for x,y > 7 twins(x+y) <= twins(x) + twins(y)
in [Math]
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From: Jesse F. Hughes on 13 May 2010 21:16 Transfer Principle <lwalke3(a)lausd.net> writes: > If Hughes is trying to imply that brevity is seldom a > bad thing, and that my posts will be so much better if > I could be briefer as MR is brief, then maybe the next > time someone asks me to explain a theory, I'll give a > brief MR-like response. No, that's *not* what I was trying to imply. You said that you wish MR would be more verbose. Mitch is certainly not on my list of people I'd like to hear more from. I know this is foreign to you, but I don't value everyone's contributions equally. -- "So now, The Hammer is here, and with it, the end of days. The world will be destroyed, and then remade, as foretold. You will be lost, with your children, and then there will be others, and one day they will be tested, and will pass, but that is another story." --James S Harris gets a bit excited.
From: Jesse F. Hughes on 13 May 2010 21:18 Transfer Principle <lwalke3(a)lausd.net> writes: > To me, simply by posting "1 > 0.999...," it is clear he > _doesn't_ mean to use those symbols in the standard way > at all. It's clear that Poster Y means to use the > symbols in a nonstandard way -- a way in which 0.999... > differs by 1 by a nonzero infinitesimal. Yes, because no one is ever simply wrong. > And so it's up to me to _find_ a way to use it that's rigorous and > not ad hoc. Wow. I really can't imagine how *that* follows! -- "I'd step through arguments in such detail that it was like I was teaching basic arithmetic and some poster would come back and act like I hadn't said anything that made sense. For a while I almost started to doubt myself." -- James S. Harris, so close and yet....
From: James Burns on 14 May 2010 12:34 Transfer Principle wrote: > On May 12, 7:36 pm, Jim Burns <burns...(a)osu.edu> wrote: >>master1729 wrote: >> >>>but then something strange happens , >>>since he didnt talk about infinitesimals from >>>the beginning , he became " crank " >>>and whatever he says is wrong !! >>>people will then disagree more often >>>and respond by things like : >>>idiot (in caps) >>>or >>>there are only reals on my real number line , >>>no infinitesimals !! stupid ! >>>which is not fair. >> >>You say that "people will then disagree more often" >>once someone is considered a crank. This, >>people disagreeing with a poster >>/because he is a crank/ is unfamiliar to me. > > It's only human nature to disagree with someone more > often once they've received a five-letter insult. It may well be human nature to /expect/ to disagree with someone who you think is a "five-letter insult". /Mathematical/ arguments are not matters of opinion. /Expecting/ someone to be wrong, even someone who has raised being wrong to an artform, is not a /mathematical/ argument. I would be shocked to read such an /opinion/ passed off as a mathematical argument by any poster in sci.logic or sci.math for whom I had the least bit of respect (that is, nearly all of them). I vaguely remember you explaining that you took sides in these crank/crank-buster disputes according to which side was less popular. I would say that that is the kind of thing you say happens all the time -- correctness or incorrectness being called because of who spoke, not what was said. When you said that, didn't you realize how much you got laughed at? Not for which side you picked, but for the reason you picked it. This is my point: no matter what human nature might or might not be like, that sort of behavior -- by anyone -- is frowned on "here". (I briefly tried to google the post I was thinking of. Maybe you would have more luck at that.) > I've once seen a newbie poster make some claim about > something -- it might have been something about a > factoring method faster than the known methods. Some > of the posters thought that the method was promising > though unlikely to work. Then another poster (not the > OP, and not myself) pointed out that had the OP been > JSH instead of a newbie, writing an identical thread, > then he wouldn't have been given the time of day, and > there would have been more ad hominem than actual > considerations of the proof. What have you got here? A third poster "pointed out" that what you believed was true? Is this supposed to be evidence of some kind? I see someone agreeing with you, nothing more. (I'm curious: who is this mysterious poster?) Have you /seen/ someone disagree with a "five-letter insult" /because/ they are a "five-letter insult"? I would be interested in what you have /seen/. I suspect that what you (and your mystery poster) /imagine must be so/ are strongly influenced by what you have seen in the popular media. I am no more convinced by that than you would be by a claim that I am a good guy /because/ I wear a white hat. In particular, I disagree with that assessment of how you /imagine/ JSH would be treated. I do not see JSH around sci.math much lately, but when he was, he was given a great deal more than "the time of day". Well after the sci.math consensus was that he was a "five-letter insult", well after he was blazing new paths in crankdom, there were "standard theorists" who put a good deal more time and effort into making sense of his ideas than JSH himself did. If he returned today, I would expect him to find the same sort of analysis -- and, I would also expect the result of the analysis to be that JSH's allegedly world-changing work was either wrong, not even wrong, or (occasionally) correct but well-known for over a hundred years. This is probably why he doesn't come around here any more. Would he receive insults? Very likely. Would he pass out insults? Even more likely. It is a large part of JSH's "charm". However, I am not claiming that posters regarded as "five letter insults" will not get insults. I am just saying that they will (very likely) get serious consideration, as well. It is also very, very likely that the serious consideration will conclude that they are wrong -- but, seriously, what is the alternative? That we all (sci.math, sci.logic, and the rest of the world) agree with someone (anyone!) /because/ they /really, really/ want to be correct? > Of course, I'm trying to avoid grouping now, and so I > should _not_ group _Burns_ with such a poster. So if > _Burns_ doesn't automatically judge the mathematical > content of a post by its author, then he deserves to > be commended for not doing so. But as I said, it's > only human nature to pre-judge a post of borderline > mathematical rigor based on its author. Whoa! "A post of borderline mathematical rigor"? I certainly might judge a post of borderline mathematical rigor as being of borderline mathematical rigor. Do you have a problem with this? Would the result be incorrect? I have no way of knowing. Surely, you must know that the worst possible argument can still have a correct concusion? > (I'd like to do a Google search and actually search > for such a post -- but of course the Google search > isn't very effective, and so it would most likely be > a waste of time.) (I don't understand what kind of post you would look for.) Jim Burns
From: James Burns on 14 May 2010 13:11 Transfer Principle wrote: > On May 10, 10:09 am, James Burns <burns...(a)osu.edu> wrote: > >>[S]uppose that Poster Y asserts that 1 > 0.999... >>and it is clear he means to use those symbols in the >>standard way. > > What does Burns mean by "the standard way"? If by "the > standard way," he means the definition in classical > analysis by which it is exactly equal to 1, then Poster > Y's assertion is equivalent to "1>1" (substitution). To > me, "1>1" is a statement which even I won't defend or > attempt to find a theory for. It's not hard to find a theory where "1 > 0.999..." is true. Just re-interpret the greater-than symbol as the greater-than-or-equal-to symbol. Of course, Poster Y very likely did not intend his statement that way. However, Poster Y is out of luck. I am afraid he will have his statement re-interpeted to something only vaguely related to what he meant. As a consolation prize, though, this new statement, which he did not say, will prove he is not a crank, somehow. > To me, simply by posting "1 > 0.999...," it is clear he > _doesn't_ mean to use those symbols in the standard way > at all. It's clear that Poster Y means to use the > symbols in a nonstandard way -- a way in which 0.999... > differs by 1 by a nonzero infinitesimal. And so it's up > to me to _find_ a way to use it that's rigorous and not > ad hoc. The particular example I had in mind of a poster asserting that "1 > 0.999..." was BURT. I am not having much luck with groups.google, so I'll have to settle for the vague memory of an extended exchange with him, where he made it clear he was correcting standard usage, not creating a new system. I think that, in this case, it really is clear he _does_ mean to use those symbols in the standard way. (No, he does not /succeed/ in using them the standard way, but he is trying to do that.) >>People tell him he is mistaken. People >>go on at great length explaining why he is mistaken, >>all to no avail. Eventually, he convinces people that >>he is not merely mistaken but a crank. Suppose that >>you find another mathematical system, another >>interpretation of "1 > 0.999..." in which it is true. >>Your use of "1 > 0.999..." is, in an important sense, >>not even the same thing Poster Y is saying. > > But what exactly is Poster Y saying? I doubt that he > would be saying "1>1" -- which is what he would be saying > if he means the standard definition of 0.999.... No. You can't do this. This whole line of argument needs to be thrown out. Consider a small boy waiting to visit Santa in a department store. Unbeknownst to the boy, his father is working as the store's Santa right now. Can we say, "The boy does not know that his father is Santa"? But, the boy's father /is/ Santa. Isn't this the same as "The boy does not know that his father is his father"? Well, no, it isn't the same. > (An analogy: Poster Z writes, "The positive integral > factors of 14 are 1,2,7, and 14." If I assume that here > "integral" means "antiderivative," then the statement > makes no sense, until I find a definition, such as the > adjectival form of "integral," which does make sense, > since I assume that Poster Z is rational.) > > I start from the assumption that Poster Y is rational, > that the poster wouldn't write something as blatantly > false as "1>1" and search for a definition that does > make sense, such as "1 > 1-iota" for some nonzero > (infinitesimal) iota. Burns, on the other hand, appears > to start from the opposite assumption, namely that > Poster Y is _irrational_, when he criticizes me for > searching for an infinitesimal theory. (But I could be > wrong about Burns's assumption here.) I beg to differ. You start from the assumption that Poster Y is /correct/. I start from the assumption that a poster /may be mistaken/. I think my assumption does much less violence to their point of view than yours does. >>Why would you expect your newly introduced system to >>change anyone's mind about Poster Y's crankhood? > > Have any of my newly introduced systems changed anyone's > mind about Poster Y's "crank"-hood yet? Not yet -- but > that could be because I have yet to post any theory that > is sufficently rigorous and non-ad hoc. (The closest > that I have come is declaring that 0.999... is the > surreal 1-1/omega, but even this causes problems as > 1-1/omega doesn't work exactly as Poster Y wants it to.) And yet, none of the criticisms I have seen of your attempts to wipe away someone's crank-hood have mentioned insufficient rigor or excessive ad-hoc-ness. How do you explain this? Jim Burns
From: Transfer Principle on 15 May 2010 00:16
On May 14, 10:11 am, James Burns <burns...(a)osu.edu> wrote: > Transfer Principle wrote: > > On May 10, 10:09 am, James Burns <burns...(a)osu.edu> wrote: > > What does Burns mean by "the standard way"? If by "the > > standard way," he means the definition in classical > > analysis by which it is exactly equal to 1, then Poster > > Y's assertion is equivalent to "1>1" (substitution). To > > me, "1>1" is a statement which even I won't defend or > > attempt to find a theory for. > It's not hard to find a theory where "1 > 0.999..." > is true. Just re-interpret the greater-than symbol > as the greater-than-or-equal-to symbol. Too ad hoc. > > To me, simply by posting "1 > 0.999...," it is clear he > > _doesn't_ mean to use those symbols in the standard way > > at all. It's clear that Poster Y means to use the > > symbols in a nonstandard way -- a way in which 0.999... > > differs by 1 by a nonzero infinitesimal. And so it's up > > to me to _find_ a way to use it that's rigorous and not > > ad hoc. > The particular example I had in mind of a poster > asserting that "1 > 0.999..." was BURT. I am not having > much luck with groups.google, so I'll have to settle for > the vague memory of an extended exchange with him, > where he made it clear he was correcting standard > usage, not creating a new system. I think that, in this > case, it really is clear he _does_ mean to use those > symbols in the standard way. (No, he does not /succeed/ > in using them the standard way, but he is trying to > do that.) But what does one mean by "correcting" standard usage, in the first place? Notice that another poster, AP, does make it clear that he's creating a new system (since he explicitly names the proposed system "AP-reals" and calls the current system "Old Reals"), yet the title of his threads refer to "Correcting Math." > > But what exactly is Poster Y saying? I doubt that he > > would be saying "1>1" -- which is what he would be saying > > if he means the standard definition of 0.999.... > Consider a small boy waiting to visit Santa in a department > store. Unbeknownst to the boy, his father is working as > the store's Santa right now. Can we say, "The boy does > not know that his father is Santa"? But, the boy's > father /is/ Santa. Isn't this the same as "The boy > does not know that his father is his father"? > Well, no, it isn't the same. (This reminds me of an old paradox about knowing that Venus is the evening star, versus knowing that Venus is the morning star. A quick Google search returns discussions about "scalar" vs. "vector" logic.) I assume that the small boy in the story corresponds to MR, Santa to 0.999..., and his father to unity. Let's discuss the possibilities for MR thus far: (1) MR is discussing an alternate theory T such that T proves "1 > 0.999..." (2) MR knows that ZFC proves "1 = 0.999..." but doesn't like this result, so he talks about how 0.999... ought to be strictly less than unity without referring to any theory T in which it is provable. (3) MR doesn't know that ZFC proves "1 = 0.999..." He believes that standard theory proves "1 > 0.999..." Only in case (3) would I consider calling MR "wrong" to make any sense. In particular, case (3) entails that MR believes that "1 > 0.999..." is provable, and that those who claim that "1 = 0.999..." is provable in a standard theory need to be "corrected." Now assume Burns is correct, and let's say that from MR's subsequent posts, we can determine that case (3) is most likely, thereby justifying calling MR "wrong" and other five-letter words. Does this mean that anyone who posts in response to MR (including myself) _must_ call MR "wrong"? Does this mean that I deserve ridicule if I respond to MR in any other way than to tell him that he is wrong? If so, than I'd rather not post in a thread at all than to join five other posters and tell MR he's wrong. In fact, I'd rather call a poster wrong in a thread in which I'm the _first_ poster to call him wrong than _repeat_ that he's wrong. (And believe it or not, I've actually called posters "wrong" before, only because I happened to be the first poster in the thread to do so.) Perhaps what I ought to do is ignore posters like MR in case (3) after all, and instead focus on posters for whom case (1) or (2) are more likely. In particular, a poster is likely to be actually trying to create a new theory if one explicitly gives the proposed theory a _name_. In particular, AP calls his theory "AP-reals," tommy1729 calls his theory "TST," WM calls his theory "MathRealism," and so on. For these posters, it's clear that they _know_ their claims are refutable in the standard theory, so they wish to come up with new theories (AP-reals, TST, MathRealism) in which their claims are provable. Case (2) may be more difficult to discern, but perhaps if a poster were to write something like "I don't like ZFC because I don't like Cantor." Perhaps in this case, instead of writing that the OP is working in some alternate theory, I simply state an alternate theory (say NFU in this case) without claiming that the OP is secretly working in it -- indeed, I just post NFU without mentioning the OP at all. In that way, I don't make refutable claims about the OP, and the OP can learn that there are alternatives to ZFC. > > I start from the assumption that Poster Y is rational, > > that the poster wouldn't write something as blatantly > > false as "1>1" and search for a definition that does > > make sense, such as "1 > 1-iota" for some nonzero > > (infinitesimal) iota. Burns, on the other hand, appears > > to start from the opposite assumption, namely that > > Poster Y is _irrational_, when he criticizes me for > > searching for an infinitesimal theory. (But I could be > > wrong about Burns's assumption here.) > I beg to differ. You start from the assumption that > Poster Y is /correct/. I start from the assumption that > a poster /may be mistaken/. I think my assumption does > much less violence to their point of view than yours > does. Yet calling a poster's point of view "wrong" and calling him another five-letter word, doesn't do much "violence" to their point of view? > > Have any of my newly introduced systems changed anyone's > > mind about Poster Y's "crank"-hood yet? Not yet -- but > > that could be because I have yet to post any theory that > > is sufficently rigorous and non-ad hoc. (The closest > > that I have come is declaring that 0.999... is the > > surreal 1-1/omega, but even this causes problems as > > 1-1/omega doesn't work exactly as Poster Y wants it to.) > And yet, none of the criticisms I have seen of your > attempts to wipe away someone's crank-hood have mentioned > insufficient rigor or excessive ad-hoc-ness. How do > you explain this? Au contraire. We go right back, earlier in this thread. I had written: > > Schema 1: If phi is a one-place predicate that does not > > mention the symbol N, then all closures of: > > (phi(0) & Ax (phi(x) -> phi(xu{x}))) -> phi(N) > > are axioms. And Jesse Hughes's response was: "In particular, let phi(x) be any standard formalization of "x is finite" and we see that N is finite. Right?" And of course, by letting phi(x) be "x is infinite," we see that N is infinite. Therefore N is both finite and infinite, therefore the theory is inconsistent. So Hughes did respond that my theory lacks rigor -- since what can be less rigorous than a theory that is proved _inconsistent_? And so I eventually attempted another theory that avoided the inconsistency -- only for Hughes to imply that it's "ad hoc" when I tried to tie it to RF's theory. Therefore, my theories have been criticized as lacking rigor and being too ad hoc all the time. |