True, but not provable
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From: John Jones on 17 Apr 2010 21:24 Nam Nguyen wrote: > Nam Nguyen wrote: >> Peter Webb wrote: >>> >>> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >>> news:hqapoj$lag$1(a)news.eternal-september.org... >>>> "Some mathematical statements are true, but not provable". Does that >>>> make sense? Let's make a grammatical analysis. >>>> >>>> 1) To begin, there are no mathematical statements that are false. >> >>> OK, here is a mathematical statement: 2+2 = 5 >>> >>> It is a mathematical statement and it is false. >>> >>> So unless you can explain this counter-example, you are clearly wrong. >> >> I've never been a fan of JJ's style of reasoning and I'm not defending >> his >> op here. But what you've implied above is: >> >> (1) There's a mathematical statement that is false. >> >> while what he said is: >> >> (2) There are no mathematical statements that are false. >> >> Assuming here by a "mathematical statement" we just mean a FOL wff, >> why do you think observation (1) is correct while (2) wrong? > > The point being is without a clear reference to a context for being true, > or being false, both your (1) and his (2) would equally make no sense. That is precisely what I said.
From: John Jones on 17 Apr 2010 21:29 Zerkon wrote: > On Fri, 16 Apr 2010 23:53:33 +0100, John Jones wrote: > >> 1) To begin, there are no mathematical statements that are false. A >> false mathematical statement isn't a mathematical statement. It's a set >> of signs that merely look like a mathematical statement. > > I Don Getit. > > A falsehood can only be a statement which is then proved or known as > false? You don't 'prove' that a mathematical statement is false: you 'show' that it is not a mathematical statement.
From: Jim Burns on 18 Apr 2010 11:55 John Jones wrote: > Jim Burns wrote: >> Peter Webb wrote: >>> "John Jones" <jonescardiff(a)btinternet.com> wrote in message >>> news:hqapoj$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. 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. >>> >>> So unless you can explain this counter-example, >>> you are clearly wrong. >> >> JJ needs to re-define "mathematical statement" >> in order to make it appear that he has drawn some >> sort of useful conclusion. Compare to the explanation >> of why all Scotsmen like haggis: because anyone not >> liking haggis is no true Scotsman. >> >> One problem with JJ's re-definition -- >> the motivating notion, about something >> true but unprovable, remains just as true as >> it ever was. The only effect of JJ's tactic is >> that what we had called a "mathematical statement" >> (and which could be either true or false) we now >> need to call something else -- perhaps "woof" >> would be a good choice. Then, instead of JJ's >> sentence, we have "Some woofs are true, >> but not provable." Not much of an improvement >> that I can see. > A mathematical statement is one where general mathematical > axioms are retained in the statement. No, not the way I would use "mathematical statement". I am interpreting your "general mathematical axioms are retained in the statement" to mean "the statement follows logically from general mathematical axioms". Seen this way, it is clear that what you call "mathematical statement" is what most other interested people would call "proven mathematical statement". > There is no such retainment for a mathematical > statement that is false. This is not true, even if we agree on your re-naming of "proven mathematical statements". If any of the axioms are false, then you will have what you call "mathematical statements" -- "retaining" the false axioms -- which are false. > Hence, a mathematical statement that is false > is not a mathematical statement. True, but only true as long as you get to define "mathematical statement", in part, as true. Things that are usually called mathematical statements: 1) 1 + 1 = 2. 2) The integral from -infinity to +infinity of the function f: R -> R, where f(x) = exp(-x^2) is equal to the squareroot of pi. 3) Every even natural number larger than 2 is the sum of two primes. 4) The squareroot of 2 is rational. Statement 1 is pretty much the prototype of a mathematical statement (possibly what you are using to inform your intuitions). I find it difficult to imagine disagreeing with 1. The goal of a mathematical argument is to force anyone who understands it to agree with the conclusion, in much the same way. Statement 2 is well-known to be true, but you need to know a trick to be able to prove it. By your definition, it is a mathematical statement for some people and not for others, at least up until those others are told the trick. Statement 3 is widely /believed/ to be true, but has not yet been /proven/. By your definition, anyone working on proving 3 is trying to prove mathematically something that is not a mathematical statement. Statement 4 is well-known to be false today, but was believed to be true (by the few who cared) at one point in the distant past. Please discuss, in light of your definition of "mathematical statement". I don't know why anyone would prefer to use a definition of mathematical statement that turns something so clear into the very model of murkiness. Jim Burns
From: Jim Burns on 18 Apr 2010 12:28 Nam Nguyen wrote: > Nam Nguyen wrote: >> Peter Webb wrote: >>> "John Jones" <jonescardiff(a)btinternet.com> >>> wrote in message >>> news:hqapoj$lag$1(a)news.eternal-september.org... >>> >>>> "Some mathematical statements are true, but not >>>> provable". Does that make sense? Let's make a >>>> grammatical analysis. >>>> >>>> 1) To begin, there are no mathematical statements >>>> that are false. >> >>> OK, here is a mathematical statement: 2+2 = 5 >>> >>> It is a mathematical statement and it is false. >>> >>> So unless you can explain this counter-example, >>> you are clearly wrong. >> >> I've never been a fan of JJ's style of reasoning >> and I'm not defending his op here. But what you've >> implied above is: >> >> (1) There's a mathematical statement that is false. >> >> while what he said is: >> >> (2) There are no mathematical statements that are false. >> >> Assuming here by a "mathematical statement" we just >> mean a FOL wff, why do you think observation (1) >> is correct while (2) wrong? You direct your question to Peter Webb, but perhaps he will not mind my answering for him. It's not as though the answers are controversial in any way. Perhaps you could explain why I am wrong, if you think I am: I think (1) is correct because I know of examples of things that are called "mathematical statements" which are widely agreed to be false, such as the example Peter Webb gave. I think (2) is wrong because it is the negation of (1). > The point being is without a clear reference to > a context for being true, or being false, both > your (1) and his (2) would equally make no sense. Why do you say there is no context? Statements (1) and (2) have the context of other things that get referred to as "mathematical statements". Would you claim that "My neighbor's dog barked all night." makes no sense because you do not have a mathematical definition of "dog"? Jim Burns
From: Nam Nguyen on 18 Apr 2010 13:54
Jim Burns wrote: > Nam Nguyen wrote: >> Nam Nguyen wrote: >>> Peter Webb wrote: >>>> "John Jones" <jonescardiff(a)btinternet.com> >>>> wrote in message news:hqapoj$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? 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? |