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Sometimes users post questions in which

(1) mathematical terminology is used in grossly incorrect ways, and

(2) sentence structure is abominable, and

(3) grammar and spelling are quite bad, and

(4) nonetheless the meaning of the question is crystal-clear with next to no effort, and

(5) the posting clearly shows that the poster has thought about the question, and

(6) it clearly indicates at what point the poster had some difficulty with it, and

(7) it's just the kind of question that a good student learning the material ought to ask, and

(8) indicates what thoughts the poster had toward answering the question.

Among mathematicians who having been posting regularly to m.s.e. for a decade there are some (and they seem like a clique of about a dozen or two) who always vote to close the question, often avowing that it is incomprehensible or off topic or otherwise objectionable despite points (4) through (8) above.

Concerning one such question posted on December 26, matching points (1) through (8) perfectly, and that was closed and deleted on the grounds of its alleged incomprehensibility despite its perfect clarity, a respectable mathematician who is a professor with many publications, recently opined to me that he had difficulty understanding it. He differs from the "clique" above in that he is willing to communicate, and I know that he is honest. His claim that there is something difficult to understand in the question is incomprehensible to me, but his willingness to communicate, which is in such sharp contrast to that of the aformentioned "clique", convinces me that he is honest.

Here is a link to the question posted on December 26 (which some user here will not be able to see because it is deleted): https://math.stackexchange.com/questions/4342155/what-makes-a-number-big/4342169#4342169

Since some won't be able to see this, here is the whole question verbatim:

Limit to infinity of a function are explained as "the value of the function as $x$ gets really really big". My question is, what is big? Big and small are relative terms. So what makes a $x=1$ small and $x=100000$ big unless they are compared?

And here is my posted answer, with a net 6 upvotes and the O.P.'s acceptance:

$$ 5.1,\quad 5.01,\quad 5.001, \quad 5.0001, \ldots, \quad 5.\underbrace{000\ldots\ldots000}_{n \text{ 0s}} 1, \quad \ldots\ldots $$
The limit of the sequence as $n$ grows is $5.$

Loosely, one might say that that means it is very close to $5$ with $n$ is very big.

But more precisely, one can say that it can be made as close to $5$ as desired by making $n$ big enough. How big is big enough depends on how close one is to make the $n\text{th}$ term to $5.$

More precisely still: No matter how small the the distance one wants between the $n\text{th}$ term of this sequence and $5,$ there is a value of $n$ so big that the $n\text{th}$ term and all later terms will be within that distance from $5.$

Several persons in comments indicate that they understand the question (as expected), and one with a reputation of more than 50000 calls it "unclear" and "meaningless" and "[not] even about mathematics".

To me it is incomprehensible that a mathematician who has taught calculus, or probably even one who has not, would fail to find the meaning of the question crystal-clear, and it is not a lack of ability to understand such matters by which I achieved a reputation of more than a quarter million on m.s.e.

But it is even more incomprehensible that the members of this clique are unwilling to be minimally polite. They have been persistently unwilling to discuss things like this for about a decade—or maybe more. Their communications to me about these things consist of dogmatic assertions and orders to me to believe them and obey them. On many occasions over the past decade I have brought these matters to the attention of the moderators and they have never once replied. (In particular, there is nothing in this present posting of which the moderators have not been recently reminded.) Their persistent non-replies have been brought to the attention of the community managers several times, and they have also never replied.

Sometimes people post on subject matter of which I am ignorant, and I do not participate in what ensues. I do not complain that such questions are incomprehensible and meaningless and not about mathematics, and I do not order those who disagree with me to believe me and obey me when I say that.

Questions: Are there really mathematicians who can't understand questions like this one? Why? Why do they feel a need to order a contributor with a quarter-million reputation to believe and obey their assertions that something is incomprensible if they are not among those who understand it? Why are they unwilling to discuss anything, preferring instead to issue such orders?

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    $\begingroup$ PS: In case anyone wonders why I don't mention names: the moderators have warned me not to specify user names of any of the posters mentioned here. Those who can see the posting at the URL I gave will see some, but far from all, of them. $\endgroup$ Mar 3 at 7:12
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    $\begingroup$ You state: "a respectable mathematician who is a professor with many publications, recently opined to me that he had difficulty understanding it." So unless you or the person you mention are lying, then the answer to your first question "Are there really mathematicians who can't understand questions like this one?" is "Yes." $\endgroup$ Mar 3 at 7:28
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    $\begingroup$ I think the site has some visitors who are not mathematicians (including myself) and it is always better to add relevant context. Often if the asker is willing it is possible to make the question clear enough by having discussion. So all efforts should be made to clarify the question and add relevant context. Closure of a question is the first step in this effort and it signifies that there is a need to improve the question. Unless this is done no one will bother to improve it. $\endgroup$
    – Paramanand Singh Mod
    Mar 3 at 7:51
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    $\begingroup$ If you want to undelete and reopen the question referred in your post you may use this meta thread: math.meta.stackexchange.com/q/34447 $\endgroup$
    – Paramanand Singh Mod
    Mar 3 at 7:58
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    $\begingroup$ This FAQ is surely a dupe, so instead of posting yet another dupe answer, you should have found a good dupe target. In case you might not know, users are being suspended for continually answering PSQs & dupes, cf. recent site policy announcement here. The goal is to evolve the site into "proofs from the book". Posting hundreds or thousands of dupe answers to FAQs will make those "best" answers impossible to locate - like needles in a haystack - and will inhibit iteratively refining prior excellent answers into proofs from the book. $\endgroup$ Mar 3 at 10:11
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    $\begingroup$ Post on the the thread linked by Paramanand above, create a good argument and initiate discussion, perhaps one leading to a reopening. It can still be done. Closure is not ordering people : it can be reversed and discussed for starters. For example, in this question, I would likely have the same response as the last comment on that thread, to the effect of : without the proper definition of a limit as studied by the author, it would be difficult to express best to the same author, the meaning of "big". I'm not sure "the value of the function as x gets really really big" is proper. $\endgroup$ Mar 3 at 10:40
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    $\begingroup$ I think suggesting that Michael post to the Reopening thread is seriously missing the point of Michael's post here. $\endgroup$ Mar 3 at 12:08
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    $\begingroup$ @GerryMyerson I can understand what you mean, perhaps the point is more of "why do people close such questions?" rather than what can be done about it, which is what I suggested. I am driven to suggest this to refute the notion that closure is "to be obeyed" and "isn't up for discussion" which seemed to be the content of the last two lines. Closure shouldn't be perceived as an "issued order". Point taken, unless you want to add to this. $\endgroup$ Mar 3 at 13:48
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    $\begingroup$ @SarveshRavichandranIyer : It is indeed treated as not up for discussion. Persons who favor closing such things have for years been unwilling to discuss such matters. $\endgroup$ Mar 3 at 15:21
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    $\begingroup$ I would start my discussion by mentioning that I have experienced occasions where I have seen comments or the first answer be poorly received or, in my opinion, completely miss the point. While I can attribute this to the answerer sometimes, this could imply that there are various ways of looking at the question and I remain concerned that the user and future visitors could be misled. Looking at your question : your answer of six upvotes uses sequences instead of functions, which could be confusing for users who learn function limits before sequence limits (happens in India). $\endgroup$ Mar 3 at 16:10
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    $\begingroup$ @SarveshRavichandranIyer "The principle itself is under question, but I'm not happy with the fact that you say we can do nothing about it, when we can post to that thread." You're wrong. Posting to the thread does not result in discussion, but rather results in orders begin given to those who propose undeletion. Speaking to such persons in sentences and talking about merits or demerits of anything is considered to be beneath those who are dominant in that thread. Speaking in reasoned complete sentences to someone with a quarter-million reputation is beneath their dignity. $\endgroup$ Mar 3 at 17:38
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    $\begingroup$ @Sarvesh, the Reopen thread is for undoing closures, one question at a time. Michael's post here is for preventing the closure of hundreds of questions. [Michael, my apologies if I have misrepresented your concerns.] $\endgroup$ Mar 3 at 21:51
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    $\begingroup$ I of course know very little of the total picture but my impression was that Michael Hardy is trying to help, even if spirited at times. I wish it did not have to come to a year long suspension. (And I understand mods do not discuss such particulars.) $\endgroup$ Mar 4 at 2:50
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    $\begingroup$ I find it quite interesting on how the downvotes increased considerably relative to upvotes after the suspension $\endgroup$ Mar 5 at 8:46
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    $\begingroup$ Spurious correlations don't mean much. The post has only been up for two days, and only a few hours between that and the suspension. If all the supporters of a post voted early, it both disguises the true community attitude and makes the later votes almost all negative. $\endgroup$
    – Nij
    Mar 5 at 21:37

2 Answers 2

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I wrote a blog post on this a while back to express my puzzlement about this issue. I said:

[A] question is groping toward something sensible but won't be all the way there, and then several people will reply saying no, that is not sensible, your idea is silly, without ever admitting that there is anything to the idea at all.

In the article I suggested a sort of hierarchy of responses to half-baked questions:

  1. Yes, that is exactly what we do, only more formally. You can find out more about the details in this source…
  2. Yes, we do something very much like that, but there are some significant differences to address points you have not considered…
  3. Yes, we might like to do something along those lines, but to make it work we need to make some major changes…
  4. That seems at first like a reasonable thing to try, but if you look more deeply you find that it can't be made to work, because…
  5. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question

I gave some examples of recent Math SE questions where the answers had been from the lower tiers, but, in my opinion, better answers were available in the upper tiers. Michael Hardy's example is a another, although my classification above isn't phrased correctly for it, since it is a “what does this mean" rather than a “can we do this”. A comment at the question claims that it is “a meaningless question that isn't even about mathematics” and apparently eight people agreed. But, like Michael Hardy, I am astounded at this response. The question seems completely ordinary and straightforward to me, the sort of thing that is treated in literally every introductory calculus text. (I agree with Bill that it should have been closed-as-duplicate, but that is not what happened.)

Our high-handed treatment of questions we don't understand makes the site less useful even for people who know exactly what they are doing. Recently I've had to correct a comment that claimed that a question about minimal logic was vague or meaningless because the questioner didn't say what they meant by “minimal”. But minimal logic is a technical term with a specific meaning, of which the commenter was unaware. A couple of months ago someone asked a perfectly reasonable question that began “Color each positive integer with red and blue”, as is quite standard. But immediately someone replied “It is impossible to answer your question without a definition of "color"”. I wish the commenter had been a little more aware about the limits of their own knowledge. There are many, many examples of this.

Math SE is supposed to be “for people studying mathematics at any level”. Not all of these people will be able to formulate their question in perfectly correct mathematical language. Often it seems to me that our standards for acceptable phrasing are impossibly high, even for professionals.

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    $\begingroup$ When that color example came, I could not stop laughing. If one takes such a silly attitude like the person asking question about "define a color" then most problems faced by students who are practicing Olympiad can not be solved at all. $\endgroup$ Mar 3 at 22:01
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    $\begingroup$ I've never closed a question for using non standard phrasing. But if the ask a psq, then standard phrasing or not, no thank you, and I link them to how to ask a good question. This question was asked by a user who is now suspended for one year, because of his lack of civility already expressed elsewhere. So I'm okay if you have a concern about which you wrote, and can cite examples. But I'd suggest you post a separate meta post, because this question you answered aimed at moderators. I trust you can actually, and usually do, post reasoned and civil questions/answered. $\endgroup$
    – amWhy
    Mar 3 at 23:14
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    $\begingroup$ That half-quote is, in my opinion, a bit misleading, because some people are taking it literally. The full quote "For people studying mathematics at any level, and professionals in related fields." clarifies the intent a bit better. This is still obviously open to interpretation. The level could (should?) be that of professionals in related fields who did not make math their vocation (but can be expected to have passed a few college courses in math). Another limitation is the minimum age (13 years of age), already ruling out questions about basic arithmetic, but that is not relevant here. $\endgroup$ Mar 4 at 6:49
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    $\begingroup$ @JyrkiLahtonen If that is indeed what the full sentence is supposed to mean, then it's very unclear, in my opinion. The only "natural" way I'm able to interpret that comma is as separating two distinct types of people at whom this site is aimed. But, in any case, I feel that the proper target audience is decided by Meta discussions and the implicit consensus obtained from the kinds of questions left open/closed on Main, not from complicated interpretations of one sentence in the tour. $\endgroup$ Mar 4 at 7:39
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    $\begingroup$ I don't understand how any of that has any bearing on my point, which was ”Often it seems to me that our standards for acceptable phrasing are impossibly high, even for professionals.” Yes, even professionals in related fields and even professionals who are over the minimum age of 13. $\endgroup$
    – MJD
    Mar 4 at 10:16
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    $\begingroup$ Thank you for the hierarchy and model, it makes sense somewhat. I would like to remark that , having gone through multiple review queues for about a year, I still remain to be convinced that answers are likelier to be of type $<5$ than of type $5$ if a question is left open. As I go through the review queues, I know that questions of the kind you mention occur, but I really don't know if I can be confident that the next person to answer that question is likelier to confuse the OP or provide a good answer. I would close a question if it is likelier to attract tier $5$ answers. $\endgroup$ Mar 4 at 17:51
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    $\begingroup$ (If I perceive that it is likelier to attract tier 5 answers) I use existing poor answers and comments requesting for clarification to justify some of these closures, but ordinarily it really depends on my gut and how much I know the material. For example, I usually encounter such a question at a nascent stage , where it's maybe got one comment which asks for clarification : for me, the safe side is to close and request for details , and the unsafe side is to hope for a tier $<5$ answer. $\endgroup$ Mar 4 at 17:54
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    $\begingroup$ For anyone else wondering at the familiarity of the elegant phrasing of (5), it is due to Charles Babbage. $\endgroup$
    – LSpice
    Mar 6 at 23:19
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Questions: Are there really mathematicians who can't understand questions like this one? Why? Why do they feel a need to order a contributor with a quarter-million reputation to believe and obey their assertions that something is incomprensible if they are not among those who understand it? Why are they unwilling to discuss anything, preferring instead to issue such orders

I haven't gone through your whole post, but I think I have some understanding of why people do this.

To find the meaning out of a question which is not very well defined/ bolster up the argument in the question to a level such that it is answerable using defined mathematics concepts is a difficult task which takes mental effort.

An experienced mathematician, depending on character, may want only questions that are approachable directly in the framework he has learned previously to exist on this site. Hence, the close votes.

Also, I would like to say that if this question was posed to the top mathematicians, then I feel they would certainly entertain it and also be able to answer it. Take for instance a book like Terence Tao's analysis, in the first chapter itself he discusses the issue of how ill-defined concepts causes problem, and in the later discusses intuition/ provisional definition before giving the exact ones.

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    $\begingroup$ I am in agreement with the general ideas suggested by OP. I think that MSE could improve inclusivity of different people in the site (in a good way) by policy shifts. $\endgroup$ Mar 3 at 12:45
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    $\begingroup$ Tao is undoubtedly a master at this. I never expected anything insightful to be gained from e.g. this MO question he answered. (also, its Terence Tao or Terry Tao :) ) $\endgroup$ Mar 3 at 13:17
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    $\begingroup$ Thanks @CalvinKhor for the nice thread on MO. He has expressed the desirable features of a notation in a very elegant fashion. $\endgroup$
    – Paramanand Singh Mod
    Mar 3 at 14:29
  • $\begingroup$ The framework missing is usually a mathematical description of something, quite often. Even here, an agreement of someone with OP of what the mathematical definition of a limit is, would have been great. Unfortunately, with the Tao citation, there seems to be even more subtlety because people could know only the intuition but not the definition. Here, actually it came to notice that OP didn't know implicit differentiation but did not state it, via a meta post. So sometimes, closure can prevent incorrect answers , I think. $\endgroup$ Mar 3 at 14:37
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    $\begingroup$ The issue can be understood more by meta question "what for?" as in why would a student ask such a question whose answer is probably hidden somewhere in an Analysis text or some more complicated field? The reason why students don't go on quests to search for answers in books, I think, is usually because finding the answer through that method is too difficult for the student at the moment for whatever reasons. The idea of posting on MSE, in my opinion, is so that an experienced expert can advice him how to resolve the the logical issue using only the relevant parts while discussing alternatives $\endgroup$ Mar 3 at 15:04
  • $\begingroup$ For solving the issue $\endgroup$ Mar 3 at 15:05
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    $\begingroup$ The above was why the other solution was not taken. A question still remains, why bother about this point at all? As in for high-school, we can just accept the points as facts and get by. If a student insists, I believe they have the same spirit of mathematicians of later centuries and must be encouraged and shown that people have thought of this, done this and that to make sense of it etc $\endgroup$ Mar 3 at 15:08
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    $\begingroup$ @Buraian I have no problem in inviting students to post ambiguous questions, after all we have faculties who are very good at clarification-based discussions and can help in improvement. My trouble is that I believe answers shouldn't be stabs in the dark, as much as comments may need to be, and that closure for a lack of clarity prevents answers to questions that may have perceivable signs of being misinterpreted, either via poorly received answers or comments requesting for details. I love the questioning spirit of users, but it still means that some questions need to be reworked. $\endgroup$ Mar 3 at 16:28
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    $\begingroup$ @CalvinKhor Not sure from where the error in spelling originated, but now there are a few posts on the main site that carry that misspelling of Tao’s first name. (This comment is strictly off-topic under this answer, so please feel free to flag it as “no longer needed” once you’ve seen it, if you wish.) $\endgroup$ Mar 3 at 16:30
  • $\begingroup$ As with everything, the question ultimately lies in the edges i.e: how much ambiguity should be allowed in the question. @SarveshRavichandranIyer $\endgroup$ Mar 3 at 16:39
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    $\begingroup$ Definitely @Buraian. The reception makes it clear how on-the-edge the question is. We'll see how the discussion goes in any case. $\endgroup$ Mar 3 at 16:41
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    $\begingroup$ BEGIN QUOTE An experienced mathematician, depending on character, may want only questions that are approachable directly in the framework he has learned previously to exist on this site. END QUOTE For a teacher of mathematics, that would be unethical. $\endgroup$ Mar 3 at 17:49
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    $\begingroup$ Depends, are professional mathematics neccesarily related to some teaching role? I am not sure. But, in my personal experience, teachers I had in real life were not that great , so it is not much of surprise to me. $\endgroup$ Mar 3 at 17:51
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    $\begingroup$ ( @TheAmplitwist My original impression was that Terrence is more common than Terence, but this seems wrong on a Google, and Terrance (OP’s original spelling) is also used. There is a recent discussion about Cauchy-Schwarz / Schwartz functions on meta.MO. Since this is meta I think these 3 off-topic comments can stay here; but we should discuss more in a chat room if needed.) $\endgroup$ Mar 4 at 2:18

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