Sequel to The nature of proofs

A while back, I ranted on when a proof may hope to be correct---the import: only when its construction and verification are essentially devoid of any intuitive understanding; you need intuition only to understand a jump in reasoning, which does not a correct proof make.

However a proof need not be shown in order for it to exist.

Formal theorem provers like me always kind of look down on the lesser mortals who talk about statements and their "proofs" in English. Even when the statements can be formalized, the "proofs" can seldom be interpreted in any coherent logical framework as sound representations of arguments for their truth. I argue that there is in fact an interpretation by doing lesser, that is, by not offering a proof at all.

In intuitionistic logic, all proofs are constructive, that is, a statement is true only if there is a proof that can be shown for it. In classical logic, however, a statement is true also if there is no proof that can be shown for the contrary. A similar idea appears in mathematics under the veil of the Axiom of Choice. Essentially, it can be proved that something exists without actually showing what it is. This is as much a theory as anything else---the Axiom of Choice follows from or breaks a number of mathematical results.

The point of the argument is that classical logic, to be fair, is as good a framework to judge "proofs" as intuitionistic logic. That out of the way, now consider the following (very common) fix:

I say that statement P is true, but obstinately refuse to show you a proof for it. What can you do about it?

So. Normally, if I say that a statement is true, it is my obligation to prove it. In this fix, it is your obligation to disprove it. If the statement is indeed true, you can never disprove it. Hence I "prove" that my statement is true by staying quiet and confident.

A possible fallacy is that there is no way to verify this proof. I could write up a complete strategy to break any argument you might offer to disprove my statement; this might constitute a "proof"---in logical terms, it is a proof of not(not(P)) instead of P; in other words, it is a proof that shows that it is absurd that P is absurd. However, if it is agreed that I am not the defender, that you are the prosecutor, then I should be okay to hide my strategy. Heck, to not even have one, or come up with one on the fly. All I need is to ridicule you whenever you say something that I have already dismissed as ridiculous.

Point made? I'd better not say too much. I probably would have done a better job by not making an argument.