Torkel Franzen on truth
in [Logic]
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From: tchow on 18 Dec 2007 12:34 In article <4768044b$0$485$b45e6eb0(a)senator-bedfellow.mit.edu>, I wrote: > (*) In the standard model of the integers, there are no integers m and n > such that m^2 = 2 n^2. I meant "no nonzero integers m and n," of course. -- 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: MoeBlee on 18 Dec 2007 12:43 On Dec 17, 11:22 pm, herbzet <herb...(a)gmail.com> wrote: > "Nam D. Nguyen" wrote: > > MoeBlee wrote: > > >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > > > >> On the basis of model that "sqrt(2) is irrational" is true, of course. > > > > Maybe you mean, on the basis that there is a model in which "sqrt(2) > > > is irrational" is true. > > > "Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner > > answer and you seemed to not understand? > > > > And there is a model in which it is false also. > > > So far I don't see what your point here is! > > > > What about operations on finite strings? > > > What about them? > > > > Don't you believe, for > > > example, irrespective of any model, that the string "0011" is the same > > > as the string "0022 [with 1 substituted for 2]"? > > > In what context are you talking about "sameness", "substitute", etc... > > Sorry your question is too vague in semantic and consequently is subject > > to different interpretations. > > OK, I'll bite: Yes, the string "0011" is the same as the string > "0022 [with 1 substituted for 2]". What does that have to do > with models and "truth" and "sqrt(2) is irrational" etc.? It has to do with the bases upon which one has certain mathematical beliefs. Does one need a notion of model to believe that the two strings are the same? (And sameness of strings is a mathematical subject). MoeBlee
From: MoeBlee on 18 Dec 2007 12:44 On Dec 18, 1:10 am, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > herbzet wrote: > > > "Nam D. Nguyen" wrote: > >> MoeBlee wrote: > > >>>>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > >>>> On the basis of model that "sqrt(2) is irrational" is true, of course. > >>> Maybe you mean, on the basis that there is a model in which "sqrt(2) > >>> is irrational" is true. > >> "Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner > >> answer and you seemed to not understand? > > >>> And there is a model in which it is false also. > >> So far I don't see what your point here is! > > >>> What about operations on finite strings? > >> What about them? > > >>> Don't you believe, for > >>> example, irrespective of any model, that the string "0011" is the same > >>> as the string "0022 [with 1 substituted for 2]"? > >> In what context are you talking about "sameness", "substitute", etc... > >> Sorry your question is too vague in semantic and consequently is subject > >> to different interpretations. > > > OK, I'll bite: Yes, the string "0011" is the same as the string > > "0022 [with 1 substituted for 2]". What does that have to do > > with models and "truth" and "sqrt(2) is irrational" etc.? > > Good question. Apparently MoeBlee (not I) is the one who asked the question > about some "string". It has to do with the bases upon which one has certain mathematical beliefs. Does one need a notion of model to believe that the two strings are the same? (And sameness of strings is a mathematical subject). MoeBlee
From: Nam D. Nguyen on 18 Dec 2007 17:59 tchow(a)lsa.umich.edu wrote: > In article <vBD9j.29$Tx.18(a)pd7urf3no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> tchow(a)lsa.umich.edu wrote: >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? >> On the basis of model that "sqrt(2) is irrational" is true, of course. > > You didn't answer my first question directly. Are you convinced that sqrt(2) > is irrational? But your first question is unanswerable, *without* a context/basis! (Why did you ask "On what basis?" *immediately* after your first question?) If you asked me whether or not the sentence "President Kennedy is dead" is true/false, I'd react in the same way: no context no possible answer. If this sentence is answered in a play tomorrow about an event in 1960, the sentence could be false, in that context. If the play happens to be actually in 1961, but about an imagined event in 2010 the sentence could still be true or false, depending on the context of the plot the play-writer intended to write. If this sentence is uttered in the "normal" (i.e. historical) context, there are still other contexts to be content with: in some religion "beliefs" there would be no dead, just one life after another; and accordingly, President Kennedy would be still "alive", years after ... well, his "assassination"! Philosophical "crab" right? Not really! Especially when talking about something that looks like a FOL truth question, like is "sqrt(2) is irrational". Let's even deal with a much simpler question: is "1+1=0" true? The way DMC, PS and you seem to say is that there is certain *absolute* truth (value) about "1+1=0" we must necessarily "believe" in, in much the same way some "religiously believe" Jesus of Nazareth is only son of God. What I counter is that there's no such kind of unchangeable belief in mathematical reasoning. What would be a distinction between these 2 kinds of belief: for one thing, model-truth belief (which you seemed to allude below) is changeable/relative/subjective; given the same event, structure, sets of ... (or whatever we may want to call), we are at the liberty to change that belief 180 degree and still are correct as far as mathematical "belief" viz-a-viz reasoning is concerned. That's not the same kind of religion beliefs: there its truth values are absolute, immutable, independent of any human perception, or even ... well, belief! > Or are you convinced only of the statement, "in the standard > model of the integers, there are no integers m and n such that m^2 = 2 n^2"? 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, as in the following "The Lady's Yes" poem: "Yes," I answered you last night; "No," this morning, Sir, I say. Colours seen by candlelight, Will not look the same by day. [...] 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. Which is my whole point here. > > Let me assume that the latter formulation is the only one that you assent to. > Then let me ask this: Do you believe (note the word "believe" here) the > following statement? > > (*) In the standard model of the integers, there are no integers m and n > such that m^2 = 2 n^2. > > If so, on what basis do you believe (*)? Not on the basis of proof, according > to what you say later. So what other basis do you have?
From: george on 18 Dec 2007 13:56
> > On Dec 15, 3:41 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > > > I know that logic books occasionally talk like this, but this has always > > > seemed to me simply to be a misuse of the word "language". Herbzet defends: On Dec 18, 2:21 am, herbzet <herb...(a)gmail.com> wrote: > Well, Peter is objecting to standard usage. You do that too! This analogy is misguided to put it charitably. Prof.Smith is actually calling the whole formal approach NON-standard. "Language" ALREADY HAD a meaning before formalists decided to make it a technical term. But that is ridiculous AS an objection. There are many words in the dictionary that list several incompatible meanings. If I were even TRYING to use language in the *same* usual standard sense that Prof.Smith is talking about, then he would have a point. But I really am not, and neither is CS as a field. The point is that the study of PURELY formal languages is a WELL-respected subfield of the field. We are not "misusing" the word "language" and we are not deviating from the non-CS "standard" use EITHER! We are talking about a DIFFERENT concept! We are *intentionally* restricting our attention to a PURELY formal extensionally-conceived collection! The fact that the word we are using for it is fraught with all these other "natural" connotations IS NOT EVEN RELEVANT. We are not "misusing" the word by proscribing those connotations OUT of consideration. We are rather simply using the word in a DIFFERENT narrow technical SENSE. "Chair" doesn't always mean a thing that you sit on with a seat and 4 legs. Sometimes it means "presiding officer" of a discussion meeting. To say that calling a formal language "a language" is misusing the word is ABUSIVE, frankly. IF we were actually trying to TRADE on or invoke or invite inference from ANY of the more "normal" attributes of the more normal definition of the word "language", then, yes, he would have a point, WE would be guilty of "misuse". BUT WE'RE NOT. WE ONLY mean what WE mean. We're ONLY talking about the formal version. |