What can we know about undiscovered proofs?
Please note that my knowledge of maths extends to simple calculus. In real life I have only ever had to use arithmetic and a little algebra. Please excuse any deficiencies on my part.
I would like to get to grips with what a mathematician can know about undiscovered proofs. To make myself clear I'll cite some examples with my proposed logic.
If no-one has found a proof for proposition pX and someone who does not know the proof can prove that there must be a proof, then they have actually proved pX
If someone can prove there must be a couterexample to pX even though no-one can find that counterexample then they they have disproved pX
It is possible to prove something about the range of pX even if the general solution of pX is not known. For example by exhaustive search up to a certain limit we can show the if a counterexample exists then the counterexample must exist outside the searched range.
If the above logic is correct then I would like to open things up by considering what sort of things can be known about an undiscovered proof.
Example
If we stringently define the mathematics that we believe Fermat to have known. Can we perhaps prove that it was impossible for him to have proved his famous Last Theorem using only those methods. Or indeed prove that it would have been possible even though we don't know how he actually did it.
Question
Is there a generally accepted term for this? I thought of meta-proof but can only find joke references.
If such a discipline exists, what can we say about the parameters of a proof that can be known short of finding an actual proof?