Is it okay to use MSE as a "proof calculator"?

Question

I can understand a negative response on questions that obviously are exercises and no more. MSE isn't for cheaters. But else, if you have a question of your own, if you perhaps is a bit lazy, if you not are that good at working out proofs and don't want to invest time in improving this ability for the moment, is it okay to just post the question?

I studied math a long time ago and I remember a lot of it, but have also forgotten a lot of it. Now I'm working on a system for computational mathematics, BigZ ('big' for big integers and 'Z' for sets) and now and then, when evaluating and experimenting, find "conjectures" of all kind. I expect the degree of difficulty to prove those right or wrong to be exponentially distributed and I have no great intuition about which are which, so I frequently ask MSE and mostly get a positive reaction.

However some people seems to react negative if the solution appear to be very easy or if I don't provide efforts of my own. Of course it's a bad habit using a calculator for 35+15 or something but working out proofs for arbitrary problems is really never trivial - even if a proof (preferable) afterward is trivial to follow.

I include an example: A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$. The nice formula $$p\in\mathbb P\iff\forall m,n\in\mathbb Z^+:p=m+n\implies\gcd(m,n)=1$$ turned out to have a trivial proof, which wasn't transparent for me.

Answer 1

Yes, it is okay in my opinion. I would be very happy to see additional high quality questions (i.e., interesting mathematical questions) that you have made no attempt to solve, like the one you link.

Disclaimer: Many things here have been said before, countless times, on meta.

What is context?

An unfortunate aspect of the neverending debate on context is a focus on homework-questions (no effort shown) versus non-homework-questions (some attempt or effort), rather than high-quality versus low-quality.

• A high-quality question asks a clear mathematical question and provides some context or related background to that question. The question is not arbitrary, but is stated in a way that motivates interest in it. The context could include theorems that may be relevant, why the question may arise, or, as in your question, a computational verification for why it is expected to be true ("Tested for all $p<100000$").

  In my opinion, adding a seriously flawed attempted proof or what have you tried? to a question is just as likely to detract as it is to add. A proof attempt only adds if it illuminates an obvious attempt or method or relevant detail to the question; otherwise a good answer might just as well ignore it, and it clutters the page for future people coming to the site. The other ways of adding context are often far more important.

  Moreover, notice it doesn't matter how easy the question is in determining whether it is of good quality. Many interesting, well-motivated questions are easy to answer.

• A low-quality question seriously lacks one or more of the above: the question is arbitrary or poorly motivated; reads like a tedious exercise; lacks relevant background: definitions, theorems, or partial results.

High-quality questions, as defined above, are encouraged. Low-quality questions are discouraged.

Why all the "what have you tried?" mantra if that's not the kind of context that matters?

Unfortunately, most questions on this site (especially during the school year) are homework exercises with poor motivation and often poor mathematical relevance or generality. The site as a whole fails to provide a framework to get rid of these poor-quality questions, and those who try to get rid of them are fighting a losing battle. Thus, rather than pointing out that these questions suck, we are forced to lower the bar and simply demand, at the very very least, that questions do not simply copy in a problem statement with no attempt to solve or provide background. It's a sad state of affairs when "what have you tried?" is as close as we come to requiring context--but nevermind, it's the way things are.

In the meantime, don't let all the garbage questions and the mantra of "what have you tried?" make you lose sight of what is actually important: good questions, properly motivated and with proper context supplied, require no attempted solution by the OP.

Answer 2

You touch on something that has been discussed many times before (search context and homework here on meta). I suggest reading through older threads on this to get the community's view. But here is my view in the context that you give.

I think all questions should have context. Context can mean different things. Often I think the motivation for the problem, or where the problem comes from (textbook, course, notes, ...), and what you find interesting about the problem is enough context. Many ask questions because they have homework they need to finish. And here I think it is essential that one provides attempts at solving the problem. We are not here to do peoples homework. Now, if it isn't homework, I suspect that you should be able to provide some context fairly easily.

Now if you have a theorem you want to prove and you are just lazy, you should just spend the time and work on the proof yourself. If you post a question just telling people that you know how to do it, but you just don't want to spend the time, then I doubt this will be received well. Spend at least a couple of minutes thinking about it. I bet that you often have some ideas for a start or from the context you have some idea of what tools you need to use. Well, then just tell the community about that when you post the question. As a teacher I rarely provide substantial help to a student who is just lazy. I try to encourage the student, and I point the student in a direction, but I do not do the students work. As a student you would probably agree that you learn more when you actually spend time on a problem. If you are sincerely interested in a problem, then that motivation should cause you to spend time on the problem. Many like mathematics because it is like a puzzle. It is actually fun to try and find the solution.

If you are seriously stuck on something, then by all means just ask. Maybe explain briefly what you have tried and maybe some "feelings" about the problem. Again, context (IMO) doesn't necessarily equate with attempt.

As for your example, I think it lacks context (and I have voted to close the question for that reason). Saying that you have tested it for some cases, isn't enough for me.

Now, I actually welcome what I think are trivial exercises. I understand that I might consider something trivial, but a beginning student might find it extremely difficult. It is all relative. To I think you have misunderstood MSE if you believe that easy exercises aren't welcome.