Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84; site rtp47.UUCP Path: utzoo!watmath!clyde!bonnie!akgua!mcnc!rti-sel!rtp47!throopw From: throopw@rtp47.UUCP (Wayne Throop) Newsgroups: net.philosophy Subject: Godel and Turing Message-ID: <227@rtp47.UUCP> Date: Sat, 19-Oct-85 19:32:27 EDT Article-I.D.: rtp47.227 Posted: Sat Oct 19 19:32:27 1985 Date-Received: Mon, 21-Oct-85 00:37:27 EDT Organization: Data General, RTP, NC Lines: 87 Some points raised by Tom Tedrick with regard to Godels incompleteness theorem seem a little incorrect to me. In particular: > The issue is that humans seem to recognize that certain formal systems > are consistent, but that this consistency cannot be proved within the > system. This mysterious ability to recognize such things being something > lacking in deterministic machines, I claim there is a distinction > between the human mind and any Turing machine. First, Godel's theory didn't have anything at all to do with recognizing consistent formal systems. It just states some properties that consistant formal systems of "sufficent power" must have, in particular, incompleteness. This "mysterious ability" is something of your own invention. Second, even allowing humans this "mysterious ability", I don't see why machines "lack" it. Do you have evidence or proof that all machines must "lack" this "mysterious ability" (or is it just that the ones you are currently familiar with seem to lack it?) > Exhibit the turing machine that is claimed to be equivalent to the human > mind, and the human mind can reason about the system in ways impossible > within the system. Thus we contradict the assumption that the machine > was equivalent to the mind. This doesn't follow at all. Your statement that "the human mind can reason about the system in ways impossible within the system" is a simple assertion, with no backing (certainly not by Godel's incompleteness theorem). If you are going to make the assumption "human mind H is equivalent to turing machine T", then one possiblity is that H (if consistant) *indeed cannot* know certain things about machine T. (Or are you asserting that humans *must be* capable of perfect self-knowlege?) In any event, the key here is that you have simply made a set of contradictory assumptions, namely, 1) T is equivalent to H, 2) T is a consistent formal system, and 3) H is complete. You can throw out any of these assumptions... Godel doesn't help you choose which one to throw out. The most suspicious assumption I see there is "H is complete". By this assumption (that "humans" (aside: I'm not sure if you mean "all humans", or "any human" or "some human", but let that pass) can discover a Godel sentence for any given formal system (in this case T)), you are asserting (in essence) that "humans" can *always* tell truth from falsehood in formal mathematical systems. This doesn't seem like a tenable position to take. > I don't believe human beings are deterministic. I also don't accept the > laws of physics as absolute. I accept them as an absolutely brilliant > model but not as complete truth. I don't accept the notion that the > human being is just a very complex machine. These, however, are *assumptions*. They are not *proven* by anybody I am aware of. By the way, I'd be interested in knowing *why* you "don't accept the notion that the human being is just a very complex machine." Do you also "not accept" the notion that "the human being is just a primate", or that "the human being is just a mammal"? How about "insects are just complex machines"? "Reptiles are just complex machines"? "Mammals are just complex machines"? "The (other) primates are just complex machines"? In other words, just what *is* "machine-like" and what is not, and why do you draw the dividing line where you do. I hope you don't think I'm being nasty here, I'd really like to know. I myself don't see any definite boundaries here to point to as the reason for definitely segregating humans from "machine-like things" (that is, things that "merely" follow the "laws of physics"). > *IS THERE ANYONE THAT AGREES WITH ME THAT THE HUMAN MIND IS PROVABLY > NOT EQUIVALENT TO A TURING MACHINE?* It would help if you said what proof you are talking about. If you mean "Is Godel's incompleteness theorem such a proof?", the answer is "definitely not". If anybody *does* agree with you that the human mind is *PROVABLY* "more powerful than" a general recursive formal system, I'd be interested in hearing what they think the proof is. (In my opinion, God Himself is no more powerful than a general recursive formal system :-) > tedrick@ucbernie.ARPA -- Wayne Throop at Data General, RTP, NC !mcnc!rti-sel!rtp47!throopw