mm-2109 Anybody have any advice on how to integrate (a_0 + a_1 x + ... + a_n x^n)^{-1}, when the polynomial cannot be factored? > Anybody have any advice on how to integrate (a_0 + a_1 x + ... + a_n x^n)^{-1}, when the polynomial cannot be factored? > Every polynomial (of degree exceeding 1) can be factored (over the complex numbers) so (in theory) the problem does not arise. In practice, life is tough. E.g., Maple says > int(1/(x^5-x-1),x); ----- 183616 4 45904 3 21716 2 309 256 ) _R ln(x - ------ _R - ----- _R - ----- _R - --- _R - ---) / 625 625 625 625 625 ----- _R = %1 5 3 2 %1 := RootOf(2869 _Z + 160 _Z - 80 _Z + 15 _Z - 1) That is to say, if you want the indefinite integral of the reciprocal of x^5 - x - 1, you get a sum over the roots of 2869 x^5 + 160 x^3 - 80 x^2 + 15 x - 1. Enjoy. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) Nntp-Posting-Host: apps.cwi.nl > (And you picked on my Leibnitz, this is the third time you write Robinsohn. > Please correct that. Already the first time I gave you the correct name.) > > Abraham R. was born Robinsohn and changed his name into Robinson. Yes, so? Do you always refer to people with the names they originally had? He changed his name when he was 22, before he had any publication on his name. > > ... that the reals are not less and not equally many as > > compared tom the rationals that they are more numerous. He did not take > > into account the correct fact that a comparison of M.8achtigkeit between > > rational and real numbers is not at all possible because infinity is a > > quality, not a quantity. > > Eh? Because it is a quality we are not allowed to compare? It is a > quality, indeed. But within that quality we can distinguish between > quantities. > > If we do so, then it has no connection with the notion of infinity. *Your* notion of infinity. There are other notions of infinity. > I would think that the presence or not of a bijection is sufficient to > distinguish sets (it is, after all, an equivalence relation). > > It is just sufficient to declare sets countable. > I agree. It is sort of equivalence relation In what way is it *not* an equivalence relation? The relation is reflexive, commutative and transitive, so, by definition, it is an equivalence relation. > It is a quality, so you cannot compare. > Even when a bijection is not available, this is no reason to distinguish. > > Not even but because... you are within the continuum. What religion is that? That you have some preconceived ideas about continuum, but I can not fathom what you mean. > Yes, that is the spirit. I finally understand your harsh abuses of people > that think Cantor is correct. In your opinion they are of the wrong > religion, > > Not wrong religion but wrong logics. > oo+a=oo is still valid in mathematics and contradicts oo, oo+1, oo+2,... Yes, and that is also false with either Cantor's cardinals and his ordinals. one-point compactification of the reals you get the oo you wish. You have to clearly distinguish what you are talking about. Cantor's cardinals are equivalence classes, as are his ordinals. And the one-point compacitification is again something different. That the natural numbers can be embedded in all three, and the reals in the third, does not make a difference. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ ... > Discarding 'tertium non datur', every element d of D satisfies > ~~(d=0) but does not satisfy d=0; also one can still add the axiom > for R to be a field as ~(x=0) => x is invertible. ... > interesting, btw. (For mr. Blumschein too, i think.) But Eckard Blumschein adheres to the law of the excluded middle... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Nntp-Posting-Host: apps.cwi.nl ... > There is in fact another approach, using _nilpotent_ infinitesimals > instead of invertible ones. Basically you start with a coordinate > ring R, the subset D = { x in R | x^2 = 0 } and the axiom that for > every map g: D ---> R there is a unique b in R such that > g(d) = g(0) + d b for all d in D. I will look at it. Alas the book you refer to is not in our library. What I see is books by Baron The origins of the infinitesimal calculus, Dieudonn.8e Calcul infinit.8esimal, Gordon Infinitesimal analysis, Keisler Foundations of infinitesimal calculus, and a few much older books that are (I think) irrelevant. Gordon is from 2002, Keisler from 1987 or somesuch. Baron and Dieudonn.8e are from the sixties. Do you think any of these is good enough? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ <429d1ba5$1_1@newsfeed.slurp.net That point is debatable. An is not considered > incorrect exactly, just antiquated. A concession to dialect. But not all dialects are equally standard. Hear them down in Soho square, dropping h's everywhere, speaking English ANY way they like. > Perhaps it's more > common in the UK than the US. Dialects where h's get dropped are more common outside the US, yes. > I think Herc is Australian. THat's a REALLY insufficent excuse. He would've needed to be Hungarian, or maybe something not even Indo-European, to excuse his level&frequency of linguistic sloppiness. > I've heard an historical That is a special case. That must ALWAYS be allowed because not only is ahistorical a word in its own right, it is the EXACT ANTONYM of historical. Most of the time, however, an before h is just a pretentious effort to sound more learned. Which is fine if you ARE learned, but insufferable if you are not. leftover pride. Well, if his explanation was an effort to explain his mistake > after the fact, he did a good job. His technical explanation > seems intelligent and consistent with a basic understanding > of propositional logic. Whiich is exactly what you deny him of > having. Moreover, if this was indeed an effort on his part to come up with a > model in which his behavior would have been consistent, he has shown > definite ingenuity in finding such a model. Shows some original > thinking. Which is more than what I can say about your own > accomplishments in mathematics. > You were successful in getting me to respond, I'll give you that. Hats > off to you. But you're still a fool. And so is your alter-ego > Anzaurr-whats-his-name. > Well, I do agree with evertything Serg says. So, it would be irrelevant whether we are each other's altere egos or not. It's the content of message that counts not the identity of the messenger who put this content on paper... Unless, of course, we are talking about giving other people credit for their intellectual property. But we'll discuss that a bit later. So, I am more than willing to assume that Serg is I and I am he. What does this tell us? Serg (that is, I) is clearly having the upper hand in his intellectual altercation with you. Is it cruel to make fun of an idiot? > If I am an idiot and I am having an upper hand, what does it make you? Look, can't you see that you have made a serious mistake in judgement when you classified me as an idiot? People who live in glass houses, shouldn't throw stones, should they? And apparently not a student who has > made any progress towards his dissertation yet. > The only major scientific accomplishments that he so proudly and > painstakingly lists are 7 tutorials, conferences and festivals that he > succeeded in attending in person. I could not find a single > original thought there. > You haven't accomplished anything in life yet to make fun of other people who are double your age. No. Wait! That's a very bad way of putting it. That implies that accomplished mathematicians may and do make fun of others whom they consider to be idiots. But no accomplished mathematician would go out of his way to make fun of some lowly person whom they consider to be an idiot. They don't have time for that. If an idiot starts bothering an accomplished mathematician - the latter may tell him to get lost and stop wasting his time. If an idiot starts interrupting a lecture by an accomplished mathematician - the latter may tell him to get lost. If an accomplished mathematician comes to an announced lecture and finds out that he has just wasted 15 minutes of his time on a total garbage - he may express his grave displeasure. But an accomplished mathematician would NEVER waste even 5 minutes of his time to compose insults towards somebody who is an idiot. Yet you do. I bet it took you at least 1 hour of your time composing your two art poeces, as you call them. You call making fun of others art and devote hours of your time to it. While you haven't time to publish or even to write a single original work. And you didn't do it because I had provoked you. No. I hadn't written a single line to you prior to your going out of your way to insult me and to make cheap fun at my expense, had I? You clearly do not have the background to understand anything > in the directory you list (except maybe that Banach-Tarski > exposition I never got around to finishing) > Now that you mention this Banach-Tarski exposition... I did read it: http://www.math.ucdavis.edu/~suh/math stuff/banach-tarski/ and http://www.math.ucdavis.edu/~suh/math stuff/banach-tarski/banach-tarski.pdf It does seem the most legibly written object on your site. But where does it say that this is an exposition? What it says is: THE BANACH-TARSKI PARADOX: AN INTRODUCTION TO PARADOXICAL >DECOMPOSITIONS CHAN-HO SUH ABSTRACT. The Banach-Tarski Paradox states that the unit ball in R3 is >equidecomposable with two unit balls in R3. We explain what this means >and give a proof. > You then proceed with the sections on preliminary lemmas and theorems. The you get around to proving two main theorems: Banach-Tarski Paradox (weak version). The unit ball, B3, in R3 is >equidecomposable to two copies of itself. Our plan in proving the Banach-Tarski Paradox is this... > and Banach-Tarski Paradox (strong version). Any two bounded subsets of R3 > with nonempty interior are equidecomposable. > Nowhere does it tell the reader the names of the people who have discovered proofs to these theorems. Your paper cites no references whatsoever. Not a single one! Not even to the original paper by Banach and Tarski. Nor do you even say a word about the history of this topic. You do have a section titled: 4. HISTORY > But it is left completely blank. And it comes AFTER all your theorems and their proofs, not before. Now, a professional mathematician or most PH.D. students would be aware that these are not YOUR results. But the rest of the World - including most physicists, undergraduate math majors and quite a few graduate students! - would think that it is you who has found proofs of this seemingly startling paradox. Everything you found on my webpage is freely open to the public. > Yes. It definitely is. It is open to the whole World to read. It has a proceedings. Any non-mathematician, interested in finding out who you are, can access your paper in 2 seconds or less. I am not saying that one shouldn't use the WWWW to post half-baked papers and results. Far from it. I understand the importance of fast communication of ideas. But the latest version of your paper has been sitting there since 10-Feb-2005! What's your excuse? How long does it take to give basic references for your proofs? Here they are: [1] Banach, S. and Tarski, A. Sur la d.8ecomposition des ensembles de points en parties respectivement congruentes. Fund. Math. 6, 244-277, 1924. [2] Robinson, R. M. On the Decomposition of Spheres. Fund. Math. 34, 246-260, 1947. It took me exactly 37 seconds to find and copy them from the MathWorld site that you surely know of. How is it that you devoted at least 1 hour of your time to writing the two posts that make fun of an idiot but you couldn't find 37 seconds in almost 4 months to cite basic references in the paper that you have posted for everybody to see?! Is it the case of wrong priorities? Of worrying more about asserting your own importance at the expense of others than about your scientific reputation? Or was this omission intentional? To make fun of you - will make him feel superior to somebody. To > retract the insults - would make him lose this feeling of > superiority. > Do you begin to understand what your artistic endeavour has turned into and who is making fun of whom? You called your original post Irony. This thread is turning out to be much more ironic than you had envisioned, isn't? You like playing intellectual games? Well, let me give you a poker analogy. You don't have the hand to play against me. You never did. You bluffed. And I called your bluff. If you want to continue raising and re-raising - be my guest. You are just throwing good money after bad and making my pot larger and larger. You have even shown me all your hole cards. I have seen them. I have played poker against many graduate students, and yours is one of the weakest hands I've ever seen. Or you can draw a chess analogy. You saw me playing against others. You attacked me. You captured two of my pawns and thought that you were having an easy win at my expense. What you failed to notice is that you started your position without the queen and both rooks and your king is sitting on e4. To return to a cards analogy for a second, you are not playing with a full deck. And I mean it. Moreover, by taking my two pawns, you further scattered your existing pieces for me to capture them. If you choose not to resign, I will capture your knights, then all 8 of your pawns one by one. And you don't have the pieces or brainpower to prevent me. I will then slowly promote all 6 of my remaining pawns to rooks. And then I will give you the longest, most protracted and enjoyable checkmate in the history of international chess. You don't have the brains to fight me. You don't have the accomplishments to fight me. But what's most important, you don't have the sympathy of intelligent and decent readers to fall back on and to protect you from my attacks. You lost it all when you cruelly attacked me and then refused my plea for truce. I am not a cruel man. I don't want to go out of my way to make deliberate fun of decent people, even if they are certifiable morons. Even though you had twice deeply and cruelly insulted me, I still > Look, I am getting on in years. I am not as sharp as I used to be. It >> sometimes takes me awhile to notice mistakes unless others are kind >> enough to correctly (without mistakes) point out their nature to me. >> But when you approach 50, you may start >> slipping once in a while. I hope your students don't treat you the >> same way you have treated me by making fun of an idiot. > I hope you are man enough to withdraw your epithets towards me. > You think I didn't know how clumsy and outright pathetic that sounded?! It caused me great pain to write these servile platitudes. But I did it for only one reason: to make sure I was not about to seriously offend a decent or a semi-decent person. And you barged right into it! You went to extreme lengths to persuade me that you are the most cruel and indecent man I've met and that you deserve all that I already had in mind for you. You took my plea for mercy and truce as a sign of further weakness. And you enjoy nothing more than to make fun at those who are weak!: > Your embarassment will be even greater, I'm sure,... Get it now? You continue to show your ignorance here. I really sense in your statements, some kind of > effort to salvage some leftover pride. Get over it. You deserved everything you got. Here's another one of those old-fashioned sayings, Take it like a > man. Or an even better one, Walk it off. > I HAVE to teach you a lesson in civility if for no other reason than to make you think twice before deeply hurting other people who accidentally show weakness in front of you. As an experiment in artforms, I entitle this post Irony. ... like > some art, I have created artistic content ..... > You wanted irony? You got it. Another title for this could be Do you know the definition of > irony?,... or Is it cruel to make fun of an idiot?, > I think either of these titles is very appropriate. or even I knew Rudolph Carnap. Rudolph Carnap was a friend of mine...... > Neh... How 'bout: I knew Banach-Tarski. Banach-Tarski was a friend of mine ? I even know how you should start your memoirs: Once Banach-Tarski and I were sitting in a Paris cafe. And I told him: And he replied: - Don't mention it! Or better yet, how about a literary title? How about calling our joint art work Count Monte Cristo, part 2? most physicists, undergraduate math majors and quite a few graduate > students! - would think that it is you who has found proofs of this > seemingly startling paradox. Uh-oh. Somebody should call the Manilla Times. capture your knights, then all 8 of your pawns one by one. And you > don't have the pieces or brainpower to prevent me. I will then slowly > promote all 6 of my remaining pawns to rooks. And then I will give you > the longest, most protracted and enjoyable checkmate in the history of > international chess. Do you expect me to talk, Goldfinger? No, Mr. Bond. I expect you to die. ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`flY:3QYT$>dUwN^sm;MBV:F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/'>}Pc?@rl8cz]d9RXOt be much more ironic than you had envisioned, isn't? Hehe. Yes it is. You and Serg Stolarov (who has no papers as far as I can see, despite his claims) are causing me great amusement. Yes, I admit it: I stole Banach and Tarski and Hausdorff's theorems. Here it is in print. Yes, you have the upper hand here, take it easy on me. Yes, Serg, Anzaurres1 has shown highly original thinking here, with which I cannot hope to match. Boo hoo. ^OX9W/.#XpUmm`>TD2zNE-t}emfPkFR.Z5`flY:3QYT$>dUwN^sm;MBV:F7aL9x*q!` ln!l}>Y6_45$%R|P7DSrBkEph@1-;P*s~F_28vO@e4p/'>}Pc?@rl8cz]d9RXOt ProQuest dissertation database is for a Pierre Demers writing on > spiritual progress in T.S. Elliot's poetry. I've yet to find any > real math Ph.D. that I've not been able to find in this database. > I'll be gracious and consider this time an anomaly. Now you've got me worried. I can't find my own PhD in ProQuest. > Either I just suck at these database searches or I've been living a > lie. Well, my PhD was given by a philosophy dept., but that shouldn't > matter, should it? I can't seem to find any of the others that got > their PhD from CMU's philosophy dept. > It's rather odd. I can find other theses written by students graduated by your advisor (Scott) the years before and after your year (2001), but yours is missing. As for the philosophy thing, Scott, I see, is in the cs department also and these other students were in the cs department. It seems that the data from the philosophy department is missing. Well, now I learned something. I also noticed that non-American mathematicians are only sporadically listed also. Anyway, certainly the Harvard math department seems well-listed. Of course double d is now claiming to have gotten a dissertation from some other department. Maybe he means some other Harvard too. <87ekbnnqfh.fsf@phiwumbda.org> <874qcjnlb9.fsf@phiwumbda.org> If you're so smart, why don't you look up a list of Harvard PhDs who are inside the financial industry, or who have donated millions to his alma mater. A successful guy like MD would certainly appear on such a list, wouldn't he? Since you're such brilliant database cryptographers, MD will leave you to do the rest. > > REPLY: Prime numbers in this domain can only be whole numbers. Cantor > > does not consider any other numbers than whole numbers for his > > transfinity., because an essential theorem reads: Every set of > > so-and-sos has a smallest element. > > I do not know what domain you are talking about. > > The domain of transfinite numbers: ordinals and cardinals. Those are two different domains. There is not a single domain of ordinals and cardinals. But I do not understand your essential theorem. If it is about the reals, it is wrong. If it is about the cardinals or ordinals it is right. On the other hand, I can talk about primes in both the cardinals and the ordinals without regarding them as integers. So I still wish to know the origin of your surprise when I said that I was not surprised when Cantor talked about primes, while I did not know that he called them whole numbers. > In current mathematics > there is a distinction between integers and cardinals. Anybody can talk > about prime numbers in the cardinals. Why not? > > Of course, but in this domain there are no fractions possible as prime > numbers. This domain is not a single domain. And what is your problem that fractions are not possible as prime numbers? > Interesting. What do you think of the following mapping? (This question > has nothing to do with the above one but came just to my mind when > reading your last paragraph.) > map > 1 --> {1} > 2 --> {1,2} > 3 --> {1,2,3} > ... > n --> {1,2,3,...,n} > ... > > Although all elements of N are in the source, this mapping does not map > onto N, because all mappings are on finite sets. Your terminology is shaky. But it does not map onto N, because none of the left hand sides are elements of N. On the other hand, indeed, N is not in the image of the map. > Hessenberg's condition as you call it would > indeed be invalid here, because it is only valid if the source is N. > > ? The source in my example was N. response to it). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Nntp-Posting-Host: apps.cwi.nl > > of P(N) for any given map f from N to P(N), such that those elements > > are not in the map? ... > But I will do it step by step and ask you to tell me which of my steps > is exactly wrong, and why: > 1: Let be given any injective map g from N to N and a map f from N to P(N). > 2: Define as H(g) the subset of N such that {h in H(g) if g(h) not in f(h)}. > 3: This is a proper subset of N. Now define M(g) = g(H(g)). > 4: Because H(g) is a proper subset of N (for all g), M(g) is also a proper > subset of N, and so an element of P(N). > 5: You can verify that none of the M(g) are in the image of f, and so f is > not surjective. Because: > 6: if f is surjective, M(g) must be the image of some element k of N. > 7: Suppose that M(g) is the image of some element k of N. > 8: The question now is: is g(k) in M(g)? This can not be answered, and > so we have a contradition. ( g(k) in M(g) iff g(k) not in M(g).) > 9: The only assumption made was in (7), so that assumption is false, M(g) > is not the image of some element k of N. > 10: And so the map is not surjective. > > I don't see what your mapping g is good for. You asked me for a repeat of the proof I had of a big family of sets that are not the image of some particular map. Pray pay attention; that is what the mapping g is good for. > The mapping f is the same as that of Hessenberg. The principle does not > change, by introducing g. That is not a reply. What particular step above is wrong? > So none of the M(g) are in the image of f. If I am wrong, tell me > exactly which step above is wrong. And do not come up with a different > map f, because that would make step 6 wrong. > > I said already that the triple {m, M, g} cannot exist. That is no reply. What particular step is wrong? > But perhaps here > is a better argument to support my claim. I have not yet had the time > to make it watertight, but it sounds promising. The best method to find > a leak is to show it to you. That the triple cannot exist is irrelevant (and yes, it cannot exist, that is the whole purpose of the definition). But what particular step in my proof is wrong? > Define a relation (it is not a mapping, but that does not matter) from > N to P(N). I asked you to tell me what particular step in my proof is wrong. You still evade the question. Pray tell me. What particular step in my proof is wrong? > In this relation each n e N is related to (mapped to) two elements of > P(N). For instance 1 --> {1} and 1--> {2,3}. In effect, N is used > twice. But set theorists would argue that doesn't matter, because even > twice the naturals are not enough to map onto P(N). But in doing so, we > have no set of non-generators, because the same natural n can be a > generator and a non-generator (like 1 in above example). You are seriously hooked on the concept of non-generators. But I will play M = {n in N | n notin f1(n) and n notin f2(n)} is this set in your map? > Can you prove the non-surjectivity of that set? Don't try it with 1a > and 1b! It is the same number 1 which is related mapped to two > elements of P(N). Is the above sufficient? But pray tell me which step of my proof is wrong. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Nntp-Posting-Host: apps.cwi.nl > ... > > [*] A. Robinson recognized already 40 years ago: (i) Infinite > > totalities do not exist in any sense of the word (i.e., either really > > or ideally). More precisely, any mention, or purported mention, of > > infinite totalities is, literally, meaningless. (ii) Nevertheless, we > > should continue the business of mathematics 'as usual', i.e., we should > > act as if infinite totalities really existed. > > A. Robinson: However, in spite of this shattering rebuttal, the idea of > infinitely small or infinitesimal quantities seems to appeal naturally > to our intuition. At any rate, the use of infinitesimals was widespread > during the formative stages of the Differential and Integral Calculus. > As for the objection quoted above, that the distance between two distinct > real numbers cannot be infinitely small, G. W. Leibniz argued that the > theory of infinitesimals implies the introduction of ideal numbers which > might be infinitely small or infinitely large compared with the real > number but which were to posses the same properties as the latter. > ... > It is shown in this book that Leibniz' ideas can be fully vindicated and > that they lead to a novel and fruitful approach to classical Analysis and > to many other branches of mathematics. > > Chapter 1 of Non-standard Analysis. > > Even great mathematicians like Robinson can err. You came up with Robinson (but have failed to give a reference to the quote you supplied). I replied with an explicit quote of Robinson from his book. Now you say he errs. Where did he err, in the quote you supplied (without source) or in the quote I supplied (with source). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Somebody told me to derviate with respect to log(x) instead of just > derivating with respect to x. What does this mean? It was written > d F(x) > ------- > dlog(x) > df(x)/d(log x) = df(x)/dx * dx/d(log x) = df(x)/dx / d(log x)/dx = df(x)/dx / (1/x) = xdf(x)/dx If every atom in the earth is expanding and pushing against each >>other, causing a net acceleration at the surface of 32ft/sec/sec, >>the effect is indistinguishable from the force called gravity. In this theory of gravitation, how would one explain the fact the moon > appears to orbit the earth, remaining at a (relatively) fixed distance? Geometry. It's a little hard to imagine without diagrams. In fact, even with diagrams it's hard. But I'll try. Imagine two identically sized worlds moving past each other, and simultaneously expanding. In /absolute/ terms the path of each would appear to fill a cone, with the center line of the cone being the path of the center of world. But, in that both the observers on these worlds and their instruments are expanding at the same rate, nothing detectable concerning their sizes is evident. However, /relative/ to each other and from the /observers/ point of view the spheres have actually drawn closer to each other, *as if* attracted by some invisible force. To go further, you must discard Newton's first law, the idea of momentum in a straight line; i.e. in a Universe of expanding atoms, objects are effectively drawn to each other by their inherent expansion, resulting in a natural orbit effect as they pass each other in space. This curved trajectory is not a /deviation/ from some straight line momentum, but rather it is the /natural/ trajectory to expect of all expanding objects in motion past each other. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein type of acceleration, but not from the constant (or slowly decreasing) velocity > of expansion of the unvierse as a whole. I rather think it comes from > relativistic effects and their integrals of massles aprticles in toroidal > oscillations, but that's not quite fleshed out yet. You'll like that one, when > it replaces QM with something a little more concrete. Anything like this: http://www.singtech.com/Unification.html -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein > [...] Instrumentalism offers *no* > answers as to why predictions are correct, and apparently you and > Guenther believe that there can be no such answers to relativity > to your intellectual laziness. Predictions are correct because the equations model the process that's > being predicted. A mathematical abstraction is *not* the physical process. You don't have a real model until you have mapped your abstract process to reality. > Do the models represent some deeper reality? A mathematical abstraction is *not* a model of reality. Of course there is a 'deeper reality'. Do you actually believe that the underlying reality of the universe is simply a mathematical equation running on a metaphysical calculator? > Maybe, but that is a > question that the model or model maker cannot answer. A mathematical abstraction is *not* a model. conspiracy to exist that intended to cripple > modern science, then they could do no > better than to encourage all men to abandon > the time honored question of Science, Why?. Why? is ambiguous, as Albert's great skill in reading comprehension > should have made clear to him by now. Any single word removed from it's context is ambiguous, Dufus. BTW, I thought you had resolved to no longer communicate with me. Does replying to my posts and yet referring to me in third person allow you to violate your resolve? > His Why? is What cause is there > for this effect? But when causes are presented to him (eg, the > curvature of space causes the effect we call gravitational attraction) > he rejects them. Not out of hand. I asked for and received no explanation for how a curved space could in any way be a force of attraction, I received no meaningful answer. > When he is asked to provide some indication of what he > would accept as an answer to his Why, he weasels out of it. Please cite my posts wherein you think I 'weasel out of it'. > Occasionally, the mask slips, though: Albert wants a religious answer, > but he wants Science (note the capital) to provide it. No. I want an scientific answer, not an instrumentalist mathematicians answer. > The reason he > wants Science and not Religion to provide the answer no doubt is an > effect of his personal history, but it's pointless and presumptuous to > speculate on what that history might be. Two separate domains. I expect scientific answers to scientific questions and I expect religious answers to religious questions. What I abhor is pseudo-scientists attempting to claim the inscrutability of God to hide the fact that they have no answers, but only an abstract equation. Science and Nature are about cause and effect. You seem to be claiming that Science has finally encountered the scientifically unexplainable, the face of God himself, and therefore to continue asking 'Why?' is sacrilege. What an odd position for those who claim to be seekers after knowledge. You are a Fundamentalist with nothing left except: But the Holy Scriptures of Mathematics say in Quantum Mechanics 3:11... -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein Instrumentalism offers *no* >>answers as to why predictions are correct, and apparently you and >>Guenther believe that there can be no such answers to relativity >>to your intellectual laziness. Yes; let us thank God that chemists, for example, know the ultimate > answer to why water is of the form H20: because oxygen has 8 > electrons and hydrogen has 1 electron. No laziness there - just a solid > answer. Why can't physics work like that? Indeed, why not? Chemistry only got that far by asking Why is the composition of water 2 parts hydrogen and 1 part oxygen? Keep asking 'Why?' and you'll find even more answers. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein <1Sike.708$dZ5.203965@news20.bellglobal.com> <8i_me.48909$gc6.39245@okepread04> Instrumentalism offers *no* >>answers as to why predictions are correct, and apparently you and >>Guenther believe that there can be no such answers to relativity >>to your intellectual laziness. Yes; let us thank God that chemists, for example, know the ultimate > answer to why water is of the form H20: because oxygen has 8 > electrons and hydrogen has 1 electron. No laziness there - just a solid > answer. Why can't physics work like that? Indeed, why not? Chemistry only got that far by asking Why is > the composition of water 2 parts hydrogen and 1 part oxygen? > Keep asking 'Why?' and you'll find even more answers. > Yes, let's keep getting answers, which will lead us to even more questions! The fun goes on forever; just keep asking why?. > -- > If a lion could speak, we would not understand > him. > -- Ludwig Wittgenstein -- Oh it's the Big World of Little Adam. The finest planet of Little Adam. We're casting off. We're blasting off. It's the Big World of Little Adam. -- 60's TV Show theme >Instrumentalism offers *no* >>answers as to why predictions are correct, and apparently you and >>Guenther believe that there can be no such answers to relativity >>to your intellectual laziness. Yes; let us thank God that chemists, for example, know the ultimate >answer to why water is of the form H20: because oxygen has 8 >electrons and hydrogen has 1 electron. No laziness there - just a solid >answer. Why can't physics work like that? >>Indeed, why not? Chemistry only got that far by asking Why is >>the composition of water 2 parts hydrogen and 1 part oxygen? >>Keep asking 'Why?' and you'll find even more answers. Yes, let's keep getting answers, which will lead us to even more > questions! The fun goes on forever; just keep asking why?. You sound like you are tired of it all, Chas. Have you considered joining the Southern Baptists? -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein >So, according to you the scientific method is nothing more than >>common sense? Where exactly am I supposed to have said this? >>In your statement, I also cannot see how the scientific method >>can lead to answers to 'why' questions in areas where science has >>reached the *limits* *of* *what* *we* *can* *aprehend* *in* >>are forced to use models and metaphors and therefore in my >>opinion, we loose any hope of finding a satisfactory 'why'. But >>this does not render the scientific method useless at all. >>Recognizing its limits empowers us to make it a much more usefull >>tool in our quest for knowledge. 1.- >So, according to you common sense is the upper limit of human >>reasoning ability? Where exactly am I supposed to have said this? >>In your statement, I also cannot see how the scientific method >>can lead to answers to 'why' questions in areas where science has >>reached the *limits* *of* *what* *we* *can* *aprehend* *in* >>are forced to use models and metaphors and therefore in my >>opinion, we loose any hope of finding a satisfactory 'why'. But >>this does not render the scientific method useless at >>all.Recognizing its limits empowers us to make it a much more >>usefull tool in our quest for knowledge. 2.- >as system not based on physical objects it didn't seem to require the >use of common sense but of more abstract ways of reasoning. >>One can surely reason about abstractions, but I have never heard >>of abstract ways of reasoning. How does reasoning about >>abstractions differ from reasoning in general? Reasoning on an abstract plane frees us from the constrains imposed by > the common sensical necessity of aknowledging physical reality. I see. You give your imagination free reign. That is certainly commendable when speculating and doing gedanken. However, you have been consistently commenting on Science. What techniques of reason do you use when you attempt to map your speculations to reality;? Or, are you simply content to offer up your fictions as reality? >>Of course, there are questions for which materialism >>and reductionism cannot find answers. But if something is truly >>predictable, then the answers to why are there somewhere. You are going purely on faith here. >>Nope. The 'common sense' you left behind when you entered the >>Brave New World tells me that a phenomenon cannot be predicted if >>it not an effect having a cause that you control. It seems you mistyped something in the sentence above and I cannot >understand it. >>Why do you assume that your lack of understanding is my fault? >>Perhaps you can be more specific concerning what in that sentence >>you don't understand. Specifically the part that says: if it [is] not an effect having a cause > that you control. I have inserted the proper missing word, though I would contend that it could have been easily inferred had your reasoning been more connected to reality. >>You are the one who is introducing >>a new principle of scientific investigation that does not ask >>'why'. Where did I do this? >>In your statement, I also cannot see how the scientific method >>can lead to answers to 'why' questions in areas where science has >>reached the *limits* *of* *what* *we* *can* *aprehend* *in* >>are forced to use models and metaphors and therefore in my >>opinion, we loose any hope of finding a satisfactory 'why'. But >>this does not render the scientific method useless at >>all.Recognizing its limits empowers us to make it a much more >>usefull tool in our quest for knowledge. 3.- >You are the one claiming that the time honored scientific >>method must be discarded regarding topics such as relativity and >>QM. where did I do this? >>In your statement, I also cannot see how the scientific method >>can lead to answers to 'why' questions in areas where science has >>reached the *limits* *of* *what* *we* *can* *aprehend* *in* >>are forced to use models and metaphors and therefore in my >>opinion, we loose any hope of finding a satisfactory 'why'. But >>this does not render the scientific method useless at all. >>Recognizing its limits empowers us to make it a much more usefull >>tool in our quest for knowledge. 4.- > Amazing wagner, you attribute me four different statements I did not > make based on the same single paragraph. You do seem to suffer from a > rather strong tendency to see things implied where there are none. No > wonder you were reading all sorts of things into Hilbert's axioms. Yes, that was the paragraph upon which you attempted to base all of your subsequent arguments. Perhaps you would care to rewrite it so it more reflects your intent. >>I really tire of you making statements and then denying their >>implications. In view of the imperative conviction with which you seem to find these > implications in my statements, I will not waste time in arguing their > denial. Instead I state that I intended my statements free of those > implications. Since I am the author of those statements, I hope you > will concede and believe me that I did not intend my them to carry such > implications. Gladly. However, since you show a remarkably agility in English, I must assume that your inability to state more clearly your intent is due to an inability to reason apart from abstractions. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein To everyone here: There are two ways in which the question why? may be used: 1) To ask for the reason why this happens in this circumstance, or does not happen, as a matter of the process that produces the result we are observing. This is equivalent to asking What causes this to happen?. 2) To ask for what purpose this thing occurs. Science cannot answer number 2, but the history of science has been nothing more than attempts to answer 1. When we talk about explanations, it is 1 we mean, not 2. > To everyone here: There are two ways in which the question why? may be used: 1) To ask for the reason why this happens in this circumstance, or does not > happen, as a matter of the process that produces the result we are > observing. This is equivalent to asking What causes this to happen?. 2) To ask for what purpose this thing occurs. Science cannot answer number 2, but the history of science has been nothing > more than attempts to answer 1. When we talk about explanations, it is 1 we > mean, not 2. > arrogant and believe us to be so stupid as to fall for the deception. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein > To everyone here: There are two ways in which the question why? may be used: 1) To ask for the reason why this happens in this circumstance, or does not > happen, as a matter of the process that produces the result we are > observing. This is equivalent to asking What causes this to happen?. 2) To ask for what purpose this thing occurs. Science cannot answer number 2, but the history of science has been nothing > more than attempts to answer 1. When we talk about explanations, it is 1 we > mean, not 2. Standard characterisation, but incomplete IMO. Two other meanings are: a) By what chain of reasoning do you arrive at this conclusion? (If the chain of reasoning is a model of a process, then it's isomorphic to meaning 1) above.) b) How did this come about? (This may or may not include an implicit request for causes - depends on context.) If purpose is interpreted as a function in larger context, then 2) is also a scientific question. Eg, Why does the Arctic hare's coat change colour in winter? has two answers: a) because it reduces the odds the hare will be seen and eaten by a predator; and b) because changes in temperature (?) or light (?) trigger biochemical changes in the hair follicles. These two answers respond to two different meanings of Why. Or so it seems to me. Why? Because that's how I think about it. :-) Wolf's Principle: That there must be a cause for every observable cause for every observable phenomenon is a superstition. But that principle not original with me, so I claim credit only for the phrasing. Hah! cause for every observable phenomenon is a superstition. Well, I have just marked your post as 'Important'. I expect to refer to it many times in the future. I am sure that all of the great scientists of the past are rolling over in their graves moaning. The Great Enlightenment that began in the Renaissance is over, and it happened not with a bang, but a whimper. A new religion is born and a new dark age begins. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein > There is something MISSING from the theory -- the explanation. We don't need no steeeenking explantions. We need good predictions. You cannot get good predictions WITHOUT explanations. Explanations SAY why something is the case in one situation and not another. All you have is a catalog of experiences if you have no explanations. > You cannot get good predictions WITHOUT explanations. Explanations SAY why > something is the case in one situation and not another. All you have is a > catalog of experiences if you have no explanations. No they don't. It is the hypothetical causes behind the equations that explain (in a manner of speaking) the effects observed. And sometime there are not even causes. Quantum Theory does not postulate causes. It only computes odds and eigenvalues. Give a causal explanation for the Stern Gerlach effect, if you can. Bob Kolker > You cannot get good predictions WITHOUT explanations. Explanations >> SAY why >> something is the case in one situation and not another. All you have >> is a >> catalog of experiences if you have no explanations. > No they don't. It is the hypothetical causes behind the equations that > explain (in a manner of speaking) the effects observed. And sometime > there are not even causes. Quantum Theory does not postulate causes. It > only computes odds and eigenvalues. Give a causal explanation for the Stern Gerlach effect, if you can. Why? You have never concerned yourself with causal explanations before. I am getting with the program now, Bob. I'm learning how human sacrifice is efficacious for controlling the weather. Aren't you proud? -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein But if it has no theory explaining why things act the way they do I fail to > see how it is more useful than someone simply deriving the same equations by > observing themselves with no theory. An accidental theory may be worse, in > fact, since it would lead us astray since we would not try certain things > because the theory would tell us they can't be done. So why has technology improved? Purely accidental. I think not. And what makes you say that it isn't? >Could it > be that Nature works the way our theories says it does? So far, nature is working in a way that comports with the theory we are using. It does not mean that nature works this way, and the less the theory can explain -- and the more it relies on directly deriving its predictions from experiences -- the less useful it is since it comes close to being nothing more than a catalog of experiences and not an explanatory theory or even a theory at all. Why? is ambiguous, There are two distinct meanings to why? One meaning is for what reason or from what cause. The second is for what purpose. The second can be answered, but only in the context os goals set by sentient beings. The first can be answered hypothetically but never ultimately. Bob Kolker Yes; let us thank God that chemists, for example, know the ultimate > answer to why water is of the form H20: because oxygen has 8 > electrons and hydrogen has 1 electron. No laziness there - just a solid > answer. Why can't physics work like that? Not ultimate. Why do electrons, protons and neutrons exist? What is their cause? Why is there an atom with 8 protons in its nucleus? Etc, etc. Ultimate reasons cannot exist. Why? Because if something is given as an ultimate reason, the question as to why it is an ultimate reason arises? To say there are ultimate reasons is to say that reality can be -deduced- from some set of a priori necessary propostions. If so, what are they? Bob Kolker <1Sike.708$dZ5.203965@news20.bellglobal.com> <8i_me.48909$gc6.39245@okepread04> answer to why water is of the form H20: because oxygen has 8 > electrons and hydrogen has 1 electron. No laziness there - just a solid > answer. Why can't physics work like that? Not ultimate. Most of us are now wondering *why* Kolker's irony detector is broken. An irony detector that even misses unmistakable scare quotes? Definitely broken. <8i_me.48909$gc6.39245@okepread04> answer to why water is of the form H20: because oxygen has 8 > electrons and hydrogen has 1 electron. No laziness there - just a solid > answer. Why can't physics work like that? Not ultimate. > Most of us are now wondering *why* Kolker's irony detector > is broken. An irony detector that even misses unmistakable > scare quotes? Definitely broken. Most likely, it was dropped or bumped. Rather fragile things, irony detectors; and devilishly hard to keep calibrated. Especially for an American, such as myself. > Most of us are now wondering *why* Kolker's irony detector > is broken. An irony detector that even misses unmistakable > scare quotes? Definitely broken. I am literal minded. Mark the irony so, and I will get it. Bob Kolker > <8i_me.48909$gc6.39245@okepread04> is broken. An irony detector that even misses unmistakable > scare quotes? Definitely broken. I am literal minded. Mark the irony so, and I will get it. Btw, the real reason I posted was that I liked the (definitely ironic) answer that cbrown gave and wanted everyone else to appreciate its aptness. I shouldn't have phrased it as a putdown of you personally, particularly since I do agree with most of your positions on the philosophy of science. > So wrong methods are those which can be perceived to be wrong and >>right methods are those which can be perceived to be right? Not much >>help but I'll pass the information along to the proper authorities. I'd rather you didn't. I gave you my answer under the assumption that > we were having a private conversation here. You're on usenet, Dufus. LOL. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein >>On 31 May 2005 15:52:52 -0700, guenther vonKnakspot I've noted that you and Kolker both attempt to disallow the word >'why', claiming that it is properly asked only of theology. But >that is false. 'Why' is very context dependent. Such questions >as Why do elements combine in specific proportions? led to the >discovery of atoms and construction of the table of elements, a >major first step toward all sub-atomic 'whys'. Instrumentalists >hate the word 'why' because it is a question that they cannot >answer concerning their theories. >>Salient point, Albert. I think I might have said Instrumentalists >>hate the word 'why' because it is a question that their instruments >>cannot answer for them. Maybe instrumentalists do not hate the word 'why' at all but chose not >to waste time looking for answers through the wrong methods. Maybe >instrumentalists prefer to approach different questions through >different paths according to the circumstances. >>OK, Guenther, this is what I would call a substantive point. But there >>are two problems: first, what are wrong methods and what are right >>methods and what makes them wrong or right, and, second, what other > Wrong methods are those which can be perceived to be incapable of > leading to success. Right methods are those which can be perceived to > lead to success. Brilliant! Absolutely Brilliant! In one fell swoop you have solved the problem of the ages. Has this been published yet? >>approaches to different questions through different paths can there be >>other than explaining why things in terms of one another? You needn't Not explaining things at all is one of many such alternatives. Farmers > in old Aegypt did not need an explanation of why the Nile flooded > regularly. They certainly would have benefited them to know why the Nile failed to flood on occasion, especially when such failures took place over protracted periods and the consequent famine brought down whole kingdoms. > It sufficed that they knew it would flood regularly to > provide the nourishing basis of a great amazing and long lived culture. Aside, of course for the occasional collapse of that culture due to prolonged famine. Is this any indication of your general knowledge of history? LOL. God save us from instrumentalists. -- If a lion could speak, we would not understand him. -- Ludwig Wittgenstein > If you have more than one theory that has the same predictive > power, they are equally good as theories. Why do you object > to more than one theory being good? Such was the case with > Einstein and Newton at one point, and other authors late in > the 19th century and early in the 20th were attempting to > find theories to explain the new phenomena they were > seeing. It was just such papers that got Einstein interested > in modern physics in the first place. It isn't that more than one theory can be good, it's that ANY theory that > can explain the same observations is equally as good. Yes. Then we make a choice on aesthetic grounds. I have offered > one of those grounds as wider predictive power. If theory > A predicts everything theory B does, but it also predicts > things predicted by theory C which theory B does not > apply to, then it is better. But it is not the same sort of better as we were talking about when we called the theory good. This is important to remember since it seems like you might be backtracking later in this post ... > If the theories have > consequences beyond what can be observed, then we have an issue > of which one to accept and act on. Yes, and there is no scientific basis on which to make that > choice. I would agree with this, and think I've said it before. But this means that if a choice is made strictly on aesthetics it cannot be treated as a choice made on strong evidentiary grounds ... meaning that others should feel free to make the opposite choice should they prefer the aesthetics of that choice for some reason. Of course I can. That's precisely where Occam applies. > Why do you think choose the simpler of two equally > good theories is something I can't do? My point was that you couldn't claim that the theory was not good using > Occam's Razor. I can't? If Occam's Razor tells me that simpler is better, then > why can't I use not as simple to mean not as good? You can't mean it that way -- ie you can't mean good in the same way as we were using it in the entire rest of the post -- without backtracking and claiming that it is not the case that all theories that explain the same evidence are equally good. So make certain you are not equivocating on the term good here. > I didn't think you'd accept a gnome theory, so either you are true to > your convictions or deluding yourself [grin]. (I suspect that if you thought I was serious about the theory, you would > insist I was wrong, since that is what you have done in other areas. But I > could be wrong about that.) You are wrong about that. Most people are when they try to > apply telepathic models to me to peer into my soul. You make this accusation a lot, but not that my comment was qualified by my impression that you did so in other areas, and thus no telepathy is required. Relativity and QM have made a lot of predictions that people, > including some of their founders, found ridiculous. So far > those predictions have borne out. But until they do, I would suggest that the theory should not be considered > as good as long as the arguments for why they are ridiculous are good. That's reasonable. I don't know about the arguments in the > early 20th century. The arguments now seem to be that's > absurd because the universe doesn't act that way in my > experience. Since these are theories that do not differ > from classical theories in the realms in which most of us > experience things (low gravity, low relative velocities), > then this is not a good argument for why it's ridiculous. If that was all it was, it would not only be not a good argument, it would be a ridiculous one. There must be contradictions in how we experience things that the theory has yet to explain. I like to use this example to categorize such arguments: Imagine that all the physics worked the same way, but that we saw little gnomes pushing all objects that roll downhill down the hill. Someone may be totally correct in asserting that it really just is gravity that does it, but someone should be totally justified in claiming that they will be skeptical of that theory if it cannot explain WHY we see the gnome even though the gnome is not involved. It is my belief that the arguments you are referring to are of that type: if the theory is correct, then why do I see this and not that? > I > would also suggest that this is what people generally do. I don't disagree. But I don't think the anti-QM and anti- > relativity arguments you see around this newsgroup, nearly > a century after the experimental record bore out the most > startling predictions, count as good arguments against. > The time to put forth good arguments is BEFORE the > confirming evidence comes in, not after. There's always time for Jello ... I mean, good arguments [grin]. The argument against QM's potential causelessness is still good, since as far as I know no experiment has shown that there is no cause, just that we can't know what it is. The argument that it isn't dependent on the mind of the observer is also still good (although not all QM requires this argument). For example, if an empirical theory ever implied solipsism, it might > not be one that would be considered good (since that might refute > empiricism). Some views of QM skirt dangerously close to doing that. This is too vague for me. Do you know what solipsism is? Yes, and I can't imagine what you mean by an empirical theory > implying solipsism. An empirical theory deals with observables. > Solipsism deals with inherently non-observable things, inside > your head. Ah, but solipsism claims to explain empirical observations ... At any rate, I expanded on it below. The point of solipsism is that it claims > that all of our experiences are mentally generated. The strength of this > form of skepticism is that it can explain and incorporate all of our sensory > experiences. So experimentally there would indeed be no difference between > the two theories. I was exposed to such philosophical nonsense in freshman > year of college. It was interesting and amusing then, but > I just find it tiresome now. Some views of QM insist that the differences are > dependent on the observer's mind. Cite? I'm not completely disagreeing, but I'm not aware of > a view of QM in which observer has to be a sentient being. > Anything which interacts with a system, which has a physical > effect from it, has made an observation of that system. know about it, I'll try to find it (I might have tossed it) but it is immaterial whether or not the view is used or not, since I was not criticizing QM here. If this was true, then it would imply > solipsism since it would make physical phenomena mind dependent. Certainly. We can't > prove one either way, We can if it implies two different people should see two > different things from the same instruments at the same > time. Um, solipsism can actually explain that. Thus it is a hugely vexing problem for anyone doing epistemology. but from a scientific point of view implying solipsism > would tend to invalidate empiricism and empirical science ... and so that > theory should not be considered as good since it would tend to refute the > means used to produce it. If there is a view of QM as you describe it, I will agree it > seems a little silly. And thus it might not be considered as good, even though it explains all the evidence? That's my entire point. <0t1je.41701$gc6.16008@okepread04> <7%%je.73$Ot6.38447@news20.bellglobal.com> <2Xike.711$dZ5.204669@news20.bellglobal.com> <6Etle.9322$dZ5.798965@news20.bellglobal.com> power, they are equally good as theories. Why do you object > to more than one theory being good? Such was the case with > Einstein and Newton at one point, and other authors late in > the 19th century and early in the 20th were attempting to > find theories to explain the new phenomena they were > seeing. It was just such papers that got Einstein interested > in modern physics in the first place. It isn't that more than one theory can be good, it's that ANY theory > that > can explain the same observations is equally as good. Yes. Then we make a choice on aesthetic grounds. I have offered > one of those grounds as wider predictive power. If theory > A predicts everything theory B does, but it also predicts > things predicted by theory C which theory B does not > apply to, then it is better. But it is not the same sort of better as we were talking about when we > called the theory good. I think you mean that I have said the only test for goodness is predictive power. > This is important to remember since it seems like > you might be backtracking later in this post ... Nope, not backtracking, but I realize I'm using two different sets of criteria. Under one, I say two theories are equally good if they have equal predictive power. Yet I'm also saying that of two equally good theories, one can be better. Am I contradicting myself? No, because I'm still saying they're equally good, but the choice of better is made on pure aesthetic, non-scientific grounds. Occam's Razor says simpler is better. That's an aesthetic choice. There's no scientific reason to prefer the simpler one. I'm also saying that historically we've preferred theories with broader application to ones with narrower ones. Again, that's an aesthetic preference, and ultimately probably an appeal to Occam as well, since a single theory for two experiments is simpler than two separate and unconnected theories. > If the theories have > consequences beyond what can be observed, then we have an issue > of which one to accept and act on. Yes, and there is no scientific basis on which to make that > choice. I would agree with this, and think I've said it before. But this means that > if a choice is made strictly on aesthetics it cannot be treated as a choice > made on strong evidentiary grounds That's correct. That is my position. How can you have strong evidentiary grounds in the absence of evidence? If they have equal predictive power, that means precisely that there is no evidentiary difference. Zero. > ... meaning that others should feel free > to make the opposite choice should they prefer the aesthetics of that choice > for some reason. That is also correct. There is no grounds on which to tell those people they are wrong. > Of course I can. That's precisely where Occam applies. > Why do you think choose the simpler of two equally > good theories is something I can't do? My point was that you couldn't claim that the theory was not good using > Occam's Razor. I can't? If Occam's Razor tells me that simpler is better, then > why can't I use not as simple to mean not as good? You can't mean it that way -- ie you can't mean good in the same way as we > were using it in the entire rest of the post Right. Let's use a different adjective. How about preferable? The aesthetic, non-scientific principle called Occam's Razor is prefer the simpler one. > -- without backtracking and > claiming that it is not the case that all theories that explain the same > evidence are equally good. So make certain you are not equivocating on the > term good here. (I suspect that if you thought I was serious about the theory, you would > insist I was wrong, since that is what you have done in other areas. > But I > could be wrong about that.) You are wrong about that. Most people are when they try to > apply telepathic models to me to peer into my soul. You make this accusation a lot, but not that my comment was qualified by my > impression that you did so in other areas, and thus no telepathy is > required. My standard response to accusations of hypocrisy, lying or dishonesty is to request a citation. I have no idea what you're referring to. > Relativity and QM have made a lot of predictions that people, > including some of their founders, found ridiculous. So far > those predictions have borne out. But until they do, I would suggest that the theory should not be > considered > as good as long as the arguments for why they are ridiculous are good. That's reasonable. I don't know about the arguments in the > early 20th century. The arguments now seem to be that's > absurd because the universe doesn't act that way in my > experience. Since these are theories that do not differ > from classical theories in the realms in which most of us > experience things (low gravity, low relative velocities), > then this is not a good argument for why it's ridiculous. If that was all it was, it would not only be not a good argument, it would > be a ridiculous one. There must be contradictions in how we experience > things that the theory has yet to explain. I like to use this example to categorize such arguments: Imagine that all > the physics worked the same way, but that we saw little gnomes pushing all > objects that roll downhill down the hill. Someone may be totally correct in > asserting that it really just is gravity that does it, Saying gravity does it is not really correct. There's not really a separate actor called gravity. Gravity is the description of how things roll downhill. I wouldn't say gravity does it, I'd say that's gravity. > but someone should > be totally justified in claiming that they will be skeptical of that theory What theory? How can you be skeptical of a theory that things are rolling downhill in light of the evidence that they are rolling downhill? What is the theory other than that? > if it cannot explain WHY we see the gnome even though the gnome is not > involved. it to predict precisely how fast things roll down different hills, somebody would be justifying is being skeptical of my equation? On what grounds? > It is my belief that the arguments you are referring to are of > that type: if the theory is correct, then why do I see this and not that? You've lost me. > The time to put forth good arguments is BEFORE the > confirming evidence comes in, not after. There's always time for Jello ... I mean, good arguments [grin]. The argument against QM's potential causelessness is still good, since as > far as I know no experiment has shown that there is no cause, I can't conceive of what such an experiment would be like. Not saying there's no possible experiment, just that I can't conceive of one (by the way, another hallmark of the anti-relativity crowd is to equate I can't understand with nobody can understand and I can't conceive with it's impossible). In fact, I have no idea what you mean by causelessness. Cause of what? What is QM's potential causelessness? > Do you know what solipsism is? Yes, and I can't imagine what you mean by an empirical theory > implying solipsism. An empirical theory deals with observables. > Solipsism deals with inherently non-observable things, inside > your head. Ah, but solipsism claims to explain empirical observations ... At any rate, I expanded on it below. > Some views of QM insist that the differences are > dependent on the observer's mind. Cite? I'm not completely disagreeing, but I'm not aware of > a view of QM in which observer has to be a sentient being. > Anything which interacts with a system, which has a physical > effect from it, has made an observation of that system. know about it, I'll try to find it (I might have tossed it) I'd like to hear a relevant quote and an author's name, if it wouldn't be too much trouble. > but it is > immaterial whether or not the view is used or not, Well, existence of such views is material to the claim that such views exist. > We can if it implies two different people should see two > different things from the same instruments at the same > time. Um, solipsism can actually explain that. Thus it is a hugely vexing problem > for anyone doing epistemology. Epistemology is one of those words which causes me to fall asleep almost inst... I'm sorry, what were you saying? > but from a scientific point of view implying solipsism > would tend to invalidate empiricism and empirical science ... and so > that > theory should not be considered as good since it would tend to refute > the > means used to produce it. If there is a view of QM as you describe it, I will agree it > seems a little silly. And thus it might not be considered as good, even though it explains all the > evidence? That's my entire point. Yes. Silly theories can be as good as the more commonly accepted theories. Many years ago I read a pamphlet which, as I recall, gave a complete explanation of Newtonian gravity in terms of universal continuous expansion of all objects. It was completely self-consistent. The fact that we have aesthetic reasons for preferring certain theories over other equally good ones, does not mean such choices will always ultimately be the right ones. As you said, there's no evidentiary grounds on which to make those choices. - Randy <5sJke.58092$sy6.12481@lakeread04> <%E4le.61445$sy6.2239@lakeread04> I agree, even if most outrageous ideas are bunk, even some of the bunk can get >people thinking in new ways that lead to something. >>For example, just for the sake of poking at a beehive with a >>stick: consider Einstein's equivalency principle along with the >>expansion of the Universe. Is it only space between large masses >>also? What are the consequences of sub-atomic space expanding? Bzzz Bzzzzzz >That is an interesting question. One would imagine that the expansion of the >think so. Could it be photon emission/electron vibration? A little more likely, >but maybe still not. Any ideas on what effect you would expect? >>How about the radius of the earth expanding at about 32ft/sec/sec? Ummm, well....it seems to be the same size as it was. Yes, it /seems/ to be. Perhaps that's why it's not noticed. Of > course every atom in the universe would be expanding at the same > rate, so we and our measuring instruments have been expanding also. I don't see that that can > be an effect of the expansion of space. The same size earth, made out of > styrofoam, would have a lower G force. Only under conventional theory. What is the G force on you in a > styrofoam rocket ship accelerating at 32ft/sec/sec? It's a matter of the mass, not the size, > of the object, that causes gravitation. You just restated conventional theory to refute an unconventional > theory. So how do you know that your theory is the correct one? > We estimate the mass of large objects like planets, based on an > unproven heuristic based on size. We then /assume/ that gravity > is somehow /caused/ by the simple presence of mass, when all that > we have actually observed is that there is some relationship > between the two. Besides, 32 ft/sec^2 is a measure of > acceleration, not velocity, so a constant expansion wouldn't cause such an > effect, as far as I can see. Of course not, that why I referenced Einstein's Equivalence > principle: Acceleration due to an applied external force is > indistinguishable from an acceleration due to gravity; e.g. > A traveler in a rocket accelerated via a rocket motor feels an > apparent weight. Another person in a rocket sitting on the launch > pad feels a weight due to gravity. I think if you want to find an effect, it needs to be related to the space > contained, and probably nothing else, and manifest itself as some kind of > velocity, such as C. Not to shoot you down or anything. I just don't see how > the force of gravity at the earth's surface relates to expansion of space on > the atomic level. If every atom in the earth is expanding and pushing against each > other, causing a net acceleration at the surface of 32ft/sec/sec, > the effect is indistinguishable from the force called gravity. > In this theory of gravitation, how would one explain the fact the moon > appears to orbit the earth, remaining at a (relatively) fixed distance? BTW, if I remember correctly there was an underground comic from the > 60's/70's that used this idea to, erm, comical effect. -- > If a lion could speak, we would not understand > him. > -- Ludwig Wittgenstein -- > I could while away the hours > Conferrin' with the flowers > Consultin' with the rain > And my head, I'd be scratchin' > While my thoughts were busy hatchin' > If I only had a brain. > -- The Scarecrow. Excuse me for meddling, Brown, but before you get youself involved to deep in this discussion, you should take into account the possibility that these guys are definitely not in Kansas anymore. <5sJke.58092$sy6.12481@lakeread04> <%E4le.61445$sy6.2239@lakeread04> other, causing a net acceleration at the surface of 32ft/sec/sec, > the effect is indistinguishable from the force called gravity. > In this theory of gravitation, how would one explain the fact the moon > appears to orbit the earth, remaining at a (relatively) fixed distance? BTW, if I remember correctly there was an underground comic from the > 60's/70's that used this idea to, erm, comical effect. -- > If a lion could speak, we would not understand > him. > -- Ludwig Wittgenstein -- > I could while away the hours > Conferrin' with the flowers > Consultin' with the rain > And my head, I'd be scratchin' > While my thoughts were busy hatchin' > If I only had a brain. > -- The Scarecrow. Excuse me for meddling, Brown, but before you get youself involved to > deep in this discussion, you should take into account the possibility > that these guys are definitely not in Kansas anymore. > That much seems obvious; I'm just tossing my oar in to see if they've ever even _heard_ of Kansas. Density is given in metric units, like kilograms per cubic meter; weight-density is given in customary units, like pounds per cubic foot. Don > Density is given in metric units, like kilograms per cubic meter; > weight-density is given in customary units, like pounds per cubic foot. How uninteresting.... weight-density seems like a very silly concept... Why do you talk about such a thing? BTW, what was that equation of yours relating f, m, a, w, and g? You know... the one you like to plaster all over these newsgroups? The one with the gratuitous parentheses? And the two equal signs? Could you post that again? I just love seeing that thing! Just one more time? Please? [snip crap] Dumb Donny HEAD is ineducable. Dumb Donny HEAD cannot even carry through units. -- Uncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz.pdf Density is given in metric units, like kilograms per cubic meter; weight-density is given in customary units, like pounds per cubic foot. Don > Density is given in metric units, like kilograms per cubic meter; > weight-density is given in customary units, like pounds per cubic foot. Density is mass per unit volume. Weight density is weight or [mass]*[gravitational force downward for whatever planet you are on] per unit volume. That is the sole difference between density and weight density, and obviously just 'density' is the more general idea that doesn't confuse ideas like weight with ideas like mass, which it is useful to keep separate. In short: density is nice and clean, weight density is a kind of cluttered measure. Yet The definition of these physical quantities has nil to do with any particular system of measuring them, that is... It is also useful to keep system of measure separate from what is measured. Density has nothing to do with kilograms or cubic meters, other than the fact that one choice you have for unit measure of density at a point is a kilogram per cubic meter, and you could start with kilograms per foot or pounds per cubic meter, if you were feeling sufficiently masochotic. Weight Density has naught to do with pounds or feet, other than under one specific system of measure you can use 1 pound per cubic foot as a unit density. > Don -Mysid > Impossible, because that's not the way life works. By survival of the > fittest, only the good ideas will prevail, while the bad ones simply die > out. Nobody is interested in bad ideas, not even for historical reasons. Survival of the fittest can work only in the long run, when there is a relatively high rate of mutation and the forces of reality are allowed to operate on the variations to kill off the less-fit ones at a higher rate than the well-adapted ones. In the short run, especially with outside interference preventing natural selection from working, poorly adapted organism may survive and prosper. In the realm of ideas, notoriously bad ideas have on occasion thrived and prospered. Take Marxism for example, which spread through the intellectual culture since the early 20th century, affected many developments, and still has significant remnants. Or, perhaps more controversially, take radical kill-the-infidels Islam, which has spread widely despite being objectively a bad idea. The root problem is that people don't automatically think rationally, and training they receive along those lines isn't nearly as effective as it should be. Thus, irrational ideas can garner a substantial following. Actually, I wanted to not have vertices 'inside' segments. This seems to reduce the number by quite a lot. As you can see, the first few numbers which we actually need are: 2, 5, 13, 32, 77, 178, 399, 877,.. which are far lower compared to those for the case when we are allowed to have vertices 'inside' segments. Aditya <429DCCD5.D23429C5@iw.net Symmetry says that Reals are infinite strings rightward with a finite portion leftward such as > 2.333..... If Natural Numbers are finite-integers then the symmetry would be broken. The symmetry of WHAT would be broken? What about finite fields? Should all of them be infinite just for symmetry? > Mathematics > would be asymmetrical. To complete the symmetry then Mathematics would be a Dual system, a duality > with two number systems that are symmetrical. Why do people want to make integers and reals dual to each other? Integers aren't even a field! And what do you mean by dual? Is there a natural bilinear form on their pair? Or is it just a trivial and algebraically unimportant fact that their digital representations look like opposites? How about all other popular fields like complex numbers, rationals, the algebraic closure of rationals, etc? Should there be a lot of other dual pairs among them? > So if we have a Real such as 2.333..... then we have > the compliment of Reals is Natural-Numbers equal to Adics. Is there any signiphicant homomorphism between the two that reflects their basic structures? > Finite-Integers were just a crude and > flawed system that captured only a tiny bit of what the Natural-Numbers really are. > By Natural-Numbers, you mean what Dik would call N-adics. Right? That's hardly the point. Fermat found the FLT statement in some book. > Both Fermat and the author of that book had a very particular view of > what they meant by the word integer in the formulation: Does > X^n+Y^n=Z^n have any integer solutions for n > 2? and the axioms they > satisfy. Their axiom set may be a very wrong description of what integers > actually are. But that's what they wanted to find: solutions to the FLT > equation within these given axioms. I hope Dik reads your above for he has the same mindset as you. Both of you believe and think and > expect that someone such as Fermat is doing real mathematics when he uses a flawed set and that it > still remains as legitimate worthwhile mathematics. First of all, Fermat's set may be a totally wrong representation of what integers REALLTY are. But why is it flawed? Why is it wrong to try to solve a polynomial equation in some abstract ring? If he tried to solve his equation in some finite ring, why would that be flawed? He has some ring. So he wants to solve an equation in it. I see nothing flawed in his desire. The point is that THERE IS NOTHING MATHEMATICALLY OR PHYSICALLY SIGNIFICANT in solving the equation X^n + Y^n = Z ^n. The only reason why people have devoted themselves to solving it over the set of Fermat naturals, is because Frmat himself tried to solve it over his set of Fermat naturals and porbably failed. These are not integers? Fine. Call them fermats. So, the equation X^n + Y^n = Z^n over the ring of fermats has no solution for n>2. Does it have solutions over other rings? Yes, it does. For example, I already gave you the pretty much full description of the continuum of solutions over reals. Woopty doo! You say true integers are dual to reals under some isomorphism? Great. That means I just gave you a continuuum of solutions to FLT in your integers. Without breaking a sweat. And any high school senior could have don the same. You say true integers are more like N-adics than like reals? Fine. Dik tells us that solutions to FLT over p-adics have been known for a long time. So, what's EEE's or your original contribution to Fermat's Last Theorem. > Do we call the persons who applied leaches or > drained people of blood in the Middle Ages as doctors and people of medicine? We do not call them > medical doctors because they were not using science, likewise Fermat was no longer a mathematician > when he tried to apply finite-integers to a^n+b^n= c^n, or at best Fermat was a wrongheaded > mathematician. > If anything, Fermat was probably a wrongheaded mathematician when he tried to solve a^n+b^n= c^n at all. A totally inconsequential thing to solve. But Fermat was not a wrongheaded mathematician. He was a man who liked puzzles. He saw some ring and he saw a polynomial. So, he tried to solve this polynomial over that ring. Why? Becuase he liked difficult posers (puzzles) and solving that particular polynomial over that particular ring seemed like a fun and challenging intllectual puzzle. If he had a different ring, it wouldn't have been an interesting puzzle at all. Even if the lallter ring is the ring of true integers. Fermat would have found solutions in 5 minutes and have forgotten all about this puzzle. Solution to the same equation in any other axiom system is totally > irrelevant. Especially if, as you say, some solutions to this equation > in p-adics are already known. You, like Dik, do not understand that axiom systems and definitions can be flawed and wrong. And > that most systems and definitions have flaws and blemishes that future people have to revise and > correct. > What you don't understand is that nobody gives a serious flying damn about FLT as applied to any ring other than the flawed ring of fermats. And if they did, they would have found solutions in 2 minutes flat. So, what's YOUR original contribution to the FLT puzzle, as F has posed it to us? Nntp-Posting-Host: apps.cwi.nl > ... > > If you define them as infinite sequences of > > digits with a period placed firmly on the right, you are close to the > > 10-adics (and that is what AP uses, and I think EEE also uses that, but > > I am not sure, he has never been very prolific). > ... > > So, is his point that N-adics are better representations of integers > > that the interpretation that we, including Fermat, usually use? > > No his point (and AP's) point is that their representations of integers > are the only possible interpretations, and that our interpretations are > just plain wrong. > > Some of that is correct. But it dodges the aspect that Dik has an > interpretation of what the Natural Numbers are. Not an interpretation. I use the definition. > I just know the Natural > Numbers are not the Finite-Integers. In that case you use a different definition. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Nntp-Posting-Host: apps.cwi.nl ... > Yup. Let A be the set of ducks with fourteen legs. See, I have defined > something. And I can work with set A. > > I don't think you really believe that. Believe, even, to the extent that > the empty set is nonreality. > This is a problem that Dik and many others have-- they think that > mathematics is above, beyond and superior to physics by the allowance of > make-believe or imagination as workable reality. In a sense, they elevate > mathematics as above physics, when in truth mathematics is begot from > physics and is a minor subset of physics. Oh. In what way is cryptography related to physics? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ > So, Dik, what I do is to take that 2-adic o ...101010.1 and that 3-adic > of ...121212.1 and flip them around or flip them over into a Real of > 1.010101... and 1.212121.... and add them together to achieve > 2.22222..... and then flip it back around or over into an Adic of > ....22222.2 and ask myself is this sum still a 2-adic or 3-adic? > Sometimes the summation of a 2-adic with 3-adic maybe a 4-adic. Ok, I give you the 2-adic ...00000010 and the 3-adic ...00000010 and ask you to tell me what it is. You come back with ...0000020. But now I say the value of that 2-adic is 2, and the value of that 3 adic is 3, so their sum is 5, what you came back with is not 5 in any of the n-adics. Or still better, I have two 2-adics: a = ...111111110 and b = ...111111100, and two 3-adics: c = ...111111110 and d = ...111111100. (Note: a != c). I ask you to add, a and c and you come back with: e = ...222222220, the same with b and d and you come back with: f = ...222222200, I ask you to subtract f from e and you come back with 20. To see that this *must* be wrong consider the following actual values: a = -2, b = -4, c = -3/2, d = -9/2. a+c = -7/2, b+d = -17/2, (a+c)-(b+d) = 5. None of the numbers you return has the correct value in any of the adics. > I remember Karl Heuer saying that addition across various different > adics such as 2-adics and 3-adics and multiplication are well-defined > already and that I should not, or others have trouble with defining > adic addition and multiplication. Yes, indeed, because the 6-adics are the direct sum of the 2-adics and the 3-adics. So the 2-adics and 3-adics can be embedded in the 6-adics as direct sum, however, there are two different embeddings when you look in base 6 notation (I think). And converting a 6-adic in base 6 notation to to direct sum notation is, eh, also quite messy. Also the properties of the operations will be quite a surprise. In direct sum notation, the 6-adics are the pairs (p, q) where p is a 2-adic and q is a 3-adic, and so the first component follows 2-adic operations and the second follows 3-adic operations. Addition and multiplication are defined as component-wise addition and multiplication. Any 2-adic a can be written as (a, 0) and any 3-adic b as (0, b), so their sum is (a, b). The 1 in the 6-adics is (1, 1), which, surprise, is the sum of the 1 in the 2-adics and the 1 in the 3-adics. The product of *any* 2-adic with *any* 3-adic yields 0. You may map the 2-adic 1 to ...152221350213 in the 6-adics and the 3-adic 1 to ...403334205344 in the 6-adics (or the other way around, I think, but perhaps there is only one way to map, in that case I do not know which of the two you need). But I think you want an embedding that is value preserving for the finite adics (i.e. for the finite adics it is just a change of base). But this will show that the conversion of the infinite adics is problematical. (I put finite and infinite in quotes because there is no consistent order relation in any of the n-adics.) Consider the 2-adics: a = ...111111110 and b = ...111111100, in 6-adics there notation is: a = ...555555554 and b = ...555555552. And the 3-adics: c = ...111111110 and d = ...111111100. become: c = ...555555554.3 and d = ...5555555551.3. So a+c = ...55555552.3 and b+d = ...55555543.3 subtracting: (a+c)-(b+d) = 5 messy, but correct. But it is not really a change of base. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ of *all* of Q_p, p|n. thus quoth: I may have mentioned that the only book I know who studies the n-adics in any detail is Mahler's book on p-adic analysis. He eventually proved that the n-adics are the direct product of Q_p for p | n. >What everbody in sci.math believes is that there are sets whose >members are SO NUMEROUS that there is no surjection from the set of >integers to such a set. We believe this because it is a theorem. >You even know a couple of proofs. You just don't understand them you mean the proof there is no number for any box that holds all the numbers of the boxes that don't hold their own number? right... Herc Anyone know if there is a way to get the series of log(sum(a_k*x^k))? i.e., I'm looking for the coefficients bk( in terms of the a's) s.t. sum(bk*x^k) = log(sum(a_k*x^k)). Jon > Virgil said: > This node is it, which 1/3 is mapped upon. This node does not exist . > True, but the well defined infinite path does. If this distinguished node does not exist, then there are > other numbers which have same paths and are isomorphic to > 1/3. WRONG! Given any node on any path in any maximal binary tree, > there are infinitely many paths passing through that node, as > it is itself the root node of a maximal binary tree, with > infinitely many paths rooted at it. > But each node does not uniquely identify a distinct infinite set > of paths. It identifies that set of paths that pass through it, and in a > maximal binary tree there are always infinitely many such paths. > Yeah that's what you just said. I read it. Respond to the rest of the > paragraph. > > Just as every node is a member of an infinite number of paths, > every path includes an infinite number of branches and nodes. > Those two infinities have a ratio: paths/branches=1/2, If you are talking about the totality of paths in a maximal binary tree compared to the totality of branches in that same tree, that ratio is not finite. In a previous post I constructed a bijection between the set of nodes to the naturals in which the root node maps onto 1, since each branch uniquely precedes a non-root node that establishes a bijection from the branches to N{1}. In that same post, I constructed a bijection between the maximal paths of a maximal binary tree and the members of P(N). Both of these bijections are valid in standard mathematics, regardless of TO's objectins to them based on his intuitions. Thus the ratio is not as TO claims. > RESPOND TO THIS DUMBASS. By your logic, I can just as easily say > there are infinitely many more branches than paths, because each path > contains an infinite number of branches. Not by MY logic. By TO's logic, anything is possible, but mine is a good deal more restrictive. Infinities have a ratio? And which set of paths is being compared > to whichset of branches? > ALL paths to ALL branches. Then, as I have shown, that ratio is infinite. > That's why they are infinite sets, Dopey. > If you had paid any attention you would know by now that Bigulosity > Theory assigns actual ratios to set size comparisons when the mapping > function that relates them is linear. But, you haven't, so you don't. (1) there is no linear mapping function. (2) there are contrary proofs. as I have demonstrated using the proof involving > insertion of a node, which you failed to really comment on, > except with your mistaken impression, which you reiterated above, > that a node cannot have a single child node. It cannot in a maximal binary tree, which is the kind under > consideration. > One can insert nodes in the middle of a binary tree. Take a data > structures course. Only in non-maximal trees. In a maximal binary tree, where do you insert anything? Maximal means it is already as big as possible, which seriously inhibits adding anything more. Perhaps TO should go back to kindergarten to learn how to read ALL the words.. Go pick up a Data Structures textbook, and learn about how binary > trees are populated and used, and their relationship to linked > lists. Linked lists are irrelevant to maximal binary tree structure in the > current context. > No they're not, but I don't want to add to your confusion, so forget > it. > To have shown this fact and the necessarily opposite opinion > of set theorists is sufficient for me. Therefore, there is no reason to continue this discussion. My > purpose has been reached. Our positions are clear. As they > can remain between men of perhaps moderate but certainly not > below-average intelligence, we see mathematics in a very > deplorable state. And mathematicians, with better reason, see WM in a deplorable > state. > Don't you wish. Don't have to, it is fact! > In your dreams. Find some mathematicians to refute it then! > David Kastrup said: David Kastrup said: >> Of course, your inductive proof that all naturals are finite >> contains exactly the same flaw you point out for lim(n->oo) >> 1/n. You are proving that n is finite, that is, n> inequality, and at each iteration of the induction, n is >> increasing. Can I say lim(n->oo) n > And nobody says that. You are again fighting straw men. >> Instead of disproving nonsense you invent yourself, how about >> actually dealing with _my_ argument. Yes, that's exactly what you're saying, when you tell me that an > infinite number of naturals with a difference of 1 unit between > successors covers an overall range which is not infinite. Let's put it bluntly: liar. I never said anything like that. You > are again putting up a straw man since you are incapable of dealing > with an argument. > So you don't believe the set of finite natural numbers is infinite? > The statements are equivalent. Not without TO's contrary to fact presumption being imposed. It is quite possible, in fact necessary, to have infinitely many finite naturals each separated from its closest neighbors by 1. The only way to prove otherwise is to beg the question, as TO keeps doing. If the range is infinite, as I have proven to you, The range covered by the naturals is infinite, sure, but nobody > claimed it wasn't. > They calimed all naturals are finite. If the range of values is > infinite, that means the difference between some pair of them is > infinite False! The range is unbounded, but that does not mean that any value in that range is unbounded, only that there are values larger that any fixed bound. Whatever value TO wishes to claim is an upper bound is not an upper bound. According to TO's arguments, a sequence cannot have an infinite limit unless it contains infinite values. Consider the sequence of partial sums s_n = sum_{1 <= k <= n} 1/k. It is a monotone increasing sequence which does not have a finite limit. By TO's theories, there must be some natural number n for which s_n is infinite. Since s_n is clearly finite for small enough n, there must be a crossover point, an n such that s_n is finite but s_(n+1) is infinite, even though the difference s_)n+1) - s_n = 1/(n+1) must be very small. Nntp-Posting-Host: apps.cwi.nl > Dik T. Winter said: ... > You keep on claiming that without a correct proof. And do not come up > with your proof with binary strings of infinite length. How many strings > of infinite length are there that have the most significant '1' in a > finite position? > > A finite number, Dik. That's the point. Get it? Still nothing more than assertion. Where is the proof? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ Nntp-Posting-Host: apps.cwi.nl > Dik T. Winter said: ... > > I don't know how many bits the number 1/3 has. But if it is a number > > then it has a path in my tree. And if it has a path in my tree then it > > has a node to be mapped on. You must know, there are infinitely many > > nodes in my tree. > > so by your reasoning there is no path in your tree representing 1/3. > > Bullcrap, Dik. 0.3333333... is 1/3, which is a number. Do you ever read what you reply to? Where did I state that 1/3 was not represented by a path? > Yes. But the nodes in your tree only represent a number when after that > node the tree goes through the 0-branch only. This means that the nodes > in your tree only represent numbers that have a final digit 1. 1/3 is not > such a number. > > No, Dik. Each and every number in WM's tree is represented by an infinite > path. The finite numbers are those whose final 1 is a finite distance > from the root, followed by infinite 0's. Indeed. At what point are there an infinite number of 0's in the binary expansion of 1/3? So at what point can you conclude that 1/3 corresponds to a node? > > If there is a path (= set of nodes) which 1/3 corresponds to, then > > there is always the node required. > > There indeed is a set of nodes 1/3 corresponds to, but it is an infinite > set. There is no particular last node which 1/3 corresponds to. > > True there is no distinct node, no last branch, and no way to tell whether > the last digit is a 0 or 1, unless one assumes the same pattern from the > beginning of 01, and assumes it ends in the repeated pattern, which ends > in a 1. But, the number is there in the form of an infinite path, as you > said, which is well defined. Nice to see you contradicting WM. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ > So, you changed your mind again. Or, maybe you don't see that a range of > values is measured by the difference between the largest and smallest in > the set? Only if the range HAS a largest and smallest. -- Daniel W. Johnson panoptes@iquest.net http://members.iquest.net/~panoptes/ 039 53 36 N / 086 11 55 W I was asking you how to construct that set of natural numbers with the varying density. The set of all sets is its own powerset. In a set theory, there are only sets and quantification over sets implies a universal set or set of all sets. ZF is inconsistent. That aside, a set of all sets is a counterexample to that there exists no bijection between a set and its powerset because the set of all sets is its own powerset. By assigning basically an index or ordinal to each set, via well-ordering, that set is as well orderable by what is generally called the set of natural integers, via transfer. A question about the real orders is how to well-order them, where it is accepted that that exists and nobody has an example in the standard real numbers. One notion that many agree upon is that they have some natural well-ordering. Because the set of reals is a set of points where trichotomy holds, it's possible to consider basically those points in order, not via their Cauchy/Dedekind/etcetera formation from basically the field of rational numbers but via the alternate perspective as a contiguous, in being continuous, sequence of points, or recovery of the Newton infinitesimal, the fluxion. Via essentially exhaustion, analytical results are discovered to show that something like an infinite point set not dense in the reals can have a positive, non-infinitesimal integral. Where that is so, then it should be possible to exhibit a variety of empirical applications. I think a theory should prove everything that's true and disprove everything that's false. Ross >The set of all sets is its own powerset. Not so. For example, in NFU (which does have a universal set V), 2^V is clearly a set all of whose elements are SETS (namely all the subsets of V), but V itself also contains ur-elements (i.e. non-sets). Thus V is strictly larger than 2^V. [SNIP] -- --------------------------- | BBB b Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | ----------------------------- >: This is a fairly silly one that came up in the pub tonight .. A4 paper is, for the pure mathematician, a rectangle 250*2^(1/4) mm >by 250*2^(-1/4) mm. (If you sell the stuff, or are a devotee of nice >convergents to sqrt(2), then it is 297 mm by 210 mm.) That is, it is >a rectangle with sides in the ratio sqrt(2):1 and area 1/16 m^2. Given a _square_ sheet of paper 1m x 1m, how many entire A4 sheets can >be cut from it? 13 is easy enough (stack a 3x3 set next to a 1x4 with >the opposite orientation), 16 clearly impossible. How about 14, or 15? >> I think your 13 solution is optimal. If you increase the sheet size to >> 1x3, you can do 44 which is almost 15 per sheet. >> --Keith Lewis klewis {at} mitre.org >> The above may not (yet) represent the opinions of my employer. Of course if the 1 square metre of paper had the aspect ratio sqr(2) - > making the size known as A0 - then there would be no problem: exactly 16 > sheets could be cut. Suggestions why the ratio sqr(2) might have been chosen? It's rep-2? (A0 can be divided into 2 A1's, A1 can be divided into 2 A2's, and so on, and all are similar?) -- ------------------------ Mark Jeffrey Tilford tilford@ugcs.caltech.edu * * * Please Read And/Or Print This! * * * Press [Ctrl][P] Keys On http://115639.aceboard.net/forum2.php?rub=158&cat=61&login=115639&page=0#id9 6 << * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * >The teachers couldn't understand, my argument is stupid... >may be something wrong with set-theory pure, they don't know, how to >introduce counting. >How one can get cardinals, without knowing either, how to count, or >ordered pairs, or functions or proper indexing? And look, how they do >this in set-theory! >But it seems, some are satisfied with themselves, nothing wrong, with >what they teach. >And that was just the problem with the kids and New Math, they didn't >learn how to count. >Hero The children learned to count. The teachers could not understand sets. One can get cardinals from 1-1 mappings. The children could learn this and what it means. But not most of those teaching the children. Functions do not require ordered pairs; they can be taken as a primitive, and I do not think this is a bad idea. Ordered pairs were a fairly late introduction into set theory. As they were only dealing with finite sets, it worked, albeit taking much on faith. But THIS should not have bothered the teachers; that is how they teach. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 >No. In mathematics the word undecidable must >be used in Godel's sense. >That is, undemonstrable within a system of axioms. There is not >relation with trueness or falsity. If Riemann Hypothesis is proved >undecidable it can be, perfectly, false. >The fact that no human being, no computer, can deduct it from the >accepted mathematical axioms, >do not permits us to assert that it is true. No, one can come up with a sequence of computations which will always produce 0 if the Riemann hypothesis is true, but which willn either pass through a zero assumed not to exist, or yield a positive number if it is false. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 <3g6ohmFaufm2U1@individual.net> 2 such > that k is not the sum of two primes, and a true statement > of the form k is not the sum of two primes is provable using > only weak arithmetical axioms. This is generalizable; it applies not only to Goldbach's conjecture but to most conjectures-about-N that are universal generalizations over every element of N. When any such conjecture is false, there is a finite standard natural instance of it that is in zeroth-order PA or Kalmar elementary arithmetic or something similarly and fundamentally propositional; you could also use some form of bounded arithmetic since the calculation has to be finite. That is why it was alleged that a whole lot of problems are like this. > So if the conjecture is undecidable even in a weak > arithmetical theory, it is true. LOTS of things are undecidable in WEAK arithmetical theories. My point here is that while we may not need to speak of models, we DO still need to speak of THEORIES. NOTHING is JUST PLAIN undecidable, undecidable simpliciter. Everything that is undecidable is undecidable FROM SOME SETS of axioms. Are there truths about N that are undecidable from PA but that are decidable from ZFC? I'm fairly confident that FLT, for example, was not proved from PA; there was a LOT more axiomatic preparation brought to the beginning of THAT fight than just Peano's. Earlier in the thread there was an allegation that if it's undecidable, we'll never know that. But that seems to be critically missing an answer to the question, undecidable FROM WHAT AXIOMS? We could know that something was undecidable from PA without knowing that it was undecidable from ZFC; indeed, we might use ZFC to construct a non-standard model of PA in which the statement came up with the other truth-value. We might also know that something was undecidable from Kalmar Elementary Arithmetic while knowing that it WAS decidable from ZFC. It does NOT follow from the fact that the standard naturals are the initial segment of All models of PA that all of s,+,*, and < are isomorphic, over all those models, over that initial segment, DOES IT? If I'm ignorant and it does, then shouldn't there be more theorems about that initial segment? Last I heard, one couldn't theoretically separate the initial segment at all. But truths about all models are supposed to be provable, by the completeness theorem. Originator: tchow@MATHSTATION029.MIT.EDU.mit.edu (Timothy Chow) >Are there truths >about N that are undecidable from PA but that are decidable from ZFC? Yes, for example, PA is consistent. If you consider that unnatural, then the classic examples are Goodstein's theorem and the Paris-Harrington theorem. >I'm fairly confident that FLT, for example, was not proved from PA; >there was a LOT more axiomatic preparation brought to the beginning >of THAT fight than just Peano's. Most mathematics is not proved explicitly from any particular set of axioms, so in this sense your claim is trivially true. Less trivially, it is certainly not obvious how to formalize the known proof of FLT in PA, and it is still open whether FLT is a theorem of PA. >Earlier in the thread there was an allegation that if it's undecidable, >we'll never know that. But that seems to be critically missing an >answer to the question, undecidable FROM WHAT AXIOMS? Right. >It does NOT follow from the fact that the standard naturals are the >initial segment of All models of PA that all of s,+,*, and < are >isomorphic, over all those models, over that initial segment, DOES IT? Well, it's certainly *true* that all those operations are isomorphic over that initial segment; whether this follows from the other fact depends on what you mean by follows. >If I'm ignorant and it does, then >shouldn't there be more theorems about that initial segment? >Last I heard, one couldn't theoretically separate the initial segment >at all. >But truths about all models are supposed to be provable, by the >completeness theorem. They're provable *if* they're expressible in the first-order language of arithmetic. And you can't separate out the initial segment in the sense that x is standard is not definable by a first-order formula. But if you allow yourself to use set theory in your meta-theoretic reasoning, then you can indeed separate out the initial segment and prove (meta-)theorems about it. See for example Richard Kaye's book Models of Peano Arithmetic for an introduction to the subject. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences <3g6ohmFaufm2U1@individual.net> If every model of PA has an initial segment that is isomorphic to the >standard naturals, why isn't there some theorem of PA to that effect? The main problem is that you can't express that sentence in the first-order language of arithmetic (or maybe I should say in the language of PA, since I recall that you have some idiosyncratic objection to the term first-order language of arithmetic). The sentence makes essential reference to infinitary sets that can't be encoded as first-order statements about integers (or equivalently, finite sets). -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences <429E3459.1010502@xs4all.nl The Quaternion to Matrix conversion formula mentioned in http://www.sjbrown.co.uk/quaternions.html will exhibit distortion of the picture shown at the end of the graphics > pipeline when > (x, y, z, w) no longer form a unit quaternion due to accumulation of > rounding and truncation errors; the matrix is not orthonormal, even not > orthogonal in general, in that case. Replace the diagonal elements by r11 = ww+xx-yy-zz, r22 = ww-xx+yy-zz, r33 = ww-xx-yy+zz and the R matrix will at least be a scalar multiple of a rotation > matrix; no distortions will occur. In my software for dynamical simulation I do not normalize the rotation > quaternion at each step; even not at each 1000th step. After N > simulation steps (of about 0.1 deg of rotation) the absolute value of my > rotation quaternion differs from unity by an amount of about N/2 * the > unit of the least significant decimal. With an Intel math coprocessor > with its 18 significant decimals you won't observe a deviation of a > single pixel in the picture after a full week of running. You shouldn't need any of that nonsense if you just stick to homogeneous 3D transformations. >[snip discussion of gimbal lock] This is correct. However, there's another advantage to using >quaternions in a computational setting when many rotations >must be composed together. As described in http://www.sjbrown.co.uk/quaternions.html it is easier to check and correct for any deviations from the >orthonormality constraint. Btw that same link explains why it is generally better, when >you reach the final product, to convert the quaternion to a >matrix and use that for the transformation, rather than >transforming vectors directly by quaternion multiplication. >>You still miss half the point. Rodrigues rotation formula is _still_ >>faster than all that. Hmm, but what about composition of rotations? That, it >seems to me, is where quaternions shine. I.e. you have >many rotations composed together and don't need to see >the intermediate results. One approach would be to apply >Rodrigues sequentially to each vector you're interested >in, but that would lose if you have many vectors; also >I don't see how you enforce the orthogonality constraint. >But I'm not an expert at this, perhaps you know better? The alternatives are (1) to generate matrices and multiply >them, thus getting a matrix for the composition, or (2) to >generate quaternions, multiply them, and then convert the >product to a matrix for the composition. The link that I >provided suggests that option (2) is better in many cases >because of the orthogonality issue. (This may be the germ of truth that lies behind schoenfeld's >flawed exposition of it.) >>You did not even understand. Probably not. And my inability to understand, of course, does >not necessarily make your exposition flawed. I should have >left judgments of that sort to the experts. Sorry. > Let S be for all n in N, P(n) If S is undecidable, then for all n in N, P(n) Otherwise some n in N with ~P(n) would decide that S is false. Thus undecidable statements have the form some n in N with P(n) <429da850$0$193$edfadb0f@dread11.news.tele.dk> <429d9f8b.17028296@news.swissonline.ch> thank you. > On 1 Jun 2005 01:54:30 -0700, coolblue hi everyone, > suppose i have a line ax+by+c=0 and a point (x1, y1), what is the >projection of this point onto the line? least square estimation? > (...) What you mean by least square estim. is probably this: find where > is the minimum of square distance of (x1, y1) to a point (x, y) of > the line written in coordinates, using differential calculus. Because > this is finding an extremal point of a 2-var. function subject to > a constraint (the equation of the line), there is probably a better > way to do this than what I did, but I guess this way will still be > a good deal more complicated than the following solution (I compared > the values of the x-coord. of the projection that I obtained with both > methods, so I had confirmation of the - not so simple - value ...) After all, we know that we look for the intersection of the given line > with the line through (x1, y1) that is orthogonal to it. Now the > vector (a,b) is orthogonal to the given line (easily seen if you > consider the parallel through the origin obtained with c replaced > by 0) so the second line has the parametric form x=x1+ta, y=y1+tb, > t real var. Put this into the equation of the first line and you > immediately get t = - (ax1+by1+c) / (a^2+b^2) which BTW is very > similar to the well known formula giving the distance between > (x1, y1) and the given line - in this the above denominator is > replaced by its square root and instead of the minus sign, one > has the absolute value - not astonishing because the length of > the vector (a,b) is said square root ... Now you only have to put > the value of t into the above parametric equations to get the coord. > of the intersection (which you may arrange as you prefer)