agree that "professional mathematicians" is better than research mathematicians. $\endgroup$
    – Andy Putman
    Commented Jul 13, 2013 at 16:51
  • $\begingroup$ I think it could really help to avoid some off-topic questions if this description were used instead of just 'for mathematicians' what we have now as the official description. I agree professional is better given the context. The word 'professional' is also used in the description of other sites (when applicable). $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 17:26
  • $\begingroup$ Why should our definition be relative to math.se? $\endgroup$
    – François G. Dorais Mod
    Commented Jul 13, 2013 at 17:28
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    $\begingroup$ @FrançoisG.Dorais: In order to give people an alternative so that they won't just ask inappropriate questions here anyway. $\endgroup$
    – Noah Snyder
    Commented Jul 13, 2013 at 17:30
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    $\begingroup$ But why put that in the definition? There's other places to tell people what the alternatives are, and there is more than just one alternative. $\endgroup$
    – François G. Dorais Mod
    Commented Jul 13, 2013 at 17:33
  • $\begingroup$ The same reason you should only ever put one question in an email: because people don't like to read more than one thing. $\endgroup$
    – Noah Snyder
    Commented Jul 13, 2013 at 17:37
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    $\begingroup$ @FrançoisG.Dorais: if you do not think the mention of math.SE is a good idea (I can see reasons for or against it), I would also think this would avoid some off-topic questions if the last sentence were removed. Even if only 'professional' was added to mathematicians I think this would already have some positive effect. The plain 'mathematicians' is really misleading, IMO, in the context of the other sites' descriptions that often-times make and amateur/professional/both distinction. And if nothing is said I think the most reasoable assumption would be 'both', yet it's wrong in our case. $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 17:48
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    $\begingroup$ But to add to what Noah Snyder said: because the description is the only thing people inside the network will see in a list of sites and insider the network math.SE is the main alternative. If somebody is searching for a math site in the SE network I think they will go by this description mainly. Here is a link to the list of all sites and their descriptions: stackexchange.com/sites?view=list#traffic I think it is fair to say at the moment we have one of the least informative descriptions of all the sites. $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 18:03
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    $\begingroup$ These days I'm inclined to think that "professional mathematicians", while not exactly what we mean, would be far better for practical purposes than "research mathematicians." We can always make exceptions for those doing maths as a hobby at a sufficiently high level, no? $\endgroup$
    – Yemon Choi
    Commented Jul 13, 2013 at 18:16
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    $\begingroup$ @FrançoisG.Dorais: please have a look at the site while not being logged in. This description is in the banner that is shown then on the front page. This description is how 'about' starts. And, this banner (in non expanded form, but still displaying this descrption) is shown one each individual posts page for users not logged in. It is quite unclear to me how somebody should arrive to use the site and not see it. From inside or outside the network it does not matter. $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 18:37
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    $\begingroup$ @FrançoisG.Dorais: But also, I observed more than one instance already of somebody asking a completely off-topic question here that, as I checked, had an stackoverflow account. It seems not unlikely they arrived somehow inter-network. And would have had no problem to ask at math.SE had they known. So I continue to think changing the descrption could avoid some (obviouly not all) off-topic questions. Also, even if it would not avoid them only having the descrption handy can help to point to on closing to avoid comments like 'Hey, but this is a math site! How can this be off-topic?' $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 19:50
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    $\begingroup$ @FrançoisG.Dorais: Finally, noone wanted to discourage people from reading as the descrption talks about what questions are accepted. But if you press me, personally, I would even want to discourage random people from reading the site. What good does it do that people not knowing the first thing about math-research read the site? (I can see some good, like general publicity for the field, but this is massively outweight by the potential of getting ever more off-topic content uninformed voters and all kind of other inconvenience.) $\endgroup$
    – user9072
    Commented Jul 13, 2013 at 19:53
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    $\begingroup$ @quid: I have no problem changing the tagline; I even suggested a different one myself. I'm strongly against any reference to any other site in the tagline, though. The visit counts I mentioned above were specifically from stackexchange.com (where you see all the sites with their taglines); most of our in-network traffic comes from math.se and stackoverflow. Even if you add all these up, they don't even account for 20% of our traffic. By far the largest source of traffic is google.com. $\endgroup$
    – François G. Dorais Mod
    Commented Jul 13, 2013 at 20:27
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    $\begingroup$ @quid: I do prefer the term professional, mostly for i18n reasons. Personally, I don't think any of the suggestions are perfectly adequate but I suspect there simply isn't a perfect word to put there. $\endgroup$
    – François G. Dorais Mod
    Commented Jul 13, 2013 at 22:26
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    $\begingroup$ "Professional mathematicians" is not a bad shorthand; it's pithier than what might be more accurate, "mathematicians of professional caliber" (the difference being that some people answering to that description are not actually paid as mathematicians). $\endgroup$
    – Todd Trimble
    Commented Jul 14, 2013 at 0:20

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