Torkel Franzen on truth
in [Logic]
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From: MoeBlee on 19 Dec 2007 13:30 On Dec 18, 8:50 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Dec 18, 2:59 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > > >> A model > >> *always already* includes a *chosen* interpretation: hence a belief has been > >> "believed" already. > > > A model is a mathematical object. > > To be precise, it's more than just a mere mathematical object: It's not just a mathematical object, in the sense that it is a certain kind of mathematical object (as any mathematical object is a certain kind of mathematical object). > it's an *interpreted* > mathematical object. I'd say it IS an interpretation (or, depending on the definition, one of its components is an interpretation). > A syntactical wff on the other hand is a non-interpreted > mathematical object. A wff is not ordinarily itself an interpretation, okay. > What is the difference? Well, the entire Gestalt view of the > collection of components of a wff is supposed to be independent of any being's > interpreation or view: a FOL formufla would mean the same to all - no choice of > an alternative. The Gestalt view of the collection of components of a model-structure > is a model basically, and is subject to individual reasoner's interpretation. > Given the same structure, you could subjectively interpret the Gestalt view > differently! I'll leave that to you, but I'll read on... > For instance, given the following structure: > > xxxbxxxxxxxxxxxxxxxxxxxxxxxxxx You're using 'structure' here in what sense exactly? > Would you *interpret* that structure as a model of the blue-eye-dragon > theory T? Or would that be *not* a model of T, because you would interpret > "b" as "non-blue" (e.g. "brown"), and in such case it could be a model of > a non-blue-eye-dragon theory T' (if you so care to view it that way). > On the other hand, given an appropriate language, the formula F df= "the dragon > has a blue eye" could not be viewed/interpreted differently. For instance, > that formula could not be viewed as ~F. I have no idea what you're talking about. Sorry. > > I don't know how you would argue > > that a model requires "a belief has been "believed" already". > > Just change "a belief" to "an interpretation" and "believed" to "interpreted" > then you would know. Then I get: A model requires an interpretation has been interpreted. Sorry, I have no idea what motivates you to string those words together. If it is of any help, what I already said: A model IS an interpretation (or, depending on the definition, includes an interpretation as a component. MoeBlee P.S. Possibly a response later by you to my example about string substitution?
From: george on 19 Dec 2007 13:45 On Dec 19, 3:59 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > On 19 Dec, 01:16, george <gree...(a)cs.unc.edu> wrote: > > > The content of the axioms is what is relevant. > > Indeed. I thought that was my Fregean point, against your Hilbertian > formalism, namely that the axioms do come with semantic content :-) No, sorry. If the axioms are first-order then semantics is simply irrelevant. That is the GODELIAN point. That is the point that follows from the completeness theorem. More to the point, formalism is not opposed to content. Content by definition IS MERELY fancy form.
From: Nam D. Nguyen on 17 Dec 2007 18:55 Daryl McCullough wrote: > Nam D. Nguyen says... > >> Unfortunately mathematical reasoning isn't religion where "beliefs" >> would be much relevant. > > Well, that's what *you* believe, but I don't. Well then, Jesus of Nazareth is the only Son of God is mathematically true? Or is that *false*, logically specking? > > -- > Daryl McCullough > Ithaca, NY >
From: george on 19 Dec 2007 13:49 On Dec 19, 7:23 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > Mathematicians don't think, or do not begin by thinking, the natural numbers > -- or the reals, or sets of sets of sets of reals, or what have you -- exist > and they can state and proves stuff about them? A very curious idea. Hardly. Factual examples do just beat you over the head. Sometimes the theory REALLY DOES get discovered BEFORE the model. Non-Euclidean geometry being the (really, really) obvious case in point. Seriously, the mere fact that THAT little episode in the history of consciousness EVERY happened REALLY should have just SHUT ALL of you UP, a hundred and fifty years ago! People thought GEOMETRY had an intended model TOO, at one time, you know. But (GEO-) it turns out that the EARTH'S surface really IS non-Euclidean! The whole concept of "the intended model" IS JUST BULLSHIT. I repeat, that is NOT EVEN OPEN to DEBATE. FACTUAL EXAMPLES are relevant. The parallel postulate IS FACTUALLY FALSE over the surface of the earth, even if you idealize the sphere. The theory was able to be investigated before that was even clarified.
From: george on 19 Dec 2007 13:53
On Dec 19, 7:25 am, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > On 2007-12-18, in sci.logic, tc...(a)lsa.umich.edu wrote: > George knows very well by now that his views on many issues are quite > bizarre hardly. > and idiosyncratic. Well, unpopular, yes. > He's just convinced every one else is wrong. No, really, I'm not. The reason why I am always in caps with harsh language is that the positions I am attacking are just blatantly obviously incoherent. The people who are asserting them don't, for the most part, believe them themselves. NObody believes them. They aren't coherenty believABLE. Prof.Smith and I have been talking, for example, about the intended model vs. the formal language. He is the one who said that he didn't think formal languages should even be referred to as a language. That is considerably less defensible than anything *I* have ever said. It really is just plain obvious that anyone can posit a language and some axioms and inquire about what models of the axioms might exist. Insisting that mathematicians "usually" have an intended model to begin with instead IS SILLY. WHERE DID THEY GET this model from? HOW did they DEFINE it?? My point is simply that there were axioms involved THERE, TOO. They can't escape. It's not my way or the highway: there IS NO highway. My way IS THE way. Even the people who are claiming NOT to be on it OBVIOUSLY ARE. |