morphy@nntp-server.caltech.edu (Jones Maxime Murphy) (11/26/90)
Can someone tell me if(and where) there are archives of discussions that have taken place in this group on Godel's icompleteness theorem and it's impact on AI? Jones
minsky@media-lab.MEDIA.MIT.EDU (Marvin Minsky) (11/26/90)
In article <1990Nov25.223132.24431@nntp-server.caltech.edu> morphy@nntp-server.caltech.edu (Jones Maxime Murphy) writes: >Can someone tell me if(and where) there are archives of discussions that have >taken place in this group on Godel's icompleteness theorem and it's impact on >AI? >Jones Interesting subject. But has anyone really got anything useful out of it? It seems to me that all the discussions have been irrelevant because of two peculiar assumptions. 1. There has been a general assumption that human reasoning is logical -- and hence is limited by the limitations implied by Godel's theorem. This is strange because people do not normally assert proofs. They make assertions. Nor do they claim, normally, to make these on the basis of a completely consistent logical or formal system. Now you might argue that whatever the brain is, it might well be approximated adequately well by some formal system, hence is subject to those limitations. But Godel's theorem doesn't apply to all formal systems, only consistent ones. Sometimes, a person may detect an inconsistency and withdraw an assertion, but that's beside the point, because there is no reason to suppose this can be always done, etc. 2. As for Godel himself, he did claim to have a peculiar nonlogical ability to sense mathematical truth. Heaven knows, he didn't make as many mistakes as some people -- but this kind of mathematical ESP should be discussed in sci.skeptic or psi.(para)psychology, shouldn't it. In other words, none of this makes sense to me. I see Godel's theorem as asserting that consistency is incompatible with heuristics, in the sense that there is no way to ensure only true assertions in suitably rich systems. Very sad, but has nothing to do with our rich, heuristic, fallible brains.
bernstei@boulder.Colorado.EDU (mwb) (11/26/90)
In article <4168@media-lab.MEDIA.MIT.EDU> minsky@media-lab.media.mit.edu (Marvin Minsky) writes: [in response to a query about Godel's incompleteness theorem...] >Interesting subject. But has anyone really got anything useful out of >it? It seems to me that all the discussions have been irrelevant >because of two peculiar assumptions. >1. There has been a general assumption that human reasoning is logical >-- and hence is limited by the limitations implied by Godel's theorem. >This is strange because people do not normally assert proofs. They >make assertions. Nor do they claim, normally, to make these on the >basis of a completely consistent logical or formal system. Now you >might argue that whatever the brain is, it might well be approximated >adequately well by some formal system, hence is subject to those >limitations. But Godel's theorem doesn't apply to all formal systems, >only consistent ones. Sometimes, a person may detect an inconsistency >and withdraw an assertion, but that's beside the point, because there >is no reason to suppose this can be always done, etc. Has there been, with the evolution of mathematical thought and formal systems, a corresponding evolution in the way all people view the world? It seems to me rather likely that with the development of explicit logical systems, the "scientific method" and other formalizations of ideas about the world, there would be a corresponding change in the way people think and express themselves. Consequently, the only reason that we can even attempt to express human thought in a formal manner is because humans have developed these same formal systems. Or, on the other hand, is formal thinking "hardwired" into the brain in the same way that language acquisition and the existence of syntax seem to be (according to Chomsky, at least)? Where do logic and proofs come from -- internally, as a byproduct of the structure of the brain, or as a necessary result of the structure of the universe? mwb bernstei@tramp.boulder.colorado.edu
cpshelley@violet.uwaterloo.ca (cameron shelley) (11/26/90)
In article <30217@boulder.Colorado.EDU> bernstei@tramp.Colorado.EDU (mwb) writes: > >Has there been, with the evolution of mathematical thought and formal >systems, a corresponding evolution in the way all people view the world? >It seems to me rather likely that with the development of explicit logical >systems, the "scientific method" and other formalizations of ideas about >the world, there would be a corresponding change in the way people think >and express themselves. Consequently, the only reason that we can even >attempt to express human thought in a formal manner is because humans have >developed these same formal systems. > I'm not sure this is really an answer to your question, so much as a related observation, but here goes. In the medieval languages that I have studied (all european and not that many really) multiple negatives can be interpreted simply as 'really negative' or 'really, really negative' etc. rather than as we usually interpret them now - which is in accord with formal negation. I believe this is still the case in colloquial speech. I recall seeing as many as five negatives in one clause in "Tristan und Isolde" (GvS) which is made out to mean just the same as if it had been four, or three and so on. With the revival of prescriptivist grammars on the latin model in the 18th century, we were told 'the truth' about these constructions and every educated generation since then has been trained to think of natural language negation as being equivalent to formal negation. Whether we actually *think* that way is another issue, but it is certainly an example of how developments in formal language have influenced natural languages. -- Cameron Shelley | "Logic, n. The art of thinking and reasoning cpshelley@violet.waterloo.edu| in strict accordance with the limitations and Davis Centre Rm 2136 | incapacities of the human misunderstanding..." Phone (519) 885-1211 x3390 | Ambrose Bierce