You're challenging what it means for something to be a proof, which I think is always subject to philosophical debate. Intuitively, something is a proof of a proposition if it convinces someone else that the proposition is true, and the proposition actually is true.

Obviously, for difficult concepts, it's possible that every expert in the world could be convinced that a proposed proof is correct, only to later find out that it's not. The way I see it, you're always only proving that something has a high probability of being correct, and showing how high the probability is. For computer software, you might be showing that your program is correct with the same probability that the underlying hardware is working as specified (so cosmic rays shifting bits or faulty hardware are possible exceptions baked into your proof).

To get a bit silly, a mathematical proof delivered from person A to person B is really only showing that a proposition is true with the same probability that person B is sane, educated, and is understanding the concepts correctly,