Who did not graph such equations as $y^2-x^2=1$ in secondary schoool?

THEREFORE: Algebraic geometry (since it is only about zero-sets of polynomials in several variables) is high-school-level stuff, not research-level. Right?

I asked if what graphs of polynomial functions of sine and cosine look like. This was closed as "high-school level" stuff. Is it? There was an intelligent comment from Neil Strickland, and also a comment that it's an imprecise question and should therefore be closed. Obviously anyone maintaining that imprecise questions have generally been unwelcome on mathoverflow would lose any debate about whether that's true. Maybe the other point is more substantial.

Postscript: Angelo draws parallels to the proposition that every question about zero sets of polynomials is on topic. That proposition is plainly false.

Angelo commits a logical error: The correct parallel is to the proposition that every question about zero sets of polynomials is off topic. That proposition is also plainly false.

PPS: The absence of Andy Putman, Mariano Suárez-Alvarez, David Roberts, Misha, and Gerry Myerson from this discussion would be impolite if they knew it was happening. I've just notified them by email. However, the fact that mathoverflow's notification system has no way to handle such a matter is a flaw. Some users' email addresses cannot readily be found.

PPPS: I wonder if anyone anywhere knows the answer to the question I posted, which was held not to be a research-level question. Just this morning I derived a simple result: The function $\theta\mapsto\tan(\theta/2)$ is the identity element in a structure that one naturally considers when thinking about this question. Maybe high-school-level in the sense that if one phrased it as a precise question, a bright high-school student would prove it. But I wouldn't be surprised it hasn't been noticed before. And it would take an even brighter high-school student to think of asking that question. Which raises a question that I'll ask on "main".

PPPPS: One should ask a "focused question" with a "specific goal". So says Anton Geraschenko below, and I agree. But he suggests that imprecise questions cannot also be "focused" questions having a "specific goal". Then he retreats from that position. Just to be clear, here are some counterexamples: i.e

Examples of imprecise questions that are focused and have a specific goal and have large numbers of up-votes on mathoverflow:

Proofs that require fundamentally new ways of thinking

nontrivial theorems with trivial proofs

Not especially famous, long-open problems which anyone can understand

Sexy vacuity ....

Examples of common false beliefs in mathematics

(307 votes for this last.)

Most intricate and most beautiful structures in mathematics

(This last was asked by Richard P. Stanley, perhaps one of mathoverflow's most respected contributors.)

Examples of seemingly elementary problems that are hard to solve?

Theorems with unexpected conclusions

(Also asked by Richard P. Stanley, 55 votes.)


closed as off-topic by Andrés E. Caicedo, Angelo, Mark Sapir, Yemon Choi, Steven Landsburg Jun 27 '13 at 21:01

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about MathOverflow or the software that powers the Stack Exchange network within the scope defined in the help center." – Angelo, Mark Sapir, Steven Landsburg
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Let's see, what you are saying is: algebraic geometry is about zero-sets of polynomials, and is non-trivial, hence no question about zero-sets of polynomials is off-topic. Am I misinterpreting? This does not strike me as a particularly sensible argument. $\endgroup$ – Angelo Jun 26 '13 at 6:32
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    $\begingroup$ Aside from being grumpy, I think this question is perfect for this meta. Threads of the form "should this question be closed?" are core to what meta should be. For us, they used to be very long discussions, but the arguments are now well-worn enough that the Q&A format will work well most of the time. The better searchability and not having to register a separate account are huge advantages of this meta over tea, so we should prefer to put stuff here whenever it makes sense. $\endgroup$ – Anton Geraschenko Jun 26 '13 at 12:58
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    $\begingroup$ (Why the rudeness? Please remove the quotation marks.) $\endgroup$ – Andrés E. Caicedo Jun 26 '13 at 16:51
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    $\begingroup$ Apologies - I happened to be sleeping and other real-world things. $\endgroup$ – David Roberts Jun 26 '13 at 20:34
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    $\begingroup$ @MichaelHardy : I saw no reason to participate in this discussion. I think that the union of Anton's post and Angelo's comment says all that I would want to say. If there were some actual support for reopening the question, then there would be a reason for me to join the discussion, but as it is you have been told clearly why the question is not appropriate. If you choose not to listen, then that is not my problem and I see no reason to get dragged into a morass of negativity. $\endgroup$ – Andy Putman Jun 26 '13 at 21:26
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    $\begingroup$ @MichaelHardy : I don't want to get dragged into this, but I do want to correct your mischaracterization of my comment on your original post. I said that the topic of the question (namely, the visual appearance of graphs of functions) was high-school level. And I stand by this. But I cannot answer your question myself because it isn't actually a well-posed mathematical question. If anything, this is even more pertinent to its closing than the level. See Anton's answer for what I mean by that. $\endgroup$ – Andy Putman Jun 26 '13 at 22:14
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    $\begingroup$ That "intricate and beautiful" question, or rather, all the answers which ignored the wording and preferred to talk about OMG COOL, really got my goat $\endgroup$ – Yemon Choi Jun 26 '13 at 22:35
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    $\begingroup$ You seem to enjoy pointing out logical errors made by others, but it seems that you have made one of your own. While algebraic geometry is often about the structure of zero sets of polynomials, this does not imply that algebraic geometry is only about the appearance of graphs of zero sets. Your conclusion that algebraic geometry is high-school-level is therefore not supported by your sequence of logical leaps. More importantly, I would prefer if you could find a way to make your point without insulting all algebraic geometers. $\endgroup$ – S. Carnahan Jun 27 '13 at 4:17
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    $\begingroup$ In general, you seem to be addressing the closure of your question with an overly confrontational writing style. For example, if you removed all instances of "plainly", "obvious", and "obviously" from your question, it would communicate the same point, but with a "lower temperature". Similarly, the people who closed your question have no obligation to participate in this discussion. $\endgroup$ – S. Carnahan Jun 27 '13 at 4:26
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    $\begingroup$ Just to make sure that the record is straight here, said email was in response to your unsolicited emails to me. Its main purpose was to make clear that I do not welcome further emails and do not wish to argue with you. $\endgroup$ – Andy Putman Jun 27 '13 at 4:54
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    $\begingroup$ Under 35? Where does this come from, given that neither Angelo Vistoli nor Gerry Myerson fit that bracket? This does not seem a productive line of argument, unless you feel ad hominem and mindreading is productive $\endgroup$ – Yemon Choi Jun 27 '13 at 6:52
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    $\begingroup$ Moreover, I do think that down-votes and closure are one of the main ways to maintain quality on MathOverflow. Many is the idle question or philosophical musing that I have not put on MO. $\endgroup$ – Yemon Choi Jun 27 '13 at 6:53
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    $\begingroup$ "I've just notified them by email. [...] Some users' email addresses cannot readily be found." Thank you for making one of the reasons why using a pseodonym can be a good idea more well-known. $\endgroup$ – user9072 Jun 27 '13 at 19:52
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    $\begingroup$ @MichaelHardy: while I think I agree it could be a good addition to the notification system that people that voted to close/reopen or perhaps also edited a post can be notified (which as far as I know is not possible, though I am completely new to this notification thing so this might be wrong) and while personally I am also (I think) pretty good about following up on MO matters on MO, under no circumstances would I want to receive email about such matters and sending them strikes me (but one might see this differenlty) as quite inapproriate . $\endgroup$ – user9072 Jun 27 '13 at 20:12
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    $\begingroup$ @MichaelHardy: the point to me is that while often times I am very active on MO, I have complete control over when I am active and when not. If I do not visit the site it is simply "gone" from my daily routine. If now I would receive emails, even more so when it would be to my general address, related to it this would stop being the case (that I can simply 'turn off MO'). I think I would also feel like this if I were posting under my real name. $\endgroup$ – user9072 Jun 27 '13 at 21:12

Obviously anyone maintaining that imprecise questions have generally been unwelcome on mathoverflow would lose any debate about whether that's true.

There's where your problem is. I maintain that imprecise questions have been (and should be) unwelcome on MathOverflow. See the very first recommendation about how to ask a good question on the old how to ask page (moved since the migration): "Ask a focused question that has a specific goal."

If it is not clear what constitutes an answer to your question, then you haven't done enough work to put it on a Q&A forum. Even if your question is just idle speculation, it's really important to make up some clear conditions for what constitutes an answer so that people don't burn a lot of cycles trying to understand what you want. If it turns out you want an answer to some other precise question afterwards, you can ask that other question later.

  • $\begingroup$ You seem to be saying that a focused question that has a specific goal cannot also be an imprecise question. I think I can probably find counterexamples to that among questions asked and answered on mathoverflow. $\endgroup$ – Michael Hardy Jun 26 '13 at 12:27
  • $\begingroup$ I agree that a focused question can still be imprecise. I feel the problem with your original question is that it was not focused. For example, I'm not sure if Baby Dragon's comment (or my follow up to it) answer your question. $\endgroup$ – Anton Geraschenko Jun 26 '13 at 12:31
  • $\begingroup$ I've posted a list of the aforementioned counterexamples as a postscript to my question. Doubtless it's incomplete, but it should be enough. $\endgroup$ – Michael Hardy Jun 26 '13 at 17:42
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    $\begingroup$ @MichaelHardy: Please stay on topic. I thought you wanted to know about why your question was closed. I believe I've explained that in this answer. Why are you arguing with me about the relative definitions of "imprecise" and "unfocused" and editing your question to put words in my mouth? Berating me with examples or angry postscripts will not make your original question better. Making it clear what constitutes an answer will. $\endgroup$ – Anton Geraschenko Jun 26 '13 at 17:55
  • $\begingroup$ I was not angry, and I don't think I misrepresented what you said. I think I was on topic because it was suggested that being imprecise is contrary to the desiderata that were stated. $\endgroup$ – Michael Hardy Jun 26 '13 at 17:58
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    $\begingroup$ It's hard for me to read your emotion (I have reason to believe that other people also think your behavior is angry). And I obviously didn't get my meaning across. Anyway, I feel like you're misplacing your energy by harping on the words "precise" and "focused". As far as I can tell, everybody agrees about the fine distinction, but they were treated as synonyms, which lead to confusion. Even Andy Putman's original comment, which started that confusion, points to the fact that you have not made clear what constitutes an answer to your question. This is the key point you should address. Do it. $\endgroup$ – Anton Geraschenko Jun 26 '13 at 18:26

I'm not interested in weighing in on the question itself, so this answer exploits the "idempotence of meta" principle that Anton G. has previously espoused.

One of the consequences of the carryover to the SE2.0 model is that non-moderators can now vote to close meta questions. I think however that this should be done very sparingly, if at all: if a question is truly off-topic for meta then a moderator will see it and have no qualms about closing it.

But I don't think you should vote to close a meta question because you think it is "bad", as long as it is on-topic. In particular I consider this question of Michael Hardy's to be obviously on-topic for meta: it's asking about the closure of a question on the main site. So I think it is wrong-headed and possibly even a bit obnoxious to vote to close this meta question. I suspect that people are doing this as an extension of the fact that they don't like the MO question that this meta question is about, but this seems like a clear "category error". Right?

  • $\begingroup$ Not really.${}$ $\endgroup$ – Andrés E. Caicedo Jun 27 '13 at 0:46
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    $\begingroup$ @Andres: Could you be a bit more specific as to what you disagree with? $\endgroup$ – Pete L. Clark Jun 27 '13 at 0:49
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    $\begingroup$ +1 I can't see the close votes, but my guess is that both down-votes and close-votes are partly because of too many edits to the question and the tone. One of the negative aspects of voting in meta discussions is that it is hard to understand what votes really mean. Is a downvote because of disagreeing with a particular behavior or is it because of disagreeing with the suitability of the topic or is it because of disagreeing with the content? $\endgroup$ – Kaveh Jun 27 '13 at 0:55
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    $\begingroup$ @Kaveh: There are currently 4 votes to close. I don't think either reason you give would have merited closure under the older meta system. (As Anton said the tone is somewhat "grumpy", but I think it would have to be much, much worse for this to be a reason to close.) $\endgroup$ – Pete L. Clark Jun 27 '13 at 0:57
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    $\begingroup$ I agree this shouldn't be closed. Maybe we should have a meta discussion about etiquette for the new meta? $\endgroup$ – Kaveh Jun 27 '13 at 0:59
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    $\begingroup$ @Kaveh: I agree with you that up/downvoting meta questions is a confusing enterprise at best. The SE party line is that upvoting is meant to express agreement and downvoting is meant to express disagreement, but since at least formally (and in many cases, actually) each meta post is a question, this has always been logically shaky. But it is easy to just ignore votes on meta except when they have been given an explicit meaning ("upvote this if..."). Closure is a different story. $\endgroup$ – Pete L. Clark Jun 27 '13 at 1:00
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    $\begingroup$ "I suspect that people are doing this as an extension of the fact that they don't like the MO question that this meta question is about." $\endgroup$ – Andrés E. Caicedo Jun 27 '13 at 1:01
  • $\begingroup$ @Andres: No, I really do suspect that, so you're certainly mistaken. :) But seriously, thanks for the clarification. $@$Kaveh: "Maybe we should have a meta discussion about etiquette for the new meta?" I think that would be a good idea. $\endgroup$ – Pete L. Clark Jun 27 '13 at 1:02
  • $\begingroup$ :-)${}{}{}{}{}$ $\endgroup$ – Andrés E. Caicedo Jun 27 '13 at 1:09
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    $\begingroup$ I don't think keeping this open is likely to lead to a decrease in entropy, especially from what I've observed of the OP's interactions. Hence my vote to close $\endgroup$ – Yemon Choi Jun 27 '13 at 3:06
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    $\begingroup$ @PeteL.Clark : I have not voted to close this question. However, I've been sorely tempted to not because of the post to which it refers, but rather because of the obnoxious behavior of the OP (not listening to the many good answers he's gotten, arguing, sending me [and I assume the other closers of the original question] multiple argumentative emails, etc.). If this behavior continues, then I will likely vote to close so as to put an end to things. $\endgroup$ – Andy Putman Jun 27 '13 at 3:08
  • $\begingroup$ @Andy: sending multiple emails could indeed be problematic. Without wanting to wade into the issue, I'll just mention my personal policy: if I reply to someone telling them that further emails (or further emails on a clearly defined subject) are absolutely unwanted, then I had better not get any further emails from that person on that subject. Exactly once I had a problem with this, so I got the IT person in my department to block all further emails from that address. $\endgroup$ – Pete L. Clark Jun 27 '13 at 5:58
  • $\begingroup$ But having excluded that, one of the main designed purposes of meta was to create an outlet for people to vent their spleens, the better for this not to occur on the main site. The current change in meta is technical; to my knowledge, the philosophy behind it has not changed, and I for one am inclined to try to honor Anton's founding principles and push back a little against changes caused by changing site mechanics. $\endgroup$ – Pete L. Clark Jun 27 '13 at 6:01
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    $\begingroup$ P.S.: Not to be pessimistic, but: is it clear that closing the question will "put an end to things"? It isn't to me. $\endgroup$ – Pete L. Clark Jun 27 '13 at 6:02
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    $\begingroup$ I can't help feeling that spleen-venting such as this would belong better on 'tea', because at least in one thread one doesn't get this splintering of conversation, and because the down/up vote stuff is silly IMHO in this context $\endgroup$ – Yemon Choi Jun 27 '13 at 6:58

I actually think that your original question on MSE is more interesting: https://math.stackexchange.com/q/427116/51970, where you look at trigonometrical functions as curves on the torus.

I think a more specific way of asking your question is, "is there any way of singling out the curves on the torus corresponding to trig functions"? Some easy questions are, what are the possible homotopy classes, how many max's and min's can it have, etc. But you can ask deeper questions, like, is the set of trigonometric curves dense in some larger set of curves? Fourier analysis says yes, but now you're including reciprocal trig functions, we can ask, What is the closure of multiples of $\sec$, $\csc$, $\tan$, etc.

  • $\begingroup$ I think my "original question on MSE" is the same question. I just gave more concrete details of one example and of one general rule. (The "general rule" is that the graphs live on a torus.) $\endgroup$ – Michael Hardy Jun 26 '13 at 22:37

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