Torkel Franzen on truth
in [Logic]
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From: george on 14 Nov 2007 14:13 On Nov 12, 6:37 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > Given that the quoted passage is perfectly clear No, it isn't. Gee, that was easy. Even talking about the clarity of *a* passage in the context of an overall take on a global issue is silly to begin with. The meaning and the clarity of the passage have obvious prior dependencies on the coherence of the context. The quoted piece involved 2 different statements on p.105 and a longer prior one from p.55. That triple does NOT fall under the definition of "the quoted passage". "Passages", instead. And once there are 3 of them then they get to have 3 different degrees of clarity. Not to mention relevance. The mere fact that the passages purport to talk about truth at all is unfortunate. That is necessary if you are trying to debunk other people's misconceptions but there is an underlying hubris here, an underlying claim to gnosis, that is far more objectionable for being SIMPLY IRRELEVANT than it is for being conceited. > it seems you're suggesting > Franzén managed to write something eminently comprehensible without himself > understanding any of it. "True" in natural language is complicated. The mere existence of "this sentence is NOT true" proves THAT. Choosing to talk about some of this stuff as though it were straightforward is part of the disease, not of the cure. > This is a curious suggestion -- perhaps you have > something more sensible in mind? You're being entirely too charitable. Just the usual cantankerous nihilism.
From: george on 14 Nov 2007 14:17 On Nov 12, 6:43 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > Gentzen's proof and the existence of the set of natural numbers don't really > have much to do with obviousness of consistency of PA. Are you saying that PA continues to remain obviously consistent even in the context where it is insisted that N is a proper classm and just plain can't be a set? Given that the model existence theorem is a theorem of SET theory, that is going to be a little complicated. There is a context in which we can PROVE that a theory is consistent if and only if it has a model. If N doesn't exist then it isn't a model. So is it always obvious how to prove that some other model exists? Or are you insisting that some context-for-debating-consistency-in- which- models-in-general-are-not-relevant has *higher*, *prior* claim or status than THE USUAL context, in which models of consistent theories MUST exist?
From: Daryl McCullough on 14 Nov 2007 19:56 Newberry says... > >On Nov 13, 9:14 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >> >> >> >On Nov 13, 1:49 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: >> >> To know that theory (PA or M) is consistent we don't necessarily need a >> >> "proof". >> >> >Right. So it means that Lucas and Penrose are right, and Franzen is >> >wrong? >> >> That doesn't follow at all. The question that Penrose asked was: >> Is it possible for there to be a computer program P(x) such that >> >> P(x) halts and returns true >> <-> >> x is the Godel number of a statement that human >> mathematicians can become convinced is an absolutely >> unassailable truth >> >> It is irrelevant whether the set of "unassailable truths" >> are provable or not. > >His point was that the human mind surpasses any machine. If the human >mind can comprehend a truth that cannot be formally proven then the >human mind surpasses any computer. No, that doesn't follow at all. You're applying a double standard. You're only requiring that the human be able to "comprehend" a truth, while you're requiring that the computer be able to *prove* it. To make it a fair comparison, you either require both to prove the statement, or require neither to prove the statement. >Example: "if a set of axioms is manifestly true then the theory is >consistent" is just as compelling as e.g. "~A, A v B |- B." It is not >clear how any machine can prove that a theory based on manifestly true >axioms is consistent. Why does it matter whether the machine can prove it, if the human can't prove it, either? As I said, for a computer program to be as powerful as a human in recognizing truth, all that's necessary is for the program to be able to *recognize* true statements. It's not necessary that the program be able to *prove* them. Having said that, there is actually no problem in proving that a true theory must be consistent: Truth is preserved by logical deduction. A contradiction cannot be true. Therefore, it is impossible to deduce a contradiction from any true theory. -- Daryl McCullough Ithaca, NY
From: Newberry on 14 Nov 2007 23:13 On Nov 14, 4:56 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Newberry says... > > > > > > > > >On Nov 13, 9:14 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > >> Newberry says... > > >> >On Nov 13, 1:49 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > >> >> To know that theory (PA or M) is consistent we don't necessarily need a > >> >> "proof". > > >> >Right. So it means that Lucas and Penrose are right, and Franzen is > >> >wrong? > > >> That doesn't follow at all. The question that Penrose asked was: > >> Is it possible for there to be a computer program P(x) such that > > >> P(x) halts and returns true > >> <-> > >> x is the Godel number of a statement that human > >> mathematicians can become convinced is an absolutely > >> unassailable truth > > >> It is irrelevant whether the set of "unassailable truths" > >> are provable or not. > > >His point was that the human mind surpasses any machine. If the human > >mind can comprehend a truth that cannot be formally proven then the > >human mind surpasses any computer. > > No, that doesn't follow at all. You're applying a double standard. > You're only requiring that the human be able to "comprehend" a truth, > while you're requiring that the computer be able to *prove* it. To > make it a fair comparison, you either require both to prove the > statement, or require neither to prove the statement. > > >Example: "if a set of axioms is manifestly true then the theory is > >consistent" is just as compelling as e.g. "~A, A v B |- B." It is not > >clear how any machine can prove that a theory based on manifestly true > >axioms is consistent. > > Why does it matter whether the machine can prove it, > if the human can't prove it, either? As I said, for > a computer program to be as powerful as a human in > recognizing truth, all that's necessary is for the > program to be able to *recognize* true statements. > It's not necessary that the program be able to > *prove* them. How can a computer recognize that PA is consistent? > > Having said that, there is actually no problem in proving > that a true theory must be consistent: Truth is preserved > by logical deduction. A contradiction cannot be true. > Therefore, it is impossible to deduce a contradiction > from any true theory. > > -- > Daryl McCullough > Ithaca, NY- Hide quoted text - > > - Show quoted text -
From: Daryl McCullough on 15 Nov 2007 00:13
Newberry says... > >On Nov 14, 4:56 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> Why does it matter whether the machine can prove it, >> if the human can't prove it, either? As I said, for >> a computer program to be as powerful as a human in >> recognizing truth, all that's necessary is for the >> program to be able to *recognize* true statements. >> It's not necessary that the program be able to >> *prove* them. > >How can a computer recognize that PA is consistent? By producing an output "true" when given the input question "Do you believe that PA is consistent?". Are you asking how one would go about *programming* a computer program that would emulate a human's mathematical reasoning? If so, I have no idea. The issue is not whether we *currently* know how to make an artificial intelligent computer program. The issue is whether Godel's theorem implies that it is impossible. It doesn't imply any such thing. There is what I think is a pretty air-tight argument that no single human can do any mathematical reasoning that is noncomputable: A real human has a finite memory capacity, and so there are only finitely many different statements of mathematics that we can ever hold in our heads at one time. So the collection of all statements that any *actual* human would ever claim to be "unassailably true" is a finite set. Every finite set of formulas is computable. Now, you could argue about what an idealized human could do, where we idealize the human to have an infinite memory capacity. Could such an idealized human do something that no Turing machine could do? Well, it depends on the details of how the "ideal human" is idealized. -- Daryl McCullough Ithaca, NY |