Torkel Franzen on truth
in [Logic]
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From: Nam D. Nguyen on 24 Dec 2007 00:46 Nam D. Nguyen wrote: > On the account of history, there were times in the past we believed the > 5th postulate be "absolutely" true or that there being no "number" whose > square is 1 be "absolutely" not false. Sorry for a typo. I meant "whose square is -1".
From: LordBeotian on 24 Dec 2007 09:33 "Nam D. Nguyen" <namducnguyen(a)shaw.ca> ha scritto >>>> <> (3) In every model of PA, there are no nonzero integers m and n such >>>> that >>>> <> m^2 = 2 n^2. >> [...] >>> For another example, if we change by what we mean by "PA" then (3) is >>> not necessarily true. >> >> But then what kind of distinction is there between mathematics and >> religion? >> Consider the following religious statement: >> >> (4) Jesus is the son of God. >> >> By changing the meaning of "Jesus," "son," and "God," the truth value of >> (4) >> could change. Maybe I have a cat named "God" and one of its litter is >> named >> "Jesus." Or maybe I have a dog named "God" with no offspring. > > The distinction is, in religion, a statement like (4) has only *one > semantic* > and is of *one truth*. It depends on the language we are using to speak about religion. If we are speaking spanish (4) has no meaning. If we are speaking "Gunglish", a new language that is equal to english for everything but the word "son", that means "father", then (4) is false. So the truth value of (4) is still relative. To fix one meaning you have to fix the language, but this would also fix the meaning of "PA" and mathematics wouldn't also be relative anymore.
From: Nam D. Nguyen on 24 Dec 2007 11:47 LordBeotian wrote: > > "Nam D. Nguyen" <namducnguyen(a)shaw.ca> ha scritto > >>>>> <> (3) In every model of PA, there are no nonzero integers m and n >>>>> such that >>>>> <> m^2 = 2 n^2. >>> [...] >>>> For another example, if we change by what we mean by "PA" then (3) is >>>> not necessarily true. >>> >>> But then what kind of distinction is there between mathematics and >>> religion? >>> Consider the following religious statement: >>> >>> (4) Jesus is the son of God. >>> >>> By changing the meaning of "Jesus," "son," and "God," the truth value >>> of (4) >>> could change. Maybe I have a cat named "God" and one of its litter >>> is named >>> "Jesus." Or maybe I have a dog named "God" with no offspring. >> >> The distinction is, in religion, a statement like (4) has only *one >> semantic* >> and is of *one truth*. > > It depends on the language we are using to speak about religion. If we > are speaking spanish (4) has no meaning. If we are speaking "Gunglish", > a new language that is equal to english for everything but the word > "son", that means "father", then (4) is false. So the truth value of (4) > is still relative. > > To fix one meaning you have to fix the language, but this would also fix > the meaning of "PA" and mathematics wouldn't also be relative anymore. > Consider: (4') Jesus ist der [einzige] Sohn Gottes. (4) and (4)' are supposed to have the identical *religious* "meaning" and "truth", which are beyond isomorphism of wordings and languages, so to speak. Otoh, "1+1=0" and "1' +' 1' = 0'" not only could differ in meaning and truth per languages, but also could differ in the same language. E.g., "1+1=0" could differ between the natural "arithmetic" and the modulo counterpart!. (And that's the difference between absolute meaning of religous truths and mathematical truths, imho).
From: tchow on 26 Dec 2007 10:21 In article <aFHbj.7280$vd4.540(a)pd7urf1no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >But it's perfectly normal >for us to speak of, say, "1+1=0" as true, or false, relative to whatever the >context that we choose. (But there's *no absolute context* that everyone must >accept that statement as true or otherwise). It still seems to me that your view suffers from a problem of infinite regress. You don't think that statements such as "1+1=0" or "PA is consistent" have a determinate truth value; one must fix the context. However, what if we fix the context? *Then* do we get a determinate truth value? So for example, consider (*) "1+1=0" is false in the standard model of PA. Does (*) have a determinate truth value? I've fixed the context of "1+1=0", haven't I? Doesn't this thereby fix the truth value of (*)? Doesn't it make the "belief having been believed already" (or whatever locution it was you used)? But when I pressed this point before, you said no; "PA," you said, was itself indeterminate. So somehow we have to fix the context of *which* PA we're talking about. But now an infinite regress looms. Won't any attempt to fix the context of PA itself use concepts whose context needs fixing? Does the process ever bottom out? If not, then it never makes sense to say something is true or false, *even relative to a context*, because there's no way to specify a context determinately enough to fix a truth value for the original sentence. If on the other hand the process *does* bottom out, then doesn't the "bottomed-out" statement have a determinate truth value in an absolute sense? (There are many choices of context, and perhaps there is no privileged choice, but once a context is fixed, the truth value of the fixed-context statement is *absolute*.) -- 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 15:25
tchow(a)lsa.umich.edu wrote: > In article <aFHbj.7280$vd4.540(a)pd7urf1no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> But it's perfectly normal >> for us to speak of, say, "1+1=0" as true, or false, relative to whatever the >> context that we choose. (But there's *no absolute context* that everyone must >> accept that statement as true or otherwise). > > It still seems to me that your view suffers from a problem of infinite > regress. The meaning of relativity is that *we* (all finite beings) would have truth-views that suffer from this infinite regress, and there isn't much we could do about changing it. > You don't think that statements such as "1+1=0" or "PA is > consistent" have a determinate truth value; one must fix the context. > However, what if we fix the context? *Then* do we get a determinate > truth value? I'd think "No". If we've relativized a truth to a context then we'd have a relativized truth, not an absolute (i.e. "determinate") one. > So for example, consider > > (*) "1+1=0" is false in the standard model of PA. > > Does (*) have a determinate truth value? I've fixed the context of > "1+1=0", haven't I? Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? If not, then you haven't fixed a precise context for (*). > Doesn't this thereby fix the truth value of (*)? 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. > Doesn't it make the "belief having been believed already" (or whatever > locution it was you used)? But when I pressed this point before, you > said no; "PA," you said, was itself indeterminate. So somehow we have > to fix the context of *which* PA we're talking about. That phrase "belief having been believed already" refers to a relative truth, one that has been already "relativised". The point I'm trying to make is that there is no absolute truth-value for (*), because, among other things, *we* *can not spell out all* the axioms: each of us might think of different set of axioms that could be named as "PA" set! And the moment each of us thinks "PA"-set has been fixed, the moment that truth value of (*) has been a relative one! > But now an infinite regress looms. May I say "Welcome to the relativity nature of (human) mathematical reasoning"! > Won't any attempt to fix the context of PA itself use concepts whose context > needs fixing? Does the process ever bottom out? "Yes" to the 1st question; and "No" to the 2nd one. I might add though there are 2 kinds of contexts where relativity of mathematical truths (in general) could take place: inheritance (parental) context, and lateral (sibling) context. Whether or not it's FOL= we're choosing is the first kind of context, while whether or not (within say FOL=) we agree which *specific* axiom-set to be referred as "PA" is the 2nd kind of context. Absolutism in mathematical truth values is impossible because both the the semantic and truth of a statement (FOL or meta) could in general be altered in either one of these 2 contexts (or even in a combination of both!). > If not, then it never makes sense to say something is > true or false, *even relative to a context*, because there's no way to > specify a context determinately enough to fix a truth value for the > original sentence. Agree. Only if we *insist* on "absolute sense" (whatever that might or might not mean), would we have to be agonized on what anything would "absolutely" make sense or be true. If, otoh, we contend that meaning and truth of any grouping of symbols is relative, we might go a long way in using mathematical language as a descriptive tool, describing what we'd (relatively) perceive as "the" reality, imho. > If on the other hand the process *does* bottom out, > then doesn't the "bottomed-out" statement have a determinate truth value > in an absolute sense? (There are many choices of context, and perhaps > there is no privileged choice, but once a context is fixed, the truth > value of the fixed-context statement is *absolute*.) As mentioned, that would be a "fixed" relativized truth. |