What is a proof? The question has two answers. The right wing ("right-or-wrong", "rule-of-law") definition is that a proof is a logically correct argument that establishes the truth of a given statement. The left wing answer (fuzzy, democratic, and human centered) is that a proof is an argument that convinces a typical mathematician of the truth of a given statement. While valid in an idealistic sense, the right wing definition of a proof has the problem that, except for trivial examples, it is not clear that anyone has ever seen such a thing. The traditional examples of correct proofs that...