mm-4459 === Subject: Re: Software to create math lessons was ich immer schon mal sagen wollte: Deine Beitr.8age und Deine freundlichen Gr.9f¤e erhellen den oft etwas bleiernen Himmel in sci.math erheblich. Im namen aller dort Beteiligten (hier: ungefragt ;-) ) - Gru¤ - Gottfried -- --- Gottfried Helms, Kassel === Subject: Re: Real life, real world > Most people in the developed world get to live in a kind of bubble > where they feel security based on what has happened, or not happened, > to them before. So you live in a Bubble. About the size of your fat head? Balloon-head! It is a sense of security and expectation that you know what can > happen. Yours therefore must be very, very small, a micro bubble. But it is only an illusion. Did someone tell you youlook like Pinhead on Puppet Master? http://www.angelfire.com/pa3/PuppetMaster/pinbck.jpg As the people of Hiroshima and Nagasaki learned when many of them were > shredded in an instant. They shreadded Pearl Harbor first. But without that illusion human beings do not do what they do, get up, > go to work, keep making babies and continue because they feel safe. Is that your life? an illusion ? You don't work, don't make babies, and don't feel safe, do you Pinhead? > Many of you have replied to me and argued with me from your bubble not > understanding how this story has to go, and most importantly, how it > must end because it is how you are built. Has your bubble been losing air ? Is it collapsing in a micro burst? Is a green goo leaking out of your micro-bubble, Pinhead? > You cannot do anything else but follow your programming. You follow your hacking, you are not a programmer yet. > Revolutions require luck, persistence, and odd juxtapositions of > crucial events. Revolutions require circular motion, dumbass. > Here I have George W. Bush, global warming, and the reality that for > some reason number theory simply stopped over a hundred years ago plus > a few little things more, just to make it cleaner. You are way way out of touch with number theory, go read a book, Dover has them for $10 new paperback. Or try ebay for $3 > What puzzles me that I keep writing posts like this one is not > questions about what will happen but an unending curiosity about > people like you. I know what is going to happen to you. And it > amazes me looking at it from this distance that you keep doing what > you do, going about your business and even arguing with me, as if you > are totally clueless. You are in outter spaced man, put down that joint, you smoke too much of that wacky weed. > So then, are you truly totally clueless or do you just wish to make it > a grander tale on some level? You should not drive stoned on the internet, goof-ball. Is there not some need to be sacrificed in you? Isn't that why you > keep up the resistance, and keep arguing and help set the stage in > this way? You talking to your mommie now? She is tired as of talking to you she is the one that called and got them to put a net over you. > You want what is coming next. You must. I just cannot believe you > are ignorant any more. Known Fact: JSH is ignorant of any/ all Number Theory > It is too wild already at this point. The drama around the world is > too fantastic. And on some level you must know there are few limits > to how incredible what is about to happen may be. Yep. You are a stoner. a Drama Queen, alone with your computer. > And that is about assuaging my guilt. I have no guilt here. I do not > make your decisions. Too bad you make decisions for yourself, no training there. > I am not here to save your life from you. Pinhead, your bubble now has zero volume, therefore your IQ is now Zero. > So then as the countdown begins in earnest and a few months separate > us from one more crucial event, and then as only a few years separate > us from even more incredible and fantastic events that will decide the > fate of life on this planet, I say, yes! keep licking them china painted toys, and backs of frogs and toads, and stay off the bong, friend. > James Harris > === Subject: sense, existence, knowledge <2ju4i3dbtil6qqujqf3sba50cq7lsvqhn4@4ax.com> <5qh6i3t5gvctdh2k295p8r4j2rh7cgu239@4ax.com> but look at your everyday usage as well >because it is very often a folk style > and can be summarised before the depth you don't say something is true > when you don't know That's true. :-) i think this is important it speaks of the meaning in words meaning is temporal in common language when statements are queried response depends upon the current knowledge state of the agent replying the semantics of all languages vary over time even formal ones like c++ >(i mean - if you're trying to convey your understanding > and don't have other motives) you instead may give a summary of what you know that summary is basically what truth values are in constructivism when you haven't derived or perceived a result > you only query truth or falsity the query hasn't returned yet [...] right now > today >i don't think you would call the riemann hypothesis a theorem it might > it could > we can talk of possibility but _is_ it? what truth value would we assign it today? we have many indications > much information about the density of zeroes > number theoretic structure > confirmed to good confidence for finite samples but still the possibility it could be refuted we acknowledge that everyday in speech that is the construcivist position on truth values > that accumulation of evidence and leads > there are more than any finite number > of such truth values and eventually things do get proven > and we can look at those statements-with-proofs > as maximal elements of its logical lattice [...] so you accept possibility > as a property relative to a knower? Definitely. :-) (Is this leading me into a trap?) possibility truth proof knowledge these are modalities of semantics there are many well known modal logics expressing syntax for these modal queries kripke models interpret modalities inside primitive data structures preorders (S4) finite strict partial orders (GL) .. being directed many of these have a natural interpretation in time these successions of frames these possible worlds change and time all relative to a given knower you once asked about the royal we in the quote tossed around i don't know the positions of the quoted but i think a separate knower can be assigned to any subcollection of matter with stable computational control of information flows collective computational mechanisms are suitable societies and their long term memory in libraries calculational phenomena in a universe can cooperate in larger and larger calculational structures through protocol negotiation symbolic exchanges building languages these are dynamic control phenomena and the semantics of an utterance must be relative to the particular state of knowledge when the utterance was produced >if all truths are truths always > what does possibility even mean? If your generalised truth values are not derived from truth and > falsehood, what does possibility even mean? when one tries to define truth counterfactually it loses epistemic import if X was true before it was proven it doesn't mean it could be used in another theorem because nobody knew it its the knowing it that gets used in the actual proofs its what makes it available on the free store available for actual use so in trying to push this truth back in time make it everlasting you end up with a truth that doesn't mean you can use it in a proof it's a modality without a knower > For definiteness, let P be the proposition gamma is rational. We are, of course, uncertain about P. But our uncertainty is > /as to/ whether P is true or false. Can you name a specific truth value which P might have (but, of > course, which it also might not have) other than true or false? (At the back of my mind, I'm worried by having occasionally seen > suggestions that even some quite mainstream-looking propositions > such as this one might actually be indeterminate. But although > I am really worried by this possibility, I do not understand it. > And even indeterminacy is indeterminacy /as to/ truth or false- > hood.) there is an important difference between a constructive truth value and an indeterminate truth value (like those of L3) constructive truth values carry information they can carry extremely large amounts of information unbounded above one way to formalise the truth values is with the data structures i mentioned earlier beth models often use binary ditrees kripke models may use other transition digraphs the truth value of a statement is built from the current state of knowledge ( you can think of the truth value as a tree built from the branching steps in current proof attempts but any data structure that allows equivalent semantics can be used ) > (I can only learn to think differently by thinking from where I > am now. One cannot unlearn what one cannot even express. So, > I'm trying to say what it is that I have been thinking without > saying.) never trust unlearning rewriting should occur only when you have learned a good reason to > Also, for any specific one of your generalised truth values, v, > can you say what truth values are possible for the proposition > P has truth value v? all of them (depending of course on P and v) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === Subject: Re: Implementable Set Theory and Consistency of ZFC <877ilbnsg6.fsf@phiwumbda.org> <1b79f$47259eca$82a1e228$3068@news2.tudelft.nl> <87myu2f9om.fsf@phiwumbda.org> <74e91$47272327$82a1e228$24521@news1.tudelft.nl> <87ve8owwr5.fsf@phiwumbda.org> <87wst48qh9.fsf@phiwumbda.org> <2aed4$47283a1c$82a1e228$16577@news1.tudelft.nl> <87y7dj7kts.fsf@phiwumbda.org> <876fa$472843f6$82a1e228$18666@news1.tudelft.nl> <87k5p37eat.fsf@phiwumbda.org> <68e5c$472864e2$82a1e228$24499@news1.tudelft.nl> <87y7dj5xf8.fsf@phiwumbda.org> <87bqafph1e.fsf@phiwumbda.org> <8569e$47288065$82a1e228$30100@news1.tudelft.nl> <873avrpga7.fsf@phiwumbda.org> <4ed3f$472897c3$82a1e228$4409@news1.tudelft.nl Whatever. Again, I ask: Do you agree that There is at most one empty > set is a theorem of ZFC, *even though* it uses only one of the axioms > of ZFC? Yes. The proof of existence uses an instance of separation and uniqueness uses extensionality. Or, if you have an empty set axiom, then, yes, just one axiom. But aside from those pedantic matters, now you see our point, right? The proof of a theorem might use only certain premises but it is still a theorem from any larger set of premises, just as this is an example: the proof of the existence of a unique empty set needs to use only certain axioms, not all of them, but the existence of a unique empty set is also a theorem of the entire set of axioms. MoeBlee === Subject: Re: Implementable Set Theory and Consistency of ZFC <728b6$4720895d$82a1e228$26020@news1.tudelft.nl> <37b63$4725a9c9$82a1e228$6935@news2.tudelft.nl> <87abq1srev.fsf@phiwumbda.org> <6ebdf$4726eaa4$82a1e228$11563@news1.tudelft.nl> <87d4uwykyv.fsf@phiwumbda.org> <294d6$4727212b$82a1e228$23945@news1.tudelft.nl> <87sl3swwl7.fsf@phiwumbda.org> <87y7dk8qrr.fsf@phiwumbda.org> <873avr8zu0.fsf@phiwumbda.org> <21a82$47284306$82a1e228$18523@news1.tudelft.nl> <87odef7egh.fsf@phiwumbda.org> <3feb9$47286470$82a1e228$24499@news1.tudelft.nl> <877il37cf3.fsf@phiwumbda.org> <84a53$47287a89$82a1e228$28335@news1.tudelft.nl> <878x5jpgf4.fsf@phiwumbda.org Here's my statement of ~Infinity: Every set has a finite number of members. In other words, every set > can be put in one-to-one correspondence with some set {0,1,...,n}. Or 1-1 with the empty set. But I don't know how to derive that as an equivalent of the axiom of infinity, where the axiom of infinity is Ex(0ex & Anex nu{n) e x). I see that, in Z-I, every set is 1-1 with a natural number entails the negation of the axiom of infinity. But how, in Z-I, do you derive every set is 1-1 with a natural number from the negation of the axiom of infinity? MoeBlee === Subject: Re: Implementable Set Theory and Consistency of ZFC <87myu2f9om.fsf@phiwumbda.org> <74e91$47272327$82a1e228$24521@news1.tudelft.nl> <87ve8owwr5.fsf@phiwumbda.org> <87wst48qh9.fsf@phiwumbda.org> <2aed4$47283a1c$82a1e228$16577@news1.tudelft.nl> <87y7dj7kts.fsf@phiwumbda.org> <876fa$472843f6$82a1e228$18666@news1.tudelft.nl> <873avr7ca7.fsf@phiwumbda.org> So, even if I don't make use of (5-8), a proof of A from (1-4) is a >proof from (1-8) ? Of course. >So, even if I say there exists a Foo, then such a statement is a >valid premise for proving that the integral of 1/t from 1 to x is >ln(x) ? Weird .. The statement can be proved in the theory consisting of the usual > axioms for real analysis and there exists a Foo, yes. Do you think > that every theorem of ZFC uses every axiom of ZFC in its proof? It seens we have a different picture in our mind about the meaning of an > implication A => B , as has been pinpointed by Ullrich as well. This is Another philosophical note is in place, when we are saying that we make with > an axiom and denote this as an implication A => B. In common mathematics, > the implication => just means what is de ned by a truth table in propositional > logic. But there is another form of mathematics, called constructivism. Within > constructivist mathematics, an implication has a more operational meaning, > like: given A, we can construct B from A. So if we say make with an axiom, > then it is expressed herewith that we adhere to the constructivist meaning of > an implication. End of philosophical note. I think that you and Ullrich adhere to the common material implication > of mathematical logic, where there is no place for axioms that cause a > theorem (so to speak). In the latter sense there is no room for premises > like there exists a Foo. The axiom of Infinity is of the latter kind. Then please specify an exact constructivist logic. Because, for example, I am not aware that intuitionistic logic (even with its semantics for '->') contradicts monotonicity of deduction (if someone informs me that intuitionisitc logic does contradict monotonicity of deduction, then I'll look into that). So, if your logic is not intutionistic, please specify your system of logic. MoeBlee === Subject: Re: Configurational Entropy Problem Eric === Subject: Re: Epistemology 501: Angular Mechanics Lester Zick: > You seem to consider that angular momentum is effected by material > things like sticks and strings. (BTW, everything as affected by _material_ things, and physics onlt describes the interaction of material things, not vice versa!) It was just an example of why angular momentum increases with radius. Here's another example: when a figure skater performs a rotation, he starts revolving a relatively slow angular speed but with his armes drawns aside (big R), then he suddenly presses his arms to his body (lesses R) and starts to revolve faster! The angular momentum being the same, the lesses the radius, the faster the speed (and hence monetum). I actually don't see any prioblems. This definition and theory describes such phenomena perfectly. If you don't like it, provide us a situation wher the theory disagrees with experiment, or where the theory is self-contradicting. Until you have demostrated of these arguments, all your swear at angular mechanics is nothing else than mere lack of undersatnding, rater that some fault of the treory! and constant but the product L=r x p is infinite because r=00 since there is no transverse acceleration and no rotation. > and constant but the product L=r x p is infinite because r=00 since > there is no transverse acceleration and no rotation. Huh! If you choose some infinitely distant axis (R=inf), then of course angular momentum will be infinite. But at normal (finite) values of R angular momentum will be also normal (finite). It is impossible to turn a material point using an infinitely long lever! That'd require an infinite moment of force (torque). > is with transverse centripetal acceleration a=00. Yeag, there are no pointa in reality. If a point has a finite size it can't have an infinite centripedal acceleration. However, the smaller the object, the easier it is to set it into rotation about it's own axis. As a logical consequence, an abstract point as infinitely easy to set into rotation. This easiness is reflected by zero angular momentum. > Sure. But the problem is transverse centripetal acceleration and r are > inversely related in mechanical terms. And that's pretty correct if are calculating centripedal force. And it has been proffed experimentally. > You don't have any problem with infinite angular momentum for motion > in a straight line without any transverse centripetal acceleration to > produce rotation? I repeat, you're free to choose the axis of rotation. Don't choose it excessively far and it'll be alrigt. Linear motion contains no real rotation, so you can't represnt it as rotation (around a motionless axis) with a finite ang. momentum! > Yes but the origin is defined in terms of linear momentum p=mv and > transverse centripetal acceleration and without that there is no > rotation. Ok, but the ifiniteness of ang. momentum in this case shows only one thing: you'll need an infinite torque to apply near the infinitely distant axis to affect the object's linear motion. However, you may choose a finite R and represent linear motion as rotation, but you have to define the origin separately for every point of the trajectory. FINAL QUESTIONS 1. Is there any self-contradiction? 2. In there any disagreement with reality (experiment)? Hope, your answer will be mathematically correct, and you won't get off with enumarating various facts about angular mechanics and saying you don't like them. You may or may may not like facts, but they're facts! P.S.: All this angular stuff _directly_ follows from the linear mechanics (laws of Newton) and is nothing more than a human-oriented and useful application of them to rotational motion. If there were angular mechanics you'd have to use the fundamental equations, which would be more laborous, and would get the same results! That's all. P.P.S: For what purpose are writing all this? If angular mechanics didn't work it it would have been discovered right away, not 300 years later! === Subject: Re: Converting deg to grads > Hi folks, > How can i convert degs and radians to grads . What is > the formula. > Hi folks, > How can i convert degs and radians to grads . What is > the formula. http://en.wikipedia.org/wiki/Grad_(angle) === Subject: Re: a problem in elementary number theory <2007103113283550073-kirakun@earthlinknet Since p^2+1 always have a factor of 2, it remains to find a p such that > p^2+1 cannot be divisible by 2^2 x 5. This is easy, pick the case for > p=9. Then > 9^2+1 = 82 = 2 x 41. > This concludes that the answer is 2^4 x 3 x 5 = 240. But 9 is composite! Stan === Subject: Re: a problem in elementary number theory >On 2007-10-31 13:24:36 -0400, Rainer Rosenthal said: > > I am stuck solving this problem from GRE Math training booklet: > Find the maximal integer x such that x divides p^4-1 for all prime > numbers p > 5. > [they actually have a list to choose from: 12, 30, 48, 120, 240] > Do you have any ideas? > > ... > So, the answer must be divisible by 2^4 x 3 x 5 = 240. This answer is > one of the choices on the list. Right, so you've proved that if x is the required maximum common divisor, x must be a positive integer multiple of 240. Thus, in the context of the multiple choice test, x = 240 is the only choice that qualifies. But you haven't proved that x = 240. How do know it's not more? > And there is no larger x dividing all these p^4-1 since for > p=7 and p=11 we have gcd(7^4-1,11^4-1) = 240. Still exploiting test cases (and there is nothing wrong with that), but >this is still nice and succinct to argue conclusively. No -- you need a test case to finish the proof -- to show that x is not more than 240. quasi === Subject: Re: Can we find the function? > Let (f o f)(x) = -2x + 3 Can we find the original function f ? How about f(x) = Sqrt(-2) x + 1 - Sqrt(-2) ? (Of course, I'm not saying that is the only such function.) David Can we answer this question instead: Is it possible for the > composition of two non-polynomial functions, to be a polynomial > function? Sure: log(exp(x)) = x. David > > Ok, you're right, but let me change my question. Can the composition > of a non-polynomial function with itself be a polynomial, excluding > monomials? > f(x) = x^sqrt(2), f(f(x)) = ? -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === Subject: Re: Can we find the function? <20071031085520.848$r8@newsreader.com> <20071031094457.518$nF@newsreader.com > Let (f o f)(x) = -2x + 3 Can we find the original function f ? How about f(x) = Sqrt(-2) x + 1 - Sqrt(-2) ? (Of course, I'm not saying that is the only such function.) David Can we answer this question instead: Is it possible for the > composition of two non-polynomial functions, to be a polynomial > function? Sure: log(exp(x)) = x. David Ok, you're right, but let me change my question. Can the composition > of a non-polynomial function with itself be a polynomial, excluding > monomials? f(x) = t^sqrt(2) + t^(-sqrt(2)) where t = (x + sqrt(x^2 - 4))/2 f(f(x)) = x^2 - 2