Torkel Franzen on truth
in [Logic]
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From: tchow on 26 Dec 2007 17:37 In article <gNycj.14626$vd4.7541(a)pd7urf1no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? >If not, then you haven't fixed a precise context for (*). PA isn't finitely axiomatizable; is that important to you? If so, let's use NBG instead of PA. NBG is finitely axiomatizable, and so we can indeed spell out *all* the axioms of NBG. >And suppose we did manage to fix the context of (*), we've just fixed >a *relative truth*, relative to the context of exactly which "PA" we've >chosen. Fine. So do you agree that there exist relativized truths? For example: (***) In every model of NBG, there do not exist nonzero integers a and b such that a^2 = 2 b^2. Because I'm using NBG, there is no problem with spelling out all its axioms explicitly. Do you agree that (***) is a relativized truth? If so, then do you *believe* (***)? If so, then on what basis do you believe (***)? Most people would say that it's because of the *proof* that sqrt(2) is irrational. But not you, I suppose? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Nam D. Nguyen on 26 Dec 2007 18:02 tchow(a)lsa.umich.edu wrote: > In article <gNycj.14626$vd4.7541(a)pd7urf1no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? >> If not, then you haven't fixed a precise context for (*). > > PA isn't finitely axiomatizable; is that important to you? As long as there is a distinct possibility ~GC isn't provable in Q, then "Yes" it seems important to me. > If so, let's use > NBG instead of PA. NBG is finitely axiomatizable, and so we can indeed spell > out *all* the axioms of NBG. Or Q. I don't know about every intelligent beings in the world (or universe) but relatively to me I could spell out all axioms of Q. And I suspect that relatively to you, a complete spell-out of NBG/Q can be done easily. So relatively speaking, the complete spell-out here has a context it seems. > >> And suppose we did manage to fix the context of (*), we've just fixed >> a *relative truth*, relative to the context of exactly which "PA" we've >> chosen. > > Fine. So do you agree that there exist relativized truths? For example: > > (***) In every model of NBG, there do not exist nonzero integers a and b > such that a^2 = 2 b^2. > > Because I'm using NBG, there is no problem with spelling out all its axioms > explicitly. Do you agree that (***) is a relativized truth? > > If so, then do you *believe* (***)? I don't much about *all* beings there could exist, I do believe in (***) as a relative truth, relatively to me as an being who could do some mathematical reasoning and who's *assuming* certain contexts, e.g. FOL= rules of inference, etc... > > If so, then on what basis do you believe (***)? Most people would say that > it's because of the *proof* that sqrt(2) is irrational. But not you, I > suppose? Oh it's because of proof too, relative to my reasoning. But of course *proof* is always relative to rules of inference, among other things. And there simply are *no "absolute" rules of inference*!
From: tchow on 26 Dec 2007 20:10 In article <I3Bcj.14806$vd4.11334(a)pd7urf1no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >tchow(a)lsa.umich.edu wrote: >> (***) In every model of NBG, there do not exist nonzero integers a and b >> such that a^2 = 2 b^2. [...] >I don't much about *all* beings there could exist, I do believe in (***) >as a relative truth, relatively to me as an being who could do some >mathematical reasoning and who's *assuming* certain contexts, e.g. >FOL= rules of inference, etc... Good. Now we can return to the point where I entered the discussion, when you said, `Unfortunately mathematical reasoning isn't religion where "beliefs" would be much relevant.' But you state here plainly that you believe in (***) as a relative truth, relative to blah blah blah. Therefore "belief," properly understood and qualified, *is* relevant to mathematics, contrary to what you asserted previously. I rest my case. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: Nam D. Nguyen on 26 Dec 2007 21:27 tchow(a)lsa.umich.edu wrote: > In article <I3Bcj.14806$vd4.11334(a)pd7urf1no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> tchow(a)lsa.umich.edu wrote: >>> (***) In every model of NBG, there do not exist nonzero integers a and b >>> such that a^2 = 2 b^2. > [...] >> I don't much about *all* beings there could exist, I do believe in (***) >> as a relative truth, relatively to me as an being who could do some >> mathematical reasoning and who's *assuming* certain contexts, e.g. >> FOL= rules of inference, etc... > > Good. Now we can return to the point where I entered the discussion, when > you said, `Unfortunately mathematical reasoning isn't religion where "beliefs" > would be much relevant.' But you state here plainly that you believe in (***) > as a relative truth, relative to blah blah blah. Therefore "belief," properly > understood and qualified, *is* relevant to mathematics, contrary to what you > asserted previously. What I've asserted from the beginning and through out is *religious belief* is supposed to be absolute and is not the relative kind of beliefs we employ in mathematical reasoning. That's all I've meant to say! In any rate, since truth and no truth could be equated to consistency and inconsistency, respectively, of syntactical axioms, truth and no truth is *relative* to *which set* the axioms are in! > > I rest my case. I might open the case again if anyone believes there is absolute truth in reasoning!
From: tchow on 27 Dec 2007 10:28
In article <G4Ecj.18314$DP1.3340(a)pd7urf2no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >What I've asserted from the beginning and through out is *religious belief* is >supposed to be absolute and is not the relative kind of beliefs we employ in >mathematical reasoning. That's all I've meant to say! Perhaps that's all you *meant* to say, but what you *actually* said was: Unfortunately mathematical reasoning isn't religion where "beliefs" would be much relevant. in response to Daryl McCullough's comment: So, for example, a proof in PA + the negation of Goldbach's conjecture would not be very convincing, because we have no reason to believe that the negation of Goldbach's conjecture is true. The "belief" in question, understood as a relative belief in a relative truth relative to blah blah blah, is entirely relevant. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences |