'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 27 Mar 2010 15:32 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Daryl McCullough wrote: >>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>>> of the language of PA. >>>> So you've agreed "G(PA) can be arithmetically false"? >>> It is false in nonstandard models of PA. >> Why don't we make it more precise. When we say F, a formula written >> in the language of arithmetic, is true or false _by default_ we >> mean it's being arithmetically true or false: i.e. true or false >> in the natural numbers. So we're *not* talking about F is being >> true/false in any general kind of models here. > > You haven't (and can't) give me an effective way to use > this definition to decide truth or falsity in the natural numbers. You and I have nothing to argue here, since I just repeated the convention about what "being true" means that virtually all of us would use. > >> The point is Alan said he wouldn't know what an absolute truth >> of G(PA) would mean >> and I've implicitly defined it for him, and >> here is the explicit version: >> >> There's _no other_ context in which the meta statement >> "G(PA) is arithmetically true in the natural number" would >> be false. > > I'm not obliged to accept any particular definition of yours. Sure. You could label/name the definition "Republican" or "Democrat" if you'd like. But first of all _you_ are the one who asked for what it might mean by G(PA) be an absolute truth; and I just gave you an definition that I think is reasonable. Secondly in spite of whatever definition might be, can you answer the relevant question I asked before (in some form): whether or not in the meta level there's a valid context in which the statement "G(PA) is arithmetically false" is true? (That's a sound technical question - independent of any definition of "absolute truth" or of any naming of any such definition you might protest.) > > FWIW you can look at Bourbaki's account of Goedel's incompleteness > theorem to note that they studiously avoid saying that the goedel > sentence is true (arithaally, absolutely, or in any other way). > >> The question I was hoping you'd answer one way or another is >> whether or not there's a context in which the meta statement: >> >> "G(PA) is arithmetically true in the natural number" >> >> would be _false_ ? >> >> If your answer is "yes", then the [arithmetically-in-the-natural- >> number] truth of G(PA) is a relative notion. Otherwise it's an >> absolute notion. > > Why should I believe the only answers are no or yes? > Your realist assumptions are showing. > >> Which answer would you have? And perhaps why? > > you need a bit more Zen. You need to answer some direct technical question presented to you.
From: Alan Smaill on 27 Mar 2010 15:53 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Daryl McCullough wrote: >>>> Nam Nguyen says... >>>>> Daryl McCullough wrote: >>>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>>>> of the language of PA. >>>>> So you've agreed "G(PA) can be arithmetically false"? >>>> It is false in nonstandard models of PA. >>> Why don't we make it more precise. When we say F, a formula written >>> in the language of arithmetic, is true or false _by default_ we >>> mean it's being arithmetically true or false: i.e. true or false >>> in the natural numbers. So we're *not* talking about F is being >>> true/false in any general kind of models here. >> >> You haven't (and can't) give me an effective way to use >> this definition to decide truth or falsity in the natural numbers. > > You and I have nothing to argue here, since I just repeated the > convention about what "being true" means that virtually all of us > would use. Well, I do disagree with you here. You will find some negative comments from Torkel Franzen, for example, about the waving of the magic wand of "standard model" as somehow settling the question of the meaning of all arithemtic statements (that's my paraphrase, not his wording). >>> The point is Alan said he wouldn't know what an absolute truth >>> of G(PA) would mean >>> and I've implicitly defined it for him, and >>> here is the explicit version: >>> >>> There's _no other_ context in which the meta statement >>> "G(PA) is arithmetically true in the natural number" would >>> be false. >> >> I'm not obliged to accept any particular definition of yours. > > Sure. You could label/name the definition "Republican" or "Democrat" > if you'd like. But first of all _you_ are the one who asked for what > it might mean by G(PA) be an absolute truth; and I just gave you > an definition that I think is reasonable. Please, I did *not* ask for a definition! > Secondly in spite of whatever definition might be, can you answer the > relevant question I asked before (in some form): whether or not in the > meta level there's a valid context in which the statement "G(PA) is > arithmetically false" is true? > > (That's a sound technical question - independent of any definition of > "absolute truth" or of any naming of any such definition you might > protest.) First, I addressed this in my previous response -- (see below) which you have ignored. Second, if you want this to be a sound technical question, we need to know what a "valid context" is, and what the metalogic in question is. Finally I do not find it convincing that the answer to this question corresponds to a notion of *absolute* truth, regardess of whether the answer is yes or no. >> FWIW you can look at Bourbaki's account of Goedel's incompleteness >> theorem to note that they studiously avoid saying that the goedel >> sentence is true (arithaally, absolutely, or in any other way). >> >>> The question I was hoping you'd answer one way or another is >>> whether or not there's a context in which the meta statement: >>> >>> "G(PA) is arithmetically true in the natural number" >>> >>> would be _false_ ? >>> >>> If your answer is "yes", then the [arithmetically-in-the-natural- >>> number] truth of G(PA) is a relative notion. Otherwise it's an >>> absolute notion. >> >> Why should I believe the only answers are no or yes? >> Your realist assumptions are showing. >> >>> Which answer would you have? And perhaps why? >> >> you need a bit more Zen. > > You need to answer some direct technical question presented to you. I do not accept that your question has a yes/no answer, as I already said. Why do you think it has? -- Alan Smaill email: A.Smaill at ed.ac.uk
From: Nam Nguyen on 27 Mar 2010 16:06 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Daryl McCullough wrote: >>>>> Nam Nguyen says... >>>>>> Daryl McCullough wrote: >>>>>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation >>>>>>> of the language of PA. >>>>>> So you've agreed "G(PA) can be arithmetically false"? >>>>> It is false in nonstandard models of PA. >>>> Why don't we make it more precise. When we say F, a formula written >>>> in the language of arithmetic, is true or false _by default_ we >>>> mean it's being arithmetically true or false: i.e. true or false >>>> in the natural numbers. So we're *not* talking about F is being >>>> true/false in any general kind of models here. >>> You haven't (and can't) give me an effective way to use >>> this definition to decide truth or falsity in the natural numbers. >> You and I have nothing to argue here, since I just repeated the >> convention about what "being true" means that virtually all of us >> would use. > > Well, I do disagree with you here. You will find some negative comments > from Torkel Franzen, for example, about the waving of the magic wand of > "standard model" as somehow settling the question of the meaning of all > arithemtic statements (that's my paraphrase, not his wording). > >>>> The point is Alan said he wouldn't know what an absolute truth >>>> of G(PA) would mean >>>> and I've implicitly defined it for him, and >>>> here is the explicit version: >>>> >>>> There's _no other_ context in which the meta statement >>>> "G(PA) is arithmetically true in the natural number" would >>>> be false. >>> I'm not obliged to accept any particular definition of yours. >> Sure. You could label/name the definition "Republican" or "Democrat" >> if you'd like. But first of all _you_ are the one who asked for what >> it might mean by G(PA) be an absolute truth; and I just gave you >> an definition that I think is reasonable. > > Please, I did *not* ask for a definition! > >> Secondly in spite of whatever definition might be, can you answer the >> relevant question I asked before (in some form): whether or not in the >> meta level there's a valid context in which the statement "G(PA) is >> arithmetically false" is true? >> >> (That's a sound technical question - independent of any definition of >> "absolute truth" or of any naming of any such definition you might >> protest.) > > First, I addressed this in my previous response -- > (see below) which you have ignored. > > Second, if you want this to be a sound technical question, we need to > know what a "valid context" is, and what the metalogic in question is. > > Finally I do not find it convincing that the answer to this question > corresponds to a notion of *absolute* truth, regardess of whether the answer > is yes or no. > >>> FWIW you can look at Bourbaki's account of Goedel's incompleteness >>> theorem to note that they studiously avoid saying that the goedel >>> sentence is true (arithaally, absolutely, or in any other way). >>> >>>> The question I was hoping you'd answer one way or another is >>>> whether or not there's a context in which the meta statement: >>>> >>>> "G(PA) is arithmetically true in the natural number" >>>> >>>> would be _false_ ? >>>> >>>> If your answer is "yes", then the [arithmetically-in-the-natural- >>>> number] truth of G(PA) is a relative notion. Otherwise it's an >>>> absolute notion. >>> Why should I believe the only answers are no or yes? >>> Your realist assumptions are showing. >>> >>>> Which answer would you have? And perhaps why? >>> you need a bit more Zen. >> You need to answer some direct technical question presented to you. > > I do not accept that your question has a yes/no answer, > as I already said. Why do you think it has? Because it's a technical question in FOL reasoning that has only 2 possible answer: whether there's a valid context or there isn't. And in fact it wouldn't take one too long to com up with 2 precise examples, one for each opposite context. _You_, not Bourbaki, would need to elaborate why it's not a "Yes/No" question, given the fact that one could come up with 2 opposing context. > >
From: Jim Burns on 27 Mar 2010 18:20 Nam Nguyen wrote: > Jim Burns wrote: >> Please correct me if I misinterpret what you are saying: >> >> Marshall called you a talentless buffoon, and that >> was wrong, because you are not a talentless buffoon. > > I don't know know exactly what talent or talentless be, it's > subjective definition. But keeping attributing people with > subjective names in a middle of technical argument while not > talking about the technical matters at hand > _is insincere and is wrong_. > And that's my opinion and that's how I'd would react. I think that you and I use "insincere" differently. Whether someone is _sincere_ about technical matters seems wholly irrelevant to me. At first, I thought it would be nit-picking to mention that, but there are a couple other instances in the post to which I respond... I think a large part of the problem here is how you read and write English. I am not calling you stupid. In fact, there is very probably at least one language you speak very well which I speak much, much worse than you speak English. Also, you do pretty well in English, for the most part. However, ... Are you familiar with the movie "The Princess Bride"? Very funny movie. There is a character, a Sicilian, very smart and very proud of how smart he is. He proclaims it to be "inconceivable" that any of several events could happen, right before they do happen. Bing! Bing! Bing! Another character says to him "That word, 'inconceivable'. I do not think it means what you think it means." I feel like borrowing that line roughly every other sentence when I read your posts. > Of course you have the right to have a different opinion. > >> You called Marshall an intellectual clown, and that >> was okay, because he is an intellectual clown. >> >> I'm a little disappointed, because I thought your >> argument went a little deeper, that it was an >> objection to shouting down unpopular views by >> burying them under a pile of nasty accusations. > > Your reaction seems to be quick here. The > _only_ sentence that he posted about conversations > I had (not with him) but with another poster > about foundational issues was: Excuse me, but I said nothing about foundational issues. Your mention of the saying about mad dogs and the exchange it was a part of suggested to me that dogs can be killed and voices can be silenced by shouting "Mad dog!" and things like that. Perhaps I did not understand what you meant. That is looking more likely as I spend more time trying to understand you. This is not about the substance of what you were arguing about, whether it be foundations or anything else. I was pointing out that you were not applying the same rule ("Do not call your opponent a mad dog.") to your own rhetoric that you were applying to Marshall's. This is about how you were arguing, call it even-handed-ness, or proper rhetoric, or symmetry. Or perhaps I never understood you properly and still do not, and all this effort is wasted. > >> That Moe has failed to do so in a way that you can > >> understand is a failing, but it's not Moe's failing. > > Since that's the lone sentence in that post (meaning he > didn't even have any thing to back up what he posted), > it's a pile of accusation (my alleged "stupidity") he made. You claim that, since Marshall did not support his one-sentence claim, he did not have any support. Are you willing to apply that rule to yourself? Any support you do not demonstrate is assumed not to exist? (This is an example of what I mean by even-hand-ness or symmetry.) Also, I don't see where he alleged you were stupid. Surely there can be more than one possible reason for you failing to understand something? Your skill in using English jumps to my mind. > would that be a "deeper" argument you're looking > for? For me, I do have reason to call him an > unintellectual clown. And Iirc, this isn't the > first time he has exhibited such character. > >> That would have made you a hypocrite, of course, >> for trying to do the very same thing to Marshall. > > As I've justed mentioned, I do have reasons and > I've presented the reason (and I could present > more reasons if you would like). It looks to me as though you are saying here that you have reasons for being a hypocrite. However, I do not believe that you mean to say that. What conclusion do you think I should draw from that, Nam? Jim Burns > Your alluding me as a hypocrite is *not* warranted! > If you want to defend him you should look at his > conversations with me in this thread (if not in > other threads as well) more closely. > >> So all this is just a difference of opinion as to >> whether you are a talentless buffoon, on the one >> hand, and Marshall is an intellectual clown, >> on the other? Then I guess it doesn't matter.
From: Jim Burns on 27 Mar 2010 18:49
Nam Nguyen wrote: > There's reason why the word "cranks" has a > different meaning than "standard mathematicians, > logicians" and I believe the difference is genuine. > > It's just that the later somehow believe that > they're aways invincible in their methods of > reasoning and they'd would slam the door shut on > a slightest hint their methods could be wrong. It may be that you and I are applying the labels "cranks" and "standard mathematicians" to different people. "Slamming the door shut" is a very good way to describe crankish behavior, in my experience. And if "standard" behavior (by which I mean behavior of persons widely respected and listened to, if not necessarily agreed with) is "slamming the door shut", where did all the non-standard (in its usual sense) analysis come from? However, because you complain about such behavior in others, I assume you do not engage in it yourself. I think it would be very useful to me in understanding what you are trying to accomplish if you were to give a summary of the best arguments AGAINST your positions. Fortunately, since you do not "slam the door shut", this should be pretty easy for you to do. Thanks. Jim Burns > > They know the 1-many problem and yet somehow they > could convince themselves they'd fully understand > the infinite complexity of the natural numbers. > > Aren't there any conservative, objective, and rational > mathematicians/logicians left in these forums to > further discussions about the current state of FOL > reasoning? |