From: "Phil Roberts, Jr."
Newsgroups: sci.bio.evolution Subject: Re: Robot Evolution Date: Tue, 26 Dec 2006 00:18:54 -0500 (EST) Tim Tyler wrote: > > John Lucas's 'Godel' argument has been much-criticized - and > Penrose's views in this area are essentially a variation on it. I concede that there is a clear majority who disagree with the Lucas/Penrose position. On the other side of the equation, however, we have: a. Hofstadter, Dennett, Penrose and Chaitin, in various ways acknowledging that Godel at least SUGGESTS a disconnect between formalism and mathematical reasoning. b. Little unanimity as to what exactly is wrong with the Godel argument, with dozens and dozens of different sorts of objections, many based on impenetrable confabulations. c. Papers still being published criticizing the Godel argument against mechanism almost 80 years after Godel first published his theorem. d. The universal abandonment of Hilbert's program of formalizing mathematical reasoning by mathematicians all over the world subsequent to Godel's proof. e. Intersubjectively reproducible empirical evidence (feelings of worthlessness) suggesting that not even Mother Nature herself seems to be able to constrain rationality within a formalism (the program for "trying to stay alive"). f. Evidence (e.g., Parfit, 'Reasons and Persons', p. 12) that any theory that attempts to constrain rationality within a formal structure (e.g., a fixed objective) can be shown to sanction rational irrationality (i.e., can be shown to be self-defeating). > Brief version of what's wrong: > > ``A mathematician often makes judgments about what > mathematical statements are true. If he or she is not more > powerful than a computer, then in principle one could write > a (very complex) computer program that exactly duplicated > his or her behavior. Assumes what is being questioned. > But any program that infers Programs don't infer, they model logical relations that have been found to underly human inferences on most occasions. As to whether these relations are actually being followed or simply EMBEDDED IN our inferences remains to be seen. > mathematical statements can infer no more than can be proved > within an equivalent formal system of mathematical axioms > and rules of inference, True, but Lucas/Penrose assumes we can go beyond this, that the intuiting of mathematical truth is not simply a matter of logical proof: The immediate consequence is that truth cannot be defined in terms of provability. In any serious intellectual endeavor we shall be able to formulate simple mathematical arguments, and thus shall be subject to Godel's incompleteness theorem. However far we go in formalizing our canons of proof, we shall be able to devise propositions which are not, according to those canons, provable, but are none the less, true. So it is one thing to be provable, and a different thing to be true. Truth outruns provability. (J.R. Lucas). > > This argument won't fly if the set of axioms to which the > human mathematician is formally equivalent is too complex > for the human to understand. What is the basis for the assumption that the intuiting of mathematical truth is based on a set of axioms, let alone that they must be too complex to understand? > > These are amazing claims, which Penrose hardly bothers to > defend. Reviewers knowledgeable about Godel's work, however, > have simply pointed out that an axiom system can infer that > if its axioms are self-consistent, An axiom system can infer? > then its Godel sentence > is true. An axiom system just can't determine its own self- > consistency. But then neither can human mathematicians know > whether the axioms they explicitly favor (much less the > axioms they are formally equivalent to) are self-consistent. > Cantor and Frege's proposed axioms of set theory turned out > to be inconsistent, and this sort of thing will undoubtedly > happen again.'' Agreed. But we can nonetheless "know" them to be true in the sense that we all agree we have good reason to believe. > > As to what this has to do with evolution - if humans can > do things no machine can do - or will ever be able to do - > that may impact the hypothesis that machine-based organisms > may replace humans as the dominant life form on earth over > the next century or so. More importantly, it would mean that there is reason to suspect that E. O. Wilson may have gotten it wrong in asserting genetic determinism: Can the cultural evolution of higher ethical values gain a direction and momentum its own and completely replace genetic evolution? I think not. The genes hold culture on a leash. The leash is very long, but inevitably values will be constrained in accordance with their effects on the human gene pool (E. O. Wilson). and that Dawkins may have actually gotten it right in asserting the converse: We, alone on earth, can rebel against the tyranny of the selfish replicators" (Dawkins, 1976, p. 215). > > However, this particular argment for the qualitative > superiority of humans is simply wrong - and (IMO) rather > obviously so for anyone who knows anything about Godel's > work. Why then is one of the papers you referenced written in 2004? Shouldn't this have all been over and done with decades ago for a flaw that is so "obvious"? [quote from Penrose] The many arguments that computationalists and other people have presented for wriggling around Godel's original argument have become known to me only comparatively recently; perhaps we act and perceive according to an unknowable algorithm, perhaps our mathematical understanding is intrinsically unsound, perhaps we could know the algorithms according to which we understand mathematics, but are incapable of knowing the actual roles that these algorithms play. All right, these are logical possibilities. But are they really plausible explanations? For those who are wedded to computationalism, explanations of this nature may indeed seem plausible. But why should we be wedded to computationalism? I do not know why so many people seem to be. Yet, some apparently hold to such a view with almost religious fervour. (Indeed, they may often resort to unreasonable rudeness when they feel this position to be threatened!) Perhaps computationalism can indeed explain the facts of human mentality -- but perhaps it cannot. It is a matter for dispassionate discussion, and certainly not for abuse! I find it curious, also, that even those who argue passionately may take for granted that computationalism in some form -- at least for the objective physical universe -- HAS to be correct. Accordingly, any argument which seems to show otherwise MUST have a "flaw" in it. Even Chalmers, in his carefully reasoned commentary, seeks out "the deepest flaw in the Godelian arguments". There seems to be the presumption that whatever form of the argument is presented, it just HAS to be flawed. Very few people seem to take seriously the slightest possibility that the argument might perhaps, at root, be correct! This I certainly find puzzling;. Nevertheless, I know of many who (like myself) do find the simple "bare" form of the Godelian argument to be very persuasive. To such people, the long and sometimes tortuous arguments that I have provided in 'Shadows of the Mind' may not add much to the case -- in fact, some have told me that they think that they effectively weaken it! It might seem that if I need to go to lengths such as that, the case must surely be a flimsy one. (Even Feferman, from his own particular non-computational standpoint, argues that my diligent efforts may be "largely wasted!) Yet, I would claim that some progress has been made. I am struck by the fact that none of the present commentators has chosen to dispute my conclusion G (in 'Shadows', p. 76) that "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth". (Roger Penrose, 'Psyche' Vol 2) PR Rationology 101 How the Author of Genesis Got It Right (and the Golden Rule Got It Wrong) http://www.rationology.net