True, but not provable
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From: Nam Nguyen on 19 Apr 2010 00:46 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.
From: Nam Nguyen on 19 Apr 2010 00:59 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, 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?
From: Nam Nguyen on 19 Apr 2010 01:37 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.
From: Nam Nguyen on 20 Apr 2010 02:25 James Burns wrote: > 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? Which model of L(PA) did Peter use when he claimed 2+2=5 is false? None, right? So, no model therefore no context.
From: Nam Nguyen on 20 Apr 2010 02:38
James Burns wrote: > 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. You're so vague here: "there is no context" for what? > > 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. Huh? I don't follow what you're trying to say. The main reason why people make argument is usually because they don't agree on certain point. Why is that becoming "disturbing" to you and "everyone"? |