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I'll say upfront: I don't think we can and should find an answer to the question in the title. But we have to discuss it to see that we can't.

In the comments of this recent question, we have an example of a discussion that I have led multiple times during the life-time of this site. It concerns the question:

Given a mathematics question, is it ontopic?

How do you decide? Is your decision process suitable as general rule, or do you draw upon personal experience? Do you think there should be a policy, and if so, which?

See also this related discussion.

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EDITED: I added a second criterion below.

There are some questions which are clearly off-topic. There are also some questions that are borderline. For a borderline question between math and cs, I would use the criteria:

Is this question much more likely to get a good answer in math.SE than in cs.SE?

Is the answer to this question likely to be useful in the future to users of cs.SE?

If the first criterion is "yes" and the second is "no", then I think it should be moved.

While in some sense, this is a fairly pragmatic criterion (in that it leads to the OP getting a good answer relatively quickly), I think it also is a fairly good gauge of whether it's actually on-topic. Suppose we accept the rather tautological definition "computer science is what computer scientists do." Using this definition, a question which people on math.SE can answer much more quickly and satisfactorily than those on cs.SE, and which other computer scientists are not going to be interested in, is clearly mathematics rather than computer science, because mathematicians can do it, while computer scientists can only do it with difficulty and are not that interested in doing it.

For the question which set off this discussion, it had to wait a week before it got answered in cs.SE. There were already several duplicate and near-duplicate questions in math.SE. So by that criterion, I would consider it off-topic. There are some mathematics questions which are just as likely to be answered by computer scientists as by mathematicians; these we should leave open.

If a question is borderline between computer science and an area which doesn't have a good stackexchange site, I'd be in favor of leaving it open.

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  • $\begingroup$ I agree with your reasoning. However, I don't see how the "average" computer scientist with limited exposure to mathematics can make an informed decision according to this criterion. It seems to be very susceptible to what I observe in the first two bullets in my answer (which I try to avoid). $\endgroup$ – Raphael Aug 18 '13 at 11:08
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    $\begingroup$ @Raphael: if you can't make an informed decision whether or not to close, you shouldn't vote. This isn't the case just for borderline questions, it's true for all questions. I don't vote to close questions I don't know anything about, and I hope you don't either. If you want to leave the close-voting on questions to people who don't know anything about the topic, it's never going to work. $\endgroup$ – Peter Shor Aug 18 '13 at 12:07
  • $\begingroup$ See, that's the problem with your guideline "computer science is what computer scientists do" -- our users are knowledgeable about CS, so what they don't know about is not CS. Thus close vote. See my answer; they never realize they are not informed enough, of course. $\endgroup$ – Raphael Aug 18 '13 at 16:17
  • $\begingroup$ By the way, you do realize this discussion takes place on meta.**cs**.SE? (Your gray box seems to indicate otherwise. Force of habit?) $\endgroup$ – Raphael Aug 18 '13 at 16:19
  • $\begingroup$ @Raphael: do you have any examples of questions which have voted closed by uninformed users? (Not the one the OP is asking about ... that was voted closed by me and Yuval Filmus, and I don't believe either of us is uninformed.) Also, if questions are voted closed uninformedly, they can always be brought up for reopening. $\endgroup$ – Peter Shor Aug 18 '13 at 16:23
  • $\begingroup$ There are two problems with your request. 1) A close vote can have many reasons; I want to talk strictly about offtopic-ness. 2) I can not reliable judge the informedness of other users. And even if I were right, these users would likely not accept my conclusion, so the point is moot, really. (The fact that you and Yuval have been expressing mildly contradicting opinions about the ontopic-ness of certain ranges of questions leave me to wonder whether both of you can be equally well-informed.) $\endgroup$ – Raphael Aug 18 '13 at 16:29
  • $\begingroup$ @Raphael: I don't believe these uninformed votes to close as off-topic are happening frequently enough that we need to worry about them. Do you have any examples of questions that might have been closed by uninformed voters who thought they were off-topic, and that should have been classified on-topic? If not, I don't see why we should worry about this. $\endgroup$ – Peter Shor Aug 18 '13 at 16:30
  • $\begingroup$ (@ earlier comment) Of course. However, both of you seem to think that mathematics "relevant" to CS should be ontopic but at the same time, you seem to disagree what kinds are "relevant". So I am back at the hypothesis expressed in my answer, namely that the measure of "relevance" is too subjective to be of use here. $\endgroup$ – Raphael Aug 18 '13 at 16:33
  • $\begingroup$ Again, I link a question I think was closed wrongly (or at least for the wrong reasons) above. Again, I can not say with certainty that this happend because of a lack of information; there are many plausible reasons (honest mistake/opinion, fed up with questions like this, ignorance, ... you name it). $\endgroup$ – Raphael Aug 18 '13 at 16:36
  • $\begingroup$ @Raphael: we can (a) use a subjective criterion for closing that people will disagree on or (b) let all questions which might possibly have even the vaguest relationship to computer science remain open. To me, clearly (a) is the better option. $\endgroup$ – Peter Shor Aug 18 '13 at 16:36
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    $\begingroup$ Never mind (c) ask the user for the relationship and await their response. Also, (a) is mostly fine (you need five votes) but we still have to find some common ground among the close-voting people. I think this set of discussions clearly shows that we have fundamental lack of agreement as to which mathematics is how close to CS. $\endgroup$ – Raphael Aug 18 '13 at 16:41
  • $\begingroup$ let us continue this discussion in chat $\endgroup$ – Peter Shor Aug 18 '13 at 18:18
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    $\begingroup$ @Raphael, you make excellent points. It occurs to me that it's also an option to do both: we can vote to close and ask the user for the relationship and await their response. If they edit their question to include some explanation, we can re-open it. I don't claim to know whether that's the best option, but it's another option. Would that be better or worse? (In gray cases where we are not sure and we want to be more friendly to users, it's also possible to ask the user, wait a modest interval for a response, then vote to close if they don't respond.) $\endgroup$ – D.W. Aug 22 '13 at 17:26
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    $\begingroup$ @D.W. Definitely, and that is being done. Mind that it can be seen as "rude" by newcomers, especially if the fix is small, though. However, I think voting to close first is more stable because most users are unlikely to revisit a problematic question after a fixed time, whereas the OP and/or other interested party do have an interest in fixing and flagging a closed question. When we do that, it is important to send the correct message: not "Mathematics is offtopic here" but "We'd like to see how this connects to (core) CS". $\endgroup$ – Raphael Aug 23 '13 at 11:05
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I look at the following questions to decide if a question is on-topic (unless explicitly made off-topic for some reason by a policy):

  • Is there a good reason to think that the question is likely to receive a satisfying answer from computer scientists?

  • Is there a good reason to think that answering the question requires the expertise of computer scientists?

For combinatorics questions, the answer to the first question is yes I think: combinatorics is an area where computer scientists are knowledgeable about. The answer to the second question is more complicated and arguable: the required expertise is shared with mathematicians so strictly speaking it doesn't require the expertise of a computer scientist to answer the question, but it is close enough I think to requiring the expertise of computer scientists.

In general, I favor more welcoming approach to questions about the mathematical foundations of computer science on CS.SE and I think combinatorics is among them. Combinatorics (or discrete mathematics as it is often called in CS) is a part of CS and falls in category G of ACM's Computing Classification System. So I lean towards Raphael's answer that these questions are on-topic, being on-topic on another site doesn't make it off-topic here.

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    $\begingroup$ I think we might want to add a third criterion: "Is there a good reason to think the answers to this question will be useful to computer scientists?" $\endgroup$ – Peter Shor Aug 18 '13 at 13:13
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    $\begingroup$ @Peter, yes, but I think we should also include "computer science students" with "computer scientists", otherwise many undergraduate questions may become off-topic. $\endgroup$ – Kaveh Aug 18 '13 at 22:57
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    $\begingroup$ I totally agree; I didn't even realize that students might be considered a separate category. $\endgroup$ – Peter Shor Aug 18 '13 at 23:54
  • $\begingroup$ By that reasoning, practitioners without formal CS education may still feel excluded even though we would clearly (especially!) welcome mathy question from them (since they may be particularly unfit to pose them in a format suitable for Mathematics). Our faq describes our target group best, imho. $\endgroup$ – Raphael Aug 20 '13 at 15:57
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In my experience, most arguments for declaring a concept of mathematics to be unrelated with computer science stem from

  • limited scope of experience w.r.t. mathematics in CS and
  • assigning measures of relevance to subjects according to that limited scope.

These are natural behaviour patterns and I don't hold it against these people (a set which probably includes myself at times). But I think we can't make rules out of them.

I base my reasoning on three observations:

  1. I have yet to find an area of mathematics that is strictly not used in computer science.
  2. Many computer scientists have to understand (some part of) mathematics to do their work.
  3. Computer scientists and mathematicians speak different (if similar) languages.

Now, 1 and 2 imply that mathematics is relevant to CS while 3 implies that answers by other computer scientists can be more useful than others.

Therefore, my decision process has been:

  • Is it a poor question by the usual standards? If so, close.
  • Has the answer been crossposted and answered on math.SE? If so, close.
  • Does the question lack any apparent connection to CS? If so, inquire.
    • Does the asker explain why a CS perspective is useful? If not, migrate.
  • If you get this far, you probably have a well-written mathematics question by an adept of computer science which is motivated by something more than "I need to pass that maths exam". Leave open, answer, enjoy!

I don't think I have closed many if any such questions. Most pure mathematics questions (as in, about pure mathematics and without CS intentions) seem to be asked on math.SE directly, as it should be.

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  • $\begingroup$ Note that I have not always been of this opinion, so you can probably find older posts by me saying different things. In particular in the area of asymptotics, I have doubted a number of questions (since they were pure maths questions) but the interest among computer scientists is so high (and they are relevant to TCS) that it feels weird to shove them away. $\endgroup$ – Raphael Aug 15 '13 at 11:30
  • $\begingroup$ I think this is a great answer. If the question contains an explanation of why/how the math question is relevant to CS, that seems like a great reason to keep it here. Let's encourage folks who ask a math question to add the CS motivation and relevance/context to the question, as I think that makes it a better question and also helps us provide better answers. $\endgroup$ – D.W. Aug 15 '13 at 14:46
  • $\begingroup$ Agreed; I think the key here is that it should be reasonable to expect that experts frequenting this site could provide valuable information or perspective not easily available elsewhere. Questions that don't otherwise clearly satisfy this criterion should contain material explaining how computing experts' answers might be more helpful than others. $\endgroup$ – Patrick87 Aug 15 '13 at 15:33
  • $\begingroup$ @D.W. One concern: a regular answer may be "It's in the course/exercise" which is valid, isn't it? Can we ask of students to replace lacking motivation/connection in the lectures they attend? $\endgroup$ – Raphael Aug 15 '13 at 16:02
  • $\begingroup$ "I have yet to find an area of mathematics that is strictly not used in computer science" - set theoretic forcing. Hopf algebras. Also, at least in my case, your two explanations of why people argue that certain questions are properly math do not apply (in my view, anyway). $\endgroup$ – Yuval Filmus Aug 16 '13 at 4:07
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    $\begingroup$ @Raphael, Good point. OK, I'll modify my suggestion: if it's obvious to us how the math question is relevant to CS, then there's no need to state the motivation in the question; otherwise, encourage the asker to explain the relevance. I would not find "it's in the course/exercise" to be a good motivation -- but I don't expect this to cause problems in practice. If it's in an undergraduate course, I expect readers here will likely be able to judge for themselves whether it is relevant to computer science. $\endgroup$ – D.W. Aug 16 '13 at 4:38
  • $\begingroup$ @YuvalFilmus 1) I don't know either concept so I can't really argue the point. Are you certain they are not used in CS? In particular, I am happy to note that Wikipedia states: "Forcing [...] has proven to be an extremely powerful technique [...] within [...] recursion theory." 2) Regarding yourself, my provocatively stated bullets may or may not apply, doesn't matter. I think it's clear that the stated conditions are inherently such that an "afflicted" person is unlikely to diagnose themselves, and any third-party may be very wrong. $\endgroup$ – Raphael Aug 16 '13 at 6:20
  • $\begingroup$ @Raphael 1) That's why I said set-theoretic forcing. Other notions of forcing are useful in proof complexity, for example. $\endgroup$ – Yuval Filmus Aug 16 '13 at 6:36
  • $\begingroup$ @YuvalFilmus: So, arguably, questions about (set-theoretic) forcing, which seems to be the ancestor, can be part of a computer scientist's learning process. I honestly don't get how you get to a workable distinction along this line of thinking. $\endgroup$ – Raphael Aug 16 '13 at 6:38
  • $\begingroup$ @Raphael I apply my common sense. Otherwise every possible question can be asked in any site, and as a result would get duplicate answers. Sometimes this is good, for example if answers come from a different perspective. But in the case of (say) elementary combinatorics, it just feels silly. Regarding the forcing example, it's like when doing biology, we sometimes have to revert to biochemistry, which is not part of biology, but is connected to it. $\endgroup$ – Yuval Filmus Aug 16 '13 at 6:40
  • $\begingroup$ @Raphael: Mulmuley's program to separate P from NP is using geometric complexity theory is clearly related to computer science. Does that mean that we shouldn't move a question about irreducible representations of reductive groups that is related to Mulmuley's program to math.SE? Who here would be able to answer such a question? $\endgroup$ – Peter Shor Aug 17 '13 at 22:25
  • $\begingroup$ @PeterShor: We can always recommend other sites to users, and migrate upon their request. That is not the same as saying "Your question is offtopic here." (That said, your example does not fit the usual question we are talking about here; they are typically basic questions about things that are basics for several CS-things, whereas yours seems to be very specialized (and advanced.) $\endgroup$ – Raphael Aug 18 '13 at 11:02
  • $\begingroup$ I think the very different opinions about combinatorics expressed here (in particular, compare Kaveh with Yuval) reinforce the observation I start my answer with. $\endgroup$ – Raphael Aug 18 '13 at 11:10
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    $\begingroup$ @Raphael: this rather elementary combinatorics question received two rather unsatisfactory answers from computer scientists before I answered it. And I was a combinatorist in grad school. However, I don't think that question should be moved because it might be useful for other computer scientists, unlike the question that started this thread. So there should be two criteria ... let me change my answer. $\endgroup$ – Peter Shor Aug 18 '13 at 12:23
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    $\begingroup$ @Raphael: basic questions about things that are basics for several CS-things are clearly on-topic using my criteria. I think that's a straw man argument ... is anybody here arguing they're off-topic? $\endgroup$ – Peter Shor Aug 18 '13 at 12:34
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Everything is connected to everything else. Quantum computing, especially the arguments for and against it, have a lot to do with quantum mechanics. Quantum mechanics is related to a lot of physics, especially historically. So is a question on solving Schrödinger's equation within our scope? Perhaps, if it pertains to some paper related to computer science, like the old papers on the possibility of quantum computation. But not in general.

In the same way, I feel that in the practice of theoretical computer science some areas of mathematics are used more (some corners of probability theory, computational complexity, algorithms, parts of category theory), and some are used less. In particular, questions regarding elementary combinatorics seem out of place; or to take a more extreme example, elementary questions on induction. At any rate, these questions belong to the core area of another site on the stackexchange network (math.stackexchange), and there seems to be no good reason for cs.stackexchange to act as a parallel site for these.

Practically speaking, if a particular area (a) has been part of the core of another site on the stackexchange network, and (b) new questions on this are are not ported here, then I feel like we shouldn't answer these questions, but rather point the OP to the other site.

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  • $\begingroup$ 1) My problem with this, as explained above, is that it disqualifies many questions we have been entertaining are offtopic by your ruling (e.g. asymptotics questions). I feel like we don't want that. 2) I very consciously wrote "mathematics used in CS", not "TCS", because some areas of mathematics are used more outside of TCS (take e.g. computer graphics or AI). 3) Furthermore, your examples both use the word "elementary". How do I have to read that? Do you want to keep similar questions if they are non-elementary? $\endgroup$ – Raphael Aug 16 '13 at 6:09
  • $\begingroup$ 4) Finally, your guideline seems extremely ill-advised to me. The mere fact that another site covers a topic does not make it offtopic here. We have had the same discussion with reversed roles with/on Mathematics. Another example is the Stack Overflow <-> Computer Science relationship. As you have noted, sciences are connected (intertwined, arguably) so I don't think there is only one place for every question. $\endgroup$ – Raphael Aug 16 '13 at 6:11
  • $\begingroup$ 3) If the questions are less elementary then it should be more clear cut. If they arise in a CS context, then they're CS. Otherwise, they aren't. This is also a problem in some departments of computer science. For example, my department has a random graph theorist - is that really computer science? $\endgroup$ – Yuval Filmus Aug 16 '13 at 6:37
  • $\begingroup$ I share your concern -- at my university, it's some mathematicians who seem to do computer science. But: is this really for us to decide, or even worthwhile to discuss here? Let that "fight" be fought elsewhere. $\endgroup$ – Raphael Aug 16 '13 at 6:39
  • $\begingroup$ Random graph theory falls between "classical" mathematics and computer science, which is why you can find it in both kinds of departments. A question belonging to such a gray area should be answered in both relevant sites. I don't feel that elementary combinatorics is such a gray area. For me, it's just a mathematical trifle that has nothing to do with computer science, other than being taught in CS degrees (alongside calculus, say), and its concepts being useful later on. $\endgroup$ – Yuval Filmus Aug 16 '13 at 6:43
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    $\begingroup$ 1) "being taught in CS degrees and its concepts being useful later on" sounds like something has plenty to do with CS, even if not in its immediate form. I know that GPK's Concrete Mathematics contains a fair share of combinatorics, and I am tempted to shuffle through TAoCP to see how much "elementary" combinatorics is directly used there. 2) Again, I'd like to hear your justification for keeping all the (elementary) asymptotics questions under your reasoning. $\endgroup$ – Raphael Aug 16 '13 at 7:18
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    $\begingroup$ 2) Asymptotic analysis is a core "subject" which is actually used in TCS practice. Elementary combinatorics not so much. Contrived questions that one gets as homework, not at all. $\endgroup$ – Yuval Filmus Aug 16 '13 at 12:27
  • $\begingroup$ wookie919 seems to disagree. Besides, I have certainly posed very academic asymptotics problems to students which I don't think have any relevance to practice. So I still don't get the distinction you make. $\endgroup$ – Raphael Aug 18 '13 at 16:38
  • $\begingroup$ @Raphael: On the other hand, I suspect wookie919's question would have been more likely to get a good answer in math.SE or stats.SE (I think I gave him one, but I don't know whether he would have gotten a good answer if I hadn't seen it. And my answer in part answers his question by linking to an answer in mathoverflow.SE). $\endgroup$ – Peter Shor Aug 18 '13 at 16:54
  • $\begingroup$ @Raphael: And here is a question which sat around for a month with no good answers, and which would probably have been answered fairly quickly if somebody who knew information theory had looked at it. I realize that we don't have an information-theory stack exchange, but there's no reason to let the same thing happen with math questions. $\endgroup$ – Peter Shor Aug 18 '13 at 18:15
  • $\begingroup$ @PeterShor Again, I don't oppose redirecting/migrating answers to a place that is better for them, as long as we don't tell the users they had posted an offtopic question. $\endgroup$ – Raphael Aug 18 '13 at 19:44
  • $\begingroup$ I agree that we should phrase the migration statement so as not to offend anybody. $\endgroup$ – Peter Shor Aug 18 '13 at 19:54
  • $\begingroup$ For the record, here is another pure mathematics question that apparently has strong connections to CS (which remain unexplained by the author) and has thus been defended as ontopic by several community members. $\endgroup$ – Raphael Aug 26 '13 at 7:18

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