for x,y > 7 twins(x+y) <= twins(x) + twins(y)
in [Math]
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From: Transfer Principle on 15 May 2010 01:20 On May 14, 9:34 am, James Burns <burns...(a)osu.edu> wrote: > 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). OK, I'll accept that _Burns_ is careful enough to avoid making the type of mistake that I ascribe the human nature, but I disagree that "nearly all" sci.math and sci.logic posters avoid this type of mistake. > > It's only human nature to disagree with someone more > > often once they've received a five-letter insult. > 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. > (I briefly tried to google the post I was thinking of. > Maybe you would have more luck at that.) Here's a post of mine that I found, dated April 25, 2009 at about 4AM Greenwich time: > > OK, that's accurate enough. If I can be convinced that Nguyen is > > wrong, then I still wouldn't want to "bully" him. In this case, > > Nguyen already has _many_ posters telling him how wrong he is, > > so why would I need to tell him that he's wrong? There's nothing > > that I'd be able to add to that conversation. I'd rather tread > > ground that hasn't already been trod by many others -- and that's > > why I defend so-called "cranks." > 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. Was I? This response by Jesse Hughes sounds typical: "I suppose I can understand this intuition, as far as it goes, but in practice you seem to have stronger tendencies. It seems to me that in this case, for instance, you give Nam the benefit of the doubt precisely because others were telling him he was wrong. You entered the thread with a partial understanding of the issues and immediately decided Nam was right. As it turns out, you seem to have defended Nam here longer than Nam himself." I suppose that one can call this "laughter." Also, I never claim that I, a human being, haven't made the type of mistake that I ascribe to human nature. Indeed, I've actually _admitted_ such. I only disagree with those who claim that I'm one of the _few_ posters who make the mistakes that I ascribe to human nature. > > 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. > 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 only I could find that post to which I was referring via a Google search. (Typing in "JSH factoring" as search terms, of course, returns threads in which JSH was the OP, not that one thread with a newbie OP in which JSH was merely mentioned.) > > 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? My claim is that if someone posted a line of _borderline_ rigor, a more reputable poster is more likely to be given the benefit of the doubt (i.e., have the post treated as if it were of full mathematical rigor) than one who has received a five-letter label.
From: Transfer Principle on 15 May 2010 02:07 On May 13, 5:42 pm, "christian.bau" <christian....(a)cbau.wanadoo.co.uk> wrote: > Assume there is a set S with the following properties: > 1. On the set S four operations +, -, * and / and three relations > <, = and > are defined in a way which follows the twelve axioms of > arithmetic. > 2. The set S contains the real numbers as a proper subset, that is > every real number is an element of S, but S contains at least one > element which is not a real number. > 3. The operations +, -, * and / and the relations <, = and > are > consistent with the operations of the same name in the real numbers, > so if x and y are both real numbers, then x + y using the definition > of S's "+" operator and using the definition of "+" in the real > numbers gives the same result. > The mystical 0.999... could be an example of an element of S which is > not a real number, you just need to get the definitions for +,-,* > and / right so that they work for this element as well. But now we get > to the meat. Under the assumptions in (1) to (3) prove the following: OK, I'll bite. Here are some proof attempts: > 0. Prove that the complex numbers are _not_ an example for such a set > S. Proof: Assume that S=C. In particular, ieS. So by Trichotomy (which I assume is one of the laws of "<, = and >" to which Bau is referring), we have either i=0, i<0, or i>0. Case 1). i=0 is evidently wrong, so we discard this case. Case 2). i>0. Then we multiply both sides by i to obtain i^2 > 0. Case 3). i<0. Then we multiply both sides by i to obtain i^2 > 0 (remembering to reverse the signs, of course, since we have multiplied by i which is less than zero here in case 3.) Thus i^2 > 0 in either of the remaining cases. But of course we have i^2 = -1, so we conclude -1>0, a contradiction. Therefore we conclude ~S=C. QED > 1. Prove that S has an element which is greater than every real > number. We may call this element "infinity". Proof: By Bau's 2. above, R is a proper subset of S, so that there exists an element of S\R. Let s be such an element. Now we consider the set: X = {reR | s>r} There are three cases: Case 1). X=R Then there is nothing to prove. We can just let s=infinity and we are done. Case 2). X=0 This is almost as easy as case 1. We just let -s=infinity and we are done. Case 3). X is neither 0 nor R Then let u be an element of R\X. We note that by transitivity of "<" and ">", no real number v<u can be in X either. Thus u is a lower bound for X. Thus X has a _greatest_ lower bound. (Notice that X is a set of _standard_ reals, and so we can still apply completeness to _X_ even if we can't apply it to _S_.) So let t be the greatest lower bound of X. Unfortunately, this is where I'm stuck. I suppose that I'm eventually supposed to let infinity be something like 1/(s-t) (or its opposite, if negative) -- perhaps using the Archimedean property of the _standard_ reals R to show that 1/(s-t) can't be bounded by a real (lest t not be the greatest lower bound of X after all). And since I'm stuck with Bau's Challenge 1., I'll skip directly to one question for which I wish to guess: > 5. Find a simple model for S. In other words, find a simple set S that > actually has all the properties from (1) to (3). The bit "Write down > lots of other elements of S" should really give you an idea for this. Back in one of my responses to Burns, I did mention the set of surreal numbers (due to Conway). In this case, I wish to identify with 0.999... with 1-1/omega. But as it turns out, the surreals still fail to satisfy all of MR's desiderata (or those of the ones who claim that 1>0.999...). For one thing, since Conway often describes the surreals using a binary tree, it makes much more sense to write surreals using _binary_, not _decimal_, digits. Yet MR writes 0.999... decimal, and not 0.111... binary. So the surreals already fail on this account. Even if we tried to identify 0.999... with 1-1/omega, we still have problems. For we would then want to identify 0.333... with 0.010101... binary. Now I suspect that MR would agree that: 0.333... + 0.333... + 0.333... = 0.999... (or 0.3r + 0.3r + 0.3r = 0.9r in MR notation) But surreals correspond to _games_, and one can prove that the game: 0.010101... + 0.010101... + 0.010101... + (-1) is exactly _zero_, not -1/omega or any sort of nonzero infinitesimal. Thus, surreals don't satisfy all of MR's intuitions, and so S cannot equal the proper class of surreals. Other sets also fail MR's intuitions. For example, Robinson's hyperreals fail even worse than Conway's surreals, since there is no natural infinitesimal like 1/omega (the positive infinitesimal with the earliest birthday) that corresponds to 1-1/omega (and besides, Robinson is a controversial mathematician in debate threads). I believe that David Tribble used to post about some set which he called "suprareals," but I'm not sure whether they would work either. > Now three simple questions: What is (1 + x) / 2? In the surreals, it would be 1/(omega*2). For MR, he wants x to be the _largest_ number that's less than unity, so (1+x)/2 doesn't exist. Asking for the value of (1+x)/2 in MR numbers is as futile as asking for the value of (1+2)/2 in the _natural_ numbers, for neither can exist. > What is the square root of x? In the surreals, it would be simply sqrt(x). I'm not sure whether one can attempt to apply the Taylor series for sqrt: sqrt(1+z) = 1 + z/2 - z^2/8 + z^3/16 - 5z^4/128 + ... to obtain sqrt(x) = 1 + 1/(omega*2) - 1/(omega^2*8) + 1/(omega^3*16) - ... If this fails in the surreals, then one might try it in the hyperreals instead. (But still, Robinson is a controversial mathematician.) Once again, MR wants x and 1 to be _adjacent_, so I will not attempt to find sqrt(x) in MR numbers. Asking for the value of sqrt(x) in MR numbers is as futile as asking for the value of sqrt(2) in the _natural_ numbers, for neither can exist. > What is the difference between (1 + x) / 2 and the square > root of x? From the above, I would say: 1/(omega^2*8) - 1/(omega^3*16) + ... It is positive (since we the arithmetic mean to be larger than the geometric mean).
From: Jesse F. Hughes on 15 May 2010 09:16 Transfer Principle <lwalke3(a)lausd.net> writes: > 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." Because the "old reals" are "fake mathematics", by which I think he means inconsistent, wrong or bad in some way. You ignore this part of AP's writings. You want him to be right that the AP-reals are sensible (despite the fact that fundamental properties change monthly), but you ignore the fact that he's wrong in another fundamental respect: the theory of real analysis is perfectly respectable and not "fake", whatever that means. -- "It seems to me that in wartime Americans shouldn't be attacking each other in this way on a *worldwide* forum. Then again, I know I'm an American, but I have no way of knowing that you are, which would explain a lot." --James Harris, on why Yanks should accept his proof
From: Jesse F. Hughes on 15 May 2010 09:23 Transfer Principle <lwalke3(a)lausd.net> writes: >> 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. > > Was I? This response by Jesse Hughes sounds typical: > > "I suppose I can understand this intuition, as far as it goes, but in > practice you seem to have stronger tendencies. > It seems to me that in this case, for instance, you give Nam the > benefit of the doubt precisely because others were telling him he was > wrong. You entered the thread with a partial understanding of the > issues and immediately decided Nam was right. > As it turns out, you seem to have defended Nam here longer than Nam > himself." > > I suppose that one can call this "laughter." I wouldn't call it laughter. It was sincere criticism. I don't deny I've poked fun at you on many occasions, but this excerpt was meant to be taken seriously. -- Jesse F. Hughes "It's your choice though, if you do not believe in mathematics, in the importance of its healthiness and correctness, then you can just walk away now." -- James S Harris, on the Pythagorean Oath
From: Jesse F. Hughes on 15 May 2010 09:21
Transfer Principle <lwalke3(a)lausd.net> writes: > 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_? That's a funny use of the word "rigor". Your theory was inconsistent. That was the problem. "Lacks rigor" suggests that the theory is too informal, vague, ambiguous. Your theory was perfectly clear -- clear enough that its inconsistency was provable.[1] > 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. Yes, I *have* claimed your theories are often ad hoc. > Therefore, my theories have been criticized as lacking > rigor and being too ad hoc all the time. By which you mean: they've been criticized as inconsistent and at least once they were criticized for being ad hoc.[2] Footnotes: [1] At least I take your word for it that, in this particular theory, the proof that N is finite was an inconsistency. I don't recall the details. [2] I may well have criticized more than one theory for being ad hoc, which I think *is* a pertinent criticism, if he's trying to capture natural intuitions of the various crank theories. -- "If .999... = 1 then (.999...)/1 should equal 1 let's see (.999...)/1 = .999... [Therefore] .999... still=/= 1" -- An astonishing proof by "S. Enterprize" |