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.