The enigmatic complexity of number theory | Hacker News Reader

The enigmatic complexity of number theory

mathoverflow.net

87 pointsmonort18 comments

Is there a name for the principle that the more interesting a topic is on a Stack Exchange forum, the more likely it is to be closed?

And Scott Aaronson quits mathoverflow due to this "off-topic" policy:

https://mathoverflow.net/questions/282869/the-enigmatic-comp...

This is so sad, MathOverflow was once such a great community. Here is his comment for context:

> I regret to say I'm ending my participation in MathOverflow, for the same reason I decided a decade ago never again to edit Wikipedia. It's hard to express how disheartening it is to spend hours of volunteer labor explaining stuff---in this case, in a way that at least 19 MO users apparently found useful---only to have your work overridden by a smaller set of users, for being (part of something larger that's) "off-topic" or whatever it is. Who the hell has time for that? From now on, if I have math questions, I'll post them on my own blog. Was nice being here for 6 years; thanks everyone. – Scott Aaronson 3 hours ago

Wow that was such a great discussion too. What an infuriating choice by the mods. I wholeheartedly agree with Aaronson's decision to leave the platform for good.

"closed as interesting"

Wikipedia and stackexchange both aim at being reference works. There are different phases in the journey to that goal, and an unfortunate casualty of maturity is the loss of interestingness.

Strikes me as similar to startups, fun when small, horrible upon enterprise "maturity". Many people like the early stages, yet the latter is the goal...

OK, in the absence of any other name I am declaring it Hefferon's Law.

IMO, if you have some problem, and want to post in some Stack Exchange site, now you have 2 problems... I'm really sick to try formulate SE questions, Im most interested if the question will be closed instead of think how can I get a answer... Im too frustrated and I see a lot like me frustrated too. I'm really really tired to fight against SE Zealots...

I'd pose this as caused by the simplicity of calculating in Z as a ring. This has allowed us to experimentally find many interesting true conjectures.

Number theory is no more complicated than any other system. What differs is the ease by which we can find statements that are probably true. If the field appears more complex, that might be because we've been enticed by these probably true statements.

I'm not sure whether I agree or disagree, but I'm reminded of how it's said that Fermat's Last Theorem wasn't really the goal (or even that the truth or falsity of FLT doesn't imply anything interesting on its own), but it's an extremely vital problem from a development-of-math perspective.

Ideals were discovered by Dedekind as a solution to an error made by Gabriel Lamé in his claimed proof of Fermat's Last Theorem: he'd overlooked that not all rings have the same properties as Z. In this situation, they were missing "unique factorization", which Kummer "fixed" by hypothesizing the existence of "ideal numbers", which Dedekind later formalized as "ideals". This was a huge step forward for the development of modern algebra.

https://www.quantamagazine.org/the-crazy-number-that-almost-...

And that's setting aside a lot of algebraic number theory, class field theory, and so on.

I'm not so sure about that. Take a look at cellular automata. It's pretty trivial to screw around with them (though not as easy as with integers, I'll grant you). Where are the theorems? There seems to be an awful lot of land, but no maps.

To a lesser extent, graph theory has this problem as well. In the ~60 years that it's been a subfield in its own right, we haven't got an awful lot of fundamental new methods. Results being proven today could be easily understood by a graph theorist from 40-50 years ago transported ahead in time, perhaps after at most the equivalent of a semester-long grad course.

Cellular automata are much harder to screw around with. Partly that is because computation by hand is a lot harder, partly because there isn't much that one can combine.

What makes the integers interesting is that addition, multiplication and exponentiation are all easy, all operate on the same numbers, and yet they behave very differently. This allows one to pose very easy questions (e.g. Can I express every prime as a sum of ...) that have no obvious solution to them.

Yes

Number Theory is a tough nut to crack and I feel that, while mathematicians have advanced a lot, there hasn't been any "master theorem" to handle it

Factorization, one way or another, still is done with brute-force (even if it's done in the EC domain or with NFS or related algos). Same for modular logarithm

Most of the theorems still fell like digging a pool with a spoon. Fermat's last theorem was finally proven, with a lot of effort and diversions. It really seems we're missing something "basic" (but complex) about it

I'm not so sure that we're missing anything basic. The top rated answer by Scott Aaronson seems spot on to me. The natural numbers are just too powerful. As the resolution of Hilbert's tenth problem shows, we can at best hope to solve special cases of statements involving natural numbers, and the solvable cases are bound to require techniques of increasing complexity.

What would count as a "master theorem" for some field, anyway? I suppose something like Tarski's quantifier elimination of semialgebraic sets over R^n might be an example, though for actual computations the best algorithms have double exponential worst case complexity (i.e. even worse than integer factorization). Or for something even more basic, perhaps the fundamental theorem of linear algebra (along with Gaussian elimination). There's the saying that mathematics is the art of reducing any problem to linear algebra -- I suppose the hard problems in number theory are the ones that don't have a good linear algebra reformulation :-)

Yeah I read that comment and I agree with you

> What would count as a "master theorem" for some field, anyway?

I think you gave good examples, but it could be something like the discovery of Calculus, where suddenly a lot of problems became tractable. And in the end we even managed to solve (some) differential equations by solving polynomial equations.

So maybe you're right and it's linear algebra all the way down, but we're missing the weird trick to convert a factorization problem into a polynomial or something like that

Not a (master) theorem per se, but how about the classification of finite groups?

>Most of the theorems still fell like digging a pool with a spoon. Fermat's last theorem was finally proven, with a lot of effort and diversions.

The Fermat's last theorem equation was an incredibly difficult edge case that is illustrative of very fine boundary of a grand theorem. In the case of FLT, that was the modularity theorem.

I wouldn't call anything put towards proving FLT a diversion, the modularity theorem and other theorems that came as a result of proving FLT are the 'basic' (but complex) discoveries that these problems yield.

The closest thing number theory has to a Grand Unified Theory is the Langlands program. It's a huge, highly interconnected, often surprising bunch of conjectures relating number theory to geometry, algebra, and even[s] string theory -- and also something that I know next to nothing about. :)

I'll put the Wikipedia[w] intro here, and note with some happiness that Frenkel has used that exact phrase before:

> In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest projects in modern mathematics, Edward Frenkel described the Langlands program as “a kind of grand unified theory of mathematics.

Wiles' proof of the modularity theorem (and hence Fermat's Last Theorem) is a proof of a some of the Langlands conjectures "in dimension 2": one of the pinnacles of human mathematical achievement is just an incomplete portion of a special case of Langlands' vision.

> Andrew Wiles' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for GL(2, Q) remains unproved.

To give you a little taste, here's a 10000-miles-above-the-ground view of what this says.

A modular function is a certain kind of object in complex analysis (something like calculus with complex instead of real numbers[c]).

A Galois representation starts with a Galois group, which tells us something about the large-scale structure of a number system, and extracts[p] ("represents") a small part of the information it contains: the full "absolute" Galois group is incomprehensibly complicated. Every elliptic curve gives rise to these representations.

There's no reason these things should be related: most number thought the modularity conjecture was an improbable goal. Now think about how it's just a tiny part of a much larger whole. They don't call it "fantastically bold"[n] for nothing.

---

[w]: https://en.wikipedia.org/wiki/Langlands_program

[s]: https://physics.stackexchange.com/questions/4748/why-is-ther...

[n]: https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...

[p]: As I understand it, Wiles' proof only uses 2-dimensional representations.

[c]: ... but it's hard to overstate how much nicer and more uniform everything becomes!

I would also recommend the book by Edward Frenkel, "Love and Math" where he talks about his involvement in the Langland program.

Also, he has a very cool lecture on the subject here:

"What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?" http://online.itp.ucsb.edu/online/bblunch/frenkel/

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