Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!sdd.hp.com!spool2.mu.edu!uunet!tdatirv!sarima From: sarima@tdatirv.UUCP (Stanley Friesen) Newsgroups: comp.ai.philosophy Subject: Re: Minds, machines, and Godel Message-ID: <93@tdatirv.UUCP> Date: 23 Jan 91 21:01:40 GMT References: <1991Jan16.035058.7465@bronze.ucs.indiana.edu> <1991Jan16.182120.20961@sics.se> <1991Jan18.012527.20104@news.cs.indiana.edu> Reply-To: sarima@tdatirv.UUCP (Stanley Friesen) Organization: Teradata Corp., Irvine Lines: 44 In article <1991Jan18.012527.20104@news.cs.indiana.edu> chalmers@iuvax.cs.indiana.edu (David Q. Chalmers) writes: >OK, now we're homing in on something closer to the point. Judgments of the >truth of Godel sentences are parasitical on judgments of the consistency >of the given system. And judgments of consistency may be hard. Are not judgements (proofs) of consistancy more than just hard? Isn't consistancy one of the undecidable questions (in the general case)? [I was under the impression that it was equivalent to the halting problem] If so, then one of the premises of the arument is invalid, not even humans can actually determine the truth of all Goedel statements. >This might be responded to by saying that all the contradiction proves is >that *one* of the assumptions is false. We can only conclude that *either* >(a) we aren't that TM; or (b) we can't know that we are that TM; or (c) we >can't know that we are consistent. Benacerraf wrote a paper in the 60's >suggesting (b) (i.e. maybe it's impossible for us to know which TM we are), >but I find this a bit weak as the sole grounds for refuting the argument. But why? Isn't denying (b) equivalent to saying that consistancy is decidable? If so, then (b) is necessarily true unles consistancy is provable. >(c) is perhaps better but still a bit weak -- just say we are that super-good >race that has every reason to believe that they never make a mistake >(remembering that we are allowed to idealize here). I suppose idealization is OK, but it does leave us with an almost uninteresting result! I do not believe that a generalized resoning system can be designed that is not subect to mistakes. I.e. being able to reach arbitrary conclusions without scope limitations automatically implies the possibility of error. Of course with humans it goes even deeper than that, since we operate almost entirely on pattern matching (i.e. guessing). This is true in *all* fields of human endeavor, even (or especially) mathematics. I do not believe that any system could match human performance in mathematics except by using pattern matching to arrive at new hypotheses 'from scratch'. And pattern matching is intrinsically unreliable. What am I saying here? I am saying that humand mathemeticians arrive at conclusions based on prior experience, intuition, training &c. and only produce mathematical proofs as an final step. The real work is in the leaps of intuition. -- --------------- uunet!tdatirv!sarima (Stanley Friesen)