'When Are Relations Neither True Nor False?
in [Math]
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From: Nam Nguyen on 11 Mar 2010 00:24 Daryl McCullough wrote: > Nam Nguyen says... >> Daryl McCullough wrote: >>> Nam Nguyen says... >>>> Daryl McCullough wrote: > >>>>> 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? > > I know exactly what the naturals are. Don't say that out loud! Only super-natural being could have a chance to know "exactly" what the naturals are! > I don't know how to answer > every question about the naturals, that's not the same thing. To know something, logically speaking, is to know everything, while not to know something, not knowing only part of it is sufficient. And I only asked you no more than 2 questions; and in fact you could pick out 1 as the only question I've asked you! For example, suppose you claim to me you know your house exactly everything about it but when I ask you "Where is you house?" you'd say "I don't know how to answer this question". Wouldn't you think your saying you know "exactly" about your house wouldn't make sense here? > > If I am holding a locked box, I may be unable to answer the question: > "What's inside the box?", but that doesn't mean that there is any > ambiguity about which box we're talking about. But to know the box _exactly_ is to know its content as well, don't you think? And I'm not kidding. The box is like a countably infinite set U of elements. And "in" that set is a collection of relations that could make U the set of naturals, or of integers. It's not U per se that your _exact_ knowledge is being asked: it's the _exact_ relations in the collection that your asked about but you don't know!
From: Transfer Principle on 11 Mar 2010 00:30 On Mar 9, 8:45 am, Rotwang <sg...(a)hotmail.co.uk> wrote: > 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". OK, the first result Google gave was a JSH post from Christmas morning, back in 2006 (1:13 AM Greenwich time). As it turns out, it was Rotwang who posted: > It's strange; despite all the maths books I've read I've never seen > "proof by invocation of imaginary alien civilization" used as a proof > method before. I guess those books just weren't "extreme" enough. And then JSH describes his planet: "I think it's an excellent way to try and get people to look at what is being done in a different way, as consider some aliens who go even further than declaring sqrt(4) as only having one solution to saying it has NO solutions, and they can build Contrary Galois Theory with exactly the same machinery human beings have built up their Galois Theory with, so that it'd look exactly the same except you'd look at it and go, hey, wait, you can just evaluate things like sqrt(4) in there. Taking the silly idea that you can declare away one solution to the square root to the limit of declaring away ALL solutions shows how dumb it is, I hope. I've proven and proven and proven the mathematical truth. Yet even publication in a mathematical journal meant nothing to people like you who can explain away just about anything. But I'll make sure that on some level you understand how stupid your position is, and realize it also ultimately rests on refusing to acknowledge that sqrt(4) is 2 or -2. Real aliens contemplating the inability of human beings to get over this hurdle and start developing number theory again would probably just conclude that our species has just kind of gone as far as it can go, and might just keep going themselves rather than deal with such an obstinate and clueless species. Oh that and how we keep destroying our own world, of course." Apparently, on JSH's planet an expression such as "sqrt(4)" denotes not just the principal square root, but both square roots as well. Now I mentioned aliens and planets only because _Nguyen_ did so, not because JSH did so. I'm not sure about Nguyen, but the point that I was trying to make is that it's only an accident of history that ZFC and not some other theory is the standard theory. Had (mathematical) history turned out differently, another theory may have become standard. (By "any theory" that doesn't contradict empirical evidence, such as a theory in which at least one infinite set exists yet all infinite sets adhere to the Pigeonhole Principle, for example -- since all known physical objects are _finite_, a theory in which _infinite_ sets behave differently from ZFC can't be refuted by empirical evidence.) In this case, when I referred to alien and planets, I don't mean that one should assume that there really is a planet on which math developed differently to the extent that "crank" claims are true -- I meant that one should _consider_the_possibility_ of such a planet. But if talk about aliens and planets are considered "crank" arguments (due to their association with JSH), then I'll drop such arguments. Nonetheless, standard theorists do still come up with obscure counterexamples, such as mentioning the relativistic Doppler effect if a "crank" were to write "the sky is blue," or if a "crank" were to refer to the Law of the Excluded Middle by saying "he is alive" must be either true or false, a standard theorist might mention Schroedinger's cat, and so on. (See my response to MoeBlee for discussion of how some standard theorists contradicted "2+2=4" merely because I was the one who wrote it.) Such responses are no better than the "proof by invocation of imaginary alien civilization" fallacy made by "cranks."
From: Transfer Principle on 11 Mar 2010 00:44 On Mar 9, 9:09 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Mar 9, 1:06 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > Maybe Nguyen's formula F refers to Cantor's theorem, or the > > existence of a bijection between N and Q, or some other common > > statement that standard theorists like to defend against the > > so-called "cranks" here because ZFC proves F. Then I'd like to > > believe that there's a theory that's every bit as good as ZFC in > > ways that matter most to standard theorists (including power and > > ease of use), yet proves ~F. This is how I interpret Nguyen's > > Principle of Symmetry. > (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. Yet if someone who's been labeled a "crank" is the one to discuss such models, then he is criticized for not adhering to the standard model. > (2) As far > as calling one model 'the standard model', if you object to lack of > neutrality, then we could just as easily refer to it as 'the blandard > model' or whatever. But what would be ridiculous would be to require > every mathematician to be as interested in EVERY possible model as > much as he or she is interested in certain particular models. People > focus on certain mathematical objects, questions, etc. for a variety > of reasons. It is not, and should not be required that mathematicians > promise to do what is not even humanly, not even finitely, possible to > do, such as study EVERY SINGLE model with as much interest and > attention as every other single model. Conversely, they shouldn't ridicule _every_single_ model just because the person who mentions it has been labeled a "crank." I'd be satisfied if standard theorists just gave _some_ constructive discussion of the alternate models. > For god's sake, we've addressed this a THOUSAND times. People are not > (ordinarily) called 'cranks' merely for proposing alternative > theories. Rather, they're called 'crank' for their irrational > argumentation, for their unwillingness to define their terminology, Actions speak louder than words. Repeating that people aren't called "cranks" for proposing alternate theories doesn't erase the numerous times I've seen people who post alternate theories called "cranks," right here on sci.math, with my own eyes. > for their circular reasoning, for their poetic/imagistic rather than > rigorous use of mathematical terminology Does "poetic" refer to their use of English, rather than formal symbolic language, to describe concepts? To me, I find "<= is a total order" much easier to read and understand than: Axyz (x<=x & ((x<=y & y<=x) -> x=x) & ((x<=y & y<=z) -> x<=z)) even if the latter is considered more "rigorous."
From: Nam Nguyen on 11 Mar 2010 02:18 MoeBlee wrote: > On Mar 10, 8:47 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Daryl McCullough wrote: >>> 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. Just so you know (and you should have), Daryl challenged me a straight forward task: "give *an* example of a nontrivial theorem in such a system" which he himself believed I wouldn't be interested in. I directly responded to him with a straight forward example and through question-style I informed him he was wrong since it's interesting to me, it being an arithmetic theorem in Q. If you interpreted that straight forward answer as an "obvious fallacy" then obviously you were incapable of comprehending a short conversation between people here. > > Damn, you're not even LISTENING - as usual. Damn, your posting is as idiotic as ever!
From: David Bernier on 11 Mar 2010 02:40
Transfer Principle wrote: [...] > Nonetheless, standard theorists do still come up with obscure > counterexamples, such as mentioning the relativistic Doppler effect if > a "crank" were to write "the sky is blue," or if a "crank" were to > refer to > the Law of the Excluded Middle by saying "he is alive" must be either > true or false, a standard theorist might mention Schroedinger's cat, > and > so on. (See my response to MoeBlee for discussion of how some > standard theorists contradicted "2+2=4" merely because I was the one > who wrote it.) Such responses are no better than the "proof by > invocation of imaginary alien civilization" fallacy made by "cranks." I think most mathematicians won't come-out and say: "By the way, I'm using the law of the excluded middle here". Then the LEM is an unstated assumption. I'm not sure what the right policy should be about unstated assumptions. But at least, in what could be called (has been called?) a "collaborative enquiry" , two people can try to figure out what each other is assuming or what is meant by a given word. Wikipedia has an article: < http://en.wikipedia.org/wiki/Unstated_assumption > , listing unstated assumption as a type of propaganda. Anyway, a collaborative enquiry can help in clearing up what each person's unstated assumptions are. David Bernier |