Good point! I'll try to think about a better way to phrase this (happy to hear suggestions)
Or some such.
To top that up, it's fact that there have been "proves" that were wrong (or maybe that's just my believe? :^]) even for a long time.
Hence, I think we can say that there are 4 options for a theorem:
1) Some mathematician believes the theorem is correct (but can't prove it)
2) Some mathematician believes the theorem is incorrect (but can't prove it)
3) Some mathematician believes the proof of a theorem is correct
4) Some mathematician believes the proof of a theorem is incorrect
Proving that a proof is correct is kind of meaningless. At that point it's all believe anyways.
Mathematical poofs are either correct or false. There is no middle ground.
What is your criteria of "can be checked then"? If a proof for "sqrt(2) is not a rational number" can't be checked by a 5yo, it's still a proof no?
* The proof of the classification of simple groups[0]
* The work on topological four manifolds by M. Freedman [1]
[0]: https://en.m.wikipedia.org/wiki/Classification_of_finite_sim...
Yes.
The fact that we don't know the truth doesn't mean there isn't one.