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[Edit: to explain some things that were not so clear in the original post]

I believe in being straightforward. Without linking to the specific question, people will just treat this as another vague gripe with nothing to discuss. But to talk about the specific issues with a specific post is equivalent to pointing out the specific authors of those posts too. I give credit where it is due; I've commended and upvoted some other answers by the author in question, yet many people have judged me solely based on my blunt criticism of this answer of his/hers. Past interactions showed that this user generally refuses to admit any serious conceptual error, and this is nowhere near the first time. So please be fair if you wish to judge my statements and actions.

Secondly, someone has taken the initiative to edit the objectively incorrect post to fix the error. As I mentioned in the comments, I thought this was not according to the SE rules, which is why I did not do it myself even though I really really wanted to. If the community thinks this is the way to go I have no problems with doing it quietly without making a fuss. It would also in my opinion be a viable solution to the whole issue of 'undeserved' upvotes.

Thirdly, someone felt that I was claiming access to some kind of (absolute?) truth. No I don't. But correct mathematics with respect to modern standards is not subjective. A correct statement is one that is provable in the chosen foundations (usually assumed to be ZFC), and a correct proof is a valid sequence of deductions in that chosen system (in practice at least a description that logicians can easily see is translatable into the foundational system). That is very much objective enough to any modern logician, and that is what I mean by "correct". For example, "$1+1 = 3$" is incorrect; anyone who disagrees should provide precise descriptions of their non-mainstream foundations or notations rather than claiming it is correct from a certain point of view or saying that someone highly qualified said it.

(I'm lenient with missing assumptions or logical gaps if and only if the expected audience is estimated to be largely capable of filling them in correctly. But the two posts I am mainly referring to here are simply false and there is nothing for me to be lenient about. I talk about this only because it's relevant to another post mentioned at the end.)

Fourthly, if this feels like a complaint, it is. Sorry if it offends anyone but I want to see Math SE remain a reliable repository of mathematics, and this is what I right now feel is a necessary topic of discussion to help to achieve that outcome.


[Original post]

The two top-voted answers to this question are of such poor quality and I don't understand why none of the many upvoters have noticed. The answer by David essentially states that we use the square-root in RMS speed simply because it gives the correct units for velocity, which is not even true strictly speaking because the final quantity is not any kind of velocity at all. The answer by Yves Daoust is worse, affirming the false claim in the question that the average speed is zero. He/she even claims that his/her answer is at the level of the OP, but the falsehood is not defensible. Why does it seem like most Math SE users are carelessly upvoting answers without reading carefully?

More pertinently, what can be done about it? Downvoting as per this meta-post clearly fails, and review queues almost always fail (The Not-An-Answer flag is specifically for posts that do not even seem to address the question, and the Low-Quality flag is nearly always rejected for posts that are long). Also, these wrong answers are not as bad as fake answers, but it seems we cannot even agree on flagging to get rid of fake answers.

Note: There are even 4 completely nonsensical answers at the bottom, 2 of which are deleted. One talks about exams, as if that has anything to do with valid mathematics! It also makes the rubbish claim that $x = y$ iff $\sqrt{x} = \sqrt{y}$. Another one says that RMS "rectifies" negative to become positive, which cannot even be made sense of. I'll leave you to peruse the other 2 yourself. But these do not appear to be a problem probably because they came late and do not gather upvotes fast enough.

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    $\begingroup$ And the downvotes to this question only serve as more evidence that Math SE users don't care enough about mathematical correctness. I'm probably not going to recommend people to learn from this site if this downhill trend continues. It's really the first time I see such a heavily upvoted answer that is seriously wrong and refuses to admit the error but has no downvotes. $\endgroup$ – user21820 Jan 17 '17 at 16:10
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    $\begingroup$ Every year, somebody complains about Eternal September... ;) Note that pressing the "upvote" button does not necessarily demand that the one doing the pressing expend some amount of thinking; thus... $\endgroup$ – J. M. is a poor mathematician Jan 17 '17 at 16:20
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    $\begingroup$ @J.M.isn'tamathematician: You see, I'm not really complaining about the upvotes per se, but rather that approximately 0 out of 25 Math SE users who read the two posts did not find the serious error. $\endgroup$ – user21820 Jan 17 '17 at 16:20
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    $\begingroup$ @J.M.isn'tamathematician: Let me put it another way. Downvotes are supposed to be the mechanism that prevents mathematically incorrect answers from getting to the top. It doesn't, and one major reason is that many people don't read carefully before voting, and another major and more annoying reason is that people who can see the errors do not want to downvote for whatever reasons. See, everyone here who responds to my complaint but doesn't downvote either supports the incorrect answers or disagrees with downvoting's intended purpose. And many will tell me not to tell others how to vote... $\endgroup$ – user21820 Jan 17 '17 at 16:24
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    $\begingroup$ Could you try a title with a less clickbaity flavour? Your post seems to be a rant rather than an identification of a trend. $\endgroup$ – Pedro Tamaroff Jan 17 '17 at 17:00
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    $\begingroup$ As noted below in my answer, one of the "wrong" answers that you've mentioned is actually correct in context of the question. $\endgroup$ – Brick Jan 17 '17 at 17:07
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    $\begingroup$ Here are the current view counts on said $\rm\color{#c00}{question}$ and the $10$ questions posted before and after it: $\ 21,28,53,15,26,46,70,33,17,7,\color{#c00}{3569},27,106,60,29,23,20,31,14,8,37.\ $ So being Hot Listed likely caused the question to get $106$ times as many views at its neighbors average $ (= 33.6). $ That's probably the root of the problem. $\endgroup$ – Bill Dubuque Jan 17 '17 at 22:10
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    $\begingroup$ "It also makes the rubbish claim that $x=y$ iff $\sqrt{x}= \sqrt{y}$" Rubbish? Seems correct to me. With the obvious proviso that the quantities all make sense to begin with. $\endgroup$ – quid Jan 18 '17 at 0:26
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    $\begingroup$ @quid: You are certainly capable of interpreting it in such a way as to make it true (namely you know all the necessary assumptions). However, from years of experience it is clear that majority of students at the level of the asker do not know. And that is the problem. It's not enough for an answer to be possibly interpreted correct. Guess why tons of people believe that the sum of all positive integers is negative? $\endgroup$ – user21820 Jan 18 '17 at 2:57
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    $\begingroup$ @amWhy: I certainly understand and appreciate your advice, and I do agree that I was more blunt than other people would generally be. But I have noticed that unfortunately most users on Math SE do turn a blind eye towards incorrect mathematics. I believe it is an instance of apathy. Consider what I said to J.M about the total non-existence of downvotes before mine and my meta-post, which shows clearly the failure of the downvoting mechanism (and certainly not the first time). Part of the reason behind my meta-post is to cajole the community into taking a serious stab at the problem. $\endgroup$ – user21820 Jan 18 '17 at 3:32
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    $\begingroup$ @amWhy: In other words, I want the community to stop 'sitting on the fence' and come to a clear consensus of either doing something about the rapidly decaying quality on Math SE (it is not just HNQ) or stating clearly that they do not really care and will leave it to the voting system. If there is a viable alternative, I don't see it. I've been here since the 2nd year of Math SE, and from what I recall there was a significant decrease in overall quality even though there is also a significantly greater amount of valuable mathematics here now. I don't believe it's good to have SNR < 100:1. $\endgroup$ – user21820 Jan 18 '17 at 3:42
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    $\begingroup$ @user21820 in the context of the question there was no issue to begin with. I do not think there are tons of people that actually believe the sum of all positive integers in negative. $\endgroup$ – quid Jan 18 '17 at 7:40
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    $\begingroup$ @quid: Hmm see math.stackexchange.com/q/816250 and math.stackexchange.com/q/39802 (with the first two correct answers having ranks 4 and 7!) and physicstoday.scitation.org/do/10.1063/PT.5.8029/full (they want people debating rather than knowing math!). You'd probably have to ask those who watched the nonsense-phile video yourself to believe what I say, but consider that it's equally hard to believe that some people actually embrace other rubbish like deconstructionism. I better close my mouth before some more rants come out. $\endgroup$ – user21820 Jan 18 '17 at 8:04
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    $\begingroup$ @user21820 well if it is that users like Matt E are "incapable of giving correct answers" our problems might run quite a bit deeper than the HNQ. ;D [The quotes are scare quotes not literal ones.] I think the issue hinges on the meaning 'tons of people.' Yeah, I suppose there are a handful of people that got confused. But again that's rather a different matter. And to the extent it is similar it is the other way round. If there is a problem it is incorrect/ambigous use of the sqrt-symbol not the assertion that the equations are equivalent. $\endgroup$ – quid Jan 18 '17 at 10:23
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    $\begingroup$ @quid: You're beginning to appeal to authority. Whether or not you realize this, mathematics is unlike other fields where qualification is irrelevant. Arguably, that's what makes mathematics 'superior'. You certainly wouldn't accept someone with a few philosophy degrees as reliable source about the [non-]existence of God. By the way, it's not that users like Matt E are incapable but rather they do not write in pedagogically sound ways sometimes. As for the sqrt-symbol, you don't get my point. By your criterion "Pigs can fly" is correct and the problem is just ambiguous use of "fly". $\endgroup$ – user21820 Jan 18 '17 at 12:27
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If you identify incorrect answers to a post, you can downvote and comment, and you can also post an answer you deem better and (hopefully) correct. The site is not immune to errors, and it is the duty of all of us to make sure we preserve a database of correct and useful answers.

Another point to be considered is that posts like the one cited, which have more upvotes than usual and treat a less mathematical topic, tend to lure in more answers, which in turn implies more people missing the point, misreading, hurrying to post, and so on. This is a mere explanation of what is going on, and is not meant to justify the incorrect answers. If you look at other "hot" posts, you'll find a big junkyard of deleted or downvoted posts: it is like that.

Why does it seem like most Math SE users are carelessly upvoting answers without reading carefully?

This claim in the form of a question has little support at the moment, and I will ignore it.

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    $\begingroup$ The "Hot Nework Questions" feature of SE is a big mess. It ends up almost completely devaluing the voting in threads that were hot, because then the majority of votes tend to be placed by recreational readers with less expertise than the typical reader of this site. Such threads should be stamped with a big scarlet letter warning readers that the votes are even less useful than they normally are. $\endgroup$ – Bill Dubuque Jan 17 '17 at 18:33
  • $\begingroup$ @BillDubuque Agreed. $\endgroup$ – Pedro Tamaroff Jan 17 '17 at 18:34
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    $\begingroup$ Is there a forum where the mods can ask that this site not be included in the Hot Network Questions list? $\endgroup$ – Brick Jan 17 '17 at 18:58
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    $\begingroup$ @Brick the HNQ is an issue for many sites, this one is not among those that are worst affect by it. This was discussed plenty of times on Meta Stack Exchange The problematic side is acknowledge but overall it appears the positives are considered to outweigh the negatives. Finally, there is a hack-y way to get the question taken of HNQ on most sites. Put MathJax in the title; it'll then not be shown on sites without MJ. All that said, HNQ just amplifies the problem. If it is not intially well received locally it never ends up there. Let's not shift the "blame" elsewhere. $\endgroup$ – quid Jan 17 '17 at 19:02
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ – Pedro Tamaroff Jan 17 '17 at 21:37
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On one hand, I tend to agree with your concern generally across all SE sites. There are cases where questions are asked that clearly have fact-based answers, and the voting system is not always good about ensuring that the factually correct answers move to the top. I think that Math is better than some of the other sites in this respect because there are a lot of users who are BOTH active and qualified.

On the other hand, it's not clear what can be done about this since there's no way within the site to independently determine what's true and what's not aside from the voting system. Moreover, it's not always completely clear cut which answers are true since context may matter. For example, you've complained about the answer on this question that claims $\sqrt{x}=\sqrt{y}$ is the same as $x=y$. As a general mathematical statement, this is, of course, false. In context of this problem, however, it's absolutely true since the physics of the problem ensure that both $x>0$ and $y>0$. It might have been better if the answer pointed out this dependency, but there is certainly a sense in which this part of the answer is correct and you are wrong in your objection.

My approach to this problem is to vote, comment, and - as a last resort - leave sites that cannot maintain a minimum level of quality. I don't think Math is anywhere near that minimum threshold for me, but I have left other SE sites for this reason. Let's hope we can keep Math heading in the right direction.

I saw your comment about not answering because people don't read beyond the first few, by the way. I think that's not quite as true as you've made it out to be, especially when there are constructive comments on the top-voted answers that draw attention to the late-comers. In a case where there is a wrong answer at the top, I urge new answers even if it takes (much) longer for a late answer to move up the list.

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    $\begingroup$ What can be done? There are various possibilities, e.g. we could create a dedicated meta thread that lists posts where voting has gone awry (e.g due to questions appearing on the Hot Network list). This would at least give these warped threads better exposure in hope that the voting could be corrected to some degree (e.g to better represent opinions of folks here rather than rubber-necking drive-by users who rarely participate here). $\endgroup$ – Bill Dubuque Jan 17 '17 at 18:46
  • $\begingroup$ @BillDubuque That's an interesting suggestion to consider. I'm not sure how well it would work, however, since it's not clear that more attention is better. Just taking this example, we've got some apparently reputable and active users who have this question wrong. More attention on this one might further drive wrong answers or correct answers for the wrong reason since there are aspect of physics (all quantities are non-negative, difference between speed and velocity) that seem to be missed by the pure mathematicians. $\endgroup$ – Brick Jan 17 '17 at 18:53
  • $\begingroup$ I think you make some valid points, appreciating the context of "truth" (e.g., in the case at hand), and the frustration of the current measure of quality answer (upvotes), given that upvotes can mean can often measure factors than quality. Some upvote only because it matches what they think (in which no one has carefully read what is really being asked). Some value an alternative (even if not mainstream) perspective, Many vote for x, over y and z, simply because they like x, don't like y, and simply don't know z. $\endgroup$ – Namaste Jan 17 '17 at 20:35
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    $\begingroup$ @BillDubuque "There are various possibilities, e.g. we could create a dedicated meta thread that lists posts where voting has gone awry (e.g due to questions appearing on the Hot Network list). This would at least give these warped threads better exposure in hope that the voting could be corrected to some degree..." What criteria are we to use to determine where and when voting has gone awry? As you can see from this example, there is likely no consensus about the thresholds at which the divisions between excellent answers, good answers, okay answers, and bad answers should be drawn. $\endgroup$ – Namaste Jan 17 '17 at 20:43
  • $\begingroup$ @amWhy Nor is there generally any consensus whether or not questions should stay closed, deleted etc, yet we do have a meta thread to draw attention to cases which may benefit from exceptional handling or further attention. When I originally devised that thread the goal was for it to be devoted to all such exceptional cases.. $\endgroup$ – Bill Dubuque Jan 17 '17 at 21:00
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    $\begingroup$ @BillDubuque Don't you think that case is a bit different? It's one thing to have such a thread for an action that can be completed by a handful of privileged users and another thing to generate (in the case at hand as of now) 20+ downvotes to unseat the top answer. Moreover, to the extent that the Hot Network Question phenomenon mentioned elsewhere is real, it seems like this might be self-defeating as more action makes it more likely to get on the list, which then draws more attention from casual users as well. I don't know... Just putting it out there. $\endgroup$ – Brick Jan 17 '17 at 21:06
  • $\begingroup$ @BillDubuque I understand what you are saying. And I'd rather have a dedicated thread to address HNQ, than everyone following user 21820's choice to post a separate meta question about every disagreement about what merits the upvotes answer abc got, vs. the answer $xyz$. $\endgroup$ – Namaste Jan 17 '17 at 21:06
  • $\begingroup$ @Brick But I'm talking about questions that are/were already on the Hot list. It's not clear that attention from meta discussion would put them back onto the list. $\endgroup$ – Bill Dubuque Jan 17 '17 at 21:31
  • $\begingroup$ @BillDubuque I don't see that specified in the OP's question. In addition, the list is dynamic right? Even if already on the list, this would potentially keep it on the list longer. (I read a few of the meta questions on how the "hotness" is computed, and I was never really clear - in part because the definition seems to have changed over time.) $\endgroup$ – Brick Jan 17 '17 at 21:33
  • $\begingroup$ @Brick The question in the OP was indeed on the Hot List (I saw it there). That's how it got over 3500 views in 1.5 days. $\endgroup$ – Bill Dubuque Jan 17 '17 at 21:37
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    $\begingroup$ See my response to quid about the square-root thing. As for readers reading the top few answers and ignoring the rest, it's easy enough to verify my claim by observing the voting behaviour on popular questions. For example, if there are equally good answers the lower-voted one will gain upvotes much more slowly than the top-voted one, even if it is just the second answer! How do I know they are equally good? On some of these questions the asker sometimes accepts the lower-voted answer, and during the period of acceptance the voting disparity reverses. Unaccept and it goes back to before. $\endgroup$ – user21820 Jan 18 '17 at 3:15
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    $\begingroup$ Of course I hope that Math SE does not deteriorate into a Yahoo Answers eventually, but if it goes that way I'll just leave as it is in my personal opinion bad to mix correct and incorrect mathematics together. The correct mathematics will give an invalid impression that the site is reliable if it is actually unreliable and full of junk. $\endgroup$ – user21820 Jan 18 '17 at 3:17

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