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What is the correct and polite thing to do when someone who has more than 100 times your reputation has posted an answer you feel is incorrect? Someone who doubles your output per year in one week? Someone who is necessary for the site while you are not. Someone you admire because he is superhuman. In addition the question is on hold and will very likely be closed. Is it not 100 times more likely that I am wrong and will just make an ass out of myself? What do you do in these spots?

I decided to let it be... Was that cowardly?

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    $\begingroup$ You can write a comment of the kind "Sorry, but what happens if $X$ is...". By the way: Reputation and mathematical ability are two different things. $\endgroup$ – Michael Greinecker Aug 22 '14 at 8:57
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    $\begingroup$ I always assume that I am wrong and end with the phrase "What am I missing?" In fact, the other day I got a comment, pointing out an obvious error, from Derek Holt (seriously clever guy, and a full professor at Warwick) which began with "Perhaps I am missing something obvious, but I don't see immediately why..." $\endgroup$ – user1729 Aug 22 '14 at 10:56
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    $\begingroup$ No one is necessary for the site. $\endgroup$ – Asaf Karagila Aug 22 '14 at 11:03
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    $\begingroup$ Someone with that many posts has 100 times the opportunities to make a mistake. $\endgroup$ – Jonas Meyer Aug 22 '14 at 12:09
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    $\begingroup$ If you don't say anything, one of two things will happen: a bad answer will stand, or else the wrong idea in your head will survive. On the other hand, if you take action, one of two things will happen: the answer will improve, or else the problem with your idea will be illuminated and you learn something. The latter pair seems much more attractive :) $\endgroup$ – rschwieb Aug 22 '14 at 12:52
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    $\begingroup$ Mistakes and corrections are a daily routine here, but as user1729 suggests, it's always good to decorate these to be as polite as possible. Terse or overly direct corrections can often be perceived as curt or rude, and this sometimes causes a lot of friction for people who don't follow this good advice (user1729's advice.) $\endgroup$ – rschwieb Aug 22 '14 at 12:54
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    $\begingroup$ @Asaf, is anyone sufficient? $\endgroup$ – Gerry Myerson Aug 22 '14 at 12:55
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    $\begingroup$ link to the question? $\endgroup$ – Will Jagy Aug 22 '14 at 19:27
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    $\begingroup$ After much hesitation, I decided not to tag this with apotheosis. $\endgroup$ – user147263 Aug 22 '14 at 22:59
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    $\begingroup$ I think that you should leave a comment pointing out the error. It is important to try to make sure that the site is as error-free as possible. $\endgroup$ – André Nicolas Aug 23 '14 at 0:57
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    $\begingroup$ SE "rep" is rarely correlated with anything (including correctness). $\endgroup$ – Bill Dubuque Aug 23 '14 at 17:59
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    $\begingroup$ @BillDubuque It's usually pretty well correlated with algebra-precalculus. :) $\endgroup$ – Alexander Gruber Aug 23 '14 at 18:33
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    $\begingroup$ I suspect "rep" is mostly correlated with "hours spent on site". $\endgroup$ – Gerry Myerson Aug 23 '14 at 23:49
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    $\begingroup$ @GerryMyerson Which implies less time working on and studying math. $\endgroup$ – Rene Schipperus Aug 25 '14 at 0:29
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    $\begingroup$ @Rene: I wrote my M.Sc. thesis and developed more than a handful of ideas by answering questions here. So spending time on this site is not necessarily not studying math or working on your research in mathematics. $\endgroup$ – Asaf Karagila Aug 27 '14 at 5:25
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I know for a fact that reputation does not reflect infallibility.

As a general rule of politeness, when I see something that seems questionable, I ignore the reputation of the author and address them with respect. It is nothing more than the same that I would expect from someone else when I make a mistake.

The comments suggesting phrases like "Perhaps I am missing something, but..." are very good. Communication via text lacks many cues that convey the emotions behind a statement, so it is always best to apply a good dose of respect for the many people here who are simply trying to help others. If that is done, I think your comments will be well accepted.

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    $\begingroup$ " Tact is the art of making a point without making an enemy. Isaac Newton" $\endgroup$ – bobbym Aug 23 '14 at 10:23
  • $\begingroup$ @robjohn We had similar answers but you said it very nicely. +1. :) $\endgroup$ – Ahaan S. Rungta Aug 23 '14 at 15:19
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    $\begingroup$ @bobbym: As if Isaac Newton would know! (Did he really say that?) $\endgroup$ – TonyK Aug 23 '14 at 19:15
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    $\begingroup$ @TonyK: Google search didn't come up with any actual references; but there are two attributions of this quote to Howard W. Newton, rather than Isaac Newton. I couldn't quite find a good reference for that either. $\endgroup$ – Asaf Karagila Aug 24 '14 at 6:51
  • $\begingroup$ brainyquote.com/quotes/quotes/i/isaacnewto377678.html $\endgroup$ – bobbym Aug 24 '14 at 9:39
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    $\begingroup$ My favourite Isaac Newton quote is "Sins:.. Threatening my father and mother Smith to burn them and the house over them." Nice guy! $\endgroup$ – user1729 Aug 24 '14 at 19:53
  • $\begingroup$ He should have taught that concept to Robert Hooke, @bobbym... $\endgroup$ – Johannes Pille Aug 27 '14 at 15:36
  • $\begingroup$ "Tact is the ability to describe others as they see themselves."--Abraham Lincoln. $\endgroup$ – chiastic-security Sep 5 '14 at 9:47
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I have been in this situation before. Only one way to go about it $-$ make it clear that you are humble but make your point! Just because the person has a lot of reputation, he/she does not have to be perfect. Here's the type of response I would provide:

Hi, @X. Thank you for your response! I may be wrong, but [xyzabc...]

This applies to whoever you are writing to $-$ whether the person has $15$ reputation or $15\text{k}$ reputation.

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As the other answers say, it's of course just a matter of appropriate phrasing.

Don't just swoop in with an edit, without any discussion (maybe the high-rep user is correct, anyway!) - this was actually an issue on SO recently.

I, like you I think, immediately doubt myself in this position - more than that, if $\Delta\text{Rep}$ was really that much, I would just assume I was wrong myself.

With that in mind, I think a good way to respond is to pose a question, rather than even suggest there may be an error.

Instead of (even the rather polite):

I may have misunderstood, but isn't 2*2 = 4?

I would go for something more like:

Thank you for explaining - one thing that still confuses me though is that I was pretty sure 4*4 = 16, but if 2*2 = 5, then wouldn't 4*4 = 2*2 * 2*2 = 5*5, which again I was pretty sure is 25?

It's a silly example - but often, just the careful thought process that asking a question makes you step through is sufficient for me to answer my own question.

If after writing it out, I am still confused or even - we believe - correct, then it gives the high-rep answerer the opportunity to either realise his mistake, or if the mistake is ours, she will likely gladly explain - or edit the answer to clarify if it was likely to confuse others too.

It needn't be an issue - one way or another it leads to an improved answer.

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    $\begingroup$ $\Delta\text{Rep}$ +1 for that. Clever. $\endgroup$ – bobbym Aug 24 '14 at 9:47
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    $\begingroup$ @bobbym Actually $\Delta \mathrm{Rep}$ -1, if I read correctly ;) $\endgroup$ – AlexR Aug 28 '14 at 14:33
  • $\begingroup$ It happens that if $2*2=5$, then indeed $4*4=2*2~*~2*2$ wouldn't be true. If you think there is a misunderstanding on the part of the other person, I would prefer go straight for it, rather than try to show its absurdity through some derived argument; in other words, prefer the first to the second cited form. $\endgroup$ – Marc van Leeuwen Aug 29 '14 at 5:28
  • $\begingroup$ @MarcvanLeeuwen It was a toy example meant to demonstrate that actually I may be wrong. So personally I would choose not to jump to "I think this is an error", but to "wait, that confuses me -" $\endgroup$ – OJFord Aug 29 '14 at 12:10
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I don't have high rep on this site, but I do on Stack Overflow, and here's what I expect:

I usually can't tell the rep of people who comment on my answers. I could tell if I went and looked at their user page, but usually I can't be bothered, and so usually I can't tell.

So, I usually respond to the comments of low-rep users exactly the same way that I respond to the comments of high-rep users I don't personally recognise from previous questions. I can't do otherwise.

In the context of StackExchange I feel that a comment that's just a factual statement of my inaccuracy is perfectly polite, regardless of the rep of the person who makes it.

Provided it's a matter of correctness and not opinion, there's no real need for prevaricating language like "I may be wrong, but", "I think I've misunderstood", or whatever. If you feel like it makes you less of an ass when you're wrong, OK, have at it, so long as you make it clear exactly where the point of disagreement lies. Don't make a vague objection for the sake of politeness, leaving the questioner to figure out why you're right or wrong, maybe argue with you at cross purposes for a while.

Phrasing the correction as a question is probably the cheapest way to "back off" a bit from your claim, without wasting your time and risking confusion. "Do those two lines really meet? What if they're parallel?". This also has the advantage that if you're obviously right it looks like a humorous rhetorical question, whereas if you're wrong it can quickly be answered straight: "yes, in this geometry any two straight lines intersect, although their intersection might be the point called "infinity". See X for more information."

You might want to put a bit more work into convincing yourself that you're definitely right, the higher your regard for the person you're questioning. But that's purely a balance of probability thing. What are the odds that a mathematical genius wouldn't know what Cauchy's Theorem is? Pretty small, so if you think they've called some completely unrelated thing "Cauchy's Theorem" then you should reflect on whether maybe there's more than one. Same goes for matters of terminology, they might simply have made a different choice from what you're used to. What are the odds that a mathematical genius would make a simple arithmetic mistake in an answer on this site? Reasonably good, and quite aside from their mental arithmetic they don't necessarily type any better than anyone else and so errors can get in that way too. But once you're convinced you're right, get on with it.

If it is a matter of opinion not accuracy, that is to say the answer is true but perhaps not the most helpful way to approach the question, then just downvote and comment if you feel like it. You're probably somewhat more likely to wind people up this way, though, unless they've thoroughly misunderstood the question and can be easily put straight.

Anyway:

  • Superhumans make mistakes.
  • There is almost nothing worse for a superhuman than reaching the point where nobody corrects them any more even when they're wrong. Quality goes through the floor.
  • Nobody is entitled on StackExchange to the deference that, for the sake of not wasting time in class on your errors, you might give to an academic superior or an instructor. There is no "after class" or "in a private email", just question them now. Of course that applies in some classes too: I don't mean to say that all instructors should be deferred to.
  • Even if you're wrong, superhumans are on the site because they want to help out us mere mortals, so they quite likely won't mind at all dealing with your incorrect objection by teaching you the rights of it.
  • Even if they do object and choose to make you look an ass, well, whatever, you've learned something about them. If they're right and you're wrong, admit it and move on.
  • You will remember your humiliations far more clearly than anyone else will.
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    $\begingroup$ Not to burst your bubble, but if you hover over a username in the comments you see their reputation. $\endgroup$ – Asaf Karagila Aug 26 '14 at 20:44
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    $\begingroup$ @AsafKaragila: ah, so maybe I'm unusual in not dismissing comments from low-rep users out of hand, if I'm the last to find that out ;-) $\endgroup$ – Steve Jessop Aug 26 '14 at 22:50

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