'When Are Relations Neither True Nor False?
in [Math]
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From: MoeBlee on 22 Mar 2010 20:15 On Mar 22, 6:30 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Just as I suspected would happen, the standard theorists are > reluctant to accept the poll to which I linked as valid. Whether "standard theorist" or not, there's hardly much reason to put very much stock in such a poll. > On the other hand, the poll to which I linked wasn't any arbitrary > poll, > but a poll given at the Metamath website. As far as I can tell, the Metamath site links to the poll, but, as far as I can tell, the poll itself wasn't conducted by the Metamath site. > One could argue that the participants in the poll aren't really > physicists > and so the poll is invalid Even IF all the participants were physicists, there's little basis to put very much stock in the poll. -- but then again, why would Metamath > choose > to link to it if they thought that the poll was meaningless? Whether "meaningless" or not, you'd have to ask Norm Megill that question. However, as to the more general question, one might link to such a poll merely for the point of curiosity or whimsy or any of many reasons. > And > recall > that Metamath isn't a "crank" site, but an automatic theorem prover -- Whatevert Metamath is, and I sure don't think of Norm Megill as a crank, just the fact that he linked to the poll doesn't entail that there's any reason to think the poll is a worthy basis upon which to infer what some percentage of phyicists think. > after all, the proof in ZF of 0.999... is given right below the link. > The > fact that the web maintainers at Metamath thought that the poll was > worth mentioning does give some validity to the poll. You've never studied statistics or scientific methodology? > > Bah! Even barely educated guesses rise far above the level of the worst > > evidence. > > Aatu writes that he prefers educated guesses to random "evidence." I > wonder what educated guess Aatu himself would give as the proportion > of physicists who believe that 0.999...=1. What difference would it make? > So far, the only poster who has given any educated guess is Rotwang, > himself a physicist. Rotwang's guess above is that a supermajority > (approaching unanimity) of physicists do accept 0.999...=1. > > Even if Rotwang is correct, I doubt that physicists would accept _all_ > of the theorems of ZFC as being true -- since for one thing, most of > them > have never heard of ZFC. So what? > Of course, we should leave out theorems that > are not directly related to physics (such as the existence of, say, a > fixed > point of aleph). MoeBlee hints that physicists might not even have an > opinion about most of these. I can merely speculate, and my speculation on such things is worth less than two cents. > But say we stick to, say, the real numbers R, and mathematical objects > that do play a role in physics. So my question is, does there exist a > statement regarding such an object that a significant number of > physicists > (say at least a third of them) believe is false, yet the statement is > provable > in ZFC (assuming that ZFC is consistent)? (1) For the sciences, as far as I know, Z-"regularity" is enough. (2) Suppose there were a statement of the kind you mention. That wouldn't entail that Z(FC) (or whatever) does not provide an axiomatization of mathematics for the sciences. In the meantime, of course it is well recognized that Z(FC) has theorems that are not even of concern in the sciences and goes well beyond mathematics needed for the sciences. > I believe so -- even if 0.999...=1 isn't such a statement. To me, it > seems > hard to believe that a supermajority of physicists -- who have never > even > heard of ZFC -- would be in complete agreement with _all_ of the > theorems > of ZFC that are related to physics (including real numbers). A more interesting question would be to look for a theorem concerning only (in some sense of "concerning") real numbers that physicists apply but that is not a theorem of Z-"regularity". > And if such a statement exists, then I believe that a theory in which > its > negation is provable (and isn't trivially inconsistent) is worth > considering. Sure, but in the context of providing an axiomatization of mathematics for the sciences, such an alternative theory is worth considering only if there's some reason to believe it provides an axiomatization of mathematics for the sciences. There's no trick to just setting up some consistent theory or another that contradicts Z(FC) but that doesn't prove much of mathematics for the sciences. > And > if the proportion of physicists who disagree with the statement isn't > just a > sizeable minority but an actual majority, then I'd argue that there's > a theory > _better_ than ZFC at axiomatizing math for the sciences. It would depend on the definition or notion of "better". > Indeed, going back to the original topic of this thread, I wonder > whether a > supermajority of physicists would accept the concept of vacuous truth > (but > some might point out that this is an issue of FOL, not ZFC). Why don't you scout around for some theory of physics that doesn't (at least implicity) use mathematics that provides ~P -> (P -> Q). MoeBlee
From: Rotwang on 22 Mar 2010 20:22 Transfer Principle wrote: > >> [...] > > So far, the only poster who has given any educated guess is Rotwang, > himself a physicist. Rotwang's guess above is that a supermajority > (approaching unanimity) of physicists do accept 0.999...=1. > > Even if Rotwang is correct, I doubt that physicists would accept _all_ > of the theorems of ZFC as being true -- since for one thing, most of > them have never heard of ZFC. I don't expect that last is true (though I say this with much less confidence than my earlier statement about 0.9r = 1); physicists are not mathematicians, but all physicists have to take a certain level of interest in maths, and I would find it surprising if most of them had avoided ever hearing the term "ZFC", in much the same way that I would be surprised if most mathematicians hadn't at least heard of QFT. But putting that to one side, I don't agree with your tacit assertion that a physicist who has never heard of ZFC would not accept all its theorems. Even if a physicist has never seen the pairing axiom, for example, it's quite likely that he would find its statement utterly uncontroversial, and the same goes for the other axioms. And if a physicist accepts all the axioms of ZFC, as well as the usual rules of FOL, then he must accept all of the resulting theorems whether he realises it or not. In fact I believe that ZFC and FOL are so successful precisely because they formalise mathematical principles that physicists and almost everyone else who uses mathematics rely on, even if unwittingly. > Of course, we should leave out theorems that > are not directly related to physics (such as the existence of, say, a > fixed > point of aleph). MoeBlee hints that physicists might not even have an > opinion about most of these. > > But say we stick to, say, the real numbers R, and mathematical objects > that do play a role in physics. So my question is, does there exist a > statement regarding such an object that a significant number of > physicists > (say at least a third of them) believe is false, yet the statement is > provable > in ZFC (assuming that ZFC is consistent)? > > I believe so -- even if 0.999...=1 isn't such a statement. To me, it > seems > hard to believe that a supermajority of physicists -- who have never > even > heard of ZFC -- would be in complete agreement with _all_ of the > theorems > of ZFC that are related to physics (including real numbers). That depends on what you mean by "agreement". Certainly no physicist actively agrees with every statement about R which ZFC proves, since no physicist knows every such statement. But I /expect/ (again, with less confidence than my previous article) that the majority of physicists don't actively disagree with any theorem of ZFC about R either - there are those statements, such as 0.9r = 1, with which most would agree, and there are those statements, such as the existence of a well-order on R, about which most would have no opinion until they bothered to find out that this is a theorem of ZFC. (There certainly are physicists who actively disagree with ZFC - there are even physicists who don't accept the law of the excluded middle. But most physicists don't pay much attention to such foundational issues at all.)
From: Jesse F. Hughes on 22 Mar 2010 20:18 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 21, 8:15 am, "J. Clarke" <jclarke.use...(a)cox.net> wrote: >> On 3/21/2010 10:13 AM, Jesse F. Hughes wrote: >> >> > Nonsense! The empty set is a set of n-tuples for *every* n. Every >> > element of the empty set is an n-tuple (no matter the value of n). >> > The empty set is a relation. >> >> I thought I had you killfiled. >> Well, back you go. > > Notice that J. Clarke here is expressing similar ideas regarding the > empty set and vacuous truth as the OP of this thread, Newberry. No, he's not. He's claiming that the empty set is not a relation. He is not claiming that vacuously true statements are neither true nor false. Why you think this is even remotely related to Newberry's pet project is beyond me. Perhaps you're convinced that there are two teams, and if J. Clarke isn't on mine, he must be on Newberry's? > Although I don't necessarily agree with the idea of killfiling someone > merely because one disagrees with them, Clarke is only doing the > same thing to Hughes that Daryl McCullough, another poster in this > thread, has done to me. And Clarke has killfiled Hughes for the exact > same reason that McCullough has killfiled me -- because both Hughes > and I have repeatedly posted statements with which Clarke and > McCullough, respectively, disagree. The so-called "cranks" and the > standard theorists have more in common with each other than either > would like to admit. You have no idea why Clarke killfiled me and neither do I (nor do I care, of course). He indicated no reason at all. > When Newberry first attacked vacuous truth in this thread, I didn't > feel that his ideas were worth defending. But now that a second > poster (Clarke) has expressed agreement with Newberry, I'm starting > to consider defending their ideas after all. Yes, because what better indication of the worth of an idea than its popularity? Really good ideas are rejected by most people but accepted by a minority. The minority has to be more than one, of course. Probably two or three is optimal. Any mathematical claim believed by only two or three people -- especially two or three people with no formal training in mathematics[1] -- is almost certainly true. Don't you agree? Mind you, Clarke has not expressed agreement, but let's not dwell on unpleasant details. > So we can attempt to come up with a theory (that isn't trivially > inconsistent) in which the intuitions of Newberry and Clarke are > provable. In all honesty, Newberry has been claiming to work on this very thing for years. He's made no headway that I can see. Nonetheless, I really think he doesn't need your help. Footnotes: [1] I'm not saying that Newberry or Clarke is untrained in mathematics. I have no idea of their backgrounds, of course. -- "[I]f I could go back, [...] I would tell myself not to step into a position where the fate of the entire world could rest in my hands. I would [avoid this] path to a nightmarish and surreal world, a topsy-turvy world, where everything changes." -- James S. Harris cannot escape his destiny.
From: MoeBlee on 22 Mar 2010 20:29 On Mar 22, 7:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Although I don't necessarily agree with the idea of killfiling someone > merely because one disagrees with them, Clarke is only doing the > same thing to Hughes that Daryl McCullough, another poster in this > thread, has done to me. And Clarke has killfiled Hughes for the exact > same reason that McCullough has killfiled me -- because both Hughes > and I have repeatedly posted statements with which Clarke and > McCullough, respectively, disagree. The so-called "cranks" and the > standard theorists have more in common with each other than either > would like to admit. "the standard theorists". You say I am one. But have I ever blocked news group messages? > When Newberry first attacked vacuous truth in this thread, I didn't > feel that his ideas were worth defending. But now that a second > poster (Clarke) has expressed agreement with Newberry, I'm starting > to consider defending their ideas after all. Why? Just to play devil's advocate as you think that is called for by the agreement of another poster? Here's a clue: There is a pretty good amount of literature already proposing relevance logic. You don't have to wait for a poster to drop in here or there. > So we can attempt to come up with a theory (that isn't trivially > inconsistent) in which the intuitions of Newberry and Clarke are > provable. You might start with the literature already available on this subject. > But in doing so, we must keep the words of Marshall Spight in mind, > and recall that any theory that's "less powerful" and "more work to > use" > than ZFC, just to avoid a "harmless" counterintuition such as vacuous > truth, is a waste of time. Oh? Me must? Funny that I don't feel any such necessity in this regard. > So we must seek out a theory that's either > at least as powerful as, or at worst as hard to use as, ZFC. If those are your criteria, then they would govern your investigations. They're not my criteria though. > (Of course, "more work to use" is highly subjective. In another > thread, > some computer scientists argued that any set theory such as ZFC is > _more_ work to use than a theory of arithmetic without _sets_.) > > So we know that in ZFC, if phi(x) is a one-place predicate of the form > Ayex (psi(y)) for some one-place predicate psi, then phi(0) must hold > by vacuous truth. There are two ways to avoid this. The first would be > to change the laws of inference of FOL in order to avoid vacuous > truth, > and the other would be to change the axioms of ZFC in order to prevent > the empty set 0 from existing. But that in itself doesn't block all instantances of vacuous implication. MoeBlee
From: Jesse F. Hughes on 22 Mar 2010 20:37
"Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: >> Although I don't necessarily agree with the idea of killfiling someone >> merely because one disagrees with them, Clarke is only doing the >> same thing to Hughes that Daryl McCullough, another poster in this >> thread, has done to me. And Clarke has killfiled Hughes for the exact >> same reason that McCullough has killfiled me -- because both Hughes >> and I have repeatedly posted statements with which Clarke and >> McCullough, respectively, disagree. The so-called "cranks" and the >> standard theorists have more in common with each other than either >> would like to admit. Two points I failed to make: First, I don't see why you think Clarke is a crank. As far as I can tell, he's simply mistaken about some basic conventions in set theory. Second, it is completely unsurprising that both cranks (ignoring the first point) and others use killfiles to avoid reading posts by people who annoy them. Certainly, I avoid posts that I don't care to read (and so do you, whether you use killfiles or not[1]). Footnotes: [1] Of course, Walker doesn't use killfiles because such 1980s technology is only available in expensive newsreaders and he can't afford one. -- "If your community has been lying about my research hoping I'd never find a way to prove that with some super dramatic discovery that's almost yanked out of the clear blue because I am a great discoverer then yeah, maybe you should worry."--James S. Harris: great discoverer |