True, but not provable
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From: Nam Nguyen on 19 Apr 2010 21:12 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.) >>> >>> 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? What do you mean by that question? > >> 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! > 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"?
From: John Jones on 20 Apr 2010 20:07 Nam Nguyen wrote: > Jim Burns wrote: >> Nam Nguyen wrote: >>> Nam Nguyen wrote: >>>> Peter Webb wrote: >>>>> "John Jones" <jonescardiff(a)btinternet.com> >>>>> wrote in message news:hqapoj$lag$1(a)news.eternal-september.org... >>>>> >>>>>> "Some mathematical statements are true, but not >>>>>> provable". Does that make sense? Let's make a >>>>>> grammatical analysis. >>>>>> >>>>>> 1) To begin, there are no mathematical statements >>>>>> that are false. >>>> >>>>> OK, here is a mathematical statement: 2+2 = 5 >>>>> >>>>> It is a mathematical statement and it is false. >>>>> >>>>> So unless you can explain this counter-example, >>>>> you are clearly wrong. >>>> >>>> I've never been a fan of JJ's style of reasoning >>>> and I'm not defending his op here. But what you've >>>> implied above is: >>>> >>>> (1) There's a mathematical statement that is false. >>>> >>>> while what he said is: >>>> >>>> (2) There are no mathematical statements that are false. >>>> >>>> Assuming here by a "mathematical statement" we just >>>> mean a FOL wff, why do you think observation (1) >>>> is correct while (2) wrong? >> >> You direct your question to Peter Webb, but >> perhaps he will not mind my answering for him. >> It's not as though the answers are controversial >> in any way. Perhaps you could explain why I am >> wrong, if you think I am: >> >> I think (1) is correct because I know of examples >> of things that are called "mathematical statements" >> which are widely agreed to be false, such as the >> example Peter Webb gave. >> >> I think (2) is wrong because it is the negation >> of (1). >> >>> 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. I might > know what "My neighbor's dog barked all night" well, but how > could I be so sure it wasn't an audio file being played by > their naughty kids, for example? But now you are just changing the goalposts. You have made a bigger situation out of the smaller situation on which your original premise of knowledge was based. > > That's of course just an analogy. In Peter's case I suppose > the context be arithmetic truth, but what exactly is _his_ > definition of arithmetic that _everyone_ would agree?
From: John Jones on 20 Apr 2010 20:10 Peter Webb wrote: > > "John Jones" <jonescardiff(a)btinternet.com> wrote in message > news:hqdmlj$uag$4(a)news.eternal-september.org... >> Peter Webb wrote: >>> >>> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >>> news:hqapoj$lag$1(a)news.eternal-september.org... >>>> "Some mathematical statements are true, but not provable". Does that >>>> make sense? Let's make a grammatical analysis. >>>> >>>> 1) To begin, there are no mathematical statements that are false. A >>>> false mathematical statement isn't a mathematical statement. It's a >>>> set of signs that merely look like a mathematical statement. >>>> >>> >>> OK, here is a mathematical statement: 2+2 = 5 >>> >>> It is a mathematical statement and it is false. >> >> What is a "false" mathematical statement? Is it a mathematical >> statement? does it adhere to the Peano axioms? >> >> >>> >>> So unless you can explain this counter-example, you are clearly wrong. >> >> See above. > > You are the person who introduced the concept of false mathematical > statements, its up to you to define what that means in the context of > your argument. > > According to you, is 2+2=5 : > > a. A mathematical statement? > b. False? > > > 2+2=5 : it's not false because it isn't a mathematical statement. If it was a mathematical statement then mathematical axioms would apply to it.
From: John Jones on 20 Apr 2010 20:12 Nam Nguyen wrote: > Peter Webb wrote: >> >> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >> news:hqdmlj$uag$4(a)news.eternal-september.org... >>> Peter Webb wrote: >>> >>>> >>>> So unless you can explain this counter-example, you are clearly wrong. >>> >>> See above. >> >> You are the person who introduced the concept of false mathematical >> statements, its up to you to define what that means in the context of >> your argument. >> >> According to you, is 2+2=5 : >> > > It's certainly up to JJ to answer this in his own way. But here are > mine. > >> a. A mathematical statement? > > In L(PA) yes, it's a mathematical statement. I's exact formulation > in that language is: > > SS0 + SS0 = SSSSS0 > >> b. False? > > It a meaningless question, without a context of what model of what > formal system which is the underlying interpreting structure where > its truth or falsehood is asserted. In arithmetic modulo-1 system, > it's true but in other kind of arithmetic it might be false. It doesn't matter what model is being used. If it doesn't follow the model then it is not a manifestation of the model.
From: John Jones on 20 Apr 2010 20:18
James Burns wrote: > Nam Nguyen wrote: >> Peter Webb wrote: >>> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >>> news:hqdmlj$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 > That looked a bit like changing the goalposts. You can't make an assessment of a statement based on other statements. You have to limit yourself to the statement that is at hand. |