I wanted to ask if I can ask questions that arise only from my curiosity (example: I waste time doing random functions and create a problem out of nothing that seems interesting to me because it has a nice expression) or must all questions have rigorous context?
A large part of our purpose here (or my purpose, anyway) is to build a library of knowledge that will be useful to others. We (or I) hope to build the library in the form of a collection of high-quality questions and answers. That means that we are (or I am) hoping for questions that will be relevant to others in the future; that will be formulated in a way that can be found by others in the future; that are clear enough they can be answered, and that others can evaluate proposed answers.
Example: if your question is "What is the value of $x$ that minimizes $17x^4+2x^3 + 3\cos x$?", and you ask it out of curiosity, then that is probably not a great fit for this site. It seems unlikely that others will have the same question.
Example: if your question is "What technique can I use to find the value of $x$ that minimizes $f(x)$, where $f(x)$ is a polynomial?", that would be natural and general enough that it has the right format for this site. There is a good chance that others will run into this same problem. On the other hand, it is likely that the question is already answered here; and it is already answered in standard textbooks; so there is not much value in asking or answering it here -- it would be better to consult standard resources.
So I don't think it's the motivation, exactly, that determines whether the question is a good fit. If your motivation is curiosity about a question that seems natural and others would likely have, that could potentially be suitable here. If your motivation is curiosity about an obscure situation that it's unlikely anyone else will ever face in the future, that probably doesn't advance the site's mission much, so such a question probably won't be well received.
I also want to share a disclaimer, that these are my views only. My understanding is that there is a significant portion of the user base of this site who have different views, and who might be more open to questions based on curiosity even if no one else is ever likely to have the same question again in the future.
Here is my observation about how the Community deals with random made-up questions :
I myself have interacted with user Dan who currently has 200+ questions , mostly made up by random curiosity and excel simulations.
Most of the questions are highly upvoted , some have 40+ votes.
Just the Month of Jan 2024 , user Dan has 13 questions which have comments like the following image :
What is the Community response/reaction to these random questions ?
Even after giving away 1500+ Points via Bounties , user Dan has 20000+ Points.
What about Context ? Here is what user Dan says , in general :
OP : "I wanted to ask if I can ask questions that arise only from my curiosity"
Short Answer : "YES , you can"
If you make the question interesting enough & generic enough & tractable enough , then the Community might interact with you on that.
When that happens , you might be motivated to interact more here.
If you make the question totally uninteresting or very very specific or very untractable , then the Community might not interact & might even resorting to downvoting & deletion.
When that happens , you can add it to your experiences & move on.
As with many such questions, my personal answer is "Possibly yes, but only with care."
There is a cultural sense (mathematical culture, that is) in which certain things pique the common curiosity. It's difficult to define, obviously, but there seems to be rough consensus on many of these things. In the recreational arena, an observation that
- the common second difference of squares is $2! = 2$
- the common third difference of cubes is $3! = 6$
- the common fourth difference of fourth powers is $4! = 24$
might draw some consensus curiosity, especially if the observation is made by a younger student*. In that context, a question as to how to prove that (which might require tools the student is unaware of, but would probably understand) could potentially be well received, especially if it comes with some kind of incomplete attempt at a proof.
There's a certain creative inquisitiveness that's exhibited by such questions. Again, it's hard to define, but you sort of know it when you see it. Surprise, elegance, succinctness, a sense of relatedness—they all work into it.
Note that even if this question comes out of idle curiosity, substantial work goes into making it a good question. For better or worse, this is not the Math.SE of 2015. To put it bluntly, the community expects to be sold on the question, and it expects the OP to do the selling.
Note also that it isn't fundamentally a question of rigor. There are, however, issues of precision. What is the OP's asking? What kind of answer would they like? Leaving this kind of context out makes the question a poor one, even if the mathematics at the heart of it is potentially interesting.
In short, the matter of what motivates the question isn't sufficient on its own to determine whether the question will be well received or poorly received. It's simply one factor of many.
*I asked myself this question when I first heard about FLT as a ten-ish-year-old. As it turns out, I did not have the machinery to see how to prove it, nor did I recognize that it's relatively trivial (compared to FLT).
Here is a take from somebody who is by no means a prolific poster --- and therefore has less right to reply than those who contend with such things on a frequent basis.
If you come up with a "random" problem that is of interest to you, and you bang your head against it for quite a while but are stuck on some obstacle, I think that asking a question is not only appropriate but beneficial to everyone --- subject to three constraints: (1) you really thought about the problem long and hard before posting, (2) you formulate the question clearly and precisely, and (3) you make a reasonable effort to search for an existing post (which can be difficult, especially when the relevant subject and jargon are unknown to you).
The formulation part is critical --- not just for the benefit of others (including possible respondents) --- but because it often obviates the need to actually post the question. In precisely framing the question, the answer often becomes obvious --- or an inability to reify it may identify a flawed assumption or some other critical error in one's reasoning. I've had both happen to me numerous times.
If the three criteria above are met, then I would argue you should post the question. Chances are, others have encountered a similar obstacle --- perhaps not with regard to that precise problem but a related one. Odds also are that any answer you get will be broader than just your question and may give future readers some useful insight, even if they aren't pursuing precisely the same problem.
Many interesting developments in mathematics started as "gee, this looks like a cool problem, I wonder if there's any meat to it." Twenty treatises, many years, and a whole new field later, the answer turns out not only to be nonobvious but profoundly important.
I'm not saying this is common (or even remotely likely) or should motivate you, but it does illustrate that "random problems" are not necessarily uninteresting to others. I've had seemingly simple problems (and I mean problems a high school student could understand) lead me to Markov Decision chains, Algebraic Geometry, the classification theory of Group Extensions, and Characteristic Classes. In most of those cases, if I hadn't asked I never would have had a clue that I wasn't just missing something obvious --- and I never would have had the pleasure of exploring those fields. Worse, I probably would have lost confidence or gotten frustrated and given up. Sometimes, it really helps to have somebody tell you that you're barking up the wrong tree or that it's a very big tree with a thousand pound gorilla hidden in it.