> 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
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...
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.
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.
And that's setting aside a lot of algebraic number theory, class field theory, and so on.
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.
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.
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
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 :-)
> 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
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.
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.
[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!
Also, he has a very cool lecture on the subject here:
"What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?"