The fact that thousands of people have failed to prove that P=NP indication that it is probably not true. It has even been proven that it cannot be proven by some methods.
My completely unscientific hunch is someone will eventually prove that P=?=NP is independent of ZF(C). Maybe the universe just really wants to mess with complexity theorists
Or even a galactic algorithm-an algorithm for solving an NP-complete problem that is technically in P, but completely useless for anything in practice, e.g. O(n^10000000)
So it's P and NP. (Edit: I keep misphrasing this!)
P ?= NP is not about ease, nor even realistic efforts.
P=NP and P=!NP are both proven nor disproven. (There is redundant information in this sentence.)
History shows us that the historical / ‘effort’ argument is not applicable to mathematics. All proofs were unproven once until proven successfully for the first time. Harder problems need bigger shoulders to stand on. Sometimes this is due to new tools, sometimes it is a magically gifted individual focusing on the problem, usually some mix of both. All we know is that all before have failed. It’s one of the beauties in math.
There were many questions with no answers for literal centuries and thousands trying, and failing, to crack them. A solution was ultimately found despite that.
A new "math" might be needed, but an answer (affirming or not) will be found.
Whether the Riemann hypotesis is true or not, is not going to have any practical effect, accept for a small group of mathematisians who are working on it. Most people do not know what a Field medal is nor care about it.
What if there exists a proof that P!=NP, but the shortest possible proof of that proposition is a googolplex symbols that long? Then P!=NP would be true, and provable and knowable in theory, yet eternally unprovable and unknowable in practice
Goodstein’s theory would take more symbols than there are atoms in the observable universe to write down in "classic" maths. To "fix" this, mathematicians had to use a "new" way of thinking about infinity known as transfinite induction.
I think if we're smart enough to detect(?) a proof, we'll find a way to express it in a finite manner.