
The enigmatic complexity of number theory - monort
https://mathoverflow.net/questions/282869/the-enigmatic-complexity-of-number-theory
======
jimhefferon
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?

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

[https://mathoverflow.net/questions/282869/the-enigmatic-
comp...](https://mathoverflow.net/questions/282869/the-enigmatic-complexity-
of-number-theory#comment698104_282869)

~~~
weinzierl
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

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rocqua
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.

~~~
pmiller2
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.

~~~
rocqua
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.

------
raverbashing
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

~~~
fdej
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 :-)

~~~
raverbashing
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

------
mrkgnao
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](https://en.wikipedia.org/wiki/Langlands_program)

[s]: [https://physics.stackexchange.com/questions/4748/why-is-
ther...](https://physics.stackexchange.com/questions/4748/why-is-there-a-deep-
mysterious-relation-between-string-theory-and-number-theory)

[n]:
[https://golem.ph.utexas.edu/category/2010/08/what_is_the_lan...](https://golem.ph.utexas.edu/category/2010/08/what_is_the_langlands_programm.html)

[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!

~~~
mathgenius
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/](http://online.itp.ucsb.edu/online/bblunch/frenkel/)

