Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!samsung!sol.ctr.columbia.edu!bronze!chalmers From: chalmers@bronze.ucs.indiana.edu (David Chalmers) Newsgroups: comp.ai.philosophy Subject: Re: Minds, machines, and Godel Message-ID: <1991Jan18.005010.23943@bronze.ucs.indiana.edu> Date: 18 Jan 91 00:50:10 GMT References: <1991Jan17.170401.8536@bronze.ucs.indiana.edu> <1991Jan17.233234.17164@cs.umn.edu> Organization: Indiana University, Bloomington Lines: 69 David Thornley writes: >Further, the Godel process, as well as I can tell, results in the >proposition P reading "Proposition P cannot be proved in this system", >which is obviously unprovable in that system, and hence true, provided >the system is consistent. That's roughly correct, except that the proposition is not "Proposition P cannot be proved in this system". Rather, it's a very complex statement of arithmetic that happens to be equivalent to that proposition. >This seems, to me, to be precisely equivalent >to statement S: "Statement S cannot be asserted by Chalmers". >I will point out that that statement is true iff Chalmers is >consistent, and challenge Chalmers to stay consistent while >agreeing with me. I agree that this proposition causes me problems ("...I want to say it's true, really I do...but I just can't. Oh no, I just did! Does the world explode now?"). But it's not directly relevant to the argument, as the argument never claimed that humans have no limitations. >I'm not talking psychology here, I'm talking about assertions and >logical consistency. No system of logic can be both complete and >consistent, and neither can a human. Any arguments that human >beings are different from logical systems will have to start somewhere >else. That's true, and it does. Humans are certainly not complete and consistent when it comes to English sentences, as your argument demonstrates; and they are very likely not complete and consistent when it comes to arithmetic. But the argument did not claim that they were. Rather, it claimed that human capabilities differ from those of any given Turing Machine. >Before we continue with this discussion, I would like to ask how we >are to see that something is true. Most of the time, I don't "see" >truth, I have to work at it. We'll get an army of mathematicians to work on judging the statement for a hundred years. The process needn't be direct. Then, if the mathematicians are computationally simulable, we'll get a TM to simulate them. >If we have a program that will pass the Turing test, we presumably >can make some sort of equivalent logical system, and derive a Godel >number corresponding to an undecidable statement, given infinite >resources. It is not at all clear to me that we can see that this >statement is true; what if a slight mistake occurred during the >process? How can we tell? Well, our army of mathematicians will have plenty of time to check and recheck the process. Of course you might still maintain that there remains a tiny but non-zero chance of error. I would like to idealize away from such errors, as I don't believe they are really relevant. Appeal to error would be an "easy way out" of the Lucas argument. And if you accepted it as the only way out, you would be committed to the claim that the Lucas argument really would apply to error-free creatures. I don't believe that, and think that the argument has a more fundamental problem. So henceforth I'd like to idealize away from the possibility of human error/inconsistency, if you'll allow me. (Those who believe that human error/inconsistency is the fundamental problem with the argument, and that the argument really would apply to hypothetical error-free creatures, should feel free to ignore the discussion from here.) -- Dave Chalmers (dave@cogsci.indiana.edu) Center for Research on Concepts and Cognition, Indiana University. "It is not the least charm of a theory that it is refutable."