Some time ago I was reading about the recent Shinichi Mochizuki's proof for the famous ABC conjecture. It's enormous and so incredibly difficult that at that time virtually nobody was able to comprehend it (now it seems to get better), and some reputable mathematician complained that if it doesn't change soon, scientific community may ignore and forget it, and never acknowledge it as valid. 

This made me wonder, what actually is a proof. Even if it's *correct*, can we call it a proof if nobody besides the author understands it, let alone can verify it? Apparently not, however conscientious and rigorous it would be. 

It could also be interesting to look at the opposite extremum. When I studied mathematics for some time, a professor told us about an early-medieval Indian mathematician (unfortunately I can't remember his name), who independently from the Greeks proved Pythagorean theorem. His proof consisted only of a drawing and one word: "Look". Nevertheless, it was clear enough that it's now widely accepted by mathematicians as a valid proof. 

At this point I'm even tempted to say that the correctness itself is irrelevant here (as Wittgenstein noticed, we cannot *prove* the proof's correctness) - what matters is that we're able to use the proof to convince right amount of right people that our statement is correct. Obviously, some ways of persuasion (like e.g. some psychological or rhetorical tricks) cannot count as a proof. We expect proof to be logical, substantial and *ad rem*. I feel that this is hardly the case with showing us a drawing and asking to stare at it. However, it can indeed make us comprehend something as being obvious - something I like to call Descartes-style proof. 

What makes proof a proof then? 

(I know the formal definition of a logical proof, but this doesn't answer my question; arguably the only people who ever prove things like this are logicians, and only on special occasions.)