Torkel Franzen on truth
in [Logic]
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From: Aatu Koskensilta on 19 Dec 2007 10:18 On 2007-12-19, in sci.logic, abo wrote: > Given that inconsistent theories prove "PA is consistent", if we > merely know of some theory T that it proves "PA is consistent" nothing > interesting at all can be concluded. Right. In general, merely from the fact that this-or-that theory proves something, nothing interesting at all can be concluded. In case of particular theories immensely interesting mathematical knowledge can sometimes be obtained on knowing that P is provable in the theory, and even more information extracted from a particular proof, e.g. upper or lower numeric bounds on this-or-that. (That's one of the concerns of modern proof theory, after all.) > Yes, well I was interested in Daryl's definitive statement that no > such proof could be interesting or useful. Apparently he meant it > only to apply to theories where the conditions of the incompleteness > theorem 2 apply, in which case I can understand why he thinks that > there are proofs of "PA is consistent" which are interesting and > useful, but not proofs (in T) of "T is consistent." That would be a reasonable reading of his comments. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam D. Nguyen on 17 Dec 2007 18:27 tchow(a)lsa.umich.edu wrote: > In article <xLw9j.7491$hQ3.4060(a)pd7urf3no>, > Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >> Unfortunately mathematical reasoning isn't religion where "beliefs" >> would be much relevant. > > You don't think beliefs are relevant in mathematical reasoning? From what I gather, we don't call that "beliefs". We call it _interpretation_ which model basically is, and in which truths are true or false. The problem of this model-truth is over the same "structure" there could be opposite interpretation. Religious truth on the other hand is supposed to *believed* as true whether or not there is a model to reflect the truth. That's why belief doesn't have much of relevance in reasoning. > Then how do you become convinced that *anything* is true? As I've explained above. > Are you convinced, for example, that sqrt(2) is irrational? On what basis? On the basis of model that "sqrt(2) is irrational" is true, of course. > On the basis of the proof? No, not on the basis of proof: what is true or false is based strictly on model. Syntactical provability is actually in a different (and independent) paradigm, not withstanding Completeness. > But the proof starts with some axioms. Of course. > On what basis do you become convinced of the correctness of the axioms? What exactly does "correctness of the axioms" mean? > Or are you *not* convinced of the axioms? The only senses for which we could talk about axioms are: (a) They be independent from each other. (b) They don't contradict each other. So, again, what does it mean to be "convinced of the axioms"? > But if you're not convinced of the axioms, then what good is a proof of > "sqrt(2) is irrational" from those axioms? Proofs of course are good as a mechanism of assisting us in preventing our reasoning from being inconsistent. Of course.
From: tchow on 19 Dec 2007 11:26 In article <fhY9j.1418$Tx.1408(a)pd7urf3no>, Nam D. Nguyen <namducnguyen(a)shaw.ca> wrote: >There's a misconception here that seems to have escaped your attention. A model >*always already* includes a *chosen* interpretation: hence a belief has been >"believed" already. What's important is this interpretation could always be >reversed to the other way - at will - and a opposite "belief" would occur. >Consequently, a mathematically *stated* belief could change back and forth, [...] >In summary, for "there are no [nonzero] integers m and n such that >m^2 = 2 n^2", its truth, or its believed truth is quite subjective and >relative, like the "Colours seen by candlelight", but unlike religion >truth and belief. So let's look at these two statements: (1) There are no nonzero integers m and n such that m^2 = 2 n^2. (2) In the standard model of the integers, there are no nonzero integers m and n such that m^2 = 2 n^2. As I understand your position, (1) does not have a determinate truth value. But in (2), the phrase "in the standard model of the integers" chooses an interpretation and hence a "belief has been `believed' already." Does that mean that (2) is absolutely, unconditionally true? *After* I choose an interpretation, the subjectivism and relativism are gone, aren't they? -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: tchow on 19 Dec 2007 11:30 In article <858aj.268758$xw3.60305(a)reader1.news.saunalahti.fi>, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: >On 2007-12-18, in sci.logic, tchow(a)lsa.umich.edu wrote: >> George, you've *got* to be kidding here. Do you not have any clue how >> idiosyncratic your viewpoint is? > >George knows very well by now that his views on many issues are quite >bizarre and idiosyncratic. He's just convinced every one else is wrong. I know that much, but here he went further, by making claims about "almost everyone ten years younger than...," i.e., attributing his idiosyncratic views to a large population of other people. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
From: tchow on 19 Dec 2007 11:38
In article <cee5694a-9d7d-45dc-a91a-27ca7e08a208(a)s12g2000prg.googlegroups.com>, george <greeneg(a)cs.unc.edu> wrote: >In defense of your point I would say that I was next taught >commutativity and associativity as axioms; I was not taught how >to prove them using induction. >But there is a pedagogical question regarding what sort of "easier" >axiom-systems would be better than PA *to* start with. An induction schema that is restricted to *first-order* formulas only cannot possibly be pedagogically the right way to start. When I was taught induction in school, it was left vague what kinds of "properties" one can apply induction to. Concrete examples were given, all of which a logician could easily express as first-order formulas, but the question "what is a property?" was never directly addressed. Pedagogically, this has to be better than trying to specify 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). -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences |