## Sunday, May 30, 2021

### What is a natural question? Who should decide?

(Thanks to Timothy Chow for inspiring this post.)

My survey on Hilbert's Tenth Problem(see  here) is about variants of the problem. One of them is as follows:

For which degrees d and number-of-vars n, is Hilbert's tenth problem decidable? undecidable? unknown?

I wondered why there was not a website with this information. More generally, the problem didn't seem to be getting much attention. (My survey does report on the attention it has gotten.)

I got several emails telling me it was the wrong question. I didn't quite know what they meant until Timothy Chow emailed me the following eloquent explanation:

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One reason there isn't already a website of the type you envision is that from a number-theoretic (or decidability) point of view, parameterization by  degree and number of variables is not as natural as it might seem at first glance. The most fruitful lines of research have been geometric, and so geometric concepts such as smoothness, dimension, and genus are more natural than, say, degree. A nice survey by a number theorist is the book Rational Points on Varieties by Bjorn Poonen. Much of it is highly technical; however, reading the preface is very enlightening. Roughly speaking, the current state of the art is that there is really only one known way to prove that a system of Diophantine equations has no rational solution.

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AGAINST THE NUMBER THEORISTS VIEWPOINT:

1) ALICE: Why are you looking for your keys under the lamppost instead of where you dropped them?

BOB: The light is better here.

2) I can imagine the following conversation:

BILL: I want to know about what happens with degree 3, and number of variables 3.

MATHPERSON: That's the wrong question you moron. The real question is what happens for fixed length of cohomology subchains.

BILL: Why is that more natural?

MATHPERSON: Because that is what we can solve. And besides, I've had 10 papers on it.

FOR THE NUMBER THEORISTS VIEWPOINT

1) They are working on really hard problems so it is natural to gravitate towards those that can be solved.

2) I suspect that the math that comes out of studying classes of equations based on smoothness, dimension, genus is more interesting than what comes out of degree and number of vars. Or at least it has been so far.

META QUESTION

Who gets to decide what problems are natural?

People outside the field (me in this case) are asking the kind of questions that a layperson would ask and there is some merit to that.

People inside the field KNOW STUFF and hence their opinion of what's interesting to study has some merit. But they can also mistake I cannot solve X' for X is not interesting'