True, but not provable
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From: Peter Webb on 21 Apr 2010 22:29 "John Jones" <jonescardiff(a)btinternet.com> wrote in message news:hqlfoq$gno$3(a)news.eternal-september.org... > Peter Webb wrote: >> >> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >> news:hqdmlj$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. "2 + 2 = 5" certainly looks like a mathematical statement to me. If its not, then you better give *your* definition of a mathematical statement. If it obviously differs from most people's understanding of what a mathematical statement is, then maybe you should pick some other term for your concept, eg a "Jones Statement". Anyway, over to you, if 2+2 = 5 is not a mathematical statement, then what is?
From: Jim Burns on 21 Apr 2010 22:49 John Jones 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? >> >> 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.) > > Of course you can speak so. > And we would all know what you meant. It seems to me that knowing what is meant is all that can reasonably be required of statements, mathematical or otherwise. Everything else changes, depending upon circumstances, up to and including whether a statement "should" be true. For example, Peter Webb used "2+2=5" as an example of a "mathematical statement" (call it something else, if you feel you must) that was false. > But then we would all, by claiming that, > endorse a set of mathematical statements that > aren't true. It's not because such statements > represent the largest infinity, as the saying > might go. Rather, it is that there is no > criterion of a mathematical truth that doesn't > circuitously rely on a mathematical falsehood. I am completely at sea here. The first sentence is not so bad, I think I understand it, although I disagree. This thread started with an analysis of yours asserting, in part, that there are no false mathematical /statements/. Now you expand your claim to say one cannot admit that a statement exists without /asserting/ it. I'll just say I disagree, and leave this new can of worms shut, for now. Is the second sentence an attempt at math humor? That is a risky business. There is no largest infinity, but perhaps that is your point. Even if I assume it is your point, though, I don't get what you are trying to say. It is the third sentence that makes my head spin. It sounds like it should have been written by someone opposing your position. (However, from my own point of view, it is only irrelevant. I say that whether something is true or false is irrelevant to whether it is a mathematical statement. This makes circuitous reliance between truth and falsehood irrelevant, too.) At any rate, I am unable to come up with a plausible hypothesis for what you mean by that. >> 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. > > You are appealing to contingencies. > Mathematics has nothing to say about contingencies. > A mathematical statement is complete. If you deny that mathematical statements which are not complete are mathematical statements (or "exist"), then what you say is true, for you. This is very much like your denial that mathematical statements which are not true are mathematical statements. Are sure that "x = 1" is not a mathematical statement? I can easily imagine it showing up in a math class, as part of a math problem. Maybe x - y = 4 2*x + y = -1 (blah blah blah) x = 1 y = -3 Is the difference that "x = 1" is complete (whatever "complete" means to you) as part of that problem? But by that standard, there are theorems and axioms not included in the problem that makes it incomplete. Where does it all end? *In the usual way*, whether something is a mathematical statement is determined by very simple syntactic rules, such as "If A is a mathematical statement then ~(A) is,too." One can then argue that a particular statement is true or false, but, either way, (or any other way) it remains a statement. What is wrong with doing things this way? Jim Burns
From: John Jones on 25 Apr 2010 07:18 Jim Burns wrote: > John Jones 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? >>> >>> 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.) >> >> Of course you can speak so. >> And we would all know what you meant. > > It seems to me that knowing what is meant is > all that can reasonably be required of statements, > mathematical or otherwise. Everything else changes, > depending upon circumstances, up to and including > whether a statement "should" be true. For example, > Peter Webb used "2+2=5" as an example of a > "mathematical statement" (call it something else, > if you feel you must) that was false. > >> But then we would all, by claiming that, >> endorse a set of mathematical statements that >> aren't true. It's not because such statements >> represent the largest infinity, as the saying >> might go. Rather, it is that there is no >> criterion of a mathematical truth that doesn't >> circuitously rely on a mathematical falsehood. > > I am completely at sea here. > > The first sentence is not so bad, I think I > understand it, although I disagree. This thread > started with an analysis of yours asserting, in > part, that there are no false mathematical > /statements/. Now you expand your claim to say > one cannot admit that a statement exists without > /asserting/ it. I'll just say I disagree, and leave > this new can of worms shut, for now. > > Is the second sentence an attempt at math humor? > That is a risky business. There is no largest > infinity, but perhaps that is your point. Even > if I assume it is your point, though, I don't get > what you are trying to say. > > It is the third sentence that makes my head spin. > It sounds like it should have been written by > someone opposing your position. (However, from > my own point of view, it is only irrelevant. > I say that whether something is true or false > is irrelevant to whether it is a mathematical > statement. This makes circuitous reliance > between truth and falsehood irrelevant, too.) > At any rate, I am unable to come up with a > plausible hypothesis for what you mean by that. > >>> 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. >> >> You are appealing to contingencies. >> Mathematics has nothing to say about contingencies. >> A mathematical statement is complete. > > If you deny that mathematical statements which are > not complete are mathematical statements (or "exist"), > then what you say is true, for you. This is very much > like your denial that mathematical statements which > are not true are mathematical statements. > > Are sure that "x = 1" is not a mathematical statement? > I can easily imagine it showing up in a math class, > as part of a math problem. Maybe > x - y = 4 > 2*x + y = -1 > (blah blah blah) > x = 1 > y = -3 > > Is the difference that "x = 1" is complete (whatever > "complete" means to you) as part of that problem? > But by that standard, there are theorems and axioms > not included in the problem that makes it incomplete. > Where does it all end? > > *In the usual way*, whether something is a mathematical > statement is determined by very simple syntactic rules, > such as "If A is a mathematical statement > then ~(A) is,too." One can then argue that a > particular statement is true or false, but, either way, > (or any other way) it remains a statement. > What is wrong with doing things this way? > > Jim Burns We judge a mathematical statement to be true only when it is not false. There are no other grounds for saying a mathematical statement is true or false. So truth and falsehood aren't significant statements in mathematics. Either something is a mathematical statement or it isn't.
From: John Jones on 25 Apr 2010 08:04 Nam Nguyen wrote: > But if one can argue for a statement to be true, one can > also argue for it to be false as well. There's no intrinsic > truth value for a statement: that has to be argued for, by > citing a context! Yes, but there is a mistake there. You assume that a statement S is independent of truth and falsity, while I was arguing that there are no grounds for considering S as a statement, whether or not S is a true mathematical statement or not..
From: John Jones on 25 Apr 2010 08:12
Peter Webb wrote: > > > "2 + 2 = 5" certainly looks like a mathematical statement to me. Yes I know, but I think that we have been brought up to think that a set of signs that look mathematical are therefore mathematical. > If its > not, then you better give *your* definition of a mathematical statement. > If it obviously differs from most people's understanding of what a > mathematical statement is, then maybe you should pick some other term > for your concept, eg a "Jones Statement". > > Anyway, over to you, if 2+2 = 5 is not a mathematical statement, then > what is? > A proof will tell us indirectly when a set of signs isn't a mathematical statement. The proof does this by failing to show that it is a mathematical statement, and NOT by showing that it isn't a mathematical statement. |