From: "Kent Paul Dolan"
Newsgroups: sci.bio.evolution Subject: Re: Robot Evolution Date: Fri, 22 Dec 2006 01:08:57 -0500 (EST) Phil Roberts, Jr. wrote: > Godel demonstrated (1931) that for any formal > (logical, mathematical, rule driven, etc.) system > capable of simple arithmetic, there is at least > one well-formed sentence or theorem, usually > referred to as the Godel or G sentence, that > cannot be proven in the system. Interestingly > enough, because of our ability to attach meaning > to the symbols employed in such a proof, the G > sentence is one that we humans can quite easily > "see" to be true in that its semantic > interpretation is simply: 'This sentence can not > be proven in this system'. But the fallacy in your, and your antecedents' thinking, is right there and very obvious. Goedel proved that there are well formed sentences stating theorems that a computer cannot _prove_ to be true _or_ to be false, within the same axiomatic system of arithmetic that the theorem concerns. In the _same terms in which Goedel worked_, NEITHER CAN A HUMAN _prove_ that the theorem is true, or is false, using only the axioms of the system of arithmetic that the theorem describes. That the human can "see" the truth or falsehood of the theorem is an unrelated topic; [and arguably a fantasy as well; thinking you know something is not the same thing as knowing something; like a theisim, "knowing" something you cannot prove isn't "knowledge" at all, it's merely _faith_, the same trap into which theists so consistently fall] the human is merely working in some other demesne than the one in which Goedel's machine was working. In particular, the human is not working in the demesne of accomplishing that proof as Goedel described that the proof must be accomplished. Goedel was more than willing to admit that some theorem unprovable in one system of arithmetic might well be provable under a stronger set of axioms, but he then showed that the stronger set of axioms would form a system for which exactly the same sort of unprovable sentence could again be written. So, all you've proved is that the human mind _may_ employ a stronger set of axioms, not that it is somehow different in kind. FWIW xanthian.