'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 9 Mar 2010 09:27 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > Jesse F. Hughes says... >> >>Transfer Principle <lwalke3(a)lausd.net> writes: >> >>> 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. >> >>That's true in many endeavors, but this is not my experience in >>mathematical discussions. In typical math discussions, a sensible >>person is unconcerned with friends and enemies and instead corrects >>mistakes on both sides. > > That's mostly true. But there are gray areas. If someone is speaking > informally (which is almost always) there are many possible areas in > which what he says is literally not true, but it's usually not worth > correcting, because everyone can see that either the mistake is easily > corrected, and it's clear what was meant, or else the mistake is minor, > and irrelevant to the conclusion. An example of the sort of mistake > or sloppiness that people make informally is failing to distinguish > between a natural number and the corresponding numeral. > > In accepting an informal argument, there is almost always a > principle of charity at work: you assume that the speaker means > something nonstupid, if you can figure out what that is. You only > bother to point out errors or ambiguities if it is unclear what the > speaker meant, or if the error seems like it cannot be corrected > without affecting his conclusion. Yes, there's a certain benefit of the doubt granted in certain contexts. Generally, this should depend on whether we figure that the speaker/writer made an inconsequential error or whether it was a deeper error. This judgment shouldn't depend on whether the speaker is "on our side" or not. -- "This page contains information of a type (text/html) that can only be viewed with the appropriate Plug-in. Click OK to download Plugin." --- Netscape 4.7 error message
From: MoeBlee on 9 Mar 2010 11:32 On Mar 8, 10:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > 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. What I mean is: What mathematical theorems can you prove? Come on, if I don't have to prove any mathematical theorems, then I wouldn't need to propose any principles AT ALL! > 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_. If you say primitive recursive arithmetic is "uncertain" then you'd have to show me that your principles are any more certain. Moreover, that your priniciples have any potency to prove very much of interest. > > 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 your knowledge of your principes is rooted in what? > 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. You completely skipped the part I wrote about starting premises. > 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. (1) A truth assignment for sentences of the langauge assigns truth or falsehood to EVERY sentence in the language. We've been over this a thousand times already. (2) For the purposes I've mentioned, it is not required that a theory be negation complete. Lack of negation completeness doesn't at all impugn PRA for the purpose of deriving certain theorems about systems of finite strings of symbols. MoeBlee
From: MoeBlee on 9 Mar 2010 11:37 On Mar 9, 12:21 am, Transfer Principle <lwal...(a)lausd.net> wrote: > 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. "The standard theorists" would do that? How do you know? WHICH "standard theorists"? And would you please say exactly what you mean by "a standard theorist"? > 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). (1) I'd like to see the full context of that. (2) So because one "standard theorist" said such and such in one instance, then you conclude that "standard theorists" (whatever you mean by that) generally say such and such? MoeBlee
From: MoeBlee on 9 Mar 2010 11:43 On Mar 9, 12:30 am, Transfer Principle <lwal...(a)lausd.net> wrote: > It would be freakin' _hilarious_ if Nelson's attempt to prove > PA inconsistent were successful. The standard theorists would > then be _forced_ to come crawling back to the ultrafinitists > like RE, Y-V, and others, if they want to find a theory that > would survive his inconsistency proof! Why would a "standard theorist" have to do any crawling at all? (1) One may hold that PA is consistent, then be proven wrong, and then simply say 'I stand corrected'. No crawling required. (2) Though I think most mathematicians who you would call "standard theorists" hold that PA is consistent, I wouldn't conclude that there are NONE who don't also allow that in principle it is possible that PA is inconsistent. (3) If PA were proven inconsistent, it wouldn't thereby follow that the only viable consistent alternative would have to be an ultrafinitist theory. MoeBlee
From: Rotwang on 9 Mar 2010 11:45
Transfer Principle wrote: > > [...] > > 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. Funny example. In fact, the "alien language" argument is not one I've ever seen one of the people you call "standard theorists" use, except for the purpose of parody, but it /is/ an argument routinely used by JSH himself. Try searching GG for the string "planet contrary". |