True, but not provable
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From: James Burns on 19 Apr 2010 10:46 Nam Nguyen wrote: > Jim Burns wrote: >> Nam Nguyen wrote: >>> Nam Nguyen wrote: >>>> Jim Burns wrote: >>>>> Nam Nguyen wrote: >> >>>>>> The point being is without a clear reference to >>>>>> a context for being true, or being false, both >>>>>> your (1) and his (2) would equally make no sense. >>>>> >>>>> >>>>> Why do you say there is no context? Statements >>>>> (1) and (2) have the context of other things that >>>>> get referred to as "mathematical statements". >>>>> >>>>> Would you claim that "My neighbor's dog barked all >>>>> night." makes no sense because you do not have >>>>> a mathematical definition of "dog"? >>>> >>>> >>>> It's truth - not semantic - that's the issue here. >> >> When exactly did truth become the issue here? > > Since the very beginning of this thread that has the title > "True, but not provable" and in which JJ had the opening > questions/statements like "'Some mathematical statements are > true, but not provable'. Does that make sense?" or "1) To begin > there are no mathematical statements that are false"; since > Peter said about truth such as in "2+2=5 ... is a mathematical > statement and it is false"; etc... > > Why would you think the conversations here do not center about > truth? (I'm a bit surprised you've asked me the above question!) Why? I refer you to the top of this post: >>>> Jim Burns wrote: >>>>> Nam Nguyen wrote: >> >>>>>> The point being is without a clear reference to >>>>>> a context for being true, or being false, both >>>>>> your (1) and his (2) would equally make no sense. >>>>> >>>>> Why do you say there is no context? Statements >>>>> (1) and (2) have the context of other things that >>>>> get referred to as "mathematical statements". It looks to me as though we are talking past each other. I think I will have to let this go after I've responded to your latest batch of responses. > >> >> Does it *make sense* to speak of a mathematical >> statement being true or false? Then we can speak >> so. (I argued above that it does make sense.) >> >> Do we need to have specific mathematical statements >> in mind, with their specific contexts, in order to be >> able to speak so? I can't imagine why that might be >> true. Certainly I can make mathematical statements >> like "x = 1" without enough context to decide whether >> they are true or false. Why would I need to be >> more specific in talking /about/ mathematical >> statements? > > > Oh but we didn't just simply said about mathematical > statements: we talked about their _truth_ and _falsehood_ > as JJ and Peter started to talk about from the beginning > of this thread and conversation. > > Don't you remember that? > > For the record, had people just talked about semantics of the > mathematical statements - or anything else except truth - I > probably would have not engaged in this discussions at all. > > But that wasn't the case. And, at least by the essence of Tarksi's > concept of truth, if there's no context for saying a statement is > true, there's no sense for saying the statement is true. For you, apparently it goes without saying that there is no context. I am disputing that, so it is a little disturbing that you continue to treat the lack of context as though everyone agrees on that point. Jim Burns
From: James Burns on 19 Apr 2010 10:46 Nam Nguyen wrote: > Nam Nguyen wrote: >> Jim Burns wrote: > >>> Does it *make sense* to speak of a mathematical >>> statement being true or false? Then we can speak >>> so. (I argued above that it does make sense.) > > I think I did address your question here when I mentioned > about Tarski's concept of truth below. >> >> For the record, had people just talked about semantics of the >> mathematical statements - or anything else except truth - I >> probably would have not engaged in this discussions at all. >> >> But that wasn't the case. And, at least by the essence of Tarksi's >> concept of truth, if there's no context for saying a statement is >> true, there's no sense for saying the statement is true. You seem to imply that there is no context. Why?
From: James Burns on 19 Apr 2010 10:46 Nam Nguyen wrote: > Nam Nguyen wrote: >> Jim Burns wrote: > >>> Does it *make sense* to speak of a mathematical >>> statement being true or false? Then we can speak >>> so. (I argued above that it does make sense.) >> >> For the record, had people just talked about semantics of the >> mathematical statements - or anything else except truth - I >> probably would have not engaged in this discussions at all. >> >> But that wasn't the case. And, at least by the essence of Tarksi's >> concept of truth, if there's no context for saying a statement is >> true, there's no sense for saying the statement is true. > > To be succinct, if 2+2=5 were true without a context, Why would "2 + 2 = 5" be without a context? > the word > "axiom" would be meaningless and rules of inference would be > in oblivion: mathematical statements would be simply evidently > true or false! > > Is that what you really meant to argue for? No, it is nothing like what I am arguing for. I am not saying "2 + 2 = 5" is true or false without a context. I am saying it has a context, the same sort of context that associated with sentences like "The neighbor's dog barked all night." Jim Burns
From: James Burns on 20 Apr 2010 10:58 Nam Nguyen wrote: > James Burns wrote: >> Nam Nguyen wrote: >>> To be succinct, if 2+2=5 were true without a context, >> >> Why would "2 + 2 = 5" be without a context? > > What do you mean by that question? If you believe that "2+2=5" as Peter Webb used it upthread to be without a context, then my question means exactly what it appears to mean: why would "2+2=5" be without a context? My experience tells me that it is not, in fact, without a context. If you don't believe that "2+2=5" is without a context, then I have misunderstood you badly. >>> the word >>> "axiom" would be meaningless and rules of inference would be >>> in oblivion: mathematical statements would be simply evidently >>> true or false! >>> >>> Is that what you really meant to argue for? >> >> No, it is nothing like what I am arguing for. >> I am not saying "2 + 2 = 5" is true or false without a >> context. > > OK. Then I don't understand what it is you want to argue with > me? Because that's exactly my point that a formula can't be > true or false without a context! It looks to me as though you picked Peter Webb's use of "2+2=5" as an example of a formula without a context that is neither true nor false. (Do I have that right, at least?) I do not accept that it is without context. Perhaps you want to write of L(PA) and such matters. That is up to you, but I warn you that I understand quite well "2+2=5" /as it is usually used/, but I do not understand talk of models of L(PA), no more than vaguely. Elsethread, you seemed to think that information about models of L(PA) was necessary to understand "2+2=5". My own experience is that it makes things worse, not better. Is it possible that you are mistaken about what the context of "2+2=5" is or should be? >> I am saying it has a context, the same sort of >> context that associated with sentences like >> "The neighbor's dog barked all night." > > But then again you're not clear in what you're saying: > what did you mean by "it" here? The semantic of "2+2=5"? > Or the truth of "2+2=5"? The sentence "2+2=5", as used by Peter Webb upthread. Jim Burns
From: James Burns on 19 Apr 2010 11:20
Nam Nguyen wrote: > Peter Webb wrote: >> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >> news:hqdmlj$uag$4(a)news.eternal-september.org... >>> Peter Webb wrote: >>>> >>>> So unless you can explain this counter-example, you are clearly wrong. >>> >>> See above. >> >> You are the person who introduced the concept of false mathematical >> statements, its up to you to define what that means in the context of >> your argument. >> >> According to you, is 2+2=5 : >> > > It's certainly up to JJ to answer this in his own way. But here are > mine. > >> a. A mathematical statement? > > > In L(PA) yes, it's a mathematical statement. I's exact formulation > in that language is: > > SS0 + SS0 = SSSSS0 > >> b. False? > > > It a meaningless question, without a context of what model of what > formal system which is the underlying interpreting structure where > its truth or falsehood is asserted. In arithmetic modulo-1 system, > it's true but in other kind of arithmetic it might be false. It is not meaningless if "2+2=5" is interpreted in the usual way. Is the problem you see that "the usual way" is not formalizable? Nearly everything we say or write is not formalizable. You seemed to have no problem understanding "The neighbor's dog barked all night." But you could be wrong! What if there is a planet on the opposite side of the Sun, matching Earth in every way, up to and including a language /almost/ the same as English, except that "dog" means "irridescent green"? Then "The neighbor's dog barked all night." makes no sense! However, if the sentence is understood *in the usual way*. then there is no problem. In fact, there is very rarely a problem between people understanding one another which is caused by different assumptions about things as fundamental as what language they are speaking. *AND* when there are such problems they are among the easiest to discover and correct. Jim Burns |