Should we encourage non-professionals to ask hard questions if that is good for the future development of the mathematics? [closed]

Question

There is a story about Gauss being "unimpressed by FLT, pointing out (after finding a proof of the n=3 case) that in number theory it’s very easy to come up with problems that are very hard.".

But, it is known that there were many developments in number theory since the time of Gauss, and some of them were needed to settle FLT (I am not implying that Fermat did not have a proof, an "elementary" one, but the margin was too small).

Also, the conjecture of Goldbach surely was out of reach in the time of Goldbach and Euler, but today it stands as an important problem in the theory of additive bases, in the subfield of additive number theory.

Therefore, I am of the opinion that a "good" mathematical question, no matter how hard, could be a source for a new developments in the field, and, possibly, for some future applications, outside or inside the field itself.

Because of that, I am of an opinion that questions that generate interest and are hard should be encouraged, and should not be closed because closing of them discourages a person who asked that question.

Also, importance of a question and its impact on the field is very subjective matter, and, what to someone seems as just another question, to someone else could be a source of inspiration for a lifelong study.

So many of us are able to ask questions that are hard, but this question is more about non-professionals than professionals, for example, I, as a non-professional, must think so much about some topic to hopefully arrive at a question that could interest some of you, while many of you can in a fraction of an hour ask a couple of questions that are very hard and, maybe, non-answerable with current techniques.

Because of that, I think that it is recommendable that some of non-professionals actively participate at the site for professionals, and that they ask questions that are hard, not just hard but also those that are very hard (if possible), because, a smart question, no matter how elementary in its formulation, can generate so much of research and can be a source of new developments.

From all of this I would also like to add that I do not like the politics of putting questions "on hold", or "closing" them, just because they are elementary in formulation and very hard or hard or at least complicated to settle.

Here the rule "the harder the better" can be applied, not because of our present inability to settle some questions, but because of enrichment of the field itself.

As about me, I am more of a conjecturer in mathematics than developer of mathematics, but the act of making conjectures in mathematics and of developing the field cannot be separated, they are very intimately connected.

Also, I think that professionals should offer as much help as possible to non-professionals, if the intentions of non-professionals are good, because it is possible that some non-professional becomes a professional, but without help, it is possible even without help, but it is hard if there is no help at all.

A conclusion would be, that:

Non-professionals should be encouraged on this site if they ask a question even if it is elementary in formulation, and that generated a discussion, but did not receive an answer that settled a question, and that question should not be put "on hold" or closed, if it was asked in a clear and systematic way.

Answer 1

Hard questions (from anybody) are usually not closed unless one can demonstrate that they are just some variations of well-known "hopeless" problems. If you got discouraged by the phrase "This question does not appear to be about research level mathematics" on your "extension of primes" question, don't pay any attention to it because it is just the result of having to do a multiple choice in which no standard option applies and being too lazy to formulate and type the real reason, which is that solving it will require far more knowledge about prime distribution than the humanity currently possesses and there are several well-known problems in the same venue already starting with the twin prime conjecture (which is "almost solved" now).

I'm in the camp that thinks that currently there are way too many people who ask questions and make conjectures and way too few who provide answers, so if it were my will, I would introduce a temporary quota of $2/3$ questions one would be allowed to ask per each one he answered for professional mathematicians (not on MO, but globally) until our Augean stables are cleaned a bit. However even I realize that, once a clear and non-trivial question is asked, it cannot be unasked, and declaring it a "bad question" or "not interesting question" or anything else like that tells more about the declarer than about the question itself.

So, don't worry: the hard elementary questions won't be lost. They'll just be put in the end of a very long queue until somebody is ready to pick them up from there.

Answer 2

I consider myself a non professional, and I volunteer my time to help out this collective effort called MathOverflow, with the primary benefit that I can read posts of a certain quality from other researchers (professional and not) on topics within my interests.

Quality (for this forum) takes a lot of work to specify. I value clear and concise posts where I can get much gain from a small amount of effort put in to read and understand such posts. I am still developing the ability to write quality posts, and I imagine nonprofessionals also need training in this ability. While I appreciate the sentiment (and have voiced it myself) that people who are not professional mathematicians have much to offer MathOverflow, I do not ask for a decrease in quality to accommodate these people. One of the main purposes of meta.mathoverflow is to help those who wish to learn how to contribute quality posts. You may find it better to ask how you can do that, rather than assume that many such clear and potentially interesting submissions are quality posts for this forum.

Gerhard "Good Questions Take Much Work" Paseman, 2019.06.18.