Torkel Franzen on truth
in [Logic]
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From: Nam D. Nguyen on 29 Dec 2007 14:55 Peter_Smith wrote: > On Dec 28, 6:44 am, "Nam D. Nguyen" <namducngu...(a)shaw.ca> wrote: >> What Daryl stated: >> >> "So, for example, a proof in PA + the negation of Goldbach's conjecture >> would not be very convincing, because we have no reason to believe that >> the negation of Goldbach's conjecture is true." >> >> So his hypothesis is: >> >> H = "we have no reason to believe that the negation of Goldbach's >> conjecture is true." >> >> His (reasoning) conclusion based on H is: >> >> C = "a proof in PA + the negation of Goldbach's conjecture would not >> be very convincing". >> >> Independent of what he actually intended to say, taken on face value his >> statement's conclusion is, after being stripped from its informality: >> >> C' = "(PA + ~GC) is an inconsistent theory due to ~GC" > > It is utterly incomprehensible that you read C' into what Daryl said. "a proof" is "a general proof" which is equivalent to *any proof*. If I say to you "*Any proof* of a formal system T *would not be convincing*", and if, on the merit of the statement alone - *not* what I intended to say but *didn't clarify*, that doesn't mean T is not inconsistent, then what on Earth would that could possibly mean to *you*, logically speaking? (Remember during my conversation with TC about his own statement and up to my post above, Daryl didn't respond to what I said I all. Consequently (and I already mentioned it) I had no choice but took his statement for what it was stated, with all the "cleansing" [of informality] that one would typically do in a technical argument.) So it's not incomprehensible at all (let alone "utterly incomprehensible") that you could read C' into C, on the face value of it! Right? > > In any case not-GC is Sigma_1, PA is Sigma_1 complete, so if not-GC, > then PA proves it and (assuming PA is consistent) so is PA + ~GC. If > Daryl DID hold not-GC (and how you get from his saying that there is > no reason to believe non-GC to his assuming it is a mystery), he -- > being a sensible chap -- would be *denying* that (PA + ~GC) is > inconsistent. > > But perhaps that's a typo for > > C" = "(PA + ~GC) is an inconsistent theory to GC > > But again Daryl, being a sensible chap doesn't say that either. Sorry, without him clarifying anything - at the time all of this was debated, I couldn't possibly know what was in his mind; and what he said is what he stated. It's that simple! > He at > most says that, if GC were true (and we have no reason to deny it), > the theory PA + ~GC wouldn't be reliably truth-generating. But that OF > COURSE doesn't entail that the theory is inconsistent. Suppose for a moment I incorporated his newly revealed intention into C, that *still doesn't invalidate the point I was arguing with TC*, which is basically that Daryl's overall statement is a religion [-like] statement simply because we don't know *for certain* Q has any model: it's a mere defaulted *assumption* (i.e. *belief*) that Q/PA has a model. And a religion-like statement doesn't mean it must necessarily have any thing to do with God (though I think it's said Godel once tried to prove Him exist, mathematically!), all that means is when you believe a statement is true/false but there is no concrete supporting proof from the framework you're using to reason.
From: Nam D. Nguyen on 29 Dec 2007 16:08 Nam D. Nguyen wrote: > Daryl McCullough wrote: >> Nam D. Nguyen says... >> >>> and in fact I would have not involved in this conversation at all! >> >> And we'd all be happier. > > Somewhere I think I've heard: to kill a dog we would just label it a > "mad" dog! Seriously. In the thread "About Consistency in 1st Order Theories." there was this comment about what I posted then from DCU (Dec. 22nd, 2005): <quote> If you have a specific replacement for FOL in mind then you might get people to comment on it. But you don't, you're just sort of hunting around for the replacement that's going to make you happy. If you want people to help you with this you might start by trying to convince people that there's a _need_ for this radical new version of logic. Exactly what the objective is is not clear to me. It seems possible that you might want to call it something other than "logic". Because whatever it is, it seems that in the thing you're looking for the "logic" is going to vary from person to person, and I suspect it's going to seem to a lot of people like the whole point to _logic_ is to study _correct_ reasoning, which will _not_ vary from person to person. Now, the class of mathematical facts that a given individual is actually able to prove certainly varies from person to person. If you want to study that somehow fine, but that seems more a topic in something like psychology than pure logic. If I'm correct in thinking that in the system you have in mind the _definition_ of correct reasoning is going to vary from person to person that seems even less like "logic". </quote> For 2 years I've tried in various posts/threads to follow his suggestion: - I've pointed out the relativity nature of current FOL reasoning is upon us all. I'd be for us all - not just me alone - to see it as it is so *we'd all* feel happy to make appropriate adjustment to this obsolete post Godel reasoning foundation. - I've pointed out that there can't be global "_correct_ reasoning" that *all reasoning beings could possibly know! - I've pointed out some suggestion how a new framework could be achieved: in a nut-shell: *) through formally recognizing certain limitation of "knowing" in some new (suggested) un-knownability principles. [One could see the section "A working perspective" from http://en.wikipedia.org/wiki/Foundational_crisis_of_mathematics for certain similarity of the "un-knownability principles" I've alluded to above. - I've similarly discussed many issues related to the above. But to be honest, for the past 2 years, many times I feel like I happened to enter a time machine going to Euclid's time where all the "professionals" would only care to utter one "intrinsic" truth: "The 5th postulate is 'absolutely' true'"! Of all the "science" fields, mathematical reasoning is supposed to be a field where we should keep up the motto: keep an open mind! It's kind of sad here in this forum most of the time we'd rather like to engage in a "fight" than in an open-mined scrutiny on mistakes of the past!
From: george on 30 Dec 2007 13:41 On Dec 23, 1:58 pm, Aatu Koskensilta <aatu.koskensi...(a)xortec.fi> wrote: > > Where are the bicycles in analysis? > > Hiding behind the universal ordinal. > The idea that "the completeness theorem > is fundamentally a proof that first-order semantics > SIMPLY DOESN'T EXIST" is > entirely arbitrary and quite bizarre. This dismissal is entirely arbitrary. It has content zero. Would that it had measure zero as well. This dismissal does not analyze or assert anything. Not that anything really needs asserting. Everybody in the target audience ALREADY knows that the completeness theorem basically alleges that first-order |- "completely" covers first-order |= (that's why the theorem is named that). Given that everybody knows that, my "idea" was simply incontestable. It is no more arbitrary than "the sky is blue", and no more doubtful. This is a classic example of what I previously meant by disagreement with my allegedly idiosyncratic&bizarre opinons being, factually, neither coherent nor (therefore) possible.
From: george on 30 Dec 2007 13:45 On Dec 28, 3:46 am, Peter_Smith <ps...(a)cam.ac.uk> wrote: > It is utterly incomprehensible that you read C' into > what Daryl said. It's not *utterly* incomprehensible. It may be incomprehensible for most people, but the person who committed this particular mis-comprehension was Nam. If you find THAT comprehensible then quite a few other previously inexplicable things become understandable.
From: Aatu Koskensilta on 1 Jan 2008 09:14
On 2007-12-30, in sci.logic, george wrote: > This is a classic example of what I previously meant > by disagreement with my allegedly idiosyncratic&bizarre > opinons being, factually, neither coherent nor (therefore) > possible. Bah. We could with equal justification, and equally arbitrarily, claim that "the completeness theorem is fundamentally a proof that first-order syntactic deduction SIMPLY DOESN'T EXIST". -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |