Torkel Franzen on truth
in [Logic]
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From: tchow on 23 Dec 2007 13:49 In article <1vfbj.4808$DP1.3488(a)pd7urf2no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >tchow(a)lsa.umich.edu wrote: >> <> (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. If this is what you mean by relativity, then the technicalities of mathematical logic are irrelevant, and there is no distinction between mathematical statements and religious statements in this regard. -- 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: george on 23 Dec 2007 13:55 > > You, the distinguished Prof.Peter Smith, are saying the following: > > 2+5=7 doesn't mean anything. On Dec 22, 7:33 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > Am I? That's odd. I always thought that "2 + 5 = 7" means that two > plus five is seven ... but obviously I must have got that wrong. Well, you got the quotes wrong, for starters. I was talking about your opinion about the meaning of 2+5=7. You responded with your opinion about the meaning of "2+5=7". Obviously if we are going to interpret all 3 of "2", 2, and two, then we are not going to run out of levels of interpretation, at least not quickly. The argument was ABOUT what things NEED to be interpreted, and whether. Are you even psychologically capable of sticking to the subject? The one thing that makes the quotes acceptable is that strings are known to be abstract. But if you are going to wax realist about the two that 2 refers to, then 2 and two are abstract as well.
From: Aatu Koskensilta on 23 Dec 2007 13:58 On 2007-12-21, in sci.logic, george wrote: > Abusive as usual. As usual. > The completeness theorem is at least ABOUT first-order semantics. > Where are the bicycles in analysis? Hiding behind the universal ordinal. The idea that "the completeness theorem is fundamentally a proof that first-order semantics SIMPLY DOESN'T EXIST" is entirely arbitrary and quite bizarre. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Peter_Smith on 23 Dec 2007 15:16 On 23 Dec, 18:55, george <gree...(a)cs.unc.edu> wrote: > > > You, the distinguished Prof.Peter Smith, are saying the following: > > > 2+5=7 doesn't mean anything. > > On Dec 22, 7:33 pm, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > Am I? That's odd. I always thought that "2 + 5 = 7" means that two > > plus five is seven ... but obviously I must have got that wrong. > > Well, you got the quotes wrong, for starters. > I was talking about your opinion about the meaning of 2+5=7. Got quotes wrong too, have I? Gosh. And there I was thinking that I was conforming to normal good practice with use and mention. Ah well, ...
From: Nam D. Nguyen on 24 Dec 2007 00:42
tchow(a)lsa.umich.edu wrote: > In article <1vfbj.4808$DP1.3488(a)pd7urf2no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> tchow(a)lsa.umich.edu wrote: >>> <> (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*. Wars have been waged because someone named a dog "God" or there is a belief that (4) is not true, so to speak. 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). > If this is what you mean by relativity, then the technicalities of mathematical > logic are irrelevant, and there is no distinction between mathematical statements > and religious statements in this regard. I'd respectfully think this is a mis-characterization of "the technicalities of mathematical logic [reasoning]", on at least 2 counts: the *history* and the *essence* of mathematical reasoning. 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. The point is history seems to have shown us the value of mathematical reasoning should be with consistency of arguments, rather than with the truth of what's formally uttered or stated. For the notion of truth is always a relative notion: any statement that's considered as true in some way would always be false, in some other (legitimate) ways! On the account of essence, mathematical reasoning is based on *finite knowledge*: from finite formula length to finite proofs. Given such finite foundation of reasoning, would you think in this case we know enough about a particular *infinite* set of axioms so that when 2 persons refer to "PA", they could be certain they are referring to 2 identical sets of axioms? If not, then isn't it true that meaning of "PA" would be relative to what exact axiom set they *each* might think they know in their individual mind? And so in this context, any statement about PA is a relative one, relative to the exact individual "PA"'s axiom-set that happens in one's mind. And this relativity is a legitimate one. Unlike that, to be a *religion statement*, (4) is supposed to have one meaning and one truth that is universally acknowledged and that's immutable, *in all contexts*. |