The great mathematician David Hilbert set out twenty-three important problems in mathematics in 1900. One of the famous outstanding ones is the Riemann zeta hypothesis. One might argue that once a good problem is found someone somewhere will eventually find the answer. Even if the 'sometime' can be a long while, as in the case of Fermat's last theorem, still, I am inclined to the view that finding a good problem is sometimes more important than finding the answer to a good problem. On the other hand, perhaps good problems are becoming more abundant? Problems concerning the internet for example only came into existence with the internet. What is your view? Which is harder, finding a good problem or finding its answer?
Researchers should begin by identifying a broader subject of interest that lends itself to investigation. For example, a researcher may be interested in physical observations in nature. Therefore, the harder is finding a good problem which therefore leads to finding interesting answers.
"Choosing good problems is essential for being a good scientist"
In my opinion, its not the problem but the framing of the appropriate question representing the problem which is important. We all see thousands of problems around us but only few of us can actually frame a question for a problem which ultimately leads to the solution. For example, we all have faced the problem of loosing important documents like IDs, birth certificates, car title etc but when the question is framed that is it possible to make the documents secure by use of modern electronic chips, the answer will come some day. Similarly, talking of VIV-based energy harvesting, narrow synchronization is a common problem. The question that can bi-stability help to increase the synchronization region gave the motivation for work given in link. So I will go with framing of appropriate questions about a problem. Furthermore, there can be many questions framed on the same problem opening new avenues for research which ultimately leads to the solution of the problem by someone.
Article Piezomagnetoelastic energy harvesting from vortex-induced vi...
In the beginning, there was a real-life problem. And that problem got turned into a research question. And that research question got turned into a study. And the answer was found!
I asked this question about two weeks ago. I like all the answers to this point. Then I thought about the question in relationship to AI. Can computers find good problems? What criterion would they use? What is important to a computer? A reliable electrical supply? I added some thoughts on this today in a short essay, Distinguishing AI and Human Intelligence. And I hope one of the people who contributed an answer above will be OK with my having quoted him in the essay.
Looking at some of the problems posed over time and those today the Millennium problems posed by the Clay Mathematics Institute (which carry a million buck prize for their solution) to those posed by Steve Smale
https://en.wikipedia.org/wiki/Smale's_problems to Hilbert's original problems. Most of these are concerned with deep structural questions in mathematics. There is overlap as one would think. The Riemann hypothesis is in all three list because if true - many open questions of number theory are immediate resolved. P=NP is is a Millennium problem list and Smale's list as is Poincare conjecture which was solved in 2000.
In some respects Hilbert's problems tend to reflect the interest of Hilbert and Smale's problems tend to be more geometrical because of the interest of Smale. The Clay problems was put together by a committee and in some respect more representative across a broad area of mathematics.
The Millennium problem can be found here.
http://www.claymath.org/millennium-problems
Most of these conjectures arose because the results from many special cases have been developed and the conjectures will develop an overarching theory.
There are only old problems (and therefore solutions obtained) and new problems (ie solutions still to be obtained). Think of the horror of having all the answers!
It is difficult to find problems which are deep, beautiful and perspective in the sence of impact for developing Mathematics in the future. Much more easier to pose hard problems as well as boring ones. That is why journals are full of such kind papers even there are whole journals of such kind. Solution of a problem may occur easy or hard that is difficult to be recognized before solution. The bright example is the 3rd problem of Hilbert that was solved very quickly. Generally speaking abilities of finding difficult but attractable problems and solving them don't coincide. Who possess both qualities are classic mathematicians as Euler, Poincare, Kolmogorov and oth. Those who able to pose enough difficult and attractible or useful problems and is able to solve them belong to the cathegory of happy mathematicians. That I wish to my fellows.
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