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In this question:

What does $dx$ mean?

someone asked what the "$dx$" in an indefinite integral meant. In one of the answers, which was accepted, the answer began by saying "Formally, $dx$ does not mean anything".

This is incorrect. $dx$ can be interpreted precisely as a differential one-form or as a measure. I don't know if there are other valid interpretations. My point is that the answer begins with a bald falsehood.

Incredibly, the answer has 8 upvotes (as of 11:07 Eastern time Dec. 13, 2013). I would encourage anyone reading this who is a good mathematician to take a moment, read the answer, vote for this answer the way s/he thinks it deserves, and upvote his/her favorite answer, if s/he likes any of the given answers. I would especially encourage you to read Carl Mummert's excellent new answer to the question, which is infinitely better than the accepted one.

When I teach beginning calculus, I do not attempt to teach my students the theory of differential forms, but I do not lie to them either.

I downvoted the answer and left comments to the answer and under the question. Several others had done likewise. I upvoted three correct answers, including Carl Mummert's excellent answer from Dec. 13.

I also flagged the accepted answer, explaining why.

I would like to know if this is an appropriate use of a flag: an accepted answer containing a major conceptual error that is likely to mislead the OP. I am not talking about an answer that has a typo or something like that.

I have seen a question or two on meta related to my question but I don't recall any consensus being reached or any rules being quoted resolving the issue one way or the other.

This is the very first accepted answer I have seen in MSE containing such an egregious conceptual error, in several years of participation in MSE, and this is the only time I have even considered flagging an answer because it was wrong. If this is an acceptable use of a flag, I would probably do this once every two years or so.

NOTE: I found a similar question at http://meta.math.stackexchange.com/questions/4731/dont-flag-wrong-answers?rq=1 It contains a link to an "FAQ on flagging": http://meta.math.stackexchange.com/questions/4328/capture-the-flag-faq-on-flagging , which seems to have more to do with flagging questions than flagging answers.

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    $\begingroup$ Moivated by the above post, I upvoted Carl Mummert's answer. I also upvoted the accepted answer, though it is perhaps a little more brief than it could be. $\endgroup$ – André Nicolas Dec 13 '13 at 17:50
  • $\begingroup$ Also: meta.math.stackexchange.com/questions/10133/… $\endgroup$ – Asaf Karagila Dec 13 '13 at 19:13
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    $\begingroup$ I just want to point out that I am the moderator who handled the flag. I'm in rather broad agreement with the current answers, and think (with a few exceptions) moderators should not be judges of mathematical content. I might add a post explaining my position tomorrow. $\endgroup$ – Michael Greinecker Dec 13 '13 at 21:41
  • $\begingroup$ @MichaelGreinecker : thank you for your time. The answer to my question, based on the three answers given so far, seems to be a clear "no", so from now on I will never flag any answer because it is incorrect. Of course I will read your post. $\endgroup$ – Stefan Smith Dec 13 '13 at 23:31
  • $\begingroup$ What next, Stefan --- a suit before the International Court of Justice at The Hague? Give it up, please. There are more fruitful uses to be made of m.se. $\endgroup$ – Gerry Myerson Dec 14 '13 at 4:11
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Mods have the following canned response for such flags:

flags should not be used to indicate technical inaccuracies, or an altogether wrong answer

In general, you can leave a comment explaining why an answer is wrong, add another answer, downvote, or any combination of these.

For this particular case, I disagree that the answer is wrong. I think it's a reasonable answer given the context of the question. Yes, $dx$ can be a differential form, but how is an answer like this helpful to someone who's just starting with calculus and differential equations?

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  • $\begingroup$ Thanks for your prompt repsonse. You majored my main question. I will waste a moment arguing that the accepted answer is wrong. It states flat-out that $dx$ has no precise meaning. This is incorrect. Trying to be helpful is no excuse for lying. Better to quickly mention differential forms in passing and the explain how to use $dx$ as a bookkeeping device or maybe attempt to interpret roughly $dx$ as some kind of infinitesimal. $\endgroup$ – Stefan Smith Dec 13 '13 at 16:13
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    $\begingroup$ Honestly, I don't think the post in a mathematical sense is wrong. The symbol $dx$, in the theory of calculus as we usually teach it in first year, actually has no meaning outside of indicating the variable of integration or differentiation. The fact that the $dx$ used there is the same symbol as the $dx$ used elsewhere doesn't change the fact that they are different, defined differently within different scopes; this is just an abuse of notation. Furthermore, it's very likely that the OP's scope concerned first-year calculus or some variant of it. $\endgroup$ – user2055 Dec 13 '13 at 18:56
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I don't believe it would be appropriate for a moderator to do either of these things:

  • $*$ Remove the "accepted" flag

  • $*$ Significantly edit the answer

So all that they could do is leave a comment below the question. Because that can be done by anyone, I don't see any benefit in flagging answers because they are wrong (or "wrong"). I do think that a polite comment pointing out the conceptual issue is appropriate.

It may be that, occasionally, an OP accepts an answer that is utterly wrong, but that seems to be an unavoidable consequence of the "vote/accept" model of the site.


Separately, the current example is one of the reason why it is not appropriate for moderators to intervene. Is "dx" a one-form? Is it is a measure? Is it just a meaningless symbol that tells what the variable of integration should be? Is it some kind of infinitesimal? All of these are, in their appropriate context, reasonable interpretations of "dx". It should be possible to have answers the question reflecting all of them. But in a sense they are all "correct", and it should not be the job of the moderator to decide which interpretations are acceptable. It is often the case that "incorrect" answers (like this) are really differences of interpretations, or are written as intentionally vague hints that are not intended to be formally correct.

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    $\begingroup$ I agree that this answer should not be called "wrong". Whether a notation ($dx$, $\delta$-functions, etc.) is meaningful depends on (the semantics of) the theory in which it is used. The semantics of beginning calculus does not assign a meaning to $dx$, even though more advanced theories do. $\endgroup$ – Andreas Blass Dec 13 '13 at 15:12
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    $\begingroup$ Many times I simply clear a flag and leave a comment. The only reason I can see for someone to flag in these situations is because they don't yet have enough reputation to comment. It is debatable whether they should be flagging at that point, too. $\endgroup$ – robjohn Dec 13 '13 at 15:35
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    $\begingroup$ At any rate, moderators cannot remove the accept mark. $\endgroup$ – user642796 Dec 13 '13 at 16:07
  • $\begingroup$ @CarlMummert : Thanks for your prompt reply. You answered my main question - incorrectness is not a valid reason for flagging an answer, no matter how incorrect the answer is. I will waste a few moments expressing my surprise that anyone would not consider the answer I referred to incorrect. It begins with a blanket statement that $dx$ has no precise meaning, period. $\endgroup$ – Stefan Smith Dec 13 '13 at 16:17
  • $\begingroup$ @AndreasBlass : the answer begins with the blanket statement that "formally, $dx$ has no precise meaning". I do not attempt to precisely define $dx$ when teaching beginning calculus, either, but I don't lie to my students. $\endgroup$ – Stefan Smith Dec 13 '13 at 16:32
  • $\begingroup$ To clarify what @ArthurFischer wrote: the only way moderators can potentially remove an accept mark is if the moderators delete the answer. That we will only consider doing if the user who posted the original answer contacted us to have the answer removed for the reason that "it is incorrect". $\endgroup$ – Willie Wong Dec 16 '13 at 9:30
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In this particular case, there is a significant amount of people that consider the answer correct as can be seen from the following discussions. That alone should be reason enough for not flagging it; it should (and I'm sure will) be enough reason for moderators not to act upon such a flag.

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