> 1999: It is not clear to us in general how to avoid this sort of false proof, the problem being that the false statement seemed so natural to us that we did not think to look at it carefully.

Assuming I understand correctly, this is basically the common issue of being unable to be objective when you’ve lived and breathed a subject matter for long enough. The answer is rigorous peer review, I think.

Peer reviews are (often) also done by people who live and breathe the subject matter.