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RobinZ comments on Belief in Belief - Less Wrong
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"Those who find this confusing may find it helpful to study mathematical logic, which trains one to make very sharp distinctions between the proposition P, a proof of P, and a proof that P is provable"
This is a bit of a side question, but wouldn't a proof that P is provable be a proof of P? In fact, it sounds like a particularly elegant form of proof.
Seconded - this is an interesting question. (And I suspect that there are some interesting cases in which a proof that P is provable does not constitute a proof, but this is mainly because I've seen mathematicians break similarly intuitive propositions before.)
It wouldn't surprise me either. However such cases would have to rely on a precise definition of 'proof' differently to what I use. The result would then be filed under 'fascinating technical example' but not under 'startling revelation' and I would take note of the jargon for use when talking to other mathematicians.