Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 19 Dec 2007 14:07 On 2007-12-19, in sci.logic, george wrote: > 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. Ah, it's all explained now: you do not in fact believe the bizarre assertions you habitually sputter. You know 'tis so, deep down your heart. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: george on 19 Dec 2007 14:11 On Dec 19, 11:38 am, tc...(a)lsa.umich.edu wrote: > When I was taught induction in school, it was left vague what kinds of > "properties" one can apply induction to. Well, the whole question of when you should even start to teach the official definition of "what a first-order language is" does arise there. But I didn't personally know how to do anything more complex at the time, so in practice I personally was certainly limited to that. > Concrete examples were given, > all of which a logician could easily express as first-order formulas, Exactly. So it is NOT like YOU were doing something MORE complicated than that. Nor was I. > but the question "what is a property?" was never directly addressed. It wasn't even FORMULATED. BY definition AND PRACTICE, a property was whatEVER you KNEW HOW to WRITE down in a HOLE in which a PEG "of type property" would fit. This was very much a grammatical question. It was not (at this juncture) the philosophical question it would need to be later. > Pedagogically, this has to be better than trying to specify precisely > what properties are acceptable. This is completely misguided. You CANnot "specify" until AFTER you have some LARGE universe of "potential possible" properties out of which you could bless some and curse others. It is this universe which (at age 14, for most people) is NOT going to exis yet. What you DO have at that point is some conventions for writing things down. You have some known strings that are acceptable. You have some vague heuristics for gluing them together to make more complicated ones. THAT IS SUFFICIENT. That is NOT going to take anyone accidentally BEYOND any first-order considerations. So there is neither need NOR POSSIBILITY (at THAT juncture) of "specifying precisely what properties are acceptable". > Anyway, my main point was that your assertion that all people of a certain > age "started with PA" is plainly false (or, in GeorgeGreenese, you LIED > when you said that). No, really, it isn't. People did and do generally start with the identities and those ARE axioms of PA. As I have already said, what people really start with is calculation, is the whole 0th-order piece. The question is, what about the first- order piece? How does THAT arise/arrive? When does it become important to start saying things about all natural numbers beyond how to add 1 to them? Seriously, what is the first contentful first-order generalization about the naturals that you personally ever learned? I am thinking in my case that it was commutativity and associativity but I am pretty sure those were presented to me as axioms. As I said before, "starting WHAT?" -- my point being that at some point one starts to prove 1st-order sentences in arithmetic. Maybe in my case the first ones were about odd/even results -- the sum of two odd numbers or two even numbers is even, the sum of an odd and an even is odd; the product of odds is odd; the product of an even and anything else is even. Those all have fairly straightforward 1st-order proofs if you can use associativity. You don't actually need induction for them. In any case, none of that is the point. The point IS that ALL the axioms of PA EXCEPT induction involve/embody simple straightforward truths about the naturals that most people WERE taught early on and WERE using THE WHOLE time. And your point about what to "specify" for induction is just bullshit. When induction is introduced, the simple introductory context into which it is being introduced will guarantee that nothing beyond 1st-order comes up.
From: george on 19 Dec 2007 14:16 On Dec 19, 11:38 am, tc...(a)lsa.umich.edu wrote: > An induction schema that is restricted to *first-order* formulas only cannot > possibly be pedagogically the right way to start. This is completely missing the point. The issue is not how to phrase the induction schema. The prior issue is WHAT KIND OF LANGUAGE is math IN GENERAL being conducted in, in your classroom? If the students are used to writing 1st-order generalizations then it will automatically occur to them to be putting that kind of peg in the hole. If they are not then it gets more complicated. But my point is, it NEVER gets 2nd-order complicated, or complicated in a way that could cause theoretical problems, because the students HAVEN'T LEARNED THAT yet. My point is that your WHOLE DISCOURSE was restricted to *0th*-order formulas and NATURAL-language generalizations for far too long. Getting in the habit of writing things with quantifiers is almost un- natural. One DOES NOT SAY (or write) Ex[2*x=y]. One RATHER says, "y is even". One might even say, "since y is even, let x=y/2". This does NOT APPEAR to be invoking 1st-order ANYthing. My point is, you don't have to be thinking about phrasing it in the canonical first-order grammar TO BE PERFORMING first-order reasoning IN PA.
From: G. Frege on 19 Dec 2007 14:21 On Wed, 19 Dec 2007 10:53:59 -0800 (PST), george <greeneg(a)cs.unc.edu> wrote: >> >> George knows very well by now that his views on many issues are >> quite bizarre >> > hardly. > But they are - sometimes, george! >> >> and idiosyncratic. >> > Well, unpopular, yes. > Well, not generally accepted, right... ;-) > > 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. > Well..., Alonzo Church [you know that guy?] also mentioned such a view, in one of his papers. He writes: "We distinguish between a /logistic system/ and a /formalized language/ on the basis that the former is an abstractly formulated calculus for which no interpretation is fixed, and thus has a syntax and no semantics; but the latter is a logistic system together with an assignment of meanings to its expressions. [...] In order to obtain a formalized language it is necessary to add to these /syntactical rules/ of the logistic system, /semantical rules/ assigning meanings (in some sense) to the well-formed expressions of the system." (Alonzo Church, The Need for Abstract Entities, 1951) [ Oh right, this is an ancient text and etc. etc. *sigh* ] > > Insisting that mathematicians "usually" have an intended model to > begin with instead IS SILLY. > Nonsense. (Clearly historically most mathematical "theories" started out _without_ being formalized as axiomatic systems. At least this is true for _set theory_, as you certainly will know.) > > WHERE DID THEY GET this model from? > Mathematical intuition? F. -- E-mail: info<at>simple-line<dot>de
From: MoeBlee on 17 Dec 2007 19:01
On Dec 17, 3:27 pm, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: > tc...(a)lsa.umich.edu wrote: > > In article <xLw9j.7491$hQ3.4060(a)pd7urf3no>, > > Nam D. Nguyen <namducngu...(a)shaw.ca> wrote: > From what I gather, we don't call that "beliefs". We call it _interpretation That we have a method for interpreting formal languages does not contradict that one may believe certain things about mathematics. > which model basically is, and in which truths are true or false. "in which truths are true or false". Are you sure you've eaten breakfast this morning? > The problem > of this model-truth is over the same "structure" there could be opposite > interpretation. No, that's completely wrong. Given a structure, there is only one interpretation associated with that structure. > Religious truth on the other hand is supposed to *believed* > as true whether or not there is a model to reflect the truth. I've never seen such a description of religious belief. > That's why > belief doesn't have much of relevance in reasoning. Belief may or may not have relevance in reasoning, but the confusions you just posted don't lead to any conclusion on the matter. > > Then how do you become convinced that *anything* is true? > > As I've explained above. No you didn't. > > Are you convinced, for example, that sqrt(2) is irrational? On what basis? > > On the basis of model that "sqrt(2) is irrational" is true, of course. Maybe you mean, on the basis that there is a model in which "sqrt(2) is irrational" is true. And there is a model in which it is false also. What about operations on finite strings? Don't you believe, for example, irrespective of any model, that the string "0011" is the same as the string "0022 [with 1 substituted for 2]"? > > On the basis of the proof? > > No, not on the basis of proof: what is true or false is based strictly on model. > Syntactical provability is actually in a different (and independent) paradigm, > not withstanding Completeness. > > > But the proof starts with some axioms. > > Of course. > > > On what basis do you become convinced of the correctness of the axioms? > > What exactly does "correctness of the axioms" mean? > > > Or are you *not* convinced of the axioms? > > The only senses for which we could talk about axioms are: > > (a) They be independent from each other. > (b) They don't contradict each other. No, there are lots of other properties of axioms. One, for example, is that of a certain model being a model of the axioms. > So, again, what does it mean to be "convinced of the axioms"? > > > But if you're not convinced of the axioms, then what good is a proof of > > "sqrt(2) is irrational" from those axioms? > > Proofs of course are good as a mechanism of assisting us in preventing > our reasoning from being inconsistent. Of course. Except if the axioms are inconsistent. Actually, (first order) proof doesn't ensure consistency but rather entailment. MoeBlee |