Every proof in mathematics is an implication of the form A -> B where B is the proven statement and A is the premise (which can consist itself of many conjuncts like axioms, inference rules, theorems from axioms etc.) That means no mathemtical proof is unconditional.
But that means we'll never know that 'this and that is the case', all that we know is that 'if this and that is assumed' then 'this and that is the case'. As an example: we'll never verify unconditionally that V2 is irrational, only that within Q, certain rules and axioms, V2 is irrational and we also cannot verify unconditionally that at least 'within Q, certain rules and axioms, V2 is irrational' because we'd need some premises again to prove it.
It means that our proofs and justifications of statements (in math) can never get us to certainty to have grasped a truth. There is an eternal split between proof and truth that is so fundamental that it's even unprovable, it just reveals itself by occuring over and over again, without exception so far. Isn't that much more distressing for mathematics than Gödel's at least negative proofs of the Incompleteness Theorems?