'When Are Relations Neither True Nor False?
in [Math]
|
From: Nam Nguyen on 6 Mar 2010 12:00 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> David Bernier wrote: >>> Nam Nguyen wrote: >>>> Aatu Koskensilta wrote: >>>>> Marshall <marshall.spight(a)gmail.com> writes: >>>>> >>>>>> For me, Nam has mostly moved into the same category as AP. >>>>> Come now, even if you don't find Nam's posts worth reading comparing him >>>>> to Archimedes Plutonium is surely excessively harsh. >>>>> >>>> Thanks. But it's ok Aatu. I've been"blasted" by both the "orthodox" and >>>> the "crank" for years; nothing is new. >>>> >>>> It's hard to be in a 3rd party isn't it? In the past one "crank" alluded >>>> that I wasn't "liberal"/"open-minded" enough in my critique of the >>>> current >>>> regime of reasoning, and recently AP "lumped" me together with the >>>> "standard theorists". >>> [...] >>> >>> I respectfully disagree with your view that mathematicians should be >>> concerned with what Branson (who, it seems, debated "denotation" >>> or something with Russell) thought, when math. questions are >>> what's being discussed. >>> >>> I think you and I are in a stalemate position here on the point above. >>> I offer to make peace, in the following form: >>> >>> That you and I agree to disagree on Branson vs Russell, >>> when limited to math. questions. >>> >>> >>> David >> I think you accidentally mistook me for another poster. I've never >> mentioned Branson or Russell here. (In fact I haven't heard of Branson >> before!) > > I think he mistook you for Newberry and used the name Branson where he > meant Strawson. > > Aside from those little errors, his post was spot on, I'm sure. > Perhaps. But since I didn't quite follow that part of thread I'm not sure what is sure there. :-)
From: David Bernier on 6 Mar 2010 12:47 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> David Bernier wrote: >>> Nam Nguyen wrote: >>>> Aatu Koskensilta wrote: >>>>> Marshall <marshall.spight(a)gmail.com> writes: >>>>> >>>>>> For me, Nam has mostly moved into the same category as AP. >>>>> Come now, even if you don't find Nam's posts worth reading comparing him >>>>> to Archimedes Plutonium is surely excessively harsh. >>>>> >>>> Thanks. But it's ok Aatu. I've been"blasted" by both the "orthodox" and >>>> the "crank" for years; nothing is new. >>>> >>>> It's hard to be in a 3rd party isn't it? In the past one "crank" alluded >>>> that I wasn't "liberal"/"open-minded" enough in my critique of the >>>> current >>>> regime of reasoning, and recently AP "lumped" me together with the >>>> "standard theorists". >>> [...] >>> >>> I respectfully disagree with your view that mathematicians should be >>> concerned with what Branson (who, it seems, debated "denotation" >>> or something with Russell) thought, when math. questions are >>> what's being discussed. >>> >>> I think you and I are in a stalemate position here on the point above. >>> I offer to make peace, in the following form: >>> >>> That you and I agree to disagree on Branson vs Russell, >>> when limited to math. questions. >>> >>> >>> David >> I think you accidentally mistook me for another poster. I've never >> mentioned Branson or Russell here. (In fact I haven't heard of Branson >> before!) > > I think he mistook you for Newberry and used the name Branson where he > meant Strawson. My memory failed me. I'm now pretty sure it was Strawson, just like you say. < http://en.wikipedia.org/wiki/P._F._Strawson > . And someone in this thread referred to a scribd.com (spelling?) image of a document where the poster cited Strawson's book. Maybe I'll be second time lucky; otherwise, I'll be second time wrong. > Aside from those little errors, his post was spot on, I'm sure. >
From: David Bernier on 6 Mar 2010 13:13 Nam Nguyen wrote: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Jesse F. Hughes wrote: >>> >>>> So, you want to deny that Goedel's theorem is true. >>> The "crank" tends to assert Goedel's theorem is false. >>> The "standard theorist" would insist GIT is true. >>> >>> That leaves the "rebel" the only side who observes the >>> method in Godel's work is invalid. >>> >>> Except for the relativists, why should we care about invalid >>> truth or falsehood? >> >> >> Because the notions of "truth" and "validity" are not beyond dispute, and >> maybe we/you/I have got those wrong. > > In a high level that makes sense, imho. But I'm not after the "absolute" > truth, validity, or correctness. It's the _process and methods_ of > reasoning that I'm really ... really after. > > My posting in threads after threads is a means to re-examine, evaluate, > and revise the current methods of reasoning in FOL in general and in > Incompleteness in particular. And my motivation of the re-examination > is basically a following of what Shoenfield said about mathematical logic: > > "Logic is the study of reasoning; and mathematical logic is the study > of reasoning done by the mathematicians. > > To _discover proper approach to mathematical logic_, we must therefore > _examine the methods of the mathematicians_." > > [The highlights are mine]. > > For what it's worth, the long and short of it is what I've been doing for > years is simply trying to evaluate, for possible shortcomings, of the > current > FOL regime and of anyone's reasonings I've come across in some way: myself, > posters in this and other fora, Torkel Franzen, Godel, Hilbert, etc... > > I'm less interested for example whether or not, say, PA is consistent > but I'm interested in based from what existing and historical reasoning > backgrounds and by what methods one would logically conclude - with no > emotion or "belief" - PA is consistent - or not. There's an ultrafinitist named Nelson, who rose in professorial (USA) ranks to Associate Professor of Mathematics or higher. That's a sign of a definitely competent mathematician, generally speaking. For years, PA and/or ZFC were ok for him. Now, his belief is that PA *might* be inconsistent. With mathematicians, there's a method to their "madness" (English saying), and, hopefully, no "madness" to their methods. > For instance, here's what MoeBlee said earlier: > > >> at least we do know how to formalize PRA. And if we cannot be > >> confident that PRA is consistent then I don't know what substantive > >> mathematical theory we could be confident as to consistency. > > This passage, imho, is a typical example of where one would forget that > reasoning is a _process_ (as Shoenfield alluded to above), _not_ a mixture > of intuitive and religious-like _beliefs_. If we agree what "formalize" > means, what methods of reasoning are, what consistency and inconsistency > means, then as a fact it's either PRA is or isn't inconsistent strictly > based on the definitions and the methods, and it's either we do know *or* > don't know about the fact. Period. "Confidence" isn't an issue nor does > it have a role here, in the process of reasoning. > > The long and short of it is from what I've gathered, Godel's "proof" is > invalid as a meta proof, because the methods used in reasoning failed > in one of the few intuitive principles about the methods of logical > reasonings. Let me cite the 2 obvious principles: > > (1) Principle of Consistency: > > No methods shall lead to contradictory conclusions. > > (2) Principle of (Method) Compatibility: > > If each of any 2 equivalent conclusions is expressed in an independent > method then it shall not be the case that one method would lead to > its perspective conclusion, while the other method wouldn not lead to > the other counterpart conclusion. > > In Godel's work, the knowledge of the natural numbers as a sheer intuition > or as a model of a FOL formal system is incompatible with rules of > inference > in so far as both of them are methods of reasonings (and they are). > > The reason being is "undecidability" of formal systems is purportedly > "equivalently" defined by both methods, but the natural number would lead > to proof of consistency while the rules of inference method would not!
From: Jesse F. Hughes on 6 Mar 2010 16:13 Newberry <newberryxy(a)gmail.com> writes: > So here is the problem. We can have semantics such that a) vacuous > sentences are true, or we can have semantics such that b) vacuous > sentences are ~(T v F). Whether a) or b) does not depend on the model > or any model. People then forget that it is the logic that generates > the truth result for vacuous sentences. Then they claim that in any > model this and any model that. They talk about truth preservation, but > it is not the axioms that inject the untruths. Sorry, I haven't any clue what you're trying to say here. -- "All that 'shock and awe' stuff we've just dumped onto the Asian part of this earth - could we have fractured something? Perhaps the earth was just reacting to something that man has done to injure it. The earth is organic, you know. It can be hurt." -- A chatroom explanation of the Dec. 2004 tsunami.
From: Newberry on 6 Mar 2010 17:23
On Mar 6, 2:26 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Newberry <newberr...(a)gmail.com> writes: > > But in any case why is quoting G del's original statement pointless > > obscurantism? > > The notation and terminology of G del's original statement is > impenetrable to anyone who doesn't have the paper in front of > them. There's no reason not to state it in standard terminology and > notation. Which is the standard formulation? > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |