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On this site it is common to see questions where OP asks if their approach or their solution to a problem is correct. Often, people answer these questions by providing their own entirely different proof. Example 1 Example 2.

My opinion is that these answers do not address the question, and they should be discouraged as they do not help the OP. Often these are only a way for the answerer to show off their nifty proof or elegant counterexample, which maybe does not resemble OP's original approach at all, and is almost impossible to find for a beginner.

What should I do in this case? Should I downvote the answer and move on? Should I flag them for moderator attention as "not an answer"? Should I leave an explanatory comment? Is there a standard boilerplate comment to explain the situation?

What is the opinion of the community on this matter?

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    $\begingroup$ This post may be of interest to you: Is it acceptable to provide alternative solutions to proof verification questions $\endgroup$
    – amWhy
    Commented Dec 18, 2017 at 1:32
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    $\begingroup$ Personally, I believe that every answer should answer the question asked; if then, after answering the OP, someone wants to show an alternative approach in the same answer, I have no problem with it. $\endgroup$
    – amWhy
    Commented Dec 18, 2017 at 1:34
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    $\begingroup$ Also, this post may also be of interest: How to answer proof verification questions $\endgroup$
    – amWhy
    Commented Dec 18, 2017 at 1:40
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    $\begingroup$ In general, I think that alternative solutions are good. The answers are not just for the OP but also for other users and are a refererence for anyone who comes across it in the future. Plus, they may be useful to the OP if no one addresses the OP's proof directly. My main concern with this would be if it is homework and someone just solves it instead of giving hints. $\endgroup$
    – Qudit
    Commented Dec 18, 2017 at 5:55
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    $\begingroup$ Not too long ago it was discussed whether we should have solution verification questions at all. (I suppose that the discussion will be revised at some points in the future, especially if there are too many such questions.) $\endgroup$
    – Martin Sleziak
    Commented Dec 18, 2017 at 13:17
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    $\begingroup$ Another suggestion from a past discussion is to close question as a duplicate after the OP has received feedback on their proof. In such situations, alternative solutions would probably be better visible on the other question (duplicate target) which is not about specific proof attempt. $\endgroup$
    – Martin Sleziak
    Commented Dec 18, 2017 at 13:17
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    $\begingroup$ Anyone wishes to formulate their thoughts as an answer? $\endgroup$
    – Federico Poloni
    Commented Dec 18, 2017 at 15:09
  • $\begingroup$ @Qudit Alternative solutions may be good, but they are not an answer to the question. Why should they appear as answers? $\endgroup$
    – Federico Poloni
    Commented Dec 18, 2017 at 15:12
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    $\begingroup$ When I hover over the upvote button on an answer it displays the help text "This answer is useful." Useful to whom? The OP, hopefully, or to the community at large. That's pretty broad. Answers which are useful only to the answerer probably don't qualify. $\endgroup$
    – Matthew Leingang
    Commented Dec 18, 2017 at 17:37
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    $\begingroup$ I think the right approach is to first discuss about OP's approach (whether it is correct/wrong/needs improvement...) and if the answerer wishes he/she can add alternative approaches. Some answers straightaway jump to alternatives and you may indicate by a comment that this does not help OP. But good answers to any mathematical problem should not be discouraged via (silent) downvotes. $\endgroup$
    – Paramanand Singh Mod
    Commented Dec 18, 2017 at 18:52
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    $\begingroup$ @Qudit answers are also for the community and only the OP It's obvious from the context that you meant "not only the OP". Completely agree with this, and your previous comment. $\endgroup$
    – dxiv
    Commented Dec 19, 2017 at 6:07
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    $\begingroup$ We should down-vote and discourage these answers. They are no way helpful to the OP if not asked for and they are also not helpful to the community. Many people can write similar proofs and then they want to know if they are correct or not, tell me how is alternative proofs is helpful to them ? Remember that the question is "Is my proof correct ?" not "How will you prove of XYZ theorem ?"; People who want to see proof can ask another question. $\endgroup$
    – user312097
    Commented Dec 20, 2017 at 17:42
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    $\begingroup$ My usual way of dealing with this, if I am qualified to do so, is, if there are no answers actually addressing the OP's question, to write one myself, beginning with a sentence like, "A number of the answers provide proofs of the result, but let me specifically address what's wrong with your attempt." I sometimes also downvote answers that read to me as willfully ignoring the actual question. $\endgroup$
    – Ben Blum-Smith
    Commented Dec 21, 2017 at 11:43
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    $\begingroup$ I remember a story about Alte Selberg reviewing a 300 page manuscript that solved a longstanding problem. He stated that the paper was technically correct, and then furnished a 50 page solution to the same problem. His 50 page solution was published along side the other manuscript. When he was asked about his review, he stated that he didn't feel that the other paper took the right approach to the problem. $\endgroup$
    – Joel
    Commented Dec 24, 2017 at 22:49
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    $\begingroup$ @A-B expanding on what you've said, if someone merely wants a proof for a statement, they can go to Wikipedia or elsewhere to find it. People come here to understand why their proof isn't working, trying to build their own mathematical maturity. Not see a proof that is often beyond them. $\endgroup$
    – Myridium
    Commented Dec 27, 2017 at 9:14

2 Answers 2

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Your first example is structured like this:

[mathematical statement]

Is this statement true or false and why?

[attempt at proof]

Is this way of looking at it right or not?

It is entirely fair to answer that question with a different proof. This OP doesn't know whether the statement is true or false. Proving it false is an answer to the question.

The second OP, however, clearly wants a proof critique, and not a novel proof of the statement. We know this is the case because they accepted an answer which does not prove the statement at all. We could have also figured it out from this sentence:

Can you please review this proof, or point out the flaw in it?

TL;DR: If the OP clearly wants a proof critique, then a proof critique is what they should get. If they indicate uncertainty about the solution to the problem, show their work, and ask for help, providing a different proof is fair enough.

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  • $\begingroup$ Thanks - makes sense. What should I do in practice in the second case? Downvote or flag "not-an-answer"? $\endgroup$
    – Federico Poloni
    Commented Dec 21, 2017 at 7:23
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    $\begingroup$ Even for your second example, an alternate proof could be a valid answer. The flaw might not be technical, but rather qualitative. For example, suppose someone was trying to find the derivative of $(x-2)^5$ by first expanding the binomial product, then applying the derivative term by term. They could achieve the correct answer, but it would not be the right way to do the problem. In this case, telling the OP that they are correct will do more harm than good. $\endgroup$
    – Joel
    Commented Dec 24, 2017 at 22:53
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    $\begingroup$ @Joel: If you just show the chain rule with no context or explanation of how it relates to the OP's proof, your answer is not helpful. You have to at least minimally state that the OP's way is valid but difficult or inelegant. That's a proof critique in my book. $\endgroup$
    – Kevin
    Commented Dec 25, 2017 at 0:17
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Two aims are involved here: (1) we need discipline on the site to keep answers relevant to the questions to which they are attached; (2) we do not want to make people feel bad about posting a correct and elegant proof of a result for the enlightenment of those who, whether as OP or visitor, have evinced some interest in the result.

We would be entirely righteous if, following the first aim, we leave a comment saying "(-1) Your answer, while providing a model proof, is irrelevant—because it does not address the issue of whether the OP's [laborious and convoluted] proof is correct". However the effect of such a comment could well be that the poster of the answer would be less inclined to bother with the site altogether, rather than prompting the thought "He's got a fair point; I'll rewrite my answer to focus on the OP's exact concern".

My own view is that the first aim is worthy; but the failure to maintain it in this particular regard doesn't loom large against the background of this site's many concerns. If I were to write anything, I would be inclined to pussyfoot around by saying "Nice proof—but the OP is more concerned about whether his proof is sound. What do you think?"

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