True, but not provable
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From: Nam Nguyen on 18 Apr 2010 13:59 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, Sorry, I meant: I might know what "My neighbor's dog barked all night" means, > but how > could I be so sure it wasn't an audio file being played by > their naughty kids, for example? > > 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: Peter Webb on 18 Apr 2010 22:22 "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$1(a)news.eternal-september.org... >>> "Some mathematical statements are true, but not provable". Does that >>> make sense? Let's make a grammatical analysis. >>> >>> 1) To begin, there are no mathematical statements that are false. A >>> false mathematical statement isn't a mathematical statement. It's a set >>> of signs that merely look like a mathematical statement. >>> >> >> OK, here is a mathematical statement: 2+2 = 5 >> >> It is a mathematical statement and it is false. > > What is a "false" mathematical statement? Is it a mathematical statement? > does it adhere to the Peano axioms? > > >> >> So unless you can explain this counter-example, you are clearly wrong. > > See above. You are the person who introduced the concept of false mathematical statements, its up to you to define what that means in the context of your argument. According to you, is 2+2=5 : a. A mathematical statement? b. False?
From: Jim Burns on 18 Apr 2010 22:56 Nam Nguyen wrote: > Jim Burns wrote: >> Nam Nguyen wrote: >>> Nam Nguyen wrote: >>>> Peter Webb wrote: >>>>> "John Jones" <jonescardiff(a)btinternet.com> >>>>> wrote in message news:hqapoj$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). >> I am going to delete the stuff above in my answer to the stuff below. There doesn't seem to be anything more to say on my part, and you seem to have lost interest in it, even though it was in answer to your question. Jim Burns >>> 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? > > 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: Jim Burns on 18 Apr 2010 23:12 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? 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? Jim Burns >> I might >> know what "My neighbor's dog barked all night" well, > > Sorry, I meant: I might know what "My neighbor's dog barked > all night" means, > >> but how >> could I be so sure it wasn't an audio file being played by >> their naughty kids, for example? >> >> 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: Nam Nguyen on 19 Apr 2010 00:35
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!) > > 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. |