(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'
I think a lot of naturalness has to do with robustness. E.g. P vs NP is "natural" because P and NP are largely machine-independent. Problems about AC0 are natural because AC0 has many representations (circuits, logic, alternating TM). Problems about LINTIME are less "natural" because they get stuck in the weeds of specific constructs. So maybe questions about Diophantine equations are "natural" if equations of the given degree/have many interpretations.
ReplyDeleteAlternatively, I think if a layperson is truly invested in the answer to a math problem, then it is because it arises as the product of some natural process (e.g. physics, chemistry, biology, optimization of a real-world process). So the questions we care about would ideally be natural by definition. But maybe there's a distinction between physical naturalness and mathematical naturalness.
I would say that there are two different senses of the word "natural." What might seem natural to a person *asking* questions won't necessarily coincide with what seems natural to a person who is developing the technology to *answer* questions. I would not call your question about Hilbert's Tenth Problem "the wrong question"; it's just that you should not be surprised if the people who actively try to answer these questions slice and dice the space in a completely different way.
ReplyDeleteHere's a possible analogy from computer science. A practitioner might want to know what problems can be solved with 100 terabytes of memory and 1000 hours of computer time. A theorist might find those constraints "unnatural" compared to the question of what problems can be solved in polynomial time or polynomial space. Both perspectives are valid in their proper contexts.
This discussion reminds me of something that happened when the proof of Fermat's Last Theorem hit the headlines. Some non-mathematicians wondered out loud why mathematicians would care about such a theorem. Some number theorists reacted by saying, "Well, we don't actually care about Fermat's Last Theorem. That would be silly. What we actually care about is that the semistable Shimura-Taniyama-Weil conjecture has been proved. Now *that's* interesting!" Even though I am a mathematician with some interest in number theory, I found this reaction to be disingenuous. There is no way that people would have been so interested in the semistable case of the Shimura-Taniyama-Weil conjecture if it had not already been shown to imply Fermat's Last Theorem. In fact, I seriously doubt that Wiles himself would have worked as hard as he did on it if there had been no connection to Fermat's Last Theorem. Just because a question is not "natural" in the sense of occupying a central position from the point of view of someone trying to build machinery doesn't mean that it isn't interesting and natural in another sense. To deny this strikes me as some kind of weird snobbery, or at least a narrow view of what the mathematical enterprise is about.
Reminds me of my experience with the unsolvability of the quintic: I wanted to know why the quintic could not be solved, and one of the algebra textbooks I looked at did Galois theory but regarded the unsolvability of the quintic as a mere curiosity (and hence it was relegated to an exercise) compared to the awesomeness of the Galois correspondence between groups and fields. I call bullshit on that!- the Quintic is WHY we care about the Galois Correspondence!
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