Torkel Franzen on truth
in [Logic]
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From: Newberry on 29 Nov 2007 10:42 On Nov 28, 11:31 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > "george" <gree...(a)cs.unc.edu> ha scritto > > >> >> we instead DO know it by mere > >> >> consideration of the intuitive not-formalized arguments 1 or 2 above. > > >> > No, really, you don't know much of ANything, especially > >> > not anything infinitary, THAT way. > > >> I desagree. The majority of mathematicians usually talk about infinitary > >> things and are convinced about their arguments without even know what ZFC > >> is > > > But you said "arguments". "Argument" implies a chain of REASONING, of > > logical INFERENCE. In other words, there are some axioms and rules > > of inference being invoked to support these beliefs. > > Axioms and rules of inference are connected only to *formal* reasoning. > Informal reasoning don't imply any defined set of axioms or rules, even if it > can if course be formalized in many different formal deductive systems. > > You should also agree that we definitely don't need a reasoning to be framed > in a set of axioms and rules to be convinced. Of course it can help, but we > are convinced (for example) of infinity of primes or Pythagorean theorem long > before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. > > >> (and also knowing nothing about first order formal systems). > > > It's not like religion. It works whether you believe in it or not. > > I know. The point is that even if you could be able to formalize the > reasoning of mathematicians in ZFC, this formalization is not really relevant > for their work and their beliefs. The discrepancy is that the formalized proofs of PA's consistency have zero cogency while the intuitive informal arguments - so I am told - have 100% cogency. In this sense the claim tha that the intuitive argument is formalizable is misleading.
From: george on 30 Nov 2007 11:46 On Nov 29, 2:31 am, "LordBeotian" <pokips...(a)yahoo.it> wrote: > Axioms and rules of inference are connected only to *formal* reasoning. WRONG. *ALL* *reasoning* is rule-based. That's what MAKES it *reasoning*. > Informal reasoning don't imply any defined set of axioms or rules, even if it > can if course be formalized in many different formal deductive systems. EXACTLY. It is the fact that it CAN be formalized that MAKES it REASONING. And in fact, in every individual case, the individual reasoner IN FACT IS USING axioms and rules. The fact that he is not being overtly explicitly conscious of which ones they are -- especially when it doesn't matter -- does NOT imply that they are not there. It does not imply that his reasoning behavior is not factually OBSERVABLY CONFORMING to certain patterns and rules. > You should also agree that we definitely don't need a reasoning to be framed > in a set of axioms and rules to be convinced. That is NOT the point. The point is that IF it IS "reasoning" AT ALL, and if it is "convincing" at all, then it is convincing BECAUSE OF its CONFORMANCE to axioms and rules. > Of course it can help, but we > are convinced (for example) of infinity of primes or Pythagorean theorem long > before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. Perhaps, but the point is, whatEVER argument convinced you of them, it made use of SOME form of axioms, definitions, and rules of inference. The fact that nobody wrote them all down in advance and required you to put all the right pegs in the right holes does NOT imply that you were not, in fact, while working through the reasoning, doing exactly that. > I know. The point is that even if you could be able to formalize the > reasoning of mathematicians in ZFC, this formalization is not really relevant > for their work and their beliefs. Of course, since ANY formulation will do. The point is it is simply BETTER to KNOW what formulation you are using. For a long time I did not know (after FLT was proved) whether FLT followed from PA or not. It turns out it was proved long ago that it doesn't. But the fact that that proof got as much acceptance as it got withOUT it being clear what axioms it was being proved from IS ENTIRELY A BAD thing.
From: george on 30 Nov 2007 11:49 On Nov 29, 10:42 am, Newberry <newberr...(a)gmail.com> wrote: > The discrepancy is that the formalized proofs of PA's consistency have > zero cogency while the intuitive informal arguments - so I am told - > have 100% cogency. This is exactly the opposite of the truth. Every individual step of any formal argument has ABSOLUTELY MAXIMAL cogency. If you know that P v Q v R etc. and you also know that ~P, then the argument that from these, the shorter Q v R etc. MUST follow, has ABSOLUTELY MAXIMAL cogency. There is a version of FOL in which THIS is THE ONLY inference rule. So all the formalized proofs OF ANYthing that you CAN prove in FOL have maximal cogency. It is the informal ones that may be open to doubt. Of course, all this is meaningless if the axioms are inconsistent. But there is certainly nothing even vaguely resembling a cogent argument that ZFC is inconsistent.
From: LordBeotian on 30 Nov 2007 15:10 "george" <greeneg(a)cs.unc.edu> ha scritto >> Axioms and rules of inference are connected only to *formal* reasoning. > > WRONG. *ALL* *reasoning* is rule-based. That's what MAKES > it *reasoning*. Maybe, but we don't know which are these axioms and rules in general, so it can be a controversial point of view to assume that they always characterize every kind of reasoning. >> Informal reasoning don't imply any defined set of axioms or rules, even if >> it >> can if course be formalized in many different formal deductive systems. > > EXACTLY. It is the fact that it CAN be formalized that MAKES it > REASONING. Well, consider that the concept of "reasoning" do not belong only to mathematics. We have reasonings in philosophy, psichology, law, politics... most of them don't seem to be so easily "formalizable". Consider also that even 2nd order Peano arithmetic is not formalizable. > And in fact, in every individual case, the individual reasoner IN FACT > IS USING > axioms and rules. The fact that he is not being overtly explicitly > conscious > of which ones they are -- especially when it doesn't matter -- does > NOT imply > that they are not there. It does not imply that his reasoning > behavior is not > factually OBSERVABLY CONFORMING to certain patterns and rules. If you allow those "axioms and rules" to be not conscious than I could still be right when I say that: >> we DON'T know that PA is consistent >> because we formalized it in ZFC (*), we instead DO know it by mere >> consideration of the intuitive not-formalized arguments 1 or 2 above. because we could still think that we are convinced by these informal reasonings because of my inconscious rules and axioms. >> You should also agree that we definitely don't need a reasoning to be >> framed >> in a set of axioms and rules to be convinced. > > That is NOT the point. The point is that IF it IS "reasoning" AT ALL, > and if it is "convincing" at all, then it is convincing BECAUSE OF its > CONFORMANCE to axioms and rules. Well, I actually don't know why the human mind find things convincing or not, maybe there are some axioms and rules hidden inside it or maybe not. This was not the issue I was addressing however. >> Of course it can help, but we >> are convinced (for example) of infinity of primes or Pythagorean theorem >> long >> before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. > > Perhaps, but the point is, whatEVER argument convinced you of them, > it made use of SOME form of axioms, definitions, and rules of > inference. Yes, you can of course say that because we know that these arguments are formalizable and you are considering a weak form of "use" of the axioms and rules (it could be inconscious). I obviously can agree with your view, but my point was another: I'm saying that we don't need to actually formalize an argument in ZFC, PA or any other formal system in order to be convinced. >> I know. The point is that even if you could be able to formalize the >> reasoning of mathematicians in ZFC, this formalization is not really >> relevant >> for their work and their beliefs. > > Of course, since ANY formulation will do. > The point is it is simply BETTER to KNOW what formulation you are > using. For a long time I did not know (after FLT was proved) whether > FLT followed from PA or not. It turns out it was proved long ago that > it > doesn't. Fermat's Last Theorem has proven to be undecidable in PA? > But the fact that that proof got as much acceptance as it > got > withOUT it being clear what axioms it was being proved from > IS ENTIRELY A BAD thing. I'm not responsible :)
From: MoeBlee on 30 Nov 2007 16:06
On Nov 30, 12:10 pm, "LordBeotian" <pokips...(a)yahoo.it> wrote: > Consider also that even 2nd order Peano arithmetic is not formalizable. I don't know what you mean by '2nd order Peano arithmetic', but there is a formal theory that is second order Peano arithmetic. MoeBlee |