Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 12 Nov 2007 13:36 On 2007-11-12, in sci.logic, abo wrote: > Obviously I don't find your explanation incomprehensible, but I do > find it lacking. It is lacking at the very beginning, in that the > entire point is how or why you think you know that you can always "add > one". That's simply part of our conception of the naturals. I find the idea that some natural might -- perhaps by accident? -- lack a successor completely baffling, and can make nothing of it unless it is explained what such a thing might mean. > One other thing. Your statement at the end about "connecting such > interests to rather elusive and incomprehensible doubts is pointless" > is a subjective claim hidden as an oracular assertion about which > there can be no dispute. Anything at all can be disputed. > You think it is pointless, no problem with that. You've been to Sunday > School, and you've learned what you've been told. Good for you! Why do you think I've been to Sunday School? -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: LauLuna on 12 Nov 2007 14:16 On Nov 11, 5:31 pm, Newberry <newberr...(a)gmail.com> wrote: > On Nov 11, 5:39 am, LauLuna <laureanol...(a)yahoo.es> wrote: > > > > > > > On Nov 10, 11:38 pm, Newberry <newberr...(a)gmail.com> wrote: > > > > On Nov 10, 12:52 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > > > On 10 Nov, 16:53, Newberry <newberr...(a)gmail.com> wrote: > > > > > > b) The human mind does not surpass a machine > > > > > > TF is in favor of b. > > > > > > If b is the case then I > > > > > wonder how we can construct a machine that can generate all the truth > > > > > of PA. > > > > > Why should any machine be able to generate all the truths of (the > > > > language of) PA? After all, we can't do that either. > > > > How can we construct a machine that can generate all the truth of PA > > > that we can? > > > Good question. > > > In 'Inexhaustibility' TF poses the following question: > > > It seems that whenever human logico-mathematical reason (HLMR) sees as > > evident a set of axioms, it also sees as evident the proposition that > > those axioms are consistent (which is a kind of reflection principle). > > But, if HLMR is consistent and sufficiently rich, that proposition > > does not always follow from those axioms (by Gödel's second theorem). > > So, if there is an initial and sufficiently rich set of logico- > > mathematical truths that must be included in HLMR and HMLR is closed > > under that kind of reflection principle, there is no algorithm > > representing human logico-mathematical reason. > > > As I interpret TF, he denies the conclusion by alleging > > > 1. It might happen that there is no such thing as a definite HLMR. > > > 2. Even if HLMR exists, human finiteness precludes the possibility > > that it is closed under that reflection principle: humans will > > hesitate as things grow increasingly involved. > > > TF's position (very akin indeed to Hofstadter's) seems questionable to > > me because it fails to recognize the existence of an ideal legality in > > human reason, that is different from what humans can actually perform, > > and that he, TF, is implicitly invoking while reasoning. > > This is the part I did not understand. And this is how I interpreted > it: > > a) We are absolutely certain about the truths of PA, even those PA > cannot prove > b) The consistency of PA can be proven in ZFC > c) Therefore we can write a computer program emulating ZFC that can > generate the truths of PA > d) We are absolutely certain about the truths of ZFC, even those ZFC > cannot prove > e) There is a theory X in which we can prove the consistency of ZFC > f) Therefore we can write a computer program emulating X that can > generate the truths of ZFC > g) We are not certain about the truths of X > > Maybe I read him wrong? > > > > > Nevertheless, I think TF's arguments show clearly why Lucas's and > > Penrose's arguments fail. They both are assuming implicitly that: > > > A. HLMR is a definite object > > B. HLMR is closed under some reflection principle(s). > > > Clearly, A and B does not follow from Gödel's theorem. > > > Regards- Hide quoted text - > > > - Show quoted text -- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text - I interpret that your a), b), c),... are just an example (for actually they are not true). If so, what you say is similar to what I think TF said. TF suggests that we would eventually get to such a convoluted theory that we would hesitate in applying the reflection principles in order to get new truths. We do believe: a') The axioms of PA b') a')+ Con(a') c') b') + Con(b') etc. TF says that at some too high level things would get so difficult that we would stagger or would be simply unable to go on. What I argue is that though this is in fact so, the argument fails to distinguish the logico-mathematical legality enclosed in human reason from what humans can effectively accomplish., i.e. what is logically possible for humans from what is physically possible for them. The necessity of reducing logical impossibility to physical impossibility is one of the weak points of AI. Regards
From: abo on 12 Nov 2007 15:44 On Nov 12, 7:36 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-11-12, in sci.logic, abo wrote: > > > Obviously I don't find your explanation incomprehensible, but I do > > find it lacking. It is lacking at the very beginning, in that the > > entire point is how or why you think you know that you can always "add > > one". > > That's simply part of our conception of the naturals. This is just the ontological fallacy (from a property of a thing you can infer the existence of a thing). You say that you have a conception whereby a natural number always has a successor,. Fine; the only things that will be natural numbers are those things that have a successor. But that doesn't imply that there exist any natural numbers. Similarly, one could say that part our conception of God is that He is an absolutely perfect being. But that doesn't imply that there is any being who is absolutely perfect. I'd add that it seems to me worthwhile to distinguish our conception of what a natural number is, and our conception of what the natural- number sequence is. I think it is incorrect to say that part of our conception of what a natural number is is that it have a successor. We do have a conception of naturals, and we'd agree that 2 is a natural number; yet it just can't be, from the fact that 2 is a natural number, that 10^10^10^10 exists and is a natural number, which would in fact follow were every natural number always to have a successor (which is a natural). I'd agree with you that our conception of the natural-number sequence is that every natural in the sequence has a successor (in the sequence); but then the question just becomes whether there is any such sequence. > I find the idea that > some natural might -- perhaps by accident? -- lack a successor completely > baffling, and can make nothing of it unless it is explained what such a > thing might mean. Here's one way to picture it: after some very big point, the naturals fade away, ever so gradually. > > > One other thing. Your statement at the end about "connecting such > > interests to rather elusive and incomprehensible doubts is pointless" > > is a subjective claim hidden as an oracular assertion about which > > there can be no dispute. > > Anything at all can be disputed. I agree with you. But surely you realize - because you surely you intend it - that your style tends at times to be oracular, where you assert something as if dispute is impossible. I usually find it more appropriate, for instance, to say, "I find it obvious that..." instead of "It's obvious that...". > > > You think it is pointless, no problem with that. You've been to Sunday > > School, and you've learned what you've been told. Good for you! > > Why do you think I've been to Sunday School? > Because you have?
From: Newberry on 12 Nov 2007 23:00 On Nov 12, 9:59 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-11-12, in sci.logic, abo wrote: > > > Somebody, who was in conversation with PS about this subject, asked > > him a question, "How do you know?" That would seem to be incentive > > enough to provide at least a modicum of scrutiny or reflection. > > Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- > find PA's consistency obvious by now. On the conception that the naturals > are obtained from 0 by repeatedly applying the "add-one"-operation the > principle of induction > > Whenever P is a determinate mathematical property of naturals, if 0 has P, > and whenever n has P, n+1 also has P, all naturals have P > > is manifestly true, as is the principle of definition by primitive > recursion, that properties definable by primitive recursion are determinate > and well-defined in the relevant sense. Combining this observation with the > determinateness of properties expressible in the first order language of > arithmetic, that is, those obtainable from the primitive recursive > properties by means of the usual logical operations, leads immediately to > the conclusion that the axioms of PA are all manifestly true, and hence no > contradiction follows from them, by the soundness of the rules of inference > of first order logic. > > Now, if someone finds this explanation incomprehensible even after > elaborations, illustrations, gentle persuasion, practice, and so on, I'm > stumped. There's simply nothing I can do but shrug. Of course, people might > be interested in e.g. what can and cannot be proved without appeal to the > totality of the successor function, full induction, etc. for perfectly > sensible reasons -- often we obtain mathematical information beyond than > just that P is true if we know that P is not only true but also provable > from these or those (weak) principles -- but connecting such interests to > rather elusive and incomprehensible doubts is pointless. Right. So now the question is how do we reconcile the absolute certainty that PA is consistent with Goedel's theorem, which says that the consistency of PA is unprovable. It seems that you just proved it. You can prove it in ZFC? First of all I do not know if the ZFC proof is the same one as the manifest truth proof. Secondly, is ZFC consistent?
From: Herman Jurjus on 13 Nov 2007 04:04
Aatu Koskensilta wrote: > On 2007-11-12, in sci.logic, abo wrote: >> Somebody, who was in conversation with PS about this subject, asked >> him a question, "How do you know?" That would seem to be incentive >> enough to provide at least a modicum of scrutiny or reflection. > > Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- > find PA's consistency obvious by now. [...] > Now, if someone finds this explanation incomprehensible Who says they do? Perhaps they just find it 'too easy' as an answer? > I'm stumped. Glad that you admit that you don't understand the issue. > There's simply nothing I can do but shrug. Glad again that you admit it yourself. But the feeling might be more mutual than you think, you know. [Everything with a grain of salt, as usual.] -- Cheers, Herman Jurjus |