'When Are Relations Neither True Nor False?
in [Math]
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From: Jesse F. Hughes on 19 Mar 2010 22:18 Transfer Principle <lwalke3(a)lausd.net> writes: > On Mar 17, 5:27 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: >> Transfer Principle <lwal...(a)lausd.net> writes: >> > (even though almost 40% of >> > physicists believe that 0.999...<1) >> Do I get labelled as a bully if I ask for some evidence for your claim? > [...] > We notice at the top of the Metamath page, it reads: > > "Interestingly, about 40% of the people responding to a poll at > http://forum.physorg.com/index.php?showtopic=13177 disagree." An online poll at physorg.com and you concluded that 40% of physicists believe 0.999... < 1? That is just incredibly lame evidence. All one can conclude is that 40% of the visitors to physorg.com *who chose to answer* the question said that 0.999... < 1. You have no idea how many of those folks were physicists, whether the percentage who chose to answer are representative of physicists as a whole (or even physorg.com visitors) and so on. It's a voluntary poll, indicative of nothing much at all. -- Jesse F. Hughes "Dead men can't talk. Especially when they've been cremated." --- From the 1944 radio program "Adventures By Morse"
From: Rotwang on 19 Mar 2010 22:42 Transfer Principle wrote: > >> [...] > > But there's a huge difference between "ZF proves 0.999...=1" and > "physicists believe that 0.999...=1." The former isn't up to debate, > whereas the latter can be determined in a poll. > > We notice at the top of the Metamath page, it reads: > > "Interestingly, about 40% of the people responding to a poll at > http://forum.physorg.com/index.php?showtopic=13177 disagree." > > This is where I found that poll I mentioned in the last post. When we > click on it, we see that out of 183 respondents, 99 agreed with the > ZF conclusion, 66 disagreed, and 18 responded with the third, joke > option "Don't ask me or I'll hurt you." The number of respondents > (and keep in mind that this is a _physics_ forum) who disagreed with > the ZF conclusion amounts to about 36%. Take a look at sci.physics. If you were to poll the regular posters there on whether the special theory of relativity is nonsense, do you think that the proportion of respondents who voted "yes" would be a good indicator of how many actual physicists think that the special theory of relativity is nonsense? Here's a hint: posting on a physics forum is not the same thing as being a physicist. > The first poster in this thread who disagreed with 0.999...=1 wrote > the > following message: > > "Let's do it like this: > 1-.9=.1 > 1-.99=.01 > 1-.999=.001 > ... > 1-.999...999=.000...001 > At no step of this recursion are the two values equal, so unless you > can > pick some magic number at infinity, they're never precisely equal > .000...001 != .000...000 > "A number ending in 1 never exactly equals the same number ending in > zero, > unless you're willing to weight the digits down to the point where you > can > close your eyes and ignore the difference. So, yes, for real world > applications > with successly [sic] smaller and smaller weighted digits it might be > close > enough but technically they aren't identical and can't be manipulated > arbitrarily and see the same result apply. > "If we could treat them indentically [sic] then we couldn't do this: > x=10^y > 1-.9r = (1-.9)/x (where y is the number of 9 digits) = 0 > But: > x*(1-.9)/x = x*0 > 1-.9=0 > Is obviously incorrect. > Another way to see this is that it's a process continually approaching > 1, but > never reaching it. If it could actually reach 1 then no more 9s would > be > needed ... but that violates the repetition." > > The poster then goes on to argue that the usual proof of 0.999...=1 is > "circular" and thus invalid. The most appropriate response to these arguments takes the form of a hand gesture. I can't reproduce it in a text-only medium, so you'll have to make do with an onomatopoeia: mmmmmmmnggg > Of course, the standard theorists are free to dispute the validity of > the poll > that I gave here. They might point out that this online poll was too > informal > to be truly representative of those who are knowledgeable in physics. > > If the standard theorists would like to convince me that a > supermajority of > physicists do accept 0.999...=1, then I'd like to see another, perhaps > more > valid poll that demonstrates this. If desired, we can start that poll > right here > by crossposting to, say, sci.physics, and then have the posters there > state whether they believe that 0.999...=1 or not. I have a PhD in physics, and as such I've met a fair few physicists. I never bothered to poll them on whether they believed that 0.9r = 1, but I nonetheless feel confident when I say that the overwhelming majority of them do. > But if they fail to convince me that a supermajority of physics people > accept 0.999...=1, then that weakens the standard theorists' > insistence > that "crank" theories can't provide an axiomatization for the > sciences. Whether or not "standard theorists" can convince you of a true fact has no bearing whatsoever on the strength of their position.
From: Marshall on 20 Mar 2010 03:29 On Mar 19, 7:18 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > On Mar 17, 5:27 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote: > >> Transfer Principle <lwal...(a)lausd.net> writes: > >> > (even though almost 40% of > >> > physicists believe that 0.999...<1) > >> Do I get labelled as a bully if I ask for some evidence for your claim? > > [...] > > > We notice at the top of the Metamath page, it reads: > > > "Interestingly, about 40% of the people responding to a poll at > >http://forum.physorg.com/index.php?showtopic=13177disagree." > > An online poll at physorg.com and you concluded that 40% of physicists > believe 0.999... < 1? > > That is just incredibly lame evidence. All one can conclude is that > 40% of the visitors to physorg.com *who chose to answer* the question > said that 0.999... < 1. You have no idea how many of those folks were > physicists, whether the percentage who chose to answer are > representative of physicists as a whole (or even physorg.com visitors) > and so on. It's a voluntary poll, indicative of nothing much at all. You know, I mostly think TP is a nut, and folks like Moe and Jesse are top-drawer, but I have to point something out here. Yes, the evidence that TP has provided is weak, and we can poke all manner of holes in it and so forth. But crappy as it is, it still counts as evidence. What has been offered against it opinion. And the best opinions in the world do not, IMHO, rise to the level of the worst evidence. I'm calling round 1 for TP here. Marshall
From: Daryl McCullough on 20 Mar 2010 08:54 Marshall says... >> > "Interestingly, about 40% of the people responding to a poll at >> >http://forum.physorg.com/index.php?showtopic=3D13177disagree." >You know, I mostly think TP is a nut, and folks like Moe and Jesse >are top-drawer, but I have to point something out here. Yes, the >evidence that TP has provided is weak, and we can poke all manner >of holes in it and so forth. But crappy as it is, it still counts as >evidence. > >What has been offered against it opinion. > >And the best opinions in the world do not, IMHO, rise to the level >of the worst evidence. I'm calling round 1 for TP here. Well, it can't literally be true that the best opinions don't rise to the worst evidence, because you can turn opinion into evidence: Opinion: I think that 0.999... = 1 Evidence: Of those surveyed, (namely me), 99.999...% agreed that 0.999... = 1. Statistical evidence doesn't have any meaning at all without some model of what the population was, how were the representatives selected, etc. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 20 Mar 2010 12:15
Nam Nguyen wrote: > Daryl McCullough wrote: >> It doesn't seem nontrivial to me. It's a one-step proof from >> Ay ~(S(y) = 0) >> It's not the sort of fact about numbers that would lead anybody >> to care about number theory. You say you find it interesting, >> and I certainly can't know that you don't, but I don't find it >> interesting. > > In summary, your challenge would never make sense, because "interesting" > and non-technical "nontrivial" is a *subjective* terminologies, but we > were talking about _technical_ merits of the 4 principles. My guess is that you meant to ask me to give an example of a proof, in a reasoning systems under the guidelines of the 4 principles, that's is sufficiently complex (i.e. "nontrivial") but is of interest, e.g. GIT. If that's case, and you have to say if it is or isn't, then an example of a "nontrivial" metamathematical theorem would be the following modified GIT: (1) _If_ PA is consistent, then for any consistent formal system T sufficient strong to carry out basic notions of arithmetic, there's a formula G(T) which is syntactically undecidable in T but of which a certain encoded formula, say, encoded(G(T)) is provable in PA. Would (1) be an interesting theorem? To me it's not. But it's a result from _valid_ reasoning. Would GIT be an interesting theorem? It might be to you, but it's a result from _invalid_ reasoning, as stipulated by the 4 principles. My point is good reasoning is a matter of correctly conforming to certain clearly stated and defined guidelines, not a matter of relying on mere intuition that could turn out to be incorrect or could change be changed at will. *The natural numbers collectively is a mere intuition* that either we can't define precisely or we can re-define at any moment. It's not a fixed notion and hence can't be used as foundation of reasoning, as we tend to believe so post 1931. |