'When Are Relations Neither True Nor False?
in [Math]
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From: MoeBlee on 8 Mar 2010 11:33 On Mar 8, 10:22 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > he hasn't said WHAT system of > reasoning he would accept as proving the incompleteness theorem IF he > doesn't already accept reasoning in the system that Koskensilta > mentioned, which is an even MORE restricted version of PRA. To be fair, he does discuss some general principles later in his post and in later posts. However, I don't find anything in his principles that supply or even suggest a system that has any potency at all while embodying premises and principles of reasoning as modest as the limited version of PRA mentioned by Koskensilta. Moreover, for each principle that Nam announces (including those I may agree with), I don't see that he's provided a basis for THEM that is any less based on "intution" than is the basis for adopting the logical and very modest mathematical principles employed in the limited version of PRA mentioned by Koskensilta. MoeBlee
From: Nam Nguyen on 8 Mar 2010 23:52 MoeBlee wrote: > On Mar 8, 10:22 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > >> he hasn't said WHAT system of >> reasoning he would accept as proving the incompleteness theorem IF he >> doesn't already accept reasoning in the system that Koskensilta >> mentioned, which is an even MORE restricted version of PRA. > > To be fair, he does discuss some general principles later in his post > and in later posts. However, I don't find anything in his principles > that supply or even suggest a system that has any potency at all while > embodying premises and principles of reasoning as modest as the > limited version of PRA mentioned by Koskensilta. Of course by presenting those principles I already had in my mind the kind of potency a reasoning system (edifice) conforming to these 4 principles would have. And I think I've expressed that before in some forms. For one thing, such a system could be conducive to relativity in mathematical reasoning, which is more _realistic_ than the "absoluteness" kind of reasoning we'd see in PRA or what Godel used to prove his meta proof. For another, it'd make our reasoning _more conservative_ in the sense that if we can't possibly know some "truth" then we have to acknowledge that impossibility, rather than acting as if we could know; this is nothing more than the principle of "conservation of knowledge" in mathematical reasoning, so to speak. The knowledge of the naturals (which Godel's work and PRA is based on, via recursion and truth of the relations of numbers) is uncertain, not conservative, and too intuitive to be rigorous for reasoning sake, which is why my 4 principles don't suggest or support it. In fact, the 4 principles basically "dismantle" any foundation based on knowledge of the natural numbers. The naturals then is just one of infinitely models (if at all), nothing more nothing less, nothing special, _nothing preferential_. > Moreover, for each > principle that Nam announces (including those I may agree with), I > don't see that he's provided a basis for THEM that is any less based > on "intution" than is the basis for adopting the logical and very > modest mathematical principles employed in the limited version of PRA > mentioned by Koskensilta. As I've mentioned above, any principles based on the knowledge of the naturals would *not* be "modest" at all. For one thing, this knowledge is rooted in intuition and however appealing as a "suggestion" force, intuition should never be a foundation of reasoning. The whole reason why today we have (Hilbert style) rules of inference is because we've never been able to completely 100% trust intuition. Intuition and reasoning as as different as the heart and the mind: one should not replace the other as far as "role" is concerned. For another thing, there are very strong indications that there are formulas in L(PA) that we can't assign truth value to them, hence a reasoning foundation based on the natural numbers would most likely incomplete.
From: Transfer Principle on 9 Mar 2010 01:21 On Mar 5, 7:17 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Mar 5, 2:01 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > >> Nam Nguyen <namducngu...(a)shaw.ca> writes: > >>> 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. OK, I've resisted posting in this thread long enough. Although I'm not posting in the main Newberry subthread, I do post here in the Nam Nguyen subthread. > > On usenet, nothing is beyond dispute. > > For example, try to get > > Nam to agree that the sky is blue. > > He has contested this in > > the past, sometimes with the argument that aliens might use > > the word "blue" to mean red. > The aliens might. Who knows? Would you be able to "know" that, Marshall? But on the other hand, if a known so-called "crank," let's say JSH, were to state that the sky is blue, the _standard theorists_ would be the ones to start coming up with obscure counterexamples such as the Doppler effect at velocities approaching c, alien languages in which "blue" means "red," and so forth. Case in point -- in a thread in which the standard theorists demanded that a "crank" accept Cantor's theorem as beyond dispute, I mentioned that there are some statements, such as 2+2=4, which, unlike Cantor's theorem, I do accept as unequivocally true. Then a standard theorist immediately brought up 2+2 == 1 (mod 3). Case in point -- spelling errors. Some standard theorists call out "cranks" who make spelling errors in their posts. They rarely point out such errors when fellow non-"cranks" are the ones who are misspelling. The truth is, most people -- standard theorists and "cranks" alike -- point out errors or obscure counterexamples only when an opponent makes the statement, even though the same statement would go unchallenged if an ally makes the statement. It's simply human nature -- I admit that I am guilty of the same. I, like both the standard theorists and Nguyen, point out errors and obscure counterexamples made by opponents all the time. (Indeed, I'm doing so right here, right now, in this very post.) So the standard theorists, no matter how much they may claim otherwise, don't avoid the natural human reaction of searching for errors and obscure counterexamples in their opponents' posts. And so, my own opinion of Spight's statement is: > > On usenet, nothing [written by an opponent] is beyond dispute [whether a standard theorist or a "crank"].
From: OP on 9 Mar 2010 01:27 Transfer Principle wrote: > On Mar 5, 7:17 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >> > On Mar 5, 2:01 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >> >>> 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. > > OK, I've resisted posting in this thread long enough. Although > I'm not posting in the main Newberry subthread, I do post here > in the Nam Nguyen subthread. > >> > On usenet, nothing is beyond dispute. >> > For example, try to get >> > Nam to agree that the sky is blue. >> > He has contested this in >> > the past, sometimes with the argument that aliens might use >> > the word "blue" to mean red. >> The aliens might. Who knows? Would you be able to "know" that, Marshall? > > But on the other hand, if a known so-called "crank," let's say > JSH, were to state that the sky is blue, the _standard theorists_ > would be the ones to start coming up with obscure counterexamples > such as the Doppler effect at velocities approaching c, alien > languages in which "blue" means "red," and so forth. > > Case in point -- in a thread in which the standard theorists > demanded that a "crank" accept Cantor's theorem as beyond dispute, > I mentioned that there are some statements, such as 2+2=4, which, > unlike Cantor's theorem, I do accept as unequivocally true. Then > a standard theorist immediately brought up 2+2 == 1 (mod 3). > > Case in point -- spelling errors. Some standard theorists call > out "cranks" who make spelling errors in their posts. They rarely > point out such errors when fellow non-"cranks" are the ones who > are misspelling. > > The truth is, most people -- standard theorists and "cranks" > alike -- point out errors or obscure counterexamples only when an > opponent makes the statement, even though the same statement would > go unchallenged if an ally makes the statement. It's simply human > nature -- I admit that I am guilty of the same. I, like both the > standard theorists and Nguyen, point out errors and obscure > counterexamples made by opponents all the time. (Indeed, I'm doing > so right here, right now, in this very post.) > > So the standard theorists, no matter how much they may claim > otherwise, don't avoid the natural human reaction of searching for > errors and obscure counterexamples in their opponents' posts. > > And so, my own opinion of Spight's statement is: > >> > On usenet, nothing [written by an opponent] is beyond dispute > [whether a standard theorist or a "crank"]. Creepy, man. You paint yourself as the defender of the weak and downtrodden but in fact all you ever do is attack people. It's classic bully behavior - really vile. This "crank" thing is just 100% bullshit.
From: Don Stockbauer on 9 Mar 2010 01:27
On Mar 9, 12:21 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Mar 5, 7:17 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > Marshall wrote: > > > On Mar 5, 2:01 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > > >> Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >>> 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. > > OK, I've resisted posting in this thread long enough. Although > I'm not posting in the main Newberry subthread, I do post here > in the Nam Nguyen subthread. > > > > On usenet, nothing is beyond dispute. > > > For example, try to get > > > Nam to agree that the sky is blue. > > > He has contested this in > > > the past, sometimes with the argument that aliens might use > > > the word "blue" to mean red. > > The aliens might. Who knows? Would you be able to "know" that, Marshall? > > But on the other hand, if a known so-called "crank," let's say > JSH, were to state that the sky is blue, the _standard theorists_ > would be the ones to start coming up with obscure counterexamples > such as the Doppler effect at velocities approaching c, alien > languages in which "blue" means "red," and so forth. > > Case in point -- in a thread in which the standard theorists > demanded that a "crank" accept Cantor's theorem as beyond dispute, > I mentioned that there are some statements, such as 2+2=4, which, > unlike Cantor's theorem, I do accept as unequivocally true. Then > a standard theorist immediately brought up 2+2 == 1 (mod 3). > > Case in point -- spelling errors. Some standard theorists call > out "cranks" who make spelling errors in their posts. They rarely > point out such errors when fellow non-"cranks" are the ones who > are misspelling. > > The truth is, most people -- standard theorists and "cranks" > alike -- point out errors or obscure counterexamples only when an > opponent makes the statement, even though the same statement would > go unchallenged if an ally makes the statement. It's simply human > nature -- I admit that I am guilty of the same. I, like both the > standard theorists and Nguyen, point out errors and obscure > counterexamples made by opponents all the time. (Indeed, I'm doing > so right here, right now, in this very post.) > > So the standard theorists, no matter how much they may claim > otherwise, don't avoid the natural human reaction of searching for > errors and obscure counterexamples in their opponents' posts. > > And so, my own opinion of Spight's statement is: > > > > On usenet, nothing [written by an opponent] is beyond dispute > > [whether a standard theorist or a "crank"]. Gives people something to do. |