Torkel Franzen on truth
in [Logic]
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From: LauLuna on 11 Nov 2007 08:39 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. 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
From: Newberry on 11 Nov 2007 11:31 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 -
From: Peter_Smith on 11 Nov 2007 11:53 On 11 Nov, 16:31, Newberry <newberr...(a)gmail.com> wrote: > a) We are absolutely certain about the truths of PA, even those PA > cannot prove Eh??? Who on earth is committing themselves to that silly claim??? Suppose L_1 is the language of first-order arithmetic, then there are various classes of truths of L_1. 1. There are the truths for which we can actually give a proof in first-order PA. 2. There are truths (like the arithmetization of Goodstein's theorem, or like Con(PA)) which are provably not provable in PA, but for which we have proofs in other, richer, theories -- like suitable fragments of set theory. 3. There are other truths which it is not known whether or not they are provable in first-order PA, though we have do proofs in other richer theories. 4. And no doubt there are other truths for which we have no kind of proof (as yet: or may be there could be no humanly surveyable proof). Plainly these different sorts of truths have different epistemic status! We might be certain of the truth of the propositions in the first three classes, given we accept the relevant proofs. But we certainly not certain about the truths in the fourth class (even if we strongly suspect that some propositions like Goldbach's conjecture do indeed fall in the class of truth-but-not-yet proved).
From: Newberry on 11 Nov 2007 12:46 On Nov 11, 12:29 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 11 Nov, 03:05, Newberry <newberr...(a)gmail.com> wrote: > > > On Nov 9, 4:17 am, aatu.koskensi...(a)xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > > > > > Read "pretty good reason" to mean "at least pretty good reason, maybe > > > > conclusive reason". As it happens I think there are conclusive reasons > > > > to believe PA consistent. > > > > Yes, PA is obviously consistent. > > > OK, how do we reconcile it with this? > > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > > There is no conflict at all between what I said (something TF held > too), and that latter quote. To hold that PA is clearly consistent is > quite compatible with holding that, with some arbitrarily thrown- > together extension of Q, we won't in the general case know whether it > is consistent, and hence won't know whether its canonical Gödel > sentence is true. So let's confine ourselves to PA for now. We can prove that it is consistent, that is we have proven G. How did we manage to do that without running into a contradiction? We did not simply add Con(T) as another axiom, we proved it. I suppose we proved it in some metatheory M. How do we know that M is consistent?
From: Newberry on 11 Nov 2007 12:56
On Nov 11, 8:53 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 11 Nov, 16:31, Newberry <newberr...(a)gmail.com> wrote: > > > a) We are absolutely certain about the truths of PA, even those PA > > cannot prove > > Eh??? Who on earth is committing themselves to that silly claim??? > > Suppose L_1 is the language of first-order arithmetic, then there are > various classes of truths of L_1. > > 1. There are the truths for which we can actually give a proof in > first-order PA. > 2. There are truths (like the arithmetization of Goodstein's > theorem, or like Con(PA)) which are provably not provable in PA, but > for which we have proofs in other, richer, theories -- like suitable > fragments of set theory. There is also 5. the truths of which we are absolutely certain although they are provably unprovable in PA, like Con(T). The certainty comes from seeing that the axioms of PA as manifestly true. Perhaps 5 is identical with 2. How do we know that these richer theories, such as a suitable fragment of ZFC, are consistent? > 3. There are other truths which it is not known whether or not they > are provable in first-order PA, though we have do proofs in other > richer theories. > 4. And no doubt there are other truths for which we have no kind of > proof (as yet: or may be there could be no humanly surveyable proof). > > Plainly these different sorts of truths have different epistemic > status! We might be certain of the truth of the propositions in the > first three classes, given we accept the relevant proofs. But we > certainly not certain about the truths in the fourth class (even if we > strongly suspect that some propositions like Goldbach's conjecture do > indeed fall in the class of truth-but-not-yet proved). |