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I really think that "popular" questions are treated with inappropriate hostility here.

Do I think that "What is the rule for constructing the sequence $3,4,6,10$?" is a mathematically interesting question? No, of course not, because I already have a lot mathematical experience.

Do I agree that trending youtube videos on the sum of all natural numbersgenerate very naive questions here? Sure, I do, because I know Banach limits, Zeta functions, renormalization, etc.

But this is because I already know a lot about mathematics. Some questions about university mathematics seem not any less naive to me, but many people who will jump on closing a "guess the sequence"-question will approve the other question because they can empathize with it.

Now, for the example that motivated this thread: Number-guessing: https://math.stackexchange.com/questions/682652/complex-math-question

The question is clearly a "guess the rule of the sequence 3,4,6,10 - question". It was originally tagged "mathematical physics, complex analysis, contest- math".

I do not find any of these tags appropriate, but on the one hand, the person who edited them not only left in "complex analysis", but added in "algorithms" which is not appropriate, either, and on the other hand, this context clearly tells us that the asker has little background in mathematics, wanted to label the question "difficult" and found "complex-*" tags instead.

The questions was "put on hold as unclear what you're asking", but it is perfectly clear what is being asked. The comments asking for "are you talking about a sequence where $a_n =$ function of $n$" ignore that the OP does not have to know functions or sequences to ask or understand this question.

A proper answer to this question would be: Link to the Encyclopedia of Integer Sequence explaining in what sense it is appropriate to use, explain how the formula could be found by hand for this particular sequence.

Ideally, we would have a detailled answer for the common types of question (recognize the differences in this case, or see that repeated differences are 0, look at the binary representation, ...).

If people think that this is a good place to post "guess the number"-sequences as riddles, one can just explain to them that they should state that they know the answer, but otherwise, there is no harm at all.

I am even more discontent with the angry closures of the $1+2+3+\dots $- threads. This video generated interest and was a perfect opportunity to write very different informative answers on different ways of assigning values to divergent sums and the merits and flaws of the video. Sure, duplicates should have been closed and redirected to the thread with the model answers. But they should have been closed gracefully with enjoyment of the enthusiasm of the askers.

And I have seen no excellent comprehensive answers that would merit a link for people stumped by the video seeking background information. Since the first questions did not mention the video and the later questions were closed quickly and with hostile comments, it was impossible to actually write a good answer about the implied question "Is this video serious? How can this work?". "Because zeta functions and you lack the math background to understand it, so go away." is really not a very good answer.

So, I am strongly in favour of treating people with little background in a friendly way and answer their questions either directly or by linking to a generic question with excellent answers, especially if it is obvious that they came here to ask a question out of curiosity. I am posting this here to hear your opinions on these issues.

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    $\begingroup$ I am strongly opposed to questions of the "guess the next number"-kind. Questions at math.stackexchange should be answerable, and by the very nature these questions are not. They are also not about mathematics, since the "answer" is not really to guess the next number, but to guess what the problem poser is thinking about; thus making it as much about psychology as about mathematics. I always show this example: 2, 4, 8, 16 to my students when we are discussing sequences to demonstrate that such questions are unanswerable. $\endgroup$
    – mrf
    Feb 20, 2014 at 11:54
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    $\begingroup$ @mrf Saying that you are opposed to them is sort of beside the point. A student has a question. Just because their teacher set them a question which is mathematically bad does not mean that the OP should be punished. An ideal response would, perhaps, be to say "The solution they (your teacher) are looking for is $X$. However, by some nifty logic you could get $Y$ (provide explanation). This is an problem inherent in these types of questions." $\endgroup$
    – user1729
    Feb 20, 2014 at 14:29
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    $\begingroup$ @user1729: Just because of that I am going to assign my students a question to write a one paragraph opinion on the comparison between Blade Runner and The Matrix as movies dwelling on what it is to be a human being. As part of the mathematical logic course. Does that mean that they should ask the question on this site and tag it as logic? $\endgroup$
    – Asaf Karagila Mod
    Feb 20, 2014 at 15:27
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    $\begingroup$ @AsafKaragila Your students are mathematically mature enough to know that you are not setting them a maths question. Students who ask guess-the-next-number questions are not mathematically mature enough to know that such problems are not actually mathematical. They need help, and perhaps a better teacher. Your just need the better teacher ;-) $\endgroup$
    – user1729
    Feb 20, 2014 at 15:31
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    $\begingroup$ @Phira: I agree that the $1+2+3+\ldots$ questions can have merit. But there's little point in having many isomorphic copies. What we should have done, as a community, was to write one proper question on this, give it proper answers, then close everything else as a duplicate. If you feel that "Guess the next number" is considered mathematics by some people, by all means write a good general question, write a good answer, and we can close everything else as an abstract duplicate. $\endgroup$
    – Asaf Karagila Mod
    Feb 20, 2014 at 15:32
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    $\begingroup$ @user1729: Any teacher who forces you to watch Blade Runner, and then The Matrix, and then contemplate what makes us human, is an excellent teacher. Next stop, write about the existentialism of Antonius Block in The Seventh Seal, and how it is solved by Jons the squire. $\endgroup$
    – Asaf Karagila Mod
    Feb 20, 2014 at 15:33
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    $\begingroup$ @user1729: Sounds like you should have went for engineering. Proofs are essentially essays. $\endgroup$
    – Asaf Karagila Mod
    Feb 20, 2014 at 16:16
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    $\begingroup$ @AsafKaragila Proofs are essays with substance! $\endgroup$
    – user1729
    Feb 20, 2014 at 18:09
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    $\begingroup$ I really do not know what happens in school in your countries, but "recognize powers of two" is a very important mathematical skill set. Explaining that $1,2,4,8,16$ can continue in other wise in real problems comes after one knows to recognize powers of two. Recognizing patterns is part of what I do in my research. Algorithms that recognize sequences of numbers in certain contexts are part of current mathematical research. Pretending that a context-free "Guess the number"-question has a unique answer is not mathematics. Setting it as an exam question is unacceptable. $\endgroup$
    – Phira
    Feb 20, 2014 at 20:56
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    $\begingroup$ Calling this kind of question in general "not about mathematics" is ignorant. Pretending that recognizing powers of two is worthless in general is a lie. And yes, I show the example with the powers of two to my students, too. To teach them about pattern recognition in context, not to teach them that "the question is unmathematical". $\endgroup$
    – Phira
    Feb 20, 2014 at 21:01
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    $\begingroup$ If someone asks to continue $1,2,4,8,16$, the mathematical answer is why some continuations are better than others and how we can find them and in what mathematical context we would employ which method. The question is devalued if a teacher decides what is right by authority. $\endgroup$
    – Phira
    Feb 20, 2014 at 21:09
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    $\begingroup$ What could be a more basic question in mathematics than "I have this pattern, what might the next number be?" ? $\endgroup$
    – Phira
    Feb 20, 2014 at 21:20
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    $\begingroup$ Is "What kind of number is 196884?" "not a mathematical question", too? $\endgroup$
    – Phira
    Feb 20, 2014 at 21:24
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    $\begingroup$ I just wish people would take context into account a bit more before downvoting. I understand the idea that we shouldn't take downvoting so personally, and it's just a tool meant to objectively rate a question without necessarily being a moral attack on the asker, but we're talking about a brand new user (and, when we're talking about low-level questions, quite possibly a brand new user who's also 12 years old). A brand new user doesn't see it that way, they simply see hostility, and promptly leave and never return. $\endgroup$
    – Jack M
    Feb 20, 2014 at 22:22
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    $\begingroup$ Let $(x_1,x_2,x_3,\cdots x_n)$ be a finite sequence. Find the $x_{n+1}$ such that the following expression $Kol[(x_1,x_2,x_3,\cdots x_n,x_{n+1})]$ is minimized, where $Kol$ is the kolmogorov complexity of the sequence. en.wikipedia.org/wiki/Kolmogorov_complexity $\endgroup$
    – Baby Dragon
    Feb 23, 2014 at 4:15

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I strongly agree with the sentiment of this question.

The often-seen dismissal of guess-the-next-number/pattern-recognition questions is also strange to me.

As Phira notes in comments, recognizing powers of $2$ is a basic skill, as is looking at successive differences in a sequence, and a question about $3,4,6,10,\ldots$ is intended to help develop them.

An example of such guessing that could well come up in my own research: suppose you're computing cohomology of something, and in one example you find, in sucessive degrees, the dimensions $1, 3, 3, 1$, and in another example, the dimensions $1,4,6,4,1,$ wouldn't you suspect that there was a general dimension formula given by binomial coefficients?

Regarding $1 + 2 + 3 + \cdots$, the fact that this can be assigned a meaningful value is an amazing fact, with deep implications. It's wonderful! It's not surprising that people find it striking and want to ask about it here. There are lots of ways to explain it, too.

For professional mathematicians, edge cases/non-obvious counterexamples can be interesting, as can the detailed hypotheses necessary to make certain statements true/false. Indeed, understanding such things is part of the pleasure of mastering a theory. But not all questions have to be answered from that vantage point.

I remember an early question I answered where the OP, coming from a quantum mechanics background, asked if commuting operators were necessarily simultenously diagonalizable. Several of the initial answers emphasized the edge cases that make this literally false; on the other hand, it is typically true, and is a basic principle of quantum mechanics, and I don't think focusing on the subtleties of why it wouldn't always hold was necessarily the best answer for the OP.

In general, I would hope that people are thoughtful about where an OP is coming from, and about what kind of answer they might be looking for. Let's try to encourage people's appreciations of mathematics. I hope that our site can show an enjoyment of mathematics as something wonderful, not just as something recondite and technical, doctrinaire, full of edge cases, counterexamples, and cautions against error.

Added in response to some comments below: It's easy to find examples of (somewhat) intolerant or judgmental behaviour in any area of human activity. It would be good if we could try to aim for the highest possible standards of tolerance, acceptance, and understanding here (even knowing that we will sometimes fall short, due to natural human fallibility), rather than dwell too much on others' failings as a justification for our own.

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    $\begingroup$ I come from a physics background myself and I think something that mathematicians need to think about what additional assumptions or structures are necessary for the statement to be true or make sense. For instance I used to say "evaluate the function $\psi$ at $x$" when $\psi$ was a function in a Hilbert space. I got attacked by mathematicians saying that point evaluation has no meaning in hilbert spaces since two functions which disagree on a set of measure zero are considered equivalent. $\endgroup$
    – Spencer
    Feb 24, 2014 at 5:15
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    $\begingroup$ (continued) I later learned years later that if I had said "apply the evaluation functional to $\psi$" everybody would have been content. $\endgroup$
    – Spencer
    Feb 24, 2014 at 5:18
  • $\begingroup$ @Spencer: I don't see why this bothers you. If there is a language where asking where the toilet is sounds close enough to "your mother is a syphilitic donkey", would you expect people not to be insulted by your "beginners mistake"? $\endgroup$
    – Asaf Karagila Mod
    Feb 24, 2014 at 6:52
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    $\begingroup$ @Spencer This kind of behaviour is not restricted to mathematicians. In a physics exercise session, we (students) had to calculate the result of a mix of some water ice and some water vapor. A good friend of mine started his presentation: "Since we expect the result to be liquid and we want the temperature in Kelvin, we do the calculation with the reference point "liquid water at 0 Kelvin". " Well, the physics professor was close to froth at the mouth because "liquid water at 0 K is physically meaningless and you must not say it" and could not be made to accept this as a good way to calculate. $\endgroup$
    – Phira
    Feb 24, 2014 at 12:15
  • $\begingroup$ @Spencer I have not found out what he should have said instead because too many instances like this made me drop the physics part of my studies. $\endgroup$
    – Phira
    Feb 24, 2014 at 12:16
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    $\begingroup$ @AsafKaragila: Dear Asaf, Could we please have fewer donkey references? (Other examples could convey the same sentiment.) Regards, $\endgroup$
    – Matt E
    Feb 24, 2014 at 12:19
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    $\begingroup$ Dear Matt, I apologize if my example offended anyone. It wasn't my intention with that hyperbole. The origin is a mixture of Monty Python (where John Cleese insults King Arthur's parents) and The Big Bang Theory (where Sheldon is trying to learn Mandarin, and calls Leonard "a syphilitic donkey" due to his difficulties as a beginner in Mandarin, final episode of the first season). Mix that with a brain that had just returned to this part of the world from a very complex dream sequence, and you have the aforementioned example. $\endgroup$
    – Asaf Karagila Mod
    Feb 24, 2014 at 12:23
  • $\begingroup$ Dear Asaf, Thanks for your mitigating explanation! Cheers, $\endgroup$
    – Matt E
    Feb 24, 2014 at 12:27
  • $\begingroup$ @MattE Donkeys are actually kinda relevant. In a word: burro. $\endgroup$
    – user1729
    Feb 24, 2014 at 13:01
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    $\begingroup$ @AsafKaragila, You misunderstand me. I'm not trying to disparage mathematicians here. I am sure this quality is good for rooting out mistakes in proofs and I think every working mathematician should have this alarm system. However if a mathematician is going to consult with people from another field his/her job is to enable them. Therefore the reaction should be "if you want to do that you should make sure your space has an evaluation functional" rather than "you can't do that". Every time I have heard the latter sentence from a mathematician there was a rigorous way around the issue. $\endgroup$
    – Spencer
    Feb 24, 2014 at 14:00
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    $\begingroup$ @Spencer: Dear Spencer, I agree quite strongly with your last comment, with the caveat that it can be hard for people from one field (say math) to easily accommodate themselves to the requirements/background/etc. of another field (e.g. physics) outside their own area of expertise. I think the key thing is to not be a priori dismissive of the view-points of other fields, and also to try to focus on central points rather than more technical issues. Regards, $\endgroup$
    – Matt E
    Feb 24, 2014 at 15:03
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    $\begingroup$ @MattE, my sentiments exactly. $\endgroup$
    – Spencer
    Feb 24, 2014 at 16:40

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