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This question is not the way I supposed it was going to go.

Let me give a little contextualization before proceeding.

I tried everything which was available in my hands to properly fix the very problematic prelude of this answer, along with a comment, which states that trying to apply one of the most well-known and foundational tool of real analysis "leads us into a morass of missing definitions" etc. Namely, I:

  • Tried to address OP directly. He tacitly ignored me.
  • Downvoted the answer.
  • Gave specific references, and answered using said tool.
  • Tried gathering attention in chat, and was also ignored.

My next step was to post the answer in meta directly, but I thought it would be probably brushed aside as "off-topic", and would yet again be ignored.

I hate when big problems are ignored, and negligent spread of misinformation is among one of the most prominent intellectual problems. The fact that the answer is currently with 7 upvotes (a net positive of 4 upvotes) and is due to one user of great reputation only makes it even more prone to being taken as granted.

Feeling completely out of resources, I decided to make a test. I repeated, almost exactly, the wording of the answer alluded above here* (including the comment*). And I tried to mimic almost every problematic aspect of the aforementioned answer, as you can see.*

I don't think it is a good thing to be done, much less the optimal one. But it was the option that was left for me to do. My intention was to see what happened by the end of the day. If there were too much downvotes, too much comments or whatever, and gather conclusions and bring the discussion here.

What I did not expect is that there is one upvote. And that is it. That obscenity of an answer (which I will delete soon enough) has one upvote. No comment telling how it is atrociously misleading or etc.

I've learned from here, still do. I think we have one of the most precious havens of mathematical knowledge on the internet. My intention with this post is to open this "issue" up for reflection and debate:

  • Are these isolated problems?
  • Why do we put so much scrutiny over less-established users, and so little over well-established ones?
  • How (if even possible) can we improve?
  • Do you think this is an overreaction? Why?

*EDIT: I've already deleted the answer and comment. For the sake of completeness, those were (respectively):

Answer:

Trying to use an argument of union leads us into a morass of missing definitions (what does the union of a countable collection of sets mean, for example), but we can see something of what happens when we use Cantor's theorem: With effect, the collection of such sequences is nothing more than $2^{\mathbb{N}}$. By Cantor's theorem, it can't be in bijection with $\mathbb{N}$, and therefore can't be in bijection with $\mathbb{Z}$.

Comment:

You can't even begin to talk about the union being countable unless you have a definition for a union of a countable family. What would such a union even mean? Usually a union is something of the form $A_1 \cup \cdots \cup A_n$, but here we would have a countable collection. Do you even have a definition that gives meaning to that? (E.g., the usual definition needs the concatenation of $x$ belongs to $A$ or $B$ or etc, but here you would need an infinite number of or's).

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    $\begingroup$ I think you are overreacting a bit. A "morass of missing definitions" is not the same thing as being meaningless. One can provide the missing definition in term of the Frechet derivative, but it is not clear from the question that this would be at a level appropriate for the OP. What the answer you criticise does is explain why OPs argument does not work and therefore answer the actual questions: "So something is very wrong here, no? Can anybody explain to me what just happened?" $\endgroup$ – Michael Greinecker Dec 7 '17 at 2:04
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    $\begingroup$ It seems that some OP upvote every answers they have as "an appreciation". This is unfortunately not good for our site. $\endgroup$ – user99914 Dec 7 '17 at 3:05
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    $\begingroup$ Comparing a disagreement about the chain rule to that poem about the Nazis' extermination of their enemies leaves a bad taste in my mouth. $\endgroup$ – Gerry Myerson Dec 7 '17 at 6:29
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    $\begingroup$ @AloizioMacedo The chain rule for functions in one variable is plainly not the same thing as the chain rule for Frechet derivatives and one can prove the former in a way that does not generalize to the latter. And the chain rule for functions in one variable, which OP clearly tried to apply, does not apply to the problem under consideration.(...) $\endgroup$ – Michael Greinecker Dec 7 '17 at 10:32
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    $\begingroup$ And while explaining that a chain rule for a more general notion of a derivative is the appropriate one and does apply is a legitimate way to answer the question, so is explaining that the chain rule for functions in one variable does not apply and isn't even meaningful in this context. I hope this answer what you call the "objective problem". $\endgroup$ – Michael Greinecker Dec 7 '17 at 10:32
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    $\begingroup$ Whether this is true will depend on a number of things, mostly on whether the idea of the derivative as a linear approximation is familiar to the OP. But I see no point in the drama, just two people disagreeing on what is pedagogically appropriate here. $\endgroup$ – Michael Greinecker Dec 7 '17 at 10:53
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    $\begingroup$ Just for sake of mathematical completeness, the comment about the unions is entirely wrong. $\endgroup$ – Asaf Karagila Dec 7 '17 at 12:03
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    $\begingroup$ "GerryMyerson I understand." I can't help thinking that if you understood, you would do something about it. "That said, what is "disagreement" about the chain rule? How can one "disagree" with a theorem?" Can you not distinguish between disagreeing about something, and disagreeing with something? "one of the comments is essentially dismissing every point I made and the above aspect of this situation to 'hey, Godwin's law.'" Godwin's Law, as you know, asserts that "As an online discussion grows longer, the probability of a comparison involving Hitler approaches 1" [continued] $\endgroup$ – Gerry Myerson Dec 7 '17 at 12:21
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    $\begingroup$ @GerryMyerson Fair point (please see extended discussion). But the matter of fact still stands that you chose to ignore everything else about the post and focus solely on my lack of touch. $\endgroup$ – Aloizio Macedo Dec 7 '17 at 12:27
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    $\begingroup$ [continued] In this case, the implied comparison involving Hitler took place in the 5th paragraph of the original post – the whole thread started with a comparison involving Hitler, so I don't think Godwin is rlevant here. Finally, to deal with material issues: you're not happy with an answer and a comment; you've done everything you could to bring your dissatisfaction to the attention of the perpetrator and the community at large; what do you want? Anyone who is interested can see the other's words and your words, and draw her own conclusions. [continued] $\endgroup$ – Gerry Myerson Dec 7 '17 at 12:30
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    $\begingroup$ [continued] There is no math.stackexchange police force to arrest the miscreant; no math.stackexchange court to fine him 10,000 reputation points; no math.stackexchange House of Representatives to impeach him for high crimes and misdemeanors. You've done what you can, and you have discovered that we live in an imperfect world. My condolences. Perhaps you will be comforted by xkcd.com/386 $\endgroup$ – Gerry Myerson Dec 7 '17 at 12:34
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    $\begingroup$ @GerryMyerson I don't see how your fatalism is constructive in any shape or form. $\endgroup$ – Aloizio Macedo Dec 7 '17 at 12:41
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    $\begingroup$ Thank you for removing the reference to the poem. $\endgroup$ – Gerry Myerson Dec 7 '17 at 20:41
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    $\begingroup$ @GerryMyerson You were correct to "call me out", and I appreciate it. On a different note, I still don't see the purpose of your fatalism. $\endgroup$ – Aloizio Macedo Dec 7 '17 at 23:08
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    $\begingroup$ @AloizioMacedo : I looked over the original question. You allude to a comment that is not visible there; perhaps that comment was deleted based on your concerns. Both answers seem to identify that definitions and results should be treated carefully when extending calculus to matrix-valued functions. It seems hostility was created from actions of downvoting and perhaps from harsh words like "misleading." I wonder what would have happened with a direct and specific comment such as "Just to clarify, definitions do exist but operations should be carried out by..." or something more graceful. $\endgroup$ – Michael Dec 8 '17 at 20:43
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What is happening with the trustworthiness and reliability of answers on MSE?

I don't think overgeneralization is appropriate: your observation is based essentially on only one particular answer but you are questioning the whole site.

I tried everything which was available in my hands to properly fix the very problematic prelude of this answer, along with a comment, which states that trying to apply one of the most well-known and foundational tool of real analysis "leads us into a morass of missing definitions" etc.

Henning's intension was distorted; the following two sentences are different:

  • Trying to use a chain rules leads us into a morass of missing definitions (what does it mean to differentiate a function whose input is a matrix, for example)
  • trying to apply one of the most well-known and foundational tool of real analysis "leads us into a morass of missing definitions"

You were ignoring the words put in the parentheses.

I hate when big problems are ignored, and negligent spread of misinformation is among one of the most prominent intellectual problems. The fact that the answer is currently with 7 upvotes (a net positive of 4 upvotes) and is due to one user of great reputation only makes it even more prone to being taken as granted.

I disagree with "... is due to one user of great reputation". Do you have any evidence to support this?

Finally, I would like to repeat my comments under your linked answer:

I disagree with the downvotes: this answer does not avoid the question. OP asks "So something is very wrong here, no? Can anybody explain to me what just happened?" NOT "why is this not a counterexample to the chain rule?", which is a question that one of the downvoter subjectively assumed. Moreover, "For example, we cannot replace the $f(t)g'(t)$ term with $g'(t)f(t)$ and expect its value to stay the same." explains why OP gets "something is very wrong".

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  • $\begingroup$ With respect to one: Not that it is any much better, but it is actually based on two instances: one which was artificially created consciously to be bad and was also upvoted. Extrapolating in two is indeed dangerous, which was why I asked: "Are these isolated problems?" - an attempt to see if any other user had experiences to share. It turns out that, indirectly, there are. If you see the discussion in chat, you will see that there are instances of people having preemptive predispositions based on how "established" someone is. Although there this was only reflected in negativity (...) $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:38
  • $\begingroup$ (...) my point is that this is probably just the other end of the spectrum of a similar rationale. $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:39
  • $\begingroup$ With regards to "intentions", I have no doubt that the user has the best ones. His contributions to this site by any reasonable metric far outreach mine, and far outreach most of the other users. My point being that it does not shield him from scrutiny. I don't believe that as he wrote "What would such a derivative even mean?" he was upholding information, that is absurd (I don't know if this is what you are suggesting). In all honesty, I believe the greatest possibilities are: he didn't recall the concept at the time, or he wasn't aware of the concept. $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:43
  • $\begingroup$ Which is completely fine. However, my objection is with respect to ignoring when I tried to point out that he was mistaken: there is such a concept. As people said in the comments of this question, this is a trivial matter. In my point of view, it would amount to something like "I wasn't aware, let me clarify" or "Oh, completely forgot! Thanks" and clarification. Unfortunately, the "reaches" of this discussion reached dramatic levels, with this post having a dramatic nature. However, I don't think that there could be another solution, and let me explain by an analogy: $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:51
  • $\begingroup$ Suppose you are last in a line, and someone enters it with a friend in front of you without noticing you. It is not even ill-intentioned, just a mindless blunder. This is all a trivial situation. Now, suppose you tell the person "hey, you cut the line". As trivial as the situation is, the person acknowledges this and simply goes back. Now, suppose that instead of doing this, the person ignores you. You try to reach his friend, and he ignores you. And then you try to talk to someone else in the line, who is not interested (...) $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:54
  • $\begingroup$ Finally, you talk to a security guard in charge of the line, and he also ignores you. You have two options: realise that the world is imperfect, or continue with your inquiry. Now, when you do, it will naturally be overblown. Almost by definition. But is it fair to shoot the person doing such inquiry, instead of the means which allowed this triviality to reach its heights? $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:56
  • $\begingroup$ Continuing responding your question: I did not ignore the words in parenthesis. In fact, I emphasized them in my head, and responded (in other words):"it means what it means, and what is well-known to mean. What is the point? This only confuses the unaware". $\endgroup$ – Aloizio Macedo Dec 9 '17 at 3:58
  • $\begingroup$ "I disagree with "... is due to one user of great reputation". Do you have any evidence to support this?" I don't understand what you "disagree" with. He is one user of great reputation. In this particular instance, I was talking about the objective "points", which are not a subjective matter. Although I also agree (see chat conversation) that the user is (deservedly) well-established in the site, and this is also relevant for the discussion. You (not you in particular, the community) are talking like I am in a personal vendetta against the user. I am not. $\endgroup$ – Aloizio Macedo Dec 9 '17 at 4:00
  • $\begingroup$ I've been trying to be the most explicit, objective, neutral and logical in all arguments I put forward. The drama is not with respect to the user, it is with respect to how I handle those facts. By quickly glancing through my comments and through the conversation in chat, you will even see that the moments where I let my emotion flow somehow is when I talk about the "IGNORING" aspect, not the user. Thus, if I were to be "angry" with anyone (which, again, I am not), it would be with a whole stratum of the community. $\endgroup$ – Aloizio Macedo Dec 9 '17 at 4:03
  • $\begingroup$ Finally, with respect to your last point: I addressed that in the question itself. Last but not least, I appreciate if you read everything up to now. $\endgroup$ – Aloizio Macedo Dec 9 '17 at 4:04
  • $\begingroup$ (Sorry, where I said "upholding", I meant "withholding") $\endgroup$ – Aloizio Macedo Dec 9 '17 at 14:54

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