'When Are Relations Neither True Nor False?
in [Math]
|
From: MoeBlee on 10 Mar 2010 13:07 On Mar 10, 8:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > > >> MoeBlee wrote: > > >>> 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! > >> You still could prove a lot of theorems under an edifice that complies > >> to these principles. In fact if you take the current FOL and strip off > >> anything that are (or _depends on_) the naturals or models from the > >> role of _inference_ then you'd obtain an edifice that would conform > >> to these 4 principles (at least from a first glance at the matter). > > > Give an example of a nontrivial theorem in such a system. I don't > > think anyone would be interested in it, not even you. > > How about ExAy[~(Sy=x)], in Q (in that edifice)? It's an arithmetic > theorem, got to be interesting, isn't it? Ha! (If your question is rhetorical, which it sure appears to be), you just committed an obvious fallacy. Daryl didn't say that ALL theorems of PA are interesting. Rather, he's (implicitly) saying that SOME theorems of PA are interesting, in contrast to whatever might be your system of which he'd like to know of at least one interesting theorem, and as he didn't challenge that ALL theorems of your system should be interesting. Damn, you're not even LISTENING - as usual. > "Interesting" is subjective and *not* logical/reasoning. Of course it is. No one has claimed otherwise. > Proving is > logical/reasoning. If you're eager to claim some truths and in the > process sacrificing the rigidity of reasoning via syntactical proofs, > what's the point? We're not sacrificing any rigor. We don't claim that 'interesting' is a rigorous notion. But the point is that human beings study subjects because they are interested in certain questions. This interest may be motivated by practical concerns (such as with arithmetic we can count our sheep to sleep) or with purely abstract yet intriguing matters. Of course this is subjective; no one claimed otherwise. > Iirc, in the French Evolution it was said something to the effect that > in the name of freedom, liberty, revolution, they did commit some > crimes. Well, I think in the name of interesting, induction, countably > infinite, we've harmed the rigor of reasoning by syntactical rules > of inference, in mathematical logic. I don't see any harm. A precise and purely syntactical proof system is devised, and we've found that from certain formulas other formulas are derivable in this proof system. That we may be more interested in some derivations than in others (and in some theorems rather than others) is a separate matter. Surely you don't even hope that every person who studies formal mathematics is equally interested in all theorems and derivations in all possible systems? MoeBlee
From: MoeBlee on 10 Mar 2010 13:10 On Mar 10, 8:49 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Daryl McCullough wrote: > > Nam Nguyen says... > >> Daryl McCullough wrote: > >>> Nam Nguyen says... > >>>> MoeBlee wrote: > > >>>>> (1) Anyone who has studied mathematical logic already recognizes that > >>>>> a sentence that is neither logically true nor logically false has > >>>>> models in which it is true and models in which it is false. > >>>> Did you yourself quantifying-ly inspect uncountably many models > >>>> of, say, PA to know that? > >>> The beauty of mathematical proof is that you can be certain > >>> of the truth of a universal statement without checking every > >>> instance. > >> You meant as "certain" as the truth of GC or "There are infinitely many > >> counter examples of GC"? > > > Neither. I mean certain as the truth of "every consistent theory has a > > countable model". > > How certain is that while you don't know exactly what the naturals collectively > is? First, we know what a natural number is, by a finite definition ('finite defintion' being redundant, I would think). Second, in a certain formal theory, we've proven a theorem that may be taken in English as "there exists a set whose members are all and only the natural numbers"). Other people may have other basic notions as to the set of natural numbers, but the aforementioned is sufficient. MoeBlee
From: MoeBlee on 10 Mar 2010 13:15 On Mar 10, 9:18 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > In any rate, do you (Daryl, MoeBlee) see anything wrong with the 4 > principles? And if so why? [You both seem to have resisted their > "power". No?] I don't have time or interest to type out responses regarding all of what you've posted recently. However, I've already made some remarks that should be adequate as a starting point. Moreover, much of your posting I really can't make much sense of. I find myself continually asking what you mean by your ersatz terminology, only to find that the terminological hole just gets deeper and deeper. Moreover, you continually skip responding to some of my most vital points. Moreover, I can barely find common ground for discussion with someone who won't even admit that we can reliably distinguish the string 1 1 1 from the string 1 1 0 (or whatever the exact example was). MoeBlee
From: Transfer Principle on 10 Mar 2010 23:21 On Mar 9, 6:27 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > stevendaryl3...(a)yahoo.com (Daryl McCullough) writes: > > 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. Yes, in _theory_, the judgment shouldn't depend on whose side the speaker is on, but in _reality_, it does. It's only human nature. And both the standard theorists and the "cranks" judge posters based on whose side they are on.
From: Transfer Principle on 11 Mar 2010 00:00
On Mar 9, 8:37 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > 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"? Fine, the non-"cranks" or anti-"cranks," then. Or perhaps the word often used by galathaea is more appropriate here -- "bullies." > > 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. OK, I found a couple of old posts with examples of this behavior. One year ago almost to the day, the morning of St. Patrick's Day of last year, I wrote: > > Let's start by considering the following three statements: > > 1) 2 + 2 = 4 (mentioned by Ullrich in another thread). > > 2) There are exactly five Platonic solids (Chandler). > > 3) 0.999... cannot equal any number other than 1. > > Obviously, all three statements are true in standard mathematics. Then the standard anti-"crank" Brian Chandler responded (the same day, approximately 5AM Greenwich time) with: "No. All three statements are true if we mean by them 'the usual thing'. Nothing to do with 'standard mathematics', within which all three statements can also be false, if we are talking about "unusual things": arithmetic modulo 3, 4D space, some properly formalised system that hasn't been presented yet, respectively. " The original context was that the "crank" MR was trying to discuss the possibility of 0.999... being unity minus a nonzero infinitesimal. So I argued that MR was being called a "crank" because he believes that "0.999... = 1," so there were some statements, like "0.999... = 1" and "2+2=4," which standard theorists will defend against "cranks." Then Chandler made the above reference to "arithmetic modulo 3," implying the equation 2+2=1 rather than 2+2=4. It's hard to tell what Chandler's point was. Perhaps he was trying to make an argument similar to the one MoeBlee makes below: > People are not (ordinarily) called 'cranks' merely for proposing > alternative theories. so that MR wasn't being called a "crank" just for arguing 0.999... < 1, in the same way that someone who contradicts "2+2=4" isn't called a "crank" since there's a ring Z/3Z in which "2+2=1." Of course, I don't agree with this, since almost every time a sci.math poster writes the inequality 0.999... < 1, out come the "bullies." So the standard theorists may claim that people aren't called "cranks" just for writing 0.999... < 1, but the evidence in sci.math posts doesn't support that claim at all. An older post in which 2+2=4 leads to a modular reponse is occurred back on the morning of 19th November 2008. I had written: > > I don't believe that having a > > set theory in which, say, Cantor's theorem > > fails is equivalent to having a set theory in > > which 2+2 is anything other than 4. And at 7:32 AM Greenwich time that morning, the standard anti-"crank" Denis Feldmann gave the following response: "Your beliefs are of no concern to me. You are wrong, period. If you believe you are right, please exhibit such a theory, or, better, solve the halting problem (this is equivalent to Cantor thoerem [sic], actually) Learn to read. And, BTW, in Z/4Z, 2+2=0" In this case, the so-called "crank" tommy1729 was trying to come up with an alternate theory with a set V of all sets, which obviously can't adhere to Cantor's Theorem. Once again, I compared Cantor's Theorem to 2+2=4 in that standard theorists would defend them. Here Feldmann argued that Cantor's Theorem is even _ more_ solid than 2+2=4, since in the ring Z/4Z, 2+2=0, but Cantor's Theorem is equivalent to the _halting problem_, and of course I can't solve the halting problem (by producing a program/Turing machine that takes another program as input and determines whether it halts or not). So in a way, Feldmann implies that Cantor's Theorem has even _more_ empirical evidence supporting it than 2+2=4 does. At the time, I hadn't learned about NFU yet, or otherwise I would have told him that NFU is an example of a theory in which Cantor's Theorem actually fails. But then, what about the halting problem? One would think that simply switching from ZFC to NFU would make the halting problem suddenly solvable. It could be that when Feldmann wrote: "the halting problem (this is equivalent to Cantor thoerem [sic], actually)" he meant that _ZFC_ proves that they are equivalent -- but they aren't necessary equivalent in another theory such as NFU. So it possible that Cantor's Theorem fails in NFU without solving the halting problem. > (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? Counting the Z/4Z example as well as the Z/3Z example, that's _two_ standard theorists (anti-"cranks"/"bullies") who said that. In one case, the poster was trying to argue that merely proposing an alternate theory doesn't make one a "crank," while the other was trying to argue that contradicting Cantor's Theorem _does_ make one a "crank," even while contradicting simple arithmetic doesn't. But what I've never seen is one standard theorist declaring "2+2=4" and a fellow non-"crank" mentioning Z/3Z or Z/4Z as counterexamples. That's why I gave the generalization (which might be viewed as a "lie") that standard anti-"cranks" only mention counterexamples like Z/3Z and Z/4Z when an opponent is making claims like 2+2=4. |