For instance I've heard that said about the Kolatz Conjecture. What are some other problems (or categories of problems) suspected to fit the bill?

This was developed relatively recently in trying to prove the abc conjecture. The belief is that it will take decades for mathematicians to figure out whether this is actually a thing that works.

https://en.wikipedia.org/wiki/Inter-...%BCller_theory

Continuum hypothesis.

This is provably neither true nor false with the normal assumptions that mathematicians use. In that sense, we've already solved it.

If you a true conjecture is assumed to be untrue, I believe it has to logically eventually lead to a a conclusion that something we know to be true is false. If that never happens when we assume the conjecture is untrue then I believe it means that the conjecture is false or undecidable.

And if the above is true I believe it means that it is not absolutely necessary to invent new math to prove a true conjecture. (Which also would mean that there is another proof of FLT that doesn't require the math of elliptical curves.)

I believe the bolded is an error. Suppose you have an unprovably true statement P. Assume P is false. If you could prove that this led to a contradiction, you would have a proof that P is true. But P is an unprovably true statement.

No. If you add the negation of a provably true statement as an axiom, (ie assume it’s true) you can prove every single true statement false (and false statement true). Proving everything and anything expressable takes 1 one line.

If it is a true statement, that is negated but cannot be proven true with the other axioms you can never prove something true and also prove it false, provided the other axioms were consistent in the first place.

No. The conjecture can still be true. The classic text book example is having a set of consistent axioms 1-5 then adding a 6th axiom that “Axioms 1-5 are consistent.” Axiom 6 is undecidable in Axioms 1-5, and therefore true.

This is largely just gibberish. There is no formal notion of “new math”. You can prove FLT with just a bunch of statements in first order logic (10s of millions of them). When people talk about “new math” it’s the norformal notion of human hurestics that we use to understand something like FLT, that we can’t understand by going through 10s of millions of lines like a computer.

I was sort of Willy nilly with what is true and what is false, because it doesn’t effect consistency when dealing with undecidable statements. But if it’s bothering you:

Let’s define a new set of axioms:
1. All the axioms of PA
2. The Axioms of PA are inconsistent.

We are adding a false statement (2) and assuming it’s true. But 2 is undecidable in 1 and therefore you can never prove anything true and false. So basically your last statement should say “provably false or undecidable”. Since undecidable can be either true or false.

A concrete example of a true statement in PA which cannot be proved in PA is Goodstein's theorem:

https://en.m.wikipedia.org/wiki/Goodstein%27s_theorem

Why do cubes add up to squares?

i've heard this said about P vs NP.

the Church–Turing thesis

In a sense just asking P vs NP required new math. Similar to the Poincare Conjecture. The conjectures themselves created thriving subfields.

I would say more important than the OPs question is what solved problems can be resolved in ways that require discovering new math. The insolvability of the quintics was already done before Galois invented abstract algebra and Galois theory. Riemann Roch was already a theorem before Grothendieck reproved it by talking about the cohomology of coherent sheaves etc. Prob more new math comes out of stuff like that than from climbing a mountain nobody has for the first time.

I officially expand my question to include this.

Also, lol @me for "Kolatz Conjecture", it's Collatz.

Coming at the question from a different angle, homotopy type theory is a new branch of maths. It combines category theory, logic, topology and has applications in computer science.

It's implemented in proof assistants such as Coq.

I have no idea if many problems will result from it, but it is still a young area.

See https://homotopytypetheory.org for more details...

At first I thought there was nothing to see here. But on second thought this is kind of interesting. Suppose we are staying within the same system of axioms. Suppose conjecture C is provably false within that system but we don't know this because we can't find the proof. Then what happens when we look at conjecture D = ~C. If we assume D it will never lead to a contradiction. Does that mean D is undecidable? Well, no. D is provable but we just don't know the proof.

Maybe I should have stayed with my first thought.


PairTheBoard

Yeah, you should have stayed with your first thought. Of course if you can't prove FLT you can assume its false without contradiction. Thats because proving FLT is exactly the same thing a showing a contradiction if you add "FLT is false" as an axiom.

Gauss kept reproving quadratic reciprocity because he felt as though his proofs didn't reveal the reasons behind the underlying structure clearly enough for him to feel good about it. (Or at least, that's the story I recall being told.)