Torkel Franzen on truth
in [Logic]
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From: Nam D. Nguyen on 19 Dec 2007 16:35 tchow(a)lsa.umich.edu wrote: > In article <fhY9j.1418$Tx.1408(a)pd7urf3no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> There's a misconception here that seems to have escaped your attention. A model >> *always already* includes a *chosen* interpretation: hence a belief has been >> "believed" already. What's important is this interpretation could always be >> reversed to the other way - at will - and a opposite "belief" would occur. >> Consequently, a mathematically *stated* belief could change back and forth, > [...] >> In summary, for "there are no [nonzero] integers m and n such that >> m^2 = 2 n^2", its truth, or its believed truth is quite subjective and >> relative, like the "Colours seen by candlelight", but unlike religion >> truth and belief. > > So let's look at these two statements: > > (1) There are no nonzero integers m and n such that m^2 = 2 n^2. > > (2) In the standard model of the integers, there are no nonzero integers > m and n such that m^2 = 2 n^2. > > As I understand your position, (1) does not have a determinate truth value. > But in (2), the phrase "in the standard model of the integers" chooses > an interpretation and hence a "belief has been `believed' already." > Does that mean that (2) is absolutely, unconditionally true? No. What is a standard model of the integers might not be the "standard" model to others! > *After* I choose an interpretation, the subjectivism and relativism > are gone, aren't they? No, they're just there besides you. That the speed of a train is absolutely 100 km/hr from you point of view doesn't make the speed of train be an absolute quantity!
From: Nam D. Nguyen on 19 Dec 2007 16:48 Nam D. Nguyen wrote: > tchow(a)lsa.umich.edu wrote: >> In article <fhY9j.1418$Tx.1408(a)pd7urf3no>, >> Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >>> There's a misconception here that seems to have escaped your >>> attention. A model >>> *always already* includes a *chosen* interpretation: hence a belief >>> has been >>> "believed" already. What's important is this interpretation could >>> always be >>> reversed to the other way - at will - and a opposite "belief" would >>> occur. >>> Consequently, a mathematically *stated* belief could change back and >>> forth, >> [...] >>> In summary, for "there are no [nonzero] integers m and n such that >>> m^2 = 2 n^2", its truth, or its believed truth is quite subjective and >>> relative, like the "Colours seen by candlelight", but unlike religion >>> truth and belief. >> >> So let's look at these two statements: >> >> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >> >> (2) In the standard model of the integers, there are no nonzero >> integers >> m and n such that m^2 = 2 n^2. >> >> As I understand your position, (1) does not have a determinate truth >> value. >> But in (2), the phrase "in the standard model of the integers" chooses >> an interpretation and hence a "belief has been `believed' already." > > >> Does that mean that (2) is absolutely, unconditionally true? > > No. What is a standard model of the integers might not be the "standard" > model to others! Let me re-phrase it: what might be "integers" or "standard" to one, might not be to the others. > > >> *After* I choose an interpretation, the subjectivism and relativism >> are gone, aren't they? > > No, they're just there besides you. That the speed of a train is absolutely > 100 km/hr from you point of view doesn't make the speed of train be an > absolute > quantity!
From: tchow on 19 Dec 2007 17:40 In article <q8gaj.19679$Tx.4697(a)pd7urf3no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >tchow(a)lsa.umich.edu wrote: >> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >> >> (2) In the standard model of the integers, there are no nonzero integers >> m and n such that m^2 = 2 n^2. [...] >> Does that mean that (2) is absolutely, unconditionally true? > >No. What is a standard model of the integers might not be the "standard" >model to others! All right, then, let's try this one: (3) In every model of PA, there are no nonzero integers m and n such that m^2 = 2 n^2. Most people would agree with (3), since the proof of the irrationality of sqrt(2) can be formalized in PA, and therefore the statement holds in all models of PA, standard or nonstandard. Do you believe (3)? Is (3) absolutely true? Whether or not your "standard" model is the same as mine makes no difference, since the assertion holds in every model, so I don't see where relativism enters. -- 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: tchow on 19 Dec 2007 17:44 In article <8c15ea4d-e748-47eb-8aaa-057487d1065f(a)l1g2000hsa.googlegroups.com>, george <greeneg(a)cs.unc.edu> wrote: >And your point about what to "specify" for induction is just bullshit. >When induction is introduced, the simple introductory context into >which it is being introduced will guarantee that nothing beyond 1st-order >comes up. Even granting a lot of things in your article that I don't agree with, this *still* makes no sense. Why PA? Why not PRA, or even weaker induction schemes? These would all suffice for anything that comes up in high school. For that matter, why PA and not some two-sorted first-order system like RCA_0 or ACA_0? They would all suffice too. In fact, in many ways they're much more natural than PA. That everyone of a certain age "started with PA" is simply a lie. -- 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 19 Dec 2007 18:11
tchow(a)lsa.umich.edu wrote: > In article <q8gaj.19679$Tx.4697(a)pd7urf3no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> tchow(a)lsa.umich.edu wrote: >>> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >>> >>> (2) In the standard model of the integers, there are no nonzero integers >>> m and n such that m^2 = 2 n^2. > [...] >>> Does that mean that (2) is absolutely, unconditionally true? >> No. What is a standard model of the integers might not be the "standard" >> model to others! > > All right, then, let's try this one: > > (3) In every model of PA, there are no nonzero integers m and n such that > m^2 = 2 n^2. > > Most people would agree with (3), since the proof of the irrationality of > sqrt(2) can be formalized in PA, and therefore the statement holds in all > models of PA, standard or nonstandard. > > Do you believe (3)? Is (3) absolutely true? Whether or not your "standard" > model is the same as mine makes no difference, since the assertion holds in > every model, so I don't see where relativism enters. Apparently you've missed my last post where I made the correction: > Let me re-phrase it: what might be "integers" or "standard" to one, > might not be to the others. Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) is a relative value, and the relativity is still there! |