Dealing with possible incorrectness

Question

I have seen a few responses on MO that I think are either incorrect or possibly incorrect.

What is the best way to draw attention to possible incorrectness?

(I provide two examples below; if this is poor form, then please edit them out.)

1. A user posts that it is impossible to prove the Fundamental Theorem of Algebra using the Brouwer Fixed Point Theorem. But another user (disclaimer: me) provides a source purporting to provide such a proof. Due to timing, the former received many up-votes early on; as of this writing, it has twice as many votes at 22 to 11. For both claims, see here. [Added (quid): the answer claiming impossibility was deleted by its owner.]

2. There is a post claiming that: Most (actually almost all) of ancient Chinese mathematics literatures [sic] are written in the forms of poems. But this claim is unsubstantiated insofar as I can tell, and one might suspect that, e.g., the last example provided is not an instance of an actual poem. Again, this post has garnered many up-votes (15 as of this writing) but I'm not sure how anyone could properly evaluate it without more information or the ability to read Classical Chinese. [Added: An MO user (Marty) posts in the comments there: I asked a colleague -- an expert in Chinese poetry -- about this, and his response is below... "Well the post is fascinating, but completely wrong." etc.]

I can think of a few ways to address such possible incorrectness. One is to leave a comment on the original question or the disputed answer; however, I worry that this method has very low visibility unless it is carried out shortly after the response is posted.

Alternatively, one might edit the post, thereby bumping it up and drawing attention to it, or put up another post disputing the earlier one. But I view such approaches as somewhat aggressive and possibly overkill. (I hope that my examples above are not viewed similarly!)

Yet another route would be to draw attention to the post here in meta.MO. I suppose that is the approach I have taken for the two examples above, but I am curious as to what should be done more generally (if anything at all).

Edit: For a related feature suggestion, Stefan Kohl comments he's left a post here.

Answer 1

One possibility, in the event of having left a comment but received no response, to edit the answer itself and add a line like:

n.b. (Benjamin): this answer appears to be incorrect. Please see my separate answer.

I think when the original answerer does not address critical comments, it's perfectly reasonable to edit.

Answer 2

Mathematicians tend to care a great deal about correctness. If you bump the question by adding your own answer (or even a minor edit), this will tend to draw attention to your comments that the other answer is incorrect. In several cases, highly-upvoted answers have been voted down after it was pointed out that they were incorrect, while other much later answers were voted up. It is unfortunate that other sites like stat.stackexchange.com do not similarly value correctness. Perhaps they will in time if they reach a critical proportion of people with either a professional level of competence or the maturity not to vote up incorrect answers with pretty pictures or high reputation authors over unadorned but correct answers.

Please be extremely cautious about changing someone else's answer to say that it is incorrect. Think about how you would feel if someone who misread the question or misunderstood the relevant mathematics changed your correct answer. The reward is often not worth the risk. Even if you are used to being right 99.9% of the time, you should not assume this is the case when you disagree with someone else who is also right 99.9% of the time.

There are several answers of mine which people have insisted at length are incorrect. In one case I'm confident that the multiple people disagreeing with me are confused and I think I understand their confusion. I think it would be terrible if they defaced my accepted and highly upvoted answer just because of their confusion. In another case, the person vehemently disagreeing eventually admitted misreading the question. In a third case, my answer covers one interpretation of the question, and was accepted, but there are other interpretations other people prefer. I think their comments and alternative answers (and down votes) are fine, and more appropriate than changing my answer.

Answer 3

Slightly tangential answer:

I don't think there is anything wrong with having incorrect answers on MO. People make mistakes, and other people are free to point out the mistakes in the comments. And as long as some form of the correct exists in the record, I don't think having incorrect answers in anyway reflect poorly on the community as a whole.

A perhaps bigger problem is the situation when people upvote incorrect answers which they either have not checked themselves or that they do not understand.

Now, this problem manifests a lot less for mathematical questions and mathematical answers, since the training of mathematicians often allow us to make educated guesses about the usefulness of answers even without following it through fully. And I certainly do not advocate voting only for answers one can personally vouch for.

But for a soft (and borderline off-topic) question like the second one linked to above, which is about mathematics and poetry, the vast majority of the MO users would be mere amateurs. Therefore the voting patterns there will reflect more of the common pitfalls about the voting on other SE sites. (There's a reason that the fastest gun in the west syndrome refuses to die on most SE sites: speed is often rewarded over elegance and correctness.)

It is perhaps a happy accident that on MO that the vote counts on answers are a better proxy for their correctness than on many other SE sites. And for those of us used to this it may seem strange that sometimes incorrect answers gather lots of upvotes.

For frequent users of this forum, a work around is to mentally readjust one's interpretation of the votes when the question is a soft/off-topic one. For uses who do not frequent MO, hopefully they will have been trained by the rest of the internet to take things like "likes" "+1s" and "upvotes" with always a healthy grain of salt.

Answer 4

I think the best practice is to leave a comment, and if necessary provide more information by writing an additional answer.

A reasonable person who reads an answer, reads comments under it. You should not care too much about unreasonable people. Also, you should not care too much about votes --- if you wrote a good answer, and a bad answer receives more upvotes, it is "the community" who has a problem, not you. And, again, reasonable people who read answers do not assess their quality by the number of up/down-votes.

Once, I thought that mathematicians on MO are just so pervert that they enjoy voting up trivial questions and silly answers. Then I recalled words of a well-known professor that I heard when I was a student:

"Michal, you are mature, so I tell you the truth: teaching is like casting fake pearls before swine"

In short, to get many upvotes you do not have to write a brilliant answer, nor be reasonable --- you have to sound like you were reasonable :-)