A proof is a convincing argument that convinces us to such a degree that we can with its help convince others.

I'm reminded of advice I've seen on when, in the martial arts, you may consider yourself a master: when people come to your students, asking "Please teach us."

Although I disagree with the social concept of mathematical proof. There is something social going on, but there is also something that is not social, but is mathematical, existing independently of ourselves, outside space and time. The latter is the actual proof, without which a statement is not a theorem, and the former is people convincing other people that the latter exists and showing them where it is. Both of these things are called "proofs", and both of them are necessary to the practice of mathematics. The social part depends on the mathematical part for its value, just as prospecting for oil has no value unless, in the end, there is oil where you say there is.