I believe that (mathematical) proofs aren't easily reducible to axiomatic proofs, and that proofs have been, and still are, profoundly social by their nature, although I don't know if that will continue indefinitely. I probably won't find the time to write a large post on this topic that I've been thinking of, so I want to quote here one observation that's been on my mind recently, in case someone finds it useful.

Eliezer quotes (in the post which I, with accordance to the above, see as wrong-headed in some respects) one definition of proof that he disagrees with: "A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing."

I think this is too vague as a definition of proof and doesn't really work, but it does capture the social/communication aspect of it I consider important. A thinks X is right and P is a proof of X. It isn't enough that P convinces A that X is right; when A communicates P to B, it should convince them too.

A few days ago, I was rereading Vladimir Uspensky's essay on the philosophy of mathematics that I read many years ago and forgot. In the section about proofs, Uspensky offers this informal definition:

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

That is, it isn't enough that B is convinced by P that X is right; P should be such that B should be able to spread the gospel onwards. A proof doesn't just bridge a void between two minds; it's capable of leaping on and on. It's a virus with conviction as its payload: it instills conviction in its host and can spread itself (rather than merely conviction) to others.

I've been musing since then about this addition to what I'd thought of as an informal social definition of a proof. I've been going back and forth about how necessary and profound it is, but I think I'm converging on forth.