>Make no mistake, I'm in this for the money. If you really believe that someone is going to pay you for discovering a new mathematical fact, I'd like to help you invest that money so that it will do me, er, I mean _both_ of us some good. There's this bridge that connects Manhattan to New Jersey. No, no, not _that_ one, the other one! Got it? Good. Now I have it on excellent authority that the bridge is for sale to a qualified loony, er, I mean party, because the Port of New York Authority needs cash. The deal involves a few million up front plus a percentage of the tolls for the next three centuries. I'm sure that with your enormous arithmetical skill, you will see instantly that purchasing this bridge will make you very, very rich - far richer than mere mathematical discoveries will ever make you. For pointing you in the direction of this investment, and brokering the deal on your behalf, I'm asking a piddling 30% of net profits. blind.owl@third.tree.from.the.corner.com, and we can work out a deal. Be sure to use RSA encryption - otherwise some damn unethical mathematician will be able to sneak in and take advantage of this opportunity before we can do so. Don't delay! This is a once in an evening opportunity. Oh, and you need not send me the key - I've found a method of decrypting RSA, which I will share with the world as soon as it recognises my superior genius and provides me loadsadough and excellent babes - a genius superior even to yours, I'm sad to say, but them's the breaks of the genetic lottery. -- Wolf Kirchmeir, Blind River ON Canada Nature does not deal in rewards or punishments, but only in consequences. (Robert Ingersoll) ==== > > Well I claim that my prime formula is a great discovery, while others > keep posting that it's not important at all! > > Yes, that would be because it's not important at all. > > -jcr Really? Then clearly if you *know* that so that you can be so certain, you can explain how it works, right? Now then, why don't you try and explain how it works. For your reference, here it is again: dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, sqrt(y-1))], S(x,1) = 0. And p(x, y) = floor(x) - S(x, y) - 1, and you get S as the sum of dS from dS(x,2) to dS(x,y). The count of primes is given by p(x, sqrt(x)), now then John C. Randolph, why don't you explain why. Can you? Or are you just another Usenet loser trying to blow off steam at my expense? James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== Well I claim that my prime formula is a great discovery, while others > keep posting that it's not important at all! Yes, that would be because it's not important at all. Really? Then clearly if you *know* that so that you can be so > certain, you can explain how it works, right? Now then, why don't you try and explain how it works. Why don't *you* explain how it works? If your discovery is as important as you claim it is, the proper course of action is to write a paper describing its operation in great detail. Either publish the paper in a reputable journal or publish it yourself. But why in the world are you asking *other* people to explain how it works? You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen. -- There are two things you must never attempt to prove: the unprovable -- and the obvious. -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== > Well I claim that my prime formula is a great discovery, while others > keep posting that it's not important at all! Yes, that would be because it's not important at all. -jcr Really? Then clearly if you *know* that so that you can be so > certain, you can explain how it works, right? Now then, why don't you try and explain how it works. > > Why don't *you* explain how it works? If your discovery is as important as you claim it is, the proper > course of action is to write a paper describing its operation in great detail. Either publish the paper in > a reputable journal or publish it yourself. But why in the world are you asking *other* people to explain > how it works? I've explained in the past but noticed that other posters would just use information I provided to try and confuse others. Yup, they'd pervert the process. Here, by not explaining first in this thread, I'm showing readers that all these people trying to convince them that my prime area discovery is in fact unimportant, are in fact, lying. If they're not lying then they have the expertise to answer the question of how my partial difference equation works. > You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen. > The essential point is that I'm someone who made a nice discovery in the area of prime numbers, but rather than at a minimum acknowledge my discovery, mathematicians chose to ignore or downplay it. It's clearly a political decision at odds with claims of mathematicians about the importance to them of pure math or beauty, or math for math's sake. It's also a high-risk strategy which clearly could back-fire with stupendous consequences, so why would they do it? I want you to think about that question without me trying to explain it to you, except to remind you of others in a high stakes area engaging in seemingly strange behavior. Like consider the current president of the United States. James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== [snip] > Why don't *you* explain how it works? If your discovery is as important as you claim it is, the proper > course of action is to write a paper describing its operation in great detail. Either publish the paper in > a reputable journal or publish it yourself. But why in the world are you asking *other* people to explain > how it works? I've explained in the past but noticed that other posters would just > use information I provided to try and confuse others. Yup, they'd > pervert the process. What process? Nothing prevents you from publishing a complete, detailed exposition of your discovery with unambiguous, step-by-step examples of its operation and explanation of the operating theory. Any decent researcher would do so as a matter of course. > Here, by not explaining first in this thread, I'm showing readers that > all these people trying to convince them that my prime area discovery > is in fact unimportant, are in fact, lying. Unimportant? To whom? Each reader is entitled to judge its importance for themselves. It is decidedly unimportant to me and I'm entitled to say so. So far, it appears that your discovery is only important to *you*. You are entitled to make that judgment. Others are equally entitled to make theirs. > If they're not lying then they have the expertise to answer the > question of how my partial difference equation works. If they are not lying, then they are sincere in their judgment that your discovery is unimportant. If it is unimportant (to them) they will have little interest or motivation in doing your work for you. It is *your* job to explain how it works. It is each reader's job to determine whether your discovery is of any use to him. > You have succeeded in elevating idiocy to unprecedented heights, Wacky. Go back to your playpen. > The essential point is that I'm someone who made a nice discovery in > the area of prime numbers, but rather than at a minimum acknowledge my > discovery, mathematicians chose to ignore or downplay it. Tough luck. No one is under any obligation to value your work. Each reader has a right to ignore your work or to downplay or even condemn it. > It's clearly a political decision at odds with claims of > mathematicians about the importance to them of pure math or beauty, > or math for math's sake. Political? How does politics enter into this? The issues of pure math, beauty, or math for math's sake are philosophical matters, not political ones. The choice to ignore or downplay your work is a judgment call that each reader is entitled to make -- by right. And it stretches credulity to imagine that political motives have any role in that choice. > It's also a high-risk strategy which clearly could back-fire with > stupendous consequences, so why would they do it? Why high-risk strategy? There's no strategy at all in evidence, much less any risk. You offered a solution to a prime counting problem, others are unimpressed. That's their prerogative. > I want you to think about that question without me trying to explain > it to you, except to remind you of others in a high stakes area > engaging in seemingly strange behavior. Like consider the current > president of the United States. What high stakes. What has the president got to do with your discovery? Are you mad???..Or are you just a legend in your own mind? -- O wad some Pow'r the giftie gie us, to see oursels as others see us! (from: To A Louse, by Robert Burns.) -- Democracy: The triumph of popularity over principle. -- http://www.crbond.com ==== Already got one reply, and it's not worth discussing much. Seems computer scientists defer to mathematicians. Oh well, not really a big surprise. __JSH > Well, I've started, and like in the past, I may post responses from > editors, as they can be so amusing. And yes, for those of you who > don't know, I've got a nice collection of interesting responses from > journal editors! > > ___JSH > > Why not submit your code to a numerical analysis or computer > journal or maybe even take it to the Computer Science > Department at Vanderbilt and see what they think? > I'll tell you why not. Because we both know that > it is not at the level of original research. It's so child-like > to maintain the fantasy of being a misunderstood genius. > Your childhood glory days are over, James. Now you're a > grown-up troll and/or crank. If you can't face it hoist another > > Wait though, it seems to me that the need of the poster Uncle Al to > question the correctness of my work shows I think the popular feeling > that a valid result in the area of prime numbers *should* be worth > noting, especially by mathematicians!!! > > The math formula is > > dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, > sqrt(y-1))], > > S(x,1) = 0. > > And p(x, y) = floor(x) - S(x, y) - 1, where you get S(x,y) as the sum > of dS from dS(x,2) to dS(x,y). > > And, amazingly enough, p(x,sqrt(x)) gives the count of prime numbers > up to and including x, like p(100,10) = 25, the count of primes up to > and including 100. > > I'll also include a straightforward Java implementation, which also > prints out prime numbers found by my formula along the way! ___JSH > ==== > That's a red herring as the speed issue is secondary. I admit that a > straightforward implementation directly from the math is slow, but my > point is uniqueness, beauty and purity, as in pure math. You have written various versions, some simpler, some more complicated, some faster, some slower. I have posted code that is simpler than your simplest version, uses less memory, and runs faster. I have also published code that runs about 1000 times faster than your fastest version. And as far as beauty is concerned, any experienced Java programmer having a look at your code will just throw up. ==== > However, I have a suspicion that the people who've been boldest in > claiming that my work isn't important are ones who can't explain how > it works!!! On my webpage www.cbau.freeserve.co.uk which contains my source code for a prime counting algorithm that beats yours by a factor of about 1000 for large values, there is also a .pdf file MeisselLehmerAlgorithm.pdf which on the first one and a half pages describes the Legendre algorithm which is equivalent to your algorithm, and describes in two sentences the basics of Meissel's algorithm and Lehmer's algorithm which are far superior. It is quite sad that you never bothered to improve your algorithm by at least using Meissel's method which is really very easy. If you had done that, your algorithm wouldn't choke on memory consumption for smallish results like pi (10^16). genuinely _new_ stuff, unlike yours! ==== > Well the data are coming from a population that follows a > normal distribution, in my case is just data from a disease, > but in these data part of them are coming from another > different disease. The fact is the second disease values > are always much bigger than the mean of the previous one > and the problem is they are not so numerous to separate then > with a mixture of gaussian. Because of they are a small number > my algorithm does not works. And I need to separate them > because they will be noise in my results. Looking to the > histogram they can consider outliers(because they are > far away from the peak) so stimating the variance I can > take the 90 % of the data that belongs to the first disease > and do my results with them Well, from this description it seems clear that the observations are described well by a mixture model, with one bump for each disease. Making a hard assigment of each observation to one bump or the other is an approximation to the right solution, which is to count each observation in proportion to how well it belongs to each bump. If there is a lot of overlap, partial assignment yields substantially different results than all-or-nothing assignment. It's not very difficult to work with partial assignments, so I don't see that there's much to gain by thinking up various hacks. Another consideration is that disease #2 may be of greater importance in some slightly different context; why not get in the habit of working carefully, so that you can say something interesting about both diseases. For what it's worth, Robert Dodier -- ... much of what is called rational seems more like rationalization, to me; a ruse intended to make something desired appear necessary. -- Jeff Inman ==== > I've always understood the ideal gas law PV=nRT to refer to an ideal > gas contained in a container with a boundary, the boundary playing some > role in the definition of pressure. On the other hand, suppose you > consider an ideal gas in a compact manifold without boundary, e.g. > the 3-sphere. How would one formulate the ideal gas law in that > context or in the full generality of manifolds? Surely someone > must have worked that out, and references to relevant literature > would be helpful. > > Would it simply be used to *define* pressure in the case of a compact > manifold without boundary? You can turn the gas law from global to local by dividing through by volume: P = nRT where P is the pressure, n is the density in moles/unit volume, and T is the temperature. Pressure can be measured by inserting a small, evacuated box into the gas and measuring the force inward on its walls. Dale ==== It's well-known that compact (real) 2-manifolds have little variety, they are just spheres with zero or more toruses and cross-caps grafted on. If you look at the orientable manifold of genus 1, the torus, it is homogeneous in that there is a homeomorphism that takes any point to any other point. And it seems to be isotropic, in that there is a homeomorphism that rotates the neighborhood of any point in any way you want. If you look at manifolds with an ordinary metric structure (differentiable manifolds with metric?), this breaks down. For instance, the torus made by identifying the opposite sides of a square (R^2 mod Z^2) is homogeneous, but it isn't isotropic, because the four shortest non-trivial loops from a point to itself are aligned on the axes of R^2. Is there a torus as a manifold with a metric structure that is both homogeneous and isotropic? What about higher dimensions? Is there a general classification theory of manifolds with metrics? Dale ==== > It's well-known that compact (real) 2-manifolds have little variety, > they are just spheres with zero or more toruses and cross-caps grafted > on. If you look at the orientable manifold of genus 1, the torus, it > is homogeneous in that there is a homeomorphism that takes any point > to any other point. And it seems to be isotropic, in that there is a > homeomorphism that rotates the neighborhood of any point in any way > you want. > As you probably know, every smooth manifold admits a self-diffeomorphism that exchanges any two of its points (and more generally, that achieves an arbitrary permutation of any of its finite pointsets [that diffeo- morphism will of course depend on what subset you're looking at, and what permutation you have in mind]). However, I'm sure I don't understand your notion of isotropic, since for any manifold M, if you give a point x and a sufficiently small neighborhood U of x in M, there is self-diffeomorphism of M that rotates that neighborhood any way you want (i.e., given an element R of SO(n), there is a map f: M ---> M, that fixes x, and maps U to itself via M as follows: f U ----> U | | | | | | V V D^n --> D^n R where the vertical maps are the (same) coordinate map to a disc in R^n). That map f can also be fixed as the identity outside an epsilon neighborhood of U. So, you clearly mean much more than this. On the other hand, as far as I know, there is no effective circle action that fixes any point of T^2 (i.e., an action by S^1 = SO(2), for which every non-identity element actually moves some point). Since S^1 is connected, any circle action will consist of maps homotopic to the identity, sending the obvious generators of H_1 to themselves. This seems to be a version of the problem you identify below [viz, the four shortest non-trivial loops]: > If you look at manifolds with an ordinary metric structure > (differentiable manifolds with metric?), this breaks down. For > instance, the torus made by identifying the opposite sides of a square > (R^2 mod Z^2) is homogeneous, but it isn't isotropic, because the four > shortest non-trivial loops from a point to itself are aligned on the > axes of R^2. > > Is there a torus as a manifold with a metric structure that is both > homogeneous and isotropic? > > What about higher dimensions? Is there a general classification > theory of manifolds with metrics? > Classification of maps among manifolds, or of maps from a manifold to itself, has been an active research area: the buzz word is mapping class group meaning the group of smooth isotopy classes (i.e., homotopy through self-diffeomorphisms) of self-diffeomorphisms. It is related to study of the homotopy theory of the space of self-diffeomorphisms. Since homotopy reduces continuous questions to discrete ones, perhaps it's not what you care to look at. There are also the areas of homogeneous spaces (spaces with a transitive action by a Lie group), and group actions on manifolds. Somewhere in my distant past I recall hearing people discuss (as a measure of symmetry of a manifold M) the largest dimension of a Lie group that had an effective action on M; the upshot of the discussion was the result that among the various exotic differential structures on spheres [I don't recall whether this was specifically S^7, or N in general], only the standard, round sphere could achieve the value of N(N-1)/2, the dimension of SO(N+1). Along these lines, I came across this paper by Volker Puppe: Do manifolds have little symmetry? www.inf.uni-konstanz.de/Schriften/ papers/2002/preprint-181.pdf > Dale Dale. ==== > It's well-known that compact (real) 2-manifolds have little variety, > they are just spheres with zero or more toruses and cross-caps grafted > on. If you look at the orientable manifold of genus 1, the torus, it > is homogeneous in that there is a homeomorphism that takes any point > to any other point. And it seems to be isotropic, in that there is a > homeomorphism that rotates the neighborhood of any point in any way > you want. > > If you look at manifolds with an ordinary metric structure > (differentiable manifolds with metric?), this breaks down. For > instance, the torus made by identifying the opposite sides of a square > (R^2 mod Z^2) is homogeneous, but it isn't isotropic, because the four ^^^^ > shortest non-trivial loops from a point to itself are aligned on the ^^^^^^^^ How did you get four? I only see two... > axes of R^2. > > Is there a torus as a manifold with a metric structure that is both > homogeneous and isotropic? > > What about higher dimensions? Is there a general classification > theory of manifolds with metrics? Maybe you are looking for *Zoll manifolds*, Riemannian manifolds whose geodesics are all simple closed curves of equal length. If yes, you will find a lot of interesting results in A. L. Besse, Manifolds All of Whose Geodesics Are Closed (Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge. A Series of Modern Surveys in Mathematics Bd. 93) By a result of Bott and Samelson, every Zoll manifold is topologically a CROSS (compact rank-one symmetric space), and so must be Sn , RPn, CPn, HPn, or the Cayley plane. ==== >> Tired light. IMO tired light is the true cause of the red-shifying. [snip in agrrement ] >> Tired light. Occam's razor. >> John >Tired light doesn't work. I've repeated ad nauseam what 'will' work, and >why nothing else will. To wit, relative motion of source and observer, >or a source frequency shift over time. Any other proposals result in a >loss of information. Frequency is by definition cycles of some system >per unit time. A cycle is an absolute event and the information >associated with that cycle is just the energy-change propagating away >from the source of the 'change'. Since observed frequency is just the >result of interception of the outgoing information, then for bodies at >rest wrt each other you have simply: f * delta t = f' * delta t' Here the time intervals are just those required to 'experience' a given >absolute portion of the information stream. If f' < f, then delta t' > delta t Thus if the clocks measuring these intervals are at rest wrt each other, >then they must be ticking at different rates. If OTOH it is noted that >delta t was the time interval wrt the source at the era of emission of >the information stream, and that this was millions of years ago in our >present discussion, then it is not necessary that the clocks be >simultaneously ticking at different rates, only that the rate was >different at that time than it is now. Tired light theory assumes no >change in ticking rates of the clocks, which latter may very well be the >very radiation source and detector respectively. Neither does it assume >a simultaneous difference in ticking rates between the clocks, and thus >it cannot account for the doppler shift without requiring a loss of >objective events (cycles) wrt one of the frames. In the case that the clocks are not at rest wrt each other, then changes >in ticking rates are unnecessary in order to account for the red shift, >the relative motion is sufficient to produce ordinary doppler. Special >relativity OTOH assumes both, i.e. that clock's rates shift 'and' there >is relative motion between source and detector. Tired light in this >context still involves the loss of information, and thus is incorrect, >no question. >Richard Perry (yes question:-) I like Richards approach but disagree with his conclusion. I do agree with Richard, it is a fact that INFO = frequency * duration = invariant. where Richard calls INFO,. A cycle is an absolute event and the information... But the duration is expressed by (N = cycles) duration = N * wavelength, then INFO = N*c, (c=freq*wavelength). Because the wavelength increases there is no loss of information carried by the photon. Think about how information transfer rate increases proportional to frequency, and think about how no information is lost when a photon is reflected from a receding mirror. A receding mirror will red-shift the photon on reflection (doppler effect) but no INFO is lost. As a photon propagates across the empty voids of inter-galatic space, it is being deflected by all the matter that gravitationally influences that location, and gravitational deflection sucks photon momentum, and frequency, but INFO remains invariant. Ken S. Tucker ==== > >> Tired light. > > IMO tired light is the true cause of the > red-shifying. > [snip in agrrement ] > >> Tired light. Occam's razor. >> John > >Tired light doesn't work. I've repeated ad nauseam what 'will' work, and >why nothing else will. To wit, relative motion of source and observer, >or a source frequency shift over time. Any other proposals result in a >loss of information. Frequency is by definition cycles of some system >per unit time. A cycle is an absolute event and the information >associated with that cycle is just the energy-change propagating away >from the source of the 'change'. Since observed frequency is just the >result of interception of the outgoing information, then for bodies at >rest wrt each other you have simply: f * delta t = f' * delta t' Here the time intervals are just those required to 'experience' a given >absolute portion of the information stream. If f' < f, then delta t' > delta t Thus if the clocks measuring these intervals are at rest wrt each other, >then they must be ticking at different rates. If OTOH it is noted that >delta t was the time interval wrt the source at the era of emission of >the information stream, and that this was millions of years ago in our >present discussion, then it is not necessary that the clocks be >simultaneously ticking at different rates, only that the rate was >different at that time than it is now. Tired light theory assumes no >change in ticking rates of the clocks, which latter may very well be the >very radiation source and detector respectively. Neither does it assume >a simultaneous difference in ticking rates between the clocks, and thus >it cannot account for the doppler shift without requiring a loss of >objective events (cycles) wrt one of the frames. In the case that the clocks are not at rest wrt each other, then changes >in ticking rates are unnecessary in order to account for the red shift, >the relative motion is sufficient to produce ordinary doppler. Special >relativity OTOH assumes both, i.e. that clock's rates shift 'and' there >is relative motion between source and detector. Tired light in this >context still involves the loss of information, and thus is incorrect, >no question. >Richard Perry > > (yes question:-) > > I like Richards approach but disagree with > his conclusion. I do agree with Richard, it > is a fact that > > INFO = frequency * duration = invariant. > > where Richard calls INFO,. A cycle is an > absolute event and the information... > > But the duration is expressed by (N = cycles) > > duration = N * wavelength, then Wrong time interval source = N / frequency frequency = c/wavelength Thus time interval source = N * wavelength/ c > > INFO = N*c, (c=freq*wavelength). INFO = N > > Because the wavelength increases there is > no loss of information carried by the photon. True, but how are you going to shift the wavelength? Each end of the wave represents the information of the beginning of two separate objective events, e.g. cycles, both of these information 'bits' is propagating at the same speed wrt the observer, and thus their instantaneous displacement in space wrt each other is fixed wrt the observer from the time of its emission to its absorption, i.e. throughout its entire trip. The only way to change a wavelength in flight is to change the speed of the wave, OTOH, this will not change > > Think about how information transfer rate > increases proportional to frequency, and > think about how no information is lost when > a photon is reflected from a receding mirror. > A receding mirror will red-shift the photon on > reflection (doppler effect) but no INFO is > lost. > > As a photon propagates across the empty voids > of inter-galatic space, it is being deflected by all > the matter that gravitationally influences that > location, and gravitational deflection sucks photon > momentum, and frequency, but INFO remains > invariant. The frequency never changes wrt a given inertial frame. > > Ken S. Tucker Richard Perry ==== > Tired light. Besides, I don't think universal expansion is possible, do you? Who are you to say what's possible? > If I'm at some arbitrary center, and I'm expanding, you are also > expanding, but you must also be accelerating away from me, and > someone behind you has to accelerate away even faster, etc, etc, to > where things really far behind must be moving at incredible speeds > away from us. Tired light. Occam's razor. One of Fred Hoyle's theories was that the universe isn't expanding, the atoms inside are shrinking. That would produce an apparent red shift since the light would have been emitted by larger atoms emitting longer wavelengths. -- http://hertzlinger.blogspot.com ==== > Within a week or so, I will be releasing a free beta version download for my > new proof checking software, DC Proof 1.0. > > In the mean time, here is a sampler from the User Guide: > > http://members.allstream.net/~dchris/DCProofT.chm > > It contains a tutorial that illustrates many of the main features of DC > Proof. Readers may be interested in both theoretical and a pedagogical > aspects of this application. Example 3, is a resolution of Russell's Paradox > without the usual prohibition on self-reference. > > Enjoy. > > Dan Christensen > Toronto, Canada Your system appears to be a great aid to logicians writing proofs, and will also hopefully lend more insight into the exact nature of proofs. I will be happy to obtain a copy. While a great bookkeeping aid, your system doesn't seem to do anything that isn't being done by hand already. Indeed, the user explicitly enters the proof itself. This is a helpful tool, but I don't see how you have resolved the Russell Paradox. You have only computerized the same proof that is written out by hand. You correctly (IMHO) conclude that there is no Russell Set (the set of sets that don't contain themselves), which is the common conclusion one reaches from seeing the contradiction. However, the question remains, what do we do about it? How do we define sets to include the sets that mathematicians use, but exclude the Russell Set? Are you simply saying don't allow it? But what do we allow? Everything else? ZF and various other axiomitizations of set theory attempt to define sets in a way that that meets these two needs (completeness without contradiction.) Do you really have a solution to that problem? Charlie Volkstorf Cambridge, MA http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 http://www.arxiv.org/html/cs.lo/0003071 ==== Within a week or so, I will be releasing a free beta version download for my new proof checking software, DC Proof 1.0. In the mean time, here is a sampler from the User Guide: http://members.allstream.net/~dchris/DCProofT.chm It contains a tutorial that illustrates many of the main features of DC Proof. Readers may be interested in both theoretical and a pedagogical aspects of this application. Example 3, is a resolution of Russell's Paradox without the usual prohibition on self-reference. 46 ==== Dear Dr. Mehran Basti, please, how to solve the inverted pendulum nonlinear differential equation system by Riccati equation. The differential equations are: 2 (M+m)x««+cx«-mlcos(fi)(fi)[Lef tGuillemet]«+mlsin(fi)(fi)« =f(t) 2 -mlx««cos(fi)+(ml +I)(fi)««+b(fi)«+mglsin(fi) =0 These two second order differential equations can be transfered into four first order ones: /*- the four differential equations system -----------------*/ double TNTargetF1(double T, double *V) { return V[2]; } double TNTargetF2(double T, double *V) { return ((m*l*sin(V[3])*V[4]*V[4]+c*V[2]+F(T))*(m*l*l+I)/(m*l*cos(V[3]))+b*V[4]+m*g*l *sin(V[3])) / (-m*l*cos(V[3])+(m*l*l+I)*(M+m)/(m*l*cos(V[3]))); } double TNTargetF3(double T, double *V) { return V[4]; } double TNTargetF4(double T, double *V) { return (F(T)+(M+m)*(b*V[4]+m*g*l*sin(V[3]))/(m*l*cos(V[3]))+c*V[2]+m*l*sin(V[3])*V[4 ]*V[4]) / (-m*l*cos(V[3])+(m*l*l+I)*(M+m)/(m*l*cos(V[3])));} /*----------------------- input finish -----------------------------------*/ I am able to solve it by Adams method. How to solve it by Riccatti equations? Michal Martinu ==== > Are there any number-theoretic statemtents which are known to be > undecidable in ZFC? If so, some examples? > > This is a follow-up to some comments in the 2^pi thread, where JIP, I > believe, asked about unknown results in number theory, and GM responded > with something about decidability. > > This may depend, of course, on what you mean by number-theoretic. > With a slightly liberal interpretation, the following is an admissible > solution, I guess: If a family of sets of natural numbers cannot be > put into one-to-one correspondence with the natural numbers, can it > then be put into one-to-one correspondence with the family of all sets > of natural numbers? > > I imagine, however, that this isn't what you had in mind. A better > solution, but one I'm not entirely sure about, may a form of > Matijasevic's theorem: there is (perhaps?) a family of diophantine > equations (namely, a single equation with one of the variables treated > as a parameter) such that the set of values of the parameter for which > the equation has a solution cannot be determined within ZFC. Once > again, I'm not quite sure this statement is correct - I'll be glad to > know if it is. reviewed in the Bulletin of the AMS in January 1995 by C. Smorynski. Here's a quote from the Book review: Matiyasevich's most important contribution since solving the problem has to be his introduction of new exponential Diophantine coding techniques. With such, he improved the initial Matiyasevich-Robinson small-number-of-variables result from the algorithmic unsolvability of the general 13-variable Diophantine problem to the algorithmic unsolvability of the 9-variable problem. If ZFC could decide every 9-variable problem, then I think the 9-variable problem would be algorithmically solvable. So I think some 9-variable problem is undecidable in ZFC. Here's a link to the book review: http://www.ams.org/journals/bull/pre-1996-data/199501/199501014.html David Bernier ==== > If ZFC could decide every 9-variable problem, then I > think the 9-variable problem would be algorithmically solvable. ??? why ??? ==== > I think perhaps the question is really: Are there any independently > interesting number theoretic statements which are undecidable in ZFC? Do a search for Paris-Harrington & see what you think. -- ==== > > I think perhaps the question is really: Are there any independently > interesting number theoretic statements which are undecidable in ZFC? > > Do a search for Paris-Harrington & see what you think. Isn't this an example of independence within PA, but not ZFC? I know that Goodstein's theorem is of this kind (unprovable in PA, true in ZFC, actually in PA+some simple ordinals) and I thought that Paris-Harrington was, too; perhaps I got it wrong. ==== > > I think perhaps the question is really: Are there any independently > interesting number theoretic statements which are undecidable in ZFC? > > Do a search for Paris-Harrington & see what you think. > > Isn't this an example of independence within PA, but not ZFC? I know > that Goodstein's theorem is of this kind (unprovable in PA, true in > ZFC, actually in PA+some simple ordinals) and I thought that > Paris-Harrington was, too; perhaps I got it wrong. You're probably right, most likely I was confused. -- ==== > > I think perhaps the question is really: Are there any independently > interesting number theoretic statements which are undecidable in ZFC? > > Do a search for Paris-Harrington & see what you think. Paris-Harrington is provable in an extension by reflection of PA. ==== This question is related to the planning of electric systems. I have a sequence of functions (c_n), defined on [0, inf) and with values in this same set. These functions, usually called marginal costs, are strictly increasing but have discontinuities. They all have a same set D of discontinuities on [0, inf), which is finite. In addition, I know that, for every x>=0, the sequence (c_n(x)) is strictly increasing and converges to a c(x), but I'm not sure if the convergence c_n -> c is uniform. All I can assure is that it's pointwise. What I want is to find a good approximation of the total cost for each x>0. So, I can integrate, at least numerically, the function c_n over [0, x], getting a function C_n that, for each x, gives the total cost of supplying the load x. But, are the given condition sufficient to guarantee this provides an accurate aproximation? If I could guarantee the set of discontinuities of c is again the set D, then I could apply Dini«s theorem to guarantee the convergence c_n -> c was piecewise uniform on [0, M] for some M. Therefore, the convergence of C_n to C would be piecewise uniform and I'd get a good approximation for the total cost. Does any one have a clue if, from the given conditions, there is something interesting we can conclude? Artur ==== > > This question is related to the planning of electric systems. I have > a sequence of functions (c_n), defined on [0, inf) and with values in > this same set. These functions, usually called marginal costs, are > strictly increasing but have discontinuities. They all have a same set > D of discontinuities on [0, inf), which is finite. In addition, I know > that, for every x>=0, the sequence (c_n(x)) is strictly increasing and > converges to a c(x), but I'm not sure if the convergence c_n -> c is > uniform. All I can assure is that it's pointwise. > > What I want is to find a good approximation of the total cost for each > x>0. So, I can integrate, at least numerically, the function c_n over > [0, x], getting a function C_n that, for each x, gives the total cost > of supplying the load x. But, are the given condition sufficient to > guarantee this provides an accurate aproximation? If I could guarantee > the set of discontinuities of c is again the set D, then I could apply > Dini«s theorem to guarantee the convergence c_n -> c was piecewise > uniform on [0, M] for some M. Therefore, the convergence of C_n to C > would be piecewise uniform and I'd get a good approximation for the > total cost. > > Does any one have a clue if, from the given conditions, there is > something interesting we can conclude? For each b > 0, Cn(b) -> C(b) by the monotone convergence theorem. This implies Cn -> C uniformly on each [0,b]: For any x in [0,b], 0 <= C(x) - Cn(x) = int_[0,x] (c - cn) <= int_[0,b] (c - cn) = C(b) - Cn(b); we used the nonnegativity of c - cn to get the second inequality. ==== >I have an unsolved problem. It's about a Diophantine Equation. >The problem is ... Find, with proof, all positive integers a and b such that >a^4+(a^2-1)^2=b^2. . The only rational numbers, c/b, close enough to sqrt(2) to suffice in equation [3] are from the continued fraction expansion of sqrt(2); i.e. 1 3 7 17 41 - , - , - , -- , -- , ... 1 2 5 12 29 Where both the numerators and denominators follow the recursive rule x = 2 x + x [4] n n-1 n-2 However, we can only use every other fraction, since equation [3] needs an underestimate; i.e. 1 7 41 - , - , -- , ... 1 5 29 Thus, if there is another pair (a,b) that satisfies [1], b needs to be greater than 10^7656. This seems to indicate that there may not be any pairs other than (1,1) and (2,5). However, I don't have a proof of this yet. Rob Johnson take out the trash before replying ==== To : Jose Carlos Santos > No. It 's not help anything. I well understand in pytagorus triples. > a^2 + b^2 = c^2 > it can be prove that : (a,b,c) = (st, (s^-t^2)/2, (s^2 + t^2)/2) > for s, t is odd and s > t >= 1 and gcd(s,t) = 1 > > or (a, b,c) = (2uv, u^ - v^, u^2 + v^2) > for u - v = odd, u > v >=1, and gcd(u, v) = 1 > > I try and try ..... to solve but it's not so children. You should to > solve and you will find that it's very very difficult. > > Now. I believe that there 's only 2 solutions. (a, b) = (1,1), (2,5) > But. I can't prove. > > No one can solve it. ??? !!!!!!!!!! There are several regulars here who I'm sure could solve it with enough effort, including possibly me (although I'm somewhat hampered by being criminally careless, as you noticed ;-( Why not try the same type of approach I used in case 2 of my 'solution'? If I get time I'll look at the problem again next week. --------------------------------------------------------------------------- John R Ramsden (jr@adslate.com) --------------------------------------------------------------------------- Eternity is a long time, especially towards the end. Woody Allen ==== Larry Hammick > phongthong > I have an unsolved problem. It's about a Diophantine Equation. > The problem is ... Find, with proof, all positive integers a and b such > that > a^4+(a^2-1)^2=b^2. . > aa = mm - nn > aa - 1 = 2mn > or > aa - 1 = mm - nn > aa = 2mn. > Therefore > (m-n)^2 - 2n^2 = 1 > or > (m-n)^2 - 2n^2 = -1. > These are Pell equations, and we can write down all the solutions. Let A be > the square matrix > 1 1 > 2 1 > and consider the powers A^n. The upper row (x,y) of A^n satisfies > xx - 2yy = -1 if n is odd > = 1 if n is even > and these are the only solutions of the two Pell equations for m-n and n. > But we also need > mm - nn to be a square (a^2). Still looking... As you thought, the only solutions for (a,b) are (1,1) and (2,5). I managed to prove it using the lemma below. Lemma: The equation x^4 - 2y^2 =1 has no solutions except y=0 and x=+-1. Proof: In the ring Z[i] we have this factorization: x^4-1 = (x-1)(x+1)(x-i)(x+i) and if we have (x-1)(x+1)(x-i)(x+i) = 2yy and y>0 then the number 1+i (which is prime in Z[i]) divides the left side an odd number of times (see why?), and the right side an even number of times, which cannot be true, since factorization in Z[i] is unique. Thus y=0, as claimed. The solution a=1 corresponds to a solution of the Pythagorean triple with a=mm-nn, and the other solution a=2 is from a=mn with m=2 and n=1. Larry ==== >Larry Hammick >> phongthong >> I have an unsolved problem. It's about a Diophantine Equation. >> The problem is ... Find, with proof, all positive integers a and b such > that >> a^4+(a^2-1)^2=b^2. . [snip] >As you thought, the only solutions for (a,b) are (1,1) and (2,5). I managed >to prove it using the lemma below. Lemma: The equation >x^4 - 2y^2 =1 >has no solutions except y=0 and x=+-1. >Proof: In the ring Z[i] we have this factorization: >x^4-1 = (x-1)(x+1)(x-i)(x+i) >and if we have >(x-1)(x+1)(x-i)(x+i) = 2yy >and y>0 then the number 1+i (which is prime in Z[i]) divides the left side >an odd number of times (see why?), and the right side an even number of >times, which cannot be true, since factorization in Z[i] is unique. Thus >y=0, as claimed. Suppose n is a real integer and that n is divisible by 1+i in Z[i]. Then n/(1+i) = n(1-i)/2 is in Z[i]. This means that n must be even and n/2 = in/(1+i)^2 is also a real integer. Thus, n is divisible by 1+i twice. So any real integer is divisible by 1+i an even number of times in Z[i]. Rob Johnson take out the trash before replying ==== Soit E l'ensemble {1,2,3}. Trouvez des relations R et S sur E qui sont transitives mais telles que R o S n'est pas transitive Est ce possible? ==== > Soit E l'ensemble {1,2,3}. Trouvez des relations R et S sur E qui > sont transitives mais telles que R o S n'est pas transitive Est ce possible? Yes, for example: R = { (1,2), (2,3), (1,3) } S = { (1,1), (2,2) } RoS = { (1,2), (2,3) } Dirk Vdm ==== >Soit E l'ensemble {1,2,3}. Trouvez des relations R et S sur E qui >sont transitives mais telles que R o S n'est pas transitive Est ce possible? (Sorry, my french is lousy) R o S = {(a,b) : (a,c) in R, (c,b) in S, c in {1,2,3} } R = {(1,1), (2,3)} est transitive S = {(1,2), (3,3))} est transitive R o S = {(1,2), (2,3)} n'est pas transitive. ====================================================================== It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes) ====================================================================== Arturo Magidin magidin@math.berkeley.edu ==== > I'm doing a report on Grice for an undergraduate philosophy class I'm > taking, and I came across this post: http://philosophy.wisc.edu/920/_disc2/00000015.htm which suggests that Grice's theory might fall apart because it forces > an infinite regress, the only way out of which is to say Grice's > theory is wrong. I've heard a few arguments against Grice's theory, > but never one like this...has anyone written on this infinite regress > in Grice's work? If someone could provide any other references to > explore Grice from this point of view, I would very much appreciate > it... 9 Speaker Meaning Grice's (1956) initial account of speaker meaning appealed to a self-referential intention. Speaker Meaning: A meantNN something by x is roughly equivalent to A uttered x with the intention of inducing a belief by means of the recognition of this intention (384). Grice added, This seems to involve a reflexive paradox, but it does not really do so. Later, as various complications were noted, Grice (1969) and Schiffer (1972) replaced the self-referential analysis with ones involving a series of intentions, with later intentions about the earlier intentions. This let to issues about the existence of the potentially infinite regress of intentions required, issues that could have been avoided by staying with self-referential formulations. http://www.princeton.edu/~harman/Papers/Adler.html [the reason there are so many ways to say the same thing is that meaning is simply a (convergence) upon similar effects of particular perceptions, irrelevent of truth or falsehood being relayed between people?] [he should have also noted ealier intentions about a range of future intentions to go along with the future intentions about past intensions] [it appears to be an extension in space AND time, though a series, there be no need to discard the self-reference or iterations since the series makes the iteration float] --------------------------------- Grice's concept of speaker's meaning was an ingenious refinement of the crude idea that communication is a matter of intentionally affecting another person's psychological states. He discovered that there is a distinctive, rational means by which the effect is achieved: by way of getting one's audience to recognize one's intention to achieve it. The intention includes, as part of its content, that the audience recognize this very intention by taking into account the fact that they are intended to recognize it. A communicative intention is thus a self-referential, or reflexive, intention. It does not involve a series of nested intentions --the speaker does not have an intention to convey something and a further intention that the first be recognized, for then this further intention would require a still further intention that it be recognized, and so on ad infinitum. Confusing reflexive with iterated intentions, to which even Grice himself was prone, led to an extensive literature replete with counterexamples to ever more elaborate characterizations of the intentions required for genuine communication (see, e.g., Strawson 1964 and Schiffer 1972), and to the spurious objection that it involves an infinite regress (see Sperber and Wilson 1986, whose own RELEVANCE theory neglects the reflexivity of communicative intentions). Although the idea of reflexive intentions raises subtle issues (see the exchange between Recanati 1986 and Bach 1987), it clearly accounts for the essentially overt character of communicative intentions, namely, that their fulfillment consists their recognition (by the intended audience). This idea forms the core of a Gricean approach to the theory of speech acts, including nonliteral and indirect speech acts (Bach and Harnish 1979). Different types of speech acts (statements, requests, apologies, etc.) may be distinguished by the type of propositional attitude (belief, desire, regret etc.) being expressed by the speaker. http://libra.sfsu.edu/~kbach/grice.htm [Iterative and recursive events are circular not straight lines of proofs each depending upon the other before it. Iteration that feedback onto itself what went before can alter its course by internal reference to points along the circular chain.] -------------------------------------- A theory of meaning within a linguistic system is another goal of the New Humanism, because, in pragmatic everyday practice and in literature, we do mean things by making utterances in a language. Thus, in the spirit of the above proof of the existential significance of Self, meaning exists, as well. Structuralist and post-structuralist criticism seems to forget the true purpose of language: to communicate. The importance of meaning as a function of communication brings us to the need for a theory of communication, if the meaning of meaning (or structurality of structure?) is to be made explicit. What is it then for an utterer to mean something in a language? Our model for determining this will be a Grice-Schiffer hybrid (see Paul Grice's Studies in the Way of Words and Stephen Schiffer's Meaning). This is an intentionalist approach in that it focuses on the subject (also, the self) as the source of the utterance and meaning as an emergent property that occurs between the utter u and audience A. In the modified, logical Gricean terms, the conception can be formed simplistically as such: an utterance p means x if utterer u intended to produce some effect E, in audience A, and the E is produced by A's recognition of u's intention to utter p to mean x (Grice 88). The Schiffer side of the hybrid incorporates the notion of mutual knowledge* in order to avoid the possibility of an infinite regress of intentions and recognitions (e.g. u intends that A recognize that u intends that A recognize that u intends that A recognizes that. . .). Mutual knowledge* is the knowledge that A knows that p, and B knows that p, and A knows that B knows that p, and B knows that A knows that p. In language and a definition of meaning, it is the recognition on behalf of both the speaker, S, and the audience, A, that there has been a precedence set in which x has taken on a property in relation to a circumstance or fact that is the object of expression p, such that when S wishes to express or mean p, S will utter x (Schiffer 30-31). The * following knowledge is to make clear that the mutuality of the knowledge is dependent upon the context of the environment in which meaning is engendered. Thus, we have provisional account of what it is to mean something pragmatically in a semiotic system. http://www.janushead.org/JHSpg99/orr.cfm -------------------------------------- inary' language. Grice is also subject to criticism from the two psychologists - for example, his idea of mutual knowledge is implausible in psychological terms - for to be manifest is weaker than to know or assume something, so is less implausible. Instead Sperber and Wilson posit a theory of mutual manifestness, which does indeed sidestep the infinite regress of Grice, and appeal to ostensive acts which alter the cognitive environment of both speaker and audience. Grice's notion of mutually accepted assumptions can also be criticised - in order for both agents in a discourse to know they share mutual knowledge, these assumptions need to be drawn by them both, and they both need to know this fact as well, and so on ad infinitum. Since a cognitive environment does not get caught in an infinite number of assumptions, Sperber and Wilson state that it can explain situations where information is exploited in a theory, unlike the Gricean perspective. They also claim that communication in general has the aim of increasing the mutuality of the cognitive environment...rather than guarantee...strict duplication of thoughts..., which of course is Grice's assertion of what implicatures achieve. By claiming that the speaker only wants to alter his own cognitive environment, not the thoughts of the hearer, Sperber and Wilson escape this. Sperber and Wilson, remember, posit two models - code and inferential, both of which are essential to explain human discourse. Grice's maxims seem to use just the inferential model, by inferring a set of conclusions from a set of premises. Moreover, their principle of Relevance suggest the existence of heuristics, some innate, some acquired via experience, and it implies that there is a degree of relevance in all communication. This can be worked out from contextual effect and processing effort. Grice merely appeals to us to 'be relevant' in his maxim of relation; further, that norms are acquired and need to be known for proper communication. In contrast, Sperber and Wilson argue that communicators do not follow their principle of relevance: indeed, they could not violate it even if they so desired. The principle of relevance is always commun ... http://tinyurl.com/x8uj [The assumption above, Since a cognitive environment does not get caught in an infinite number of assumptions is very problematical and stands as no condemnation of relationships in series be they circular or recursive] -------------------------------------- Good luck gotta go..here's the page I left off on: http://tinyurl.com/x8v0 ==== > I'm doing a report on Grice for an undergraduate philosophy class I'm > taking, and I came across this post: > > http://philosophy.wisc.edu/920/_disc2/00000015.htm > > which suggests that Grice's theory might fall apart because it forces > an infinite regress, the only way out of which is to say Grice's > theory is wrong. I've heard a few arguments against Grice's theory, > but never one like this...has anyone written on this infinite regress > in Grice's work? If someone could provide any other references to > explore Grice from this point of view, I would very much appreciate > it... Just about every philosophical theory I've seen goes into an infinite regress if pushed too hard. (This is why I no longer study philosophy--it's much too easy to be destructive instead of constructive) 'cid ==== Here's the post I was talking about: A note (or hypothesis) on Grice regress, but it makes Grice wrong. I'm finding it difficult to clearly explain how the regress arises. Here goes. Grice says that the hearer gets a strong conditional via a conversational implicature on a material conditional. That is, all that is actually *said* or *uttered* is a material conditional, and the rest has to be inferred from conversational context. This works by inference to the best explanation, or abduction. The hearer thinks, That utterance is strange in that it violates some Conversational Maxim(s), and I have every reason to suspect that the speaker would follow these Maxims under normal circumstances. In order to preserve the speaker's following the Maxims, I must take him to be trying to convey something not identical with what was, strictly speaking, uttered. Example: Professor B reads a letter of recommendation from Professor A regarding grad student C. The letter reads: C showed up to every class, and has an excellent command of English. There is a clear conversational implicature that C is not a good philosopher, and A does not recommend C to B. OK, so you need abduction in order to generate an implicature. The problem is that you must have, in your brain, a representation of strong conditionality in order to make an abduction. This is because explanations operate on strong conditionals. The thing doing the explaining (explanans) is related to the thing being explained (explanandum) in the same way as the antecedent is related to the consequent in a strong conditional. You must already be able to understand and manipulate strong conditionals in order to perform an abduction. So here's the regress. The speaker utters material conditional M. In connection with context C, the hearer gets strong conditional S by implicature (by getting S as the best explanation of why M was uttered in C). This means the hearer must be able to represent a strong conditional E to himself (or use it) in order to generate the abduction. But where did E come from? If it came from a separate implicature, then we have a further abduction which needed a strong conditional E*, and so on. The way to stop the regress is to say that E is not created by implicature, that we already have a mental symbol for Îstrong if Î. But if this is the case, then why can't we say that sometimes if means strong if, that if is ambiguous? That is, if E has to be primitive (in the sense that it was not generated by implicature), isn't that immediate evidence that there's more than one sense of Îif'? Grice seems keen on using the Razor to keep senses to a minimum. But here we seem forced into admitting one. And why do we need to keep senses to a minimum? It is not as though there are SENSES floating in platonic space, and we need to keep the world of Forms as small as we can. It's not as though, in saying that Îif' is ambiguous, we're postulating some *thing*. Does this regress idea make any sense to anyone? The complaint is essentially that strong conditionality cannot depend for its generation on our already having command of strong conditionality. For then we have another strong conditional to reduce, and it is really implausible to suppose that it is reducible in the same way. More generally, suppose our account of reduction looks like this: For any strong conditional S, S is reducible to a corresponding material conditional M plus some extra considerations XYZ. What I want to say about such an account is this: our extraction (or construction, or what have you) of S from M and XYZ must not require us to use/have/represent-to-ourselves another strong conditional T. For then what does T reduce to? Some M* plus XYZ? The way out of the looming regress is to take T as already in our possession independently of the S-XYZ story. But this leaves T unaccounted for, and seems to indicate that strong conditionals are not cashable in terms of material conditionals, or at least that not all strong conditionals are so cashable. > I'm doing a report on Grice for an undergraduate philosophy class I'm > taking, and I came across this post: > > http://philosophy.wisc.edu/920/_disc2/00000015.htm > > which suggests that Grice's theory might fall apart because it forces > an infinite regress, the only way out of which is to say Grice's > theory is wrong. I've heard a few arguments against Grice's theory, > but never one like this...has anyone written on this infinite regress > in Grice's work? If someone could provide any other references to > explore Grice from this point of view, I would very much appreciate > it... Any thoughts??? ==== > Here's the post I was talking about: > [snip] > ... So here's the regress. The speaker > utters material conditional M. In connection with context C, the > hearer gets strong conditional S by implicature (by getting S as the > best explanation of why M was uttered in C). This means the hearer > must be able to represent a strong conditional E to himself (or use > it) in order to generate the abduction. But where did E come from? If > it came from a separate implicature, then we have a further abduction > which needed a strong conditional E*, and so on. [snip] Does this regress idea make any sense to anyone? The complaint is > essentially that strong conditionality cannot depend for its > generation on our already having command of strong conditionality. For > then we have another strong conditional to reduce, and it is really > implausible to suppose that it is reducible in the same way. [snip] Any thoughts??? Closest resemblance is Carroll's paradox. (Texts available via Google.) Perhaps i understand either of them incorrectly, though. Herman Jurjus ==== Advanced, and Challenge. Please visit us at http://math.smsu.edu/~les/POTW.html [I will be tackling the baclog of old problems over the Christmas break.] seems to be interested. Skip ==== Here's a little something else about Venus, regarding their rigid airships and/or of what we could do if push comes down to shove, as we could somewhat narrow the rigid airship gap, possibly even creating a hybrid shuttle/airship of which hopefully they don't have just yet. Also a little more pertaining to the utilization of good old basalt that a few too many Earthly folks don't seem to have a clue about. http://guthvenus.tripod.com/airship-01.htm http://guthvenus.tripod.com/gv-basalt.htm Lunar basalt composite applications, besides the LSE-CM/ISS tether; http://guthvenus.tripod.com/gv-lm-1.htm ==== -- www.StealthHostiing.com You rule Truman. http://tinyurl.com/iky4 Hey Trueman...love the show. YOU ARE the Truman I heard him. Very spooky! >Is the truman living in Townsville? I've been hearing stuff, yeah. Webmasters help the TRUEman by joining www.theBanner.net Current:1 Goal:1000 ---------------------------------------------------------------------------- ------ > Can't remember the rest. Anyone? ... Monday's episode was: [cheesy organ introduction music] Roger Ramjet and his Eagles fighting for our freeeeedom, Fly through in- and outer space, not to join 'em but to beat 'em. Roger Ramjet he's our man, hero of our nation. For his adventures just be sure and stay tuned to this station. So come and join us all you kids for lots of fun and laughter As Roger Ramjet and his men get all the crooks they're after. Roger Ramjet he's our man, hero of our nation. For his adventures just be sure and stay tuned to this station. VOICEOVER: As this encroaching episode starts to begin, we find Roger Ramjet and The American Eagles about to join thousands of other Americans for a fun weekend: a weekend in ... the snow! ROGER RAMJET: I've got a good idea. Let's build something out of snow. DOODLE EAGLE: What Roger? RAMJET: How about a Ramjet Man? VOICEOVER: And just as the egomaniac was about to get started a strange phenomenon took place. DOODLE: Hey, what's going on? YANK EAGLE: The snow, it's moving. RAMJET: Look out, it must be an apple lance! DAN EAGLE: That's avalanche, but who's counting. VOICEOVER: Yes, all over America it was happening. The snow was being stolen! But how? And why? SOLENOID ROBOT MEREDITH: Soon we will be ready. Soon our interplanetary snow swiping machine will have stolen all the snow in the world and piled it here . We will be the only ones with any snow! SOLENOID ROBOT STANLEY: Yes. Then all the skiers and skaters will have to come to us to play in the snow and we will charge them a lot of money . MEREDITH: We will make a fortune before taxes . VOICEOVER: The Solenoid Robots! So they were the plots behind this rat, were you!?! DOODLE: Roger, what could be happening? RAMJET: Maybe it's just an early thaw... YANK: No, Roger. The snow didn't melt, it just slid away. It .. it .. disappeared. DAN: I think I've got the answer, men. RAMJET: I think you do too. DAN: I think somebody stole the snow. And that means they'll have to pile it somewhere. RAMJET: I hope they keep it out of my driveway. YANK: Let's get to our 'planes and scout around. We ought to be able to find it from the air. VOICEOVER: And as the scrambling American Eagles take to the air... MEREDITH: That just about does it Stanley . We have stolen all of the snow in the world . Get the signs up . STANLEY: Right you are Meredith . VOICEOVER: And as the money-mad robots waited for their first customers, the American Eagles were hot on their trail... YANK: Roger look, the snow! RAMJET: Where? I don't see any? YANK: Below us Roger! RAMJET: Oh yes, of course. It *is* the snow, and it's piled in a gigantic hill. Let's land and investigate. MEREDITH: Oh look Stanley, our first customers! STANLEY: Our first customers are troublemakers. That's Roger Ramjet . RAMJET: What?!? The Solenoid Robots! So you are the ones who took all the snow, and now you're charging people to play in it. That's not fair! STANLEY: Quickly, put on your skis. We'll escape to the bottom of the hill and get in our spaceship. VOICEOVER: And, no sooner said than done, the two crafty robots quickly donned they hickories and skied away. RAMJET: Quickly boys. Put on those skis and get after them . YANK: Roger lookout, you're skiing straight for a tree! RAMJET: Straight for a what? Yiippes!!! I wonder how I did ... Oooff! ... that. MEREDITH: Quickly Stanley, let's go down this ski-jump. That should get those Eagles off our trail. RAMJET (and Tree): Woooaah!!! [Robots now standing thoughtfully at bottom of ski-jump, looking up...] MEREDITH: Look up in the sky ... STANLEY: It's a bird ... MEREDITH: It's a 'plane ... STANLEY: It's ... [Ramjet and tree land on Stanley and Meredith.] RAMJET: Ooofff!!! ... Superski! And your snow-stealing is over. You must return all this snow immediately. The people in our world should be able to play in the snow without paying all kinds of money for it. That's the American way. MEREDITH: Have you been to Squaw Valley lately? VOICEOVER: And so once aain, cunning, daring and blind luck has paid off for Roger Ramjet, and he's made the world a better one in which to ski! Who threw that? [cheesy organ finale music as credits roll] When Ramjet takes a proton pill the crooks begin to worry. They can't escape their awful fate from proton's mighty fury. Roger Ramjet he's our man, hero of our nation. For his adventures just be sure and stay tuned to this station. So come and join us all you kids for lots of fun and laughter As Roger Ramjet and his men get all the crooks they're after. Roger Ramjet he's our man, hero of our nation. For his adventures just be sure and stay tuned to this station. ==== >Topology is totally irrelevant to, e.g., projective >spaces over finite fields. It is? Surely the cohomology (or cohomologies) of projective spaces of finite fields is central to their study? -- 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 ==== >>Topology is totally irrelevant to, e.g., projective >>spaces over finite fields. It is? Surely the cohomology (or cohomologies) of projective spaces of >finite fields is central to their study? Of course, the non-visionary sticks-in-the-mud among us (making no reference to anyone or -ones in particular) would say, that ain't topology, that's just more gol-derned algebray, tricked out in a disguise! Lee Rudolph X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 01:46 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said: >For example, you can't prove that > If G = {G} and H = {H}, then G=H. Yes you can. That comes from Extesionality - you don't need Foundation. >Your set X (which is X={X}) He did not defined such a set; he gave an axiom asserting that one exists; X={X} is *NOT* a definition of X, only a property. He did not give an axiom asserting that only one such set exists. >might make the axiom of infinity unnecessary, No. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org <3fcbbc14$17$fuzhry+tra$mr2ice@news.patriot.net> ==== > at 01:46 PM, tb+usenet@becket.net (Thomas Bushnell, BSG) said: >For example, you can't prove that > If G = {G} and H = {H}, then G=H. Yes you can. That comes from Extesionality - you don't need > Foundation. No, it doesn't. Let's try to prove it together. We'll aim for using Extensionality, as you suggest. That means that we need to show that, for all x, x in G iff x in H. Now, x in G iff x = G and x in H iff x = H. Thus, if we can show that, for all x, x = G iff x = H, then we may conclude (by Extensionality) that G = H. You got any shortcuts for showing that claim? I haven't. Thomas is right. Extensionality is insufficient. -- Jesse Hughes Such behaviour is exclusively confined to functions invented by mathematicians for the sake of causing trouble. -Albert Eagle's _A Practical Treatise on Fourier's Theorem_ X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 03:35 PM, David R. MacIver said: >Has any particularily interesting work been done on this sort of >thing? That depends on what you consider interesting. Certainly Quine published a set theory without Foundation. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== > > >>For no especially good reason I recently became curious as to what >>axiomatic set theory would look like without the axiom of >>foundation. > > > The general opinion is that there isn't anything interesting that > comes about from ~Foundation. > > >> More interestingly what it would look like with some sort >>of strong negation of foundation. I'm not sure exactly what that >>should mean, but I'm sure it can be made precise in some sense. > > > Immediately you have a problem, which is to decide whether the shape > of a set suffices for uniqueness. > > For example, you can't prove that > > If G = {G} and H = {H}, then G=H. > > Which means that you might have two sets, both which are: > > {{{{{{...}}}}}} {{{{{{...}}}}}} > > and which would not be equal. > > With foundation, of course, you can prove that if two sets have the > same shape, then the must be equal. (Not by anything deep: > foundation simply excludes the cases where the axiom of extensionality > is not sufficient to answer the question.) I know I'm being vague > here about what I mean by shape, but I hope my example and > statements make the intuitive concept clear enough. That is quite interesting. I don't suppose there's an easy example of models to show independence? It seems to me that if you have a model of them without equality then you can do some clever thing with taking equivalences over 'shape' to get a model with equality of such sets (I can't immediately see how to make this precise, but it looks plausible). I don't immediately see how to get a model of existence of such sets without equality though. > Before foundation was as popular as it is today, set theorists and > logicians worried about these cases. If you drop foundation, then you > need to decide what to do. If you don't mind unequal sets with the > same shape, then there's no problem. But if you don't want that, > then when you drop foundation you need to add something to the axiom > of extensionality to cover these cases. > > As for adding something in place of foundation, that is, an axiom > asserting some non-well-founded set, that might go. > > Your set X (which is X={X}) might make the axiom of infinity > unnecessary, but I haven't thought much about it to be sure. (Just > based on the intuition that there is something infinite about it.) I see where you're coming from, but can't you use a set of finite graphs to provide a model of the theory without infinity? (Using 'x extends y' for y is an element of x'). I'm not entirely sure about that particular construction, but it seems likely that you can; I seem to recall you can construct a model of ZFC without Infinity using well-founded graphs, so if you relax the well-foundedness condition to allow loops you're probably going to get a model of this theory without infinity. Maybe, kindof, sortof. I don't know. :) I'm committing my usual crime of posting to sci.math while half-asleep (which I should really learn not to do). of some of the references you supplied shortly. David ==== >For no especially good reason I recently became curious as to what >axiomatic set theory would look like without the axiom of >foundation. >> The general opinion is that there isn't anything interesting that >> comes about from ~Foundation. > More interestingly what it would look like with some sort >of strong negation of foundation. I'm not sure exactly what that >should mean, but I'm sure it can be made precise in some sense. >> Immediately you have a problem, which is to decide whether the shape >> of a set suffices for uniqueness. >> For example, you can't prove that >> If G = {G} and H = {H}, then G=H. >> Which means that you might have two sets, both which are: >> {{{{{{...}}}}}} {{{{{{...}}}}}} >> and which would not be equal. >> With foundation, of course, you can prove that if two sets have the >> same shape, then the must be equal. (Not by anything deep: >> foundation simply excludes the cases where the axiom of extensionality >> is not sufficient to answer the question.) I know I'm being vague >> here about what I mean by shape, but I hope my example and >> statements make the intuitive concept clear enough. >That is quite interesting. I don't suppose there's an easy example of >models to show independence? Take any model of NBG set theory whatever (ZF will also work), and use any well-defined permutation F of the elements. Define a new element relation E by x E Y iff F(x) in Y. There is no problem whatever in showing that this is a model of set theory; each of the axioms goes through. It can look very odd. -- 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 ==== > |> be cancelling all of your posts later this evening > |> so that no one will have to read them. > | > |what, exactly, gives you the authority to do such thing? i said he could. > > and, who or what exactly, gives you the authority to do such thing? > > I said he could. and who made you the authority??? ==== > > |> be cancelling all of your posts later this evening > |> so that no one will have to read them. > | > |what, exactly, gives you the authority to do such thing? i said he could. and, who or what exactly, gives you the authority to do > such thing? I said he could. > > and who made you the authority??? Sorry, but I don't have the authority to tell you that. Fortunately, I do have the authority to tell you I don't have the authority to tell you that. Have a nice day. Jim Burns ==== Given a determinant with only three major diagonals nonzero it is easy to construct a three term recursion relation to evaluate the determinant. Is there any similar result for a determinant with only five major diagonals nonzero? ==== > Given a determinant with only three major diagonals nonzero it is easy > to construct a three term recursion relation to evaluate the > determinant. Is there any similar result for a determinant with only > five major diagonals nonzero? Are you referring to tridiagonal and pentadiagonal matrices, resp.? That is, are the diagonals located consecutively about the main diagonal? Do you require a linear recurrance, or will a nonlinear recurrance do as well? ==== can some one please break this down for me, im not all together clear on this con cept as a whole ==== I don't know the notation for a subscript character without the proper character set (e.g. 2^2 = two squared (superscript), but there is no down arrow to represent subscript characters). So I'm going to use |x for this example. If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 in the form a + ib. I have more examples but I need this one to start me off. TIA. ==== > I don't know the notation for a subscript character without the proper > character set (e.g. 2^2 = two squared (superscript), but there is no down > arrow to represent subscript characters) The underscore character is used for this. x_1 = x_2 + x_3 > 2z|1 + 3z|2 in the See? This is how I read the above: Two times z times the absolute value of (1 + 3 times z) times 2. Carlos -- ==== > I don't know the notation for a subscript character without the proper > character set (e.g. 2^2 = two squared (superscript), but there is no down > arrow to represent subscript characters). So I'm going to use |x for this > example. > > If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 in the > form a + ib. > > I have more examples but I need this one to start me off. > > TIA. > > > 0. If you're learning complex numbers in a class (or from a book), you should have a reference that spells all the mathematics in steps 2-3 out explicitly. That reference may be your instructor. 1. Subscripts are frequently written using underscore, as in z_1 = 2 + 3i z_2 = -1 + 2i z_3 = -3 - 4i Just in case you are trying to pick all this up by osmosis, or find your text impenetrable: 2. To form the combination 2 z_1 + 3 z_2 you need to know these things: (a) how to multiply a complex number by a real number (b) how to add two complex numbers For (a), a complex number is a sum of two quantities: the real part, and i times the imaginary part. If you treat the i as an algebraic symbol (like x), multiplied by its coefficient the imaginary part, then multiplying the complex number by a real number is no different from multiplying a polynomial by a number: multiply each term by the new factor. Then combine terms. For instance, 5*(-2 + i) = 5*(-2) + 5*i = -10 + 5i 7*(7 - 6i) = 7*7 + 7*(-6i) = 49 - 42i. For (b), addition of complex numbers is again similar to addition of polynomials. The real parts are added together, and the imaginary parts are added together. The sum of real parts is the real part of the result, and the sum of imaginary parts is the imaginary part of the result. For instance: (1 + i) + (2 - 3i) = (1 + 2) + (1 - 3)i = 3 - 2i (7 - 3i) + (12 - 7i) = (7 + 12) + (-3 - 7)i = 19 - 10i This information should be more than enough to get you through your problem. Good luck. Dale. ==== > I don't know the notation for a subscript character without the proper > character set (e.g. 2^2 = two squared (superscript), but there is no down > arrow to represent subscript characters). So I'm going to use |x for this > example. If z|1 = 2 + 3i, z|2 = -1 + 2i and z|3 = -3 - 4i, express 2z|1 + 3z|2 in the > form a + ib. I have more examples but I need this one to start me off. TIA. > 2(2+3i)+3(-1+2i) 4+6i-3+6i 1+12i David Moran <6vigsvsqpjpbbptor2nt81ft88pam673it@4ax.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== >Im saying that both INFINITY and ZERO, represent a discontinuity. What do you mean by discontinuity? Why doesn't 1 represent a discontinuity? 666? 3.14159? If you're going to use a private language, define your terms, and choose words with different spelling from those the rest of us use. >we no longer care Perhaps you meant to post to sci.psychology? We no longer care is not a Mathematical concept. >When a cardinal number gets to that grand/small a scope, What do you mean by a cardinal number? >more practically relevant and interesting, Again, that has nothing to do either with Mathematics or with Infinity. >it becomes necessary to plot them on cartesian co-ordinates Do you know what Cartesian coordinates are? >if that were not the case then we could hardly claim to have found a >discontinuity, We don't claim that - you do. >and polar coordinates Please don't use termss you don't understand. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 09:18 AM, tylerneylon@yahoo.com (Tyler Neylon) said: >However, having said that, I believe Stepan is >actually trying to express some rather valid >mathematical intuition along the lines of >nonstandard analysis. such an hypothesis to be tenable. >We can express an >extended set of numbers as vectors a+b*d, >which we could write in the shorthand (a,b). >Hence (0,b) and (1,y) can be thought of as >infinitely far apart regardless of b and y. No. But there are pairs of nonstandard reals that we can think of that way. >I'm not sure exactly how well imaginary numbers fit in here. They don't. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org <0ksgsvkktgi0fpfj2517qh6a7d9jug8fkn@4ax.com> <3f46af44.0311291209.6c04ac@posting.google.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== >My personal idea of infinity goes like: >Infinity (not a number) is a property of a partially ordered set, >that says: for every element there's a great element (wrt to that >order). So if x and y are real numbers, (x,y) is infinite and [x,y] is not? -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== >A nice profound explanation of ZERO and INFINITY: ~nice, ~profound, ~explanation. >You are standing at a point on the ground marked ZERO, A number is not a point on the ground. >EXTREMELY far away Extremely far away is not infinity. >The measured distance between the two of you can be written >mathematically in polar coordinate or as an imagenary number >s=25-i50 That would not be a distance. >ZERO is an arbitrary frame of reference on the near side of a >discontinuity. Do you imagine that your statement has any meaning? >If we decide exactly where this INFINITY point is >supposedly located, then we can start to say things like Infinity >plus 3 etc. Or we could label it eggnog and start saying things like eggnog plus 3. That would make just as much sense. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== >So you may say, fine, I'll DEFINE 1/0 for ya. It's infinity! ...or alternately define 1/infinity as well. >Then you have to invent your own infinity arithmetic. For you cannot >have infinity follow the same rules that we all agree regular numbers >follow. For example, 1 / 0 = infinity, 2 / 0 = infinity, so by >definition of division 0 * infinity has all the values of the rainbow. >So infinity * 0 is not unique. Incidentally, 0 / 0 is therefore not >unique either, since the answer is the number which multiplied by 0 >gives 0 but this is true for all numbers. So you can replace the >with and and have yourself a nice little world with your infinity >and zero arithmetic. ...so now you have two different sets of rules. Does this mean that you now have two different ways/methode for solving a real world problem which can be put to paper? If the answer is yes, then can we convert from from one system of arithmetic to the other? Does the notion of 0 x INFINITY = ? somehow make this connection impossible? Intuitively it seems to me that the number ZERO which is finite under the conventional set of rules, would be infinately small if veiewed in the context of the alternate set of rules. Conversely, the number INFINITY under the alternate rules, should be finite and behave much like OUR number 0 does. What I tried to explore with the original example that started this whole thread, is whether you can have two such systems in co-existence. In order for them to co-exist, you have to keep the numbers from one system, separate from the numbers of the other, because the rules are in fact different. I tried to achieve that separation by using the REAL numbers to represent quantities which should be manipulated by the conventional rules of arithmetic, and at the same time using the IMAGINARY numbers to represent quantities which should be manupulated using the unconventional rules. >No one says you can't do that. It just has to be consistent and >logical. You can invent your own math. If it's interesting and no one >has done it before, you can even publish it. :-) I find it deeply disturbing that no one has done this. Does anyone out there have any idea why that is? Is it because there is no real world application for the answer to what is meant by 0 x INFINITY ? When you think about it, the number 0 as a real number (in the mathematical sense) is not that meningfull for measurement, because whichever fractional quantity you are measuring eventually becomes unmeasarable at the point where it becomes so small that it gets obfuscated by noise or measurement accuracy. At this point where the measured quantity is undetectable, it becomes undefined, zero, or infinately small. At this point there is no need to make a distinction between 0 and Unmeasurably-Small, so for the purpose of solving real world problems we can define the small quantity as either 0 or alternately and equally correctly as x/infinity where x is any arbitrary finite positive value. The exact same argument holds true at the opposite end of the spectrum, ie INFINITY versus 2*INFINITY etc. In this case, the number INFINITY is any number grand enough in scope, that increasing the measured value beyond that quantity is irellevant to the real world problem we are trying to solve. In other words, if INFINITY is out of scope, then the product of INFINITY and any cardinal number is also out of scope and therefore, for all practical purposes relevant to the real world problem we are trying to solve, we can say that 6 x INFINITY = INFINITY. For example, look straight up into the sky on a clear night. How far can you see? Can you see 20 miles? Yes. Can you see 2 million miles? You see all the stars so Yes. Can you see to 37.6 X 10 ^ 387 miles out? The answer at this point is that for practical purposes of solving any real world problems, it doesn't matter whether you can see that far. If it doesn't matter whether we can see that far, then we will not bother assigning a numeric value to that distance, but we will instead call that distance INFINITY. Now it should be apparent that there is no need to make a distinction between INFINITY and 3xINFINITY etc. ...Stepan ...Stepan ==== So you may say, fine, I'll DEFINE 1/0 for ya. It's infinity! ...or alternately define 1/infinity as well. Then you have to invent your own infinity arithmetic. That's not necessary. It's already been invented. >For you cannot >have infinity follow the same rules that we all agree regular numbers >follow. For example, 1 / 0 = infinity, 2 / 0 = infinity, so by >definition of division 0 * infinity has all the values of the rainbow. >So infinity * 0 is not unique. In the system to which I refer, the answer is unique, but it is a set of numbers, rather than a single number. Specifically, the answer is the set of extended reals, [-oo, +oo]. And, as you say below, the situation is the same for 0/0. >Incidentally, 0 / 0 is therefore not >unique either, since the answer is the number which multiplied by 0 >gives 0 but this is true for all numbers. So you can replace the >with and and have yourself a nice little world with your infinity >and zero arithmetic. [snip] >No one says you can't do that. It just has to be consistent and >logical. You can invent your own math. If it's interesting and no one >has done it before, you can even publish it. :-) It's been done before, as I said. It is consistent, logical, and quite useful in the context of interval arithmetic (or, if you wish to be more abstract, in the context of algebraic structures known as wheels). One of its primary exponents is Bill (G. William) Walster, an engineer at Sun. are available on the web. > I find it deeply disturbing that no one has done this. Does anyone out > there have any idea why that is? Is it because there is no real world > application for the answer to what is meant by 0 x INFINITY ? Ah, then you need be disturbed no more! David Cantrell ==== >It's been done before, as I said. It is consistent, logical, and quite >useful in the context of interval arithmetic (or, if you wish to be more >abstract, in the context of algebraic structures known as wheels). One of >its primary exponents is Bill (G. William) Walster, an engineer at Sun. >are available on the web. > Where on the web is this stuff please. Walster does not google. ==== It's been done before, as I said. It is consistent, logical, and quite >useful in the context of interval arithmetic (or, if you wish to be more >abstract, in the context of algebraic structures known as wheels). One >of its primary exponents is Bill (G. William) Walster, an engineer at >system, are available on the web. Where on the web is this stuff please. Walster does not google. Do a Google web search using g. william walster interval arithmetic You should find plenty. Choose his most recent papers to look at. David ==== >g. william walster interval arithmetic Since +oo is the first member of the SET OF NUMBERS to the right of the SET OF REALs, and since -oo is the last member of the SET OF NUMBERS to the left of the SET OF REALs, it follows that including +oo and/or -oo in the SET OF REALS, is a formal way of declaring one or both of the following two points: 1) We are UNABLE to define an upper (or lower) bound in the context of the problem we are trying to solve. In this case we should expect an approximate result. This is the real world problem we have when we try to quantify(write down as a number) analog measurements in the presence of uncertainty, noise, and lack of time to acquire a proper measurement. 2) Or we know that there is no upper (or lower) bound in the context of the problem we are trying to solve. In this case we should expect an accurate result. This appears to be the case in calculus. ...Stepan ==== >In the system to which I refer, the answer is unique, but it is a set of >numbers, rather than a single number. Specifically, the answer is the set >of extended reals, [-oo, +oo]. And, as you say below, the situation is the >same for 0/0. > What you say makes perfect sense if the following is observed: The answer you refer to above should be an INFINATE number of UNIQUE and ORDERED sets, in other words, the theory should be placing markers(defining sets) at regular intervals in the progression from 0 to INFINITY (AND beyond). In this case, INFINITY is NOT located arbitrarily. It is a boundary value which defines the scope of relevant numeric values(the working set) for any given problem. Specifically, this boundary is defined to exist at the very first unnatainable value in a progression. In other words, INFINITY points to the FIRST element of the NEXT set, where both the sets and also the values within the sets are ordered. Then it all makes perfect sense in the world of applied mathematics, and I can see the (possibly weak) analogy to wheels. Such a grouping would allow us to perform mathematical operations upon an infinate number of UNIQUE sets, and on paper using the proper notation, it would be a single mathematical operation. It would probably feel like vector arithmetic, but instead of being 2-Ddimensional (real and imaginary numbers), or three dimensional, it would be infinate-Dimensional. However beware: The answer to which you are referring, can not be just ONE SINGLE SET of numbers. That would not be a usefull extension. ...Stepan ==== >I am of the beleif that math in and of itself is completely >meaningless, just a set of symbols (numbers) and relationships between >those symbols (operators) To make math usefull as opposed to Just A >Form Of Art (applied mathematics) you need to declare a relationship >between the numbers and some real-world >(measurable/countable/observable) properties, so that the operators >can help you solve a problem. You still need rigorous mathematical definitions as a basis for your applied mathematics. While you can define infinity as an arbitrary frame of reference on the far side of a discontinuity just as well as Euclid could define a point as that which has no dimension, neither is very helpful in developing useful mathematical frameworks that are required in real life problems like, say, predicting the asymptotic behavior of a system described by ODEs. >You are talking about INFINITY as a form of art, while I am talking >about INFINITY as a practical entity. They are two very different >worlds and I can neither agree nor disagree with you. I don't see how your definition is any more practical than any of the other quasi-mathematical definitions of infinity I've seen passed around by philosophists of various schools. <716e06f5.0311242055.78c757ee@posting.google.com> <992b156f.0311281250.e905845@posting.google.com> ==== > I'm making a huge gear-shift from the original topic. >> so part of what's going on here is this: first, all categories are in >> a sense imitations of the category of sets, the objects being >> imitations of sets and the morphisms being imitations of functions (or >> maps or mappings or whatever you call them). This is a very set-centric view. I thought the whole purpose of > bothering with category theory in the first place was to escape this > view. I think I agree with George here. One can take the set-theoretic intuitions too far. What about posets as categories? The arrows aren't imitations of functions, are they? What about a category in which the objects are formulas and the arrows are proofs? [...] > How exactly one would one even DEFINE the category of sets if one > were NOT STARTING with ZFC or some other rich set theory as a > foundation? My point is simply that if you have ZFC, what do you > need categories for? Sets are already adequate as a foundation; you > can do EVERYthing, INCLUDING categories, AS sets. The category of > sets starts to get viciously circular. But if you don't have a set > theory, if you are using categories as a foundation instead, then > the category of sets is simply nowhere in evidence: how do you > even DEFINE set? And here, I think George goes too far (or I don't get his point). Some folks want category theory as an alternate foundation, it's true. Others just like it for its unifying qualities and ability to make apparently disparate phenomena particular instances of a common structure. I don't see why there's any particular issue for talking about the category of sets as a particular category for *either* group. That said, the foundations folk may still have a good argument against taking set theory as the real foundation. Namely, the aims of structuralism seem to be much more easily attained via categorical foundations than set theoretic foundations. But here, I'm talking a bit out my ass and the interested reader should find McClarty's Numbers can be just what they have to, an eminently accessible numbers could not be. -- Destiny is a funny thing. Once I thought I was destined to become Emperor of Greenland, sole monarch over its 52,000 inhabitants. Then I thought I was destined to build a Polynesian longship in my garage. I was wrong then, but I've got it now. -- The Tick ==== | |> I'm making a huge gear-shift from the original topic. |> |> |>> so part of what's going on here is this: first, all categories are in |>> a sense imitations of the category of sets, the objects being |>> imitations of sets and the morphisms being imitations of functions (or |>> maps or mappings or whatever you call them). |> |> This is a very set-centric view. I thought the whole purpose of |> bothering with category theory in the first place was to escape this |> view. | |I think I agree with George here. One can take the set-theoretic |intuitions too far. What about posets as categories? The arrows |aren't imitations of functions, are they? sure they are; specifically, of inclusion functions between subsets (is one way to think of it). (there's things to say about why this is a special degenerate case but i don't feel like saying any of them because i think this whole thread is misguided.) |What about a category in which the objects are formulas and the |arrows are proofs? in such circumstances it's often crucial to think of such an arrow as a function from proofs of the antecedent formula to proofs of the consequent formula. again, though, i probably shouldn't be replying to this thread other than to point out that y'all are reading way too much that isn't there into imitation. -- <992b156f.0311281250.e905845@posting.google.com> <87smk4gm61.fsf@phiwumbda.org> ==== | > |> I'm making a huge gear-shift from the original topic. > | | |>> so part of what's going on here is this: first, all categories are in > |>> a sense imitations of the category of sets, the objects being > |>> imitations of sets and the morphisms being imitations of functions (or > |>> maps or mappings or whatever you call them). > | |> This is a very set-centric view. I thought the whole purpose of > |> bothering with category theory in the first place was to escape this > |> view. > | > |I think I agree with George here. One can take the set-theoretic > |intuitions too far. What about posets as categories? The arrows > |aren't imitations of functions, are they? sure they are; specifically, of inclusion functions between subsets > (is one way to think of it). (there's things to say about why this is > a special degenerate case but i don't feel like saying any of them > because i think this whole thread is misguided.) I think that the whole categories are just sets and set functions is missing the point of category theory. There is nothing about collections and elementhood which is fundamentally more basic than objects and arrows between them. To believe that *every* poset is just essentially an abstraction from sets and inclusions seems to rather miss the point I'd think. There's something to be said for the view that orderings on collections is epistemically prior to ZF. > |What about a category in which the objects are formulas and the > |arrows are proofs? in such circumstances it's often crucial to think of such an arrow as > a function from proofs of the antecedent formula to proofs of the > consequent formula. Even if we do that, your point isn't made. A formula is not a set of proofs of that formula, is it? If not, then the arrows are not functions from the set of proofs of the antecedent to the set of proofs of the consequent. Maybe things can be viewed that way, but that's not particularly natural. > again, though, i probably shouldn't be replying to this thread other > than to point out that y'all are reading way too much that isn't there > into imitation. Perhaps. I came into this thread secondhand. Why don't you tell us what you mean when you write, all categories are in a sense imitations of the category of sets, the objects being imitations of sets and the morphisms being imitations of functions. It's very plausible that I don't know what the heck you mean. You might mean merely that category theory can be interpreted in set theory. But that's obvious. You might mean something else. I haven't a clue. -- My proof has been checked very thoroughly, both by me and others. Those others apparently decided that they would not believe the proof was correct, but cannot support that position using mathematics. But hey, they're just human beings. --JSH, prover of Fermat's Last Thm <716e06f5.0311242055.78c757ee@posting.google.com> <992b156f.0311281250.e905845@posting.google.com> X-Cise: tanbanso@iinet.net.au X-CompuServe-Customer: Yes X-Coriate: admin@interspeed.co.nz X-Ecrate: tanandtanlawyers.com X-Pose: george_cox@btinternet.com X-Punge: Micro$oft ==== at 12:50 PM, greeneg@cs.unc.edu (George Greene) said: >This is a very set-centric view. >I thought the whole purpose of bothering with >category theory in the first place was to escape this view. No. Category theory is a tool for other branches of Mathematics, whether you take categories as fundamental or take sets as fundamental. >My point is simply that if you have ZFC, what do >you need categories for? Because you can prove things once in Category Theory and then apply them to various branches of Mathematics. Your question is like asking what you need groups for. >The category of sets starts to get viciously circular. Why is that an issue? You can do Set Theory without the Axiom of Foundation. -- Shmuel (Seymour J.) Metz, SysProg and JOAT not reply to spamtrap@library.lspace.org ==== |I'm making a huge gear-shift from the original topic. | | |> so part of what's going on here is this: first, all categories are in |> a sense imitations of the category of sets, the objects being |> imitations of sets and the morphisms being imitations of functions (or |> maps or mappings or whatever you call them). | |This is a very set-centric view. |I thought the whole purpose of bothering with |category theory in the first place was to escape this view. well, i don't want to get involved much in a side discussion about this; i know that you indicated that it's a shift of topic, but i think it's even much more off the topic than that. anyway, i just don't agree that the whole purpose of bothering with category theory is anything remotely like what you suggest it is. -- ==== : >> so part of what's going on here is this: first, all categories are in : >> a sense imitations of the category of sets, the objects being : >> imitations of sets and the morphisms being imitations of functions (or : >> maps or mappings or whatever you call them). : > This is a very set-centric view. : > I thought the whole purpose of bothering with : > category theory in the first place was to escape this view. : That is a bit exaggerated. : My impression is, that categorists have no problem with sets as such, : but would like to regard function as the primitive concept instead : of elementhood, because this is closer to mathematical practice. Well, maybe Marc and James should conduct competing polls of the category of all categorists; I mean, I doubt it can simultaneously be the case that both all categories are imitations of the category of sets AND that function, as opposed to element, should be the primitive concept. : the Math Intelligencer (?). It should still be available via : : > You don't have to presume a set of objects upon which to : > found categories. ... : > How exactly one would one even DEFINE the category of sets : > if one were NOT STARTING with ZFC or some other rich set theory : > as a foundation? ... : > if you are using categories as a foundation instead, then the category of : > sets is simply nowhere in evidence: how do you even DEFINE set? : You would define set like categories. Just how hard is that? Don't you need to add, to the basic axioms, some definitions describing which categories are categories- of-categories, and then some more definitions, describing which of THOSE categories have sets-as-their-objects? That's a lot of layers up, for something that was allegedly supposed to be the foundational template. : First you can add the topos axioms, then you can add more axioms : to narrow down the candidates. This is already sketched in the appendix : of the 2nd edition of MacLanes CWM; Well, sure, topoi are categories with sets in them. But how does J.Random Topos relate to the category of all sets? There are a big new bunch of topos axioms (above and beyond the handful defining a category), and isn't it simply ABSURD to allege that all categories are imitations of the category of all sets, when a great many of those categories need FAR FEWER additional axioms than are needed by topoi? I mean, topoi are complicated. The category of all sets is, in at least SOME people's opinion, simple -- simple enough to be analogous to a lot of other simple small categories. : see also the new book by Lawvere and Rosebrugh Sets for Mathematics. -- --- It's difficult ... you need to be united to have any strength, but internal issues have to be addressed. --- E. Ray Lewis, on liberalism in America ==== >Does the trace >essentially contain every other sort of information that we could get >from the complete set of latent roots of the representative matrix? Others have pointed out that if your only tool is the calculation of traces, you can still say everything you might want to about a single similarity class of matrices -- as long as you are willing to apply that tool to other (similarity classes of) matrices too, and as long as you didn't want any information which couldn't be obtained from the eigenvalues themselves. (Some information about the similarity class IS lost that way, e.g. the question of whether the matrices are diagonalizable. But non-diagonalizable matrices don't show up in the context described below anyway.) Your subject line is a little different, though: when you are looking at a character of a (finite) group, you're looking at a _function_ defined on the group, namely chi(g) = trace( R(g) ) where R(g) is a representation matrix. That is, you haven't got just one matrix but lots of them. In fact, one is not usually so interested in _a_ character of a group but rather the whole character table, which sheds light on the structure of the group. These pieces of information make a difference in your question: it's not just linear algebra now but rather group theory. For example, we have some theorems: 1. If R_1 and R_2 are two complex representations of a finite group whose characters are equal, then the representations are equivalent (that is, there is a single invertible P with P R_1(g) = R_2(g) P for every P; you might say the R1's and R2's are uniformly similar.) In other words: the trace is all you need to distinguish two representations anyway. 2. Any complex-valued function on G which is constant on conjugacy classes is a linear combination of characters. In other words, the traces of all the characters already give you all the kinds of functions you could make out of the similarity classes anyway. Taken together, (1) and (2) sort of tell us that traces are the only similarity invariant we need in classical representation theory. dave ==== > > Besides, my textbook says nothing about the traces of the exterior > powers of the linear transformation either. Do they ever really enter > group character theory? > > Yes, they give another way of constructing new representations > from a given one > (as well as adding or multiplying representations). You mean the exterior powers of the representing linear transformations are themselves representations? I rekhon you mean direct sum and direct product in the second line. I just got started...anything smartly stated beats me. > The book (Ledermann, Intro. to group > characters) characterizes groups by traces completely. As far as I > know that's what the tradition is too. > > You do take the trace in this case too. > If the transformation T acts on the vector space V, > then there is a corresponding transformation T^(r) > on the exterior product V^(r), > and you take the trace of T^(r). If this is what Chapman and Dolan are talking about, thanks for spelling it out. If I got it right, this corresponding means it respects the exterior product like f(y ^ v) = f(y) ^ f(v)? ==== |> |> | |> |> Similarity transformations preserve much more than the trace---the |> |> characteristic equation itself, |> | |> |The coefficients of the characteristic polynomial are all traces: |> |the traces of the exterior powers of the linear transformation. |> |> the traces of the exterior powers of the linear transformation by |> which a group element acts aren't among the traces directly mentioned |> in the character of a group representation, though, so wouldn't it be |> more relevant to point out that the traces of the _ordinary_ powers of |> a linear transformation contain a lot of information about the |> similarity class of the transformation? (assuming i didn't |> misunderstand the original question.) | |I think you didn't. But I didn't quite get what you said (as before). |The eigenvalues of the ordinary powers of a linear transformation are |the ordinary powers of the eigenvalues, aren't they (if Av = av, then |A^2v = Aav = aAv = a^2v and induction)? yes. |Their traces would be the sum |of all the same powered eigenvalues, right (sum_i a_i^2, sum_i |a_i^3)? yes. |Am I missing something? i don't know. do you agree that the traces of all the ordinary powers of a linear transformation gives a lot more information than just the trace of the linear transformation itself? that's the only real point i was trying to make. (in the present context i suspect the traces of the ordinary powers completely determine the linear transformation up to similarity, but offhand i forget some of the details of how that should work.) -- ==== I said *data*, as I don't agree with much of Dr. > Arp's cosmological theories. Having followed the topic for some years, I have some questions and observations; Inflation theory proposes that space is flat/Omega=1, that expansion and gravitational contraction are balanced. Observations show this to be true. If the collapsing space of gravity and expanding intergalactic space cancel each other out, where is the additional expansion for the universe as a whole to expand? My first thought in reading of this prediction was that a convective process made more sense that the apparent coincidence that BBT suggests. Big Bang theory is essentially based on the assumption that light frequencies do not deteriorate, therefore the redshift can only be explained by recession. The first problem with this is that we are willing to accept that space is not an absolute and is gravitationally malleable. Now gravity effectively collapses our measure of space, BUT it is constantly radiating the energy of which this matter consisted! So it would seem extremely logical to assume that radiation has the opposite effect and expands the measure of space. This would result in a very basic and understandable convective cycle, as matter collapses and energy expands. Say that empty space has a very low threshold for holding stable radiation, such as 2.7k. At that phase transition point it starts to condense out as hydrogen. This propels the system, as there is more radiation than space to hold it. This vital explanation for the smoothness of background radiation would be more logical than as the remnant of a 13 billion year old event. Assuming the universe is infinite and already full, local space has nowhere to expand to. The only place for the pressure to express is onto the gravitationally collapsing vortex of galaxies. This would provide a very neat explanation for the excess spin attributed to dark matter. When the light of distant sources passes through intermediate gravitational fields, it is magnified by the well known lensing effect. Suppose this compresses the light waves, blueshifting. Given the distribution of galaxies, light from distant sources is going to pass through the residual gravity field of a number of intermediate sources. This blueshifting will reduce the overall redshift, so that the average redshift of closer sources is greater, which would explain the phenomena for which dark energy is proposed. As it is, Big Bang theory proposes a universe in which ninety-seven percent of the matter and energy is invisible to everything but the math. All because we are completely convinced that we know the properties of light over distances we can never positively quantify, through a medium we can never test. NINETY_SEVEN PERCENT UNKNOWN!!?? ALL BECAUSE WE ARE SURE LIGHT IS INFLEXIBLE, EVEN THOUGH WE ACCEPT THE SPACE IT'S CROSSING ISN'T!!! Not to mention what we are willing to accept with inflation theory! Another point, the greater the distance, the greater the radius of the volume of space being considered, but according to BBT, the greater the distance, the smaller space is!!!!! I'm not an expert and when I first considered the topic I assumed the experts were right, but I just don't see it. ==== >> 1) Measure redshift of object. >> 2) Measure distance to object. We can't measure distances directly beyond parallax range (about 300 >parsecs). The 'measurements' that we use beyond the nearest few dozen >galaxies (based mostly on cepheids) are all based squarely on assuming the >big bang. Describe those methods and tell me where this assumption comes into > the method, or stop repeating this ridiculous bit of nonsense. You can easily show me my error by describing (i.e. not just naming) one method that does NOT use the big-bang assumption -- either directly or indirectly (i.e. for calibration). I simpy won't bother answering an open-ended question, from someone who snips and ignores all prior evidence in the thread. No matter how many methods I describe that do use the BB directly or for calibration, you can always complain that I missed one. SO much easier for you (and educational for everyone) to simply show me a case where I'm wrong. -- greywolf42 ubi dubium ibi libertas ==== >> 1) Measure redshift of object. >> 2) Measure distance to object. We can't measure distances directly beyond parallax range (about 300 >parsecs). The 'measurements' that we use beyond the nearest few dozen >galaxies (based mostly on cepheids) are all based squarely on assuming > the >big bang. Describe those methods and tell me where this assumption comes into > the method, or stop repeating this ridiculous bit of nonsense. > > You can easily show me my error by describing (i.e. not just naming) one > method that does NOT use the big-bang assumption -- either directly or > indirectly (i.e. for calibration). That's a very rational request for Randy Poe. Can he give a clear and direct answer to it? > I simpy won't bother answering an open-ended question, from someone who > snips and ignores all prior evidence in the thread. No matter how many > methods I describe that do use the BB directly or for calibration, you can > always complain that I missed one. Yeah, it's odd, but that's Randy Poe. He creatively deletes to create false implication, and then keeps posting repeatedly until he drives away the person he's arguing with by replying, and replying, and replying. It's very odd behavior to me as it involves a bit of energy, persistence, and continual scanning of threads. Randy Poe must do all of those things, but why? > SO much easier for you (and educational for everyone) to simply show me a > case where I'm wrong. That makes sense. Let's see if Randy Poe has the energy and persistence to reply again now--with a straight answer. James Harris ==== >> Describe those methods and tell me where this assumption comes into >> the method, or stop repeating this ridiculous bit of nonsense. You can easily show me my error by describing (i.e. not just naming) one >method that does NOT use the big-bang assumption -- either directly or >indirectly (i.e. for calibration). I posted another message asking you to explain what that last statement means, since I expect you'll use it uses the Big Bang for calibration as a catch all for everything, whether it has anything to do with the Big Bang or redshifts at all. You've already illustrated a willingness to grossly misread with your circular argument post. Meanwhile, I'll just note that Hubble's red shift data was published in 1929 (with distances measured by parallax), but the calibration curve for Cepheid variables was published by Henrietta Leavitt in 1912. Big Bang theory in its present form is mostly credited to Gamow in the 1940s with a successful prediction of the 3-degree background, though Lemaitre in 1927 did propose an explosive-origin theory. Explain to me how Leavitt managed to use Big Bang theory for her calibration in 1912, and what use Big Bang for calibration means, and I'll explain both Leavitt's calibration and the Cepheid variable method. - Randy ==== > Describe those methods and tell me where this assumption comes into >> the method, or stop repeating this ridiculous bit of nonsense. You can easily show me my error by describing (i.e. not just naming) one >method that does NOT use the big-bang assumption -- either directly or >indirectly (i.e. for calibration). I posted another message asking you to explain what that last > statement means, since I expect you'll use it uses the Big Bang for > calibration as a catch all for everything, whether it has anything to > do with the Big Bang or redshifts at all. You've already illustrated a > willingness to grossly misread with your circular argument post. Yes, you avoided the question in the parallel post, too. > Meanwhile, I'll just note that Hubble's red shift data was published > in 1929 (with distances measured by parallax), but the calibration > curve for Cepheid variables was published by Henrietta Leavitt in > 1912. Did you have a point to make? > Big Bang theory in its present form is mostly credited to Gamow > in the 1940s with a successful prediction of the 3-degree background, Which was a false claim, as the lowest temperature predicted by Gamow, prior to Penzias and Wilson was 50 degrees (a factor of 10,000 error in energy density -- which was the basis for his estimate). > though Lemaitre in 1927 did propose an explosive-origin theory. Yes. Carl Wirtz first published an empirical redshift-distance relation in 1924 (pre Cepheid variable identification). Lemaitre's publication of the 'expanding universe' theory came in 1927, and was based partly on Wirtz' empirical work. Hubble's version of the redshift relation was not published until 1929 (after Cephied variable identification made Wirtz' relationship more certain). There have been at least five major revisions of the explosive origin theory that is now called the 'big bang.' Which one are you defending? > Explain to me how Leavitt managed to use Big Bang theory for her > calibration in 1912, I never claimed that the cepheid period-luminosity relationship was based on the big bang theory. What I noted was that Hubble's law was based on the cepheid curve. > and what use Big Bang for calibration means, > and I'll explain both Leavitt's calibration and the Cepheid variable > method. Not necessary. All I've (repeatedly) asked you to do was simply describe one, modern distance estimatation method -- applicable beyond the range of cepheid variable resolution -- that does not depend upon the hubble constant, and/or is not calibrated by same. -- greywolf42 ubi dubium ibi libertas ==== >> Meanwhile, I'll just note that Hubble's red shift data was published >> in 1929 (with distances measured by parallax), but the calibration >> curve for Cepheid variables was published by Henrietta Leavitt in >> 1912. Did you have a point to make? Cepheid calibration 1912. Big Bang Theory post-1940s. Cepheid calibration can't be based on Big Bang Theory. Only understand short sentences? >> and what use Big Bang for calibration means, >> and I'll explain both Leavitt's calibration and the Cepheid variable >> method. Not necessary. All I've (repeatedly) asked you to do was simply describe >one, modern distance estimatation method -- applicable beyond the range of >cepheid variable resolution As I asked in your other post, what would beyond the range of cepheid variable resolution be, since those are as far as I know the most distant sources used for Hubble Law tests? - Randy ==== >Explain to me how Leavitt managed to use Big Bang theory for her >calibration in 1912, and what use Big Bang for calibration means, >and I'll explain both Leavitt's calibration and the Cepheid variable >method. In Fowles' book on optics I read about Michelson's stellar interferometer, which measures stellar diameters. And there was an intensity method that's supposed to have improved precision, but I don't really understand it. Fowles didn't give any numbers, but it seemed to me that when parallax fails, you can keep going if you can measure diameter, and combine that with brightness and temperature. Assuming diameter measurements can be made farther out than parallax measurements. -- And don't skimp on the mayonnaise! ==== > >Explain to me how Leavitt managed to use Big Bang theory for her >calibration in 1912, and what use Big Bang for calibration means, >and I'll explain both Leavitt's calibration and the Cepheid variable >method. > > In Fowles' book on optics I read about Michelson's stellar interferometer, > which measures stellar diameters. And there was an intensity method > that's supposed to have improved precision, but I don't really understand > it. Fowles didn't give any numbers, but it seemed to me that when > parallax fails, you can keep going if you can measure diameter, > and combine that with brightness and temperature. Assuming diameter > measurements can be made farther out than parallax measurements. What I found in reading pages on astronomical distance estimation is that many authors seem to use a technique called main sequence fitting. Google on that term and you'll learn more than you ever wanted to. The original idea is to use the brightness: based on other data about the source, how bright should it be, and then how bright does it actually appear to be. The difference tells you distance in a more or less obvious way. This idea has been modernized. Now rather than brightness it's some sort of multi-spectral measure that gives much more accurate determinations. But it's still the same basic idea: if you know how bright something is, and you see how bright it appears to be, then you know how far away it is. No Big Bang assumptions. The big unknown always is how bright is this object really? and that's where main sequence fitting comes in. I'm probably mangling this a little. I need to read those pages in more detail before responding directly to greywolf. - Randy ==== >> >>Explain to me how Leavitt managed to use Big Bang theory for her >>calibration in 1912, and what use Big Bang for calibration means, >>and I'll explain both Leavitt's calibration and the Cepheid variable >>method. >> >> In Fowles' book on optics I read about Michelson's stellar interferometer, >> which measures stellar diameters. And there was an intensity method >> that's supposed to have improved precision, but I don't really understand >> it. Fowles didn't give any numbers, but it seemed to me that when >> parallax fails, you can keep going if you can measure diameter, >> and combine that with brightness and temperature. Assuming diameter >> measurements can be made farther out than parallax measurements. What I found in reading pages on astronomical distance >estimation is that many authors seem to use a technique >called main sequence fitting. Google on that term and >you'll learn more than you ever wanted to. The original idea is to use the brightness: based on other >data about the source, how bright should it be, and then >how bright does it actually appear to be. The difference >tells you distance in a more or less obvious way. This idea has been modernized. Now rather than brightness >it's some sort of multi-spectral measure that gives much >more accurate determinations. But it's still the same >basic idea: if you know how bright something is, and you >see how bright it appears to be, then you know how far >away it is. No Big Bang assumptions. The big unknown >always is how bright is this object really? and that's >where main sequence fitting comes in. I'm probably mangling this a little. I need to read those >pages in more detail before responding directly to >greywolf. I vaguely recall main sequence charts. What could be considered a problem for distance determination is that some things are big and red, big and white, small and red, small and white So there's some more theory involved in turning brightness and temperature into a distance. I thought having a diameter measurement must be more direct. -- When the fool walks through the street, in his lack of understanding he calls everything foolish. -- Ecclesiastes 10:3, New American Bible ==== > 1) Measure redshift of object. > 2) Measure distance to object. >>We can't measure distances directly beyond parallax range (about 300 >>parsecs). The 'measurements' that we use beyond the nearest few dozen >>galaxies (based mostly on cepheids) are all based squarely on assuming >the >>big bang. >> Describe those methods and tell me where this assumption comes into >> the method, or stop repeating this ridiculous bit of nonsense. You can easily show me my error by describing (i.e. not just naming) one >method that does NOT use the big-bang assumption -- either directly or >indirectly (i.e. for calibration). Noted, for the record, that you made a blanket statement without any knowledge. Now you want me to provide the actual information so you can try to find a place to claim your silliness applies. >I simpy won't bother answering an open-ended question, from someone who >snips and ignores all prior evidence in the thread. No matter how many >methods I describe that do use the BB directly or for calibration, you can >always complain that I missed one. and I'll describe distance measurement via Cepheid variables. Found several good links. - Randy ==== >> 1) Measure redshift of object. > 2) Measure distance to object. >>We can't measure distances directly beyond parallax range (about 300 >>parsecs). The 'measurements' that we use beyond the nearest few dozen >>galaxies (based mostly on cepheids) are all based squarely on assuming >the >>big bang. >> Describe those methods and tell me where this assumption comes into >> the method, or stop repeating this ridiculous bit of nonsense. You can easily show me my error by describing (i.e. not just naming) one >method that does NOT use the big-bang assumption -- either directly or >indirectly (i.e. for calibration). Noted, for the record, that you made a blanket statement without any > knowledge. I have knowledge of several different methods of estimating distance. But they are all based directly or indirectly on assuming the hubble constant. Obviously, you think you know one. All it takes is one, to show me (and the rest of the newsgroup) my error. Go on! Don't you want to provide education? > Now you want me to provide the actual information so you > can try to find a place to claim your silliness applies. I figured you couldn't provide any. >I simpy won't bother answering an open-ended question, from someone who >snips and ignores all prior evidence in the thread. No matter how many >methods I describe that do use the BB directly or for calibration, you >can always complain that I missed one. What does use the Big Bang for calibration even mean? Every standard candle distance method requires a calibration step. There are methods used for distance estimation that are used beyond the range of cepheids. Look up the section entitled secondary distance indicators in the book The Cosmological Distance Ladder. These are defined as ... indicators which depend for their calibration on our knowing the distance to some representative nearby galaxies through primary distance indicators. Taking the first method in that section -- for no other reason than it's first in the book -- we have the HII regions method. It is based on the assumption that one can estimate the dimensions of core and halo diameters within the HII regions of a galaxy (and that these surround new O and B stars). A 'correlation' was found between the HII region diameter and the galaxy luminosity class. There are several problems with this method (including the fact that the relationship was not the one that was first identied, but was 'forced' as a secondary method when the first was found to be nearly useless), which are listed in the book. The primary one being that the method of core and halo diameters on the plates are subjective and liable to systematic errors. But this method was created by first ensuring (via extinction and selection of the 'proper' definitions for intensity) that the correlation (which has no theoretical justification) would be both linear and consistent with the hubble law. > and I'll describe distance measurement via Cepheid variables. Found > several good links. I'm not interested in links. And I don't need to know about estimating WITH cepheid variables. I'm talking about distance measurments *beyond* the distance where cepheid variables are resolvable. -- greywolf42 ubi dubium ibi libertas ==== >> Now you want me to provide the actual information so you >> can try to find a place to claim your silliness applies. I figured you couldn't provide any. You're wrong. I found more than enough detail in online sources to provide a technical explanation. Bookmarked a bunch in preparation to composing my reply. But I won't address the use the Big Bang indirectly question till I understand what the question is. >>I simpy won't bother answering an open-ended question, from someone who >>snips and ignores all prior evidence in the thread. No matter how many >>methods I describe that do use the BB directly or for calibration, you >>can always complain that I missed one. >> What does use the Big Bang for calibration even mean? Every standard candle distance method requires a calibration step. There >are methods used for distance estimation that are used beyond the range of >cepheids. Look up the section entitled secondary distance indicators in >the book The Cosmological Distance Ladder. Don't have it. I'll be replying from online sources, which include NASA pages, published papers, Hubble pages, and various sites attached to various astronomy and physics departments in the US and UK. But just one point of logic: If you are trying to plot a bunch of points to measure H0 (slope of the redshift vs. distance best-fit line), how can you possibly use a value of H0 to get your x-axis distance values? Won't you find kind of a perfect fit, whose slope is exactly the value of H0 you used? Think about that in connection with papers that have new values of H0, with error bars. > These are defined as ... >indicators which depend for their calibration on our knowing the distance to >some representative nearby galaxies through primary distance indicators. Taking the first method in that section -- for no other reason than it's >first in the book -- we have the HII regions method. It is based on the >assumption that one can estimate the dimensions of core and halo diameters >within the HII regions of a galaxy (and that these surround new O and B >stars). A 'correlation' was found between the HII region diameter and the >galaxy luminosity class. There are several problems with this method >(including the fact that the relationship was not the one that was first >identied, but was 'forced' as a secondary method when the first was found to >be nearly useless), which are listed in the book. The primary one being >that the method of core and halo diameters on the plates are subjective and >liable to systematic errors. But this method was created by first ensuring >(via extinction and selection of the 'proper' definitions for intensity) >that the correlation (which has no theoretical justification) would be both >linear and consistent with the hubble law. OK, stop right there. Since you're claiming scientific fraud, I'm going to ask for details. What is meant by extinction and selection of the proper definitions for intensity. What method did they use, and did the authors say we want to ensure compliance with the Hubble law so we fudged the data? Or what did they say? Surely they had a justification for whatever this procedure is you are alluding to. What is it? >> and I'll describe distance measurement via Cepheid variables. Found >> several good links. I'm not interested in links. I described what the links are. A number of them are papers. All of them are written by astronomers. Saying you aren't interested in links is saying you're going to dismiss out of hand all research I lay my hands on. Isn't that a little prejudicial? Why should I bother? I bookmarked at least 10 pages in the last 24 hours with plans to read the calibration procedures in detail and summarize them here. Are you saying that by virtue of being on a web page you're going to dismiss anything I write sight unseen? > And I don't need to know about estimating >WITH cepheid variables. I'm talking about distance measurments *beyond* the >distance where cepheid variables are resolvable. Cepheids are the farthest thing we've got distance measurements on. The Hubble paper involved Cepheids out to pretty extreme ranges. - RAndy ==== >> 1) Measure redshift of object. >> 2) Measure distance to object. We can't measure distances directly beyond parallax range (about 300 >parsecs). The 'measurements' that we use beyond the nearest few dozen >galaxies (based mostly on cepheids) are all based squarely on assuming the >big bang. > > Describe those methods and tell me where this assumption comes into > the method, or stop repeating this ridiculous bit of nonsense. > > - Randy Notice that when faced with a rational response that included a *detailed* explanation, Randy Poe deleted out the details, and did so creatively delete as to put things out of context so that he can insinuate to naive readers that something was left undone. However, in the post to which Randy Poe replied as well as several others in the thread greywolf42 has given all the facts needed to make his case. One of the problems with a poster like Randy Poe is that he is calculated in his irrationality, and seems rather savvy of the weaknesses of Usenet, and worst of all, he's persistent. You can answer his questions in *detail* repeatedly, yet he'll come back with some lame reply, as if you never explained a thing. And he will reply, and reply, and reply, and reply. James Harris ==== >> 1) Measure redshift of object. >> 2) Measure distance to object. We can't measure distances directly beyond parallax range (about 300 >parsecs). The 'measurements' that we use beyond the nearest few dozen >galaxies (based mostly on cepheids) are all based squarely on assuming the >big bang. > > Describe those methods and tell me where this assumption comes into > the method, or stop repeating this ridiculous bit of nonsense. > > - Randy > > Notice that when faced with a rational response that included a > *detailed* explanation, Randy Poe deleted out the details, and did so > creatively delete as to put things out of context so that he can > insinuate to naive readers that something was left undone. You made another accusation elsewhere in this thread, where you called me a frequent liar. I invited you twice to back up that claim with statements I made which you considered to be false. I note that you haven't taken up that challenge. I'll just issue it for the the third time here. > However, in the post to which Randy Poe replied as well as several > others in the thread greywolf42 has given all the facts needed to > make his case. Um, no he didn't. When I ask him for these details, his response (No, you provide the details) is a tacit admission that he has not yet done so. I have collected an array of links containing those details which have not yet appeared in this thread, by me or anyone else. I will summarize them when greywolf responds to a question I asked to make sure we're on the same page. Since he is marginally more rational than you, I expect he will do so before the end of the day and I will in turn provide the details I allude to. You, in contrast, are full of empty blather and strong accusations which you are unwilling to back up. In fact for most of your accusations I suspect you know when you make them that they are untrue. What do you call a person who makes statements they know to be untrue? - Randy ==== >> >> Yes. Very nice. And this is such an obvious observation that hey, >> anybody with half a brain should've made it, right?:-) Lets not be >> ridiculous. >> >> Physics is a progression of better an better models. There is no need >> for dramatic posturing along the lines of we've overturned the old >> evil order. This is childishness. The Greeks of Aristotle's time could make smooth inclined planes also. >The Greeks of Aristotle's time, with the exception of gun powder and the >telescope had the same technology at hand, as did Galileo. What the The Greeks had charcoal, sulfur, and saltpeter. They could have made gunpowder. The Greeks had copper, zinc, lodestones, and acids, they could have discovered Maxwell's equations. The Greeks had sand and fire, they could have invented the refracting telescope. The Greeks could work metal and distill alcohal, they could have invented the internal combustion engine. They really dropped the ball. In a metaphorical sense, I mean. -- Suppose you were an idiot... And suppose you were a member of Congress... But I repeat myself. - Mark Twain ==== > The Greeks had charcoal, sulfur, and saltpeter. They could have made > gunpowder. The Greeks had copper, zinc, lodestones, and acids, they could > have discovered Maxwell's equations. The Greeks had sand and fire, they > could have invented the refracting telescope. The Greeks could work metal > and distill alcohal, they could have invented the internal combustion > engine. Considering the acomplishments of the only decent Greek scientist, Archimedes [1], your ironic commentary borders closely on truth. What the Greek philosophs were missing was humility, not technology; the realization that one ought to submit the products of the mind to the judgement of Nature. Bob Kolker [1] See the bio of Archimedes in Plutarch's -Lives-. He put the scare into the arrogant Romans. I think it was the Death Rays (parabolic mirrors reflecting sunlight) that upset the Romans the most. ==== > Considering the acomplishments of the only decent Greek scientist, > Archimedes [1], your ironic commentary borders closely on truth. What > the Greek philosophs were missing was humility, not technology; the > realization that one ought to submit the products of the mind to the > judgement of Nature. Archimedes was a remarkable engineer and mathematician. Pretty much nothing is known about whether he was a very good scientist. By at least one canon of science, he was miserable: he kept his discoveries a secret. Thomas ==== > Considering the acomplishments of the only decent Greek scientist, >> Archimedes [1], your ironic commentary borders closely on truth. What >> the Greek philosophs were missing was humility, not technology; the >> realization that one ought to submit the products of the mind to the >> judgement of Nature. Archimedes was a remarkable engineer and mathematician. Pretty much >nothing is known about whether he was a very good scientist. By at >least one canon of science, he was miserable: he kept his discoveries >a secret. Ptui. How do you know that? I was in the computer biz. We did not document the obvious. Now that obvious is gone and today's developers are rediscovering the knowledge all over again. I spent a gazillion Joules of brain energy trying to anticipate the lossage and preserved what I knew had a high probability of getting forgotten. I was very wrong in my estimates. One of my deepest regrets is that I did not take history seriously enough so that I could use that data to anticipate what kinds of things would become lost. I can guarantee you that nobody preserves the obvious. /BAH ==== > The Greeks had charcoal, sulfur, and saltpeter. They could have made >> gunpowder. The Greeks had copper, zinc, lodestones, and acids, they could >> have discovered Maxwell's equations. The Greeks had sand and fire, they >> could have invented the refracting telescope. The Greeks could work metal >> and distill alcohal, they could have invented the internal combustion >> engine. Considering the acomplishments of the only decent Greek scientist, >Archimedes [1], your ironic commentary borders closely on truth. What >the Greek philosophs were missing was humility, not technology; the >realization that one ought to submit the products of the mind to the >judgement of Nature. And who can say what a future generation will say was right in front of our noses and should have been discovered. -- In any case, don't stress too much--cortisol inhibits muscular hypertrophy. -- Eric Dodd ==== > The Greeks had charcoal, sulfur, and saltpeter. They could have made > gunpowder. The Greeks had copper, zinc, lodestones, and acids, they could > have discovered Maxwell's equations. The Greeks had sand and fire, they > could have invented the refracting telescope. The Greeks could work metal > and distill alcohal, they could have invented the internal combustion > engine. >>Considering the acomplishments of the only decent Greek scientist, >>Archimedes [1], your ironic commentary borders closely on truth. What >>the Greek philosophs were missing was humility, not technology; the >>realization that one ought to submit the products of the mind to the >>judgement of Nature. And who can say what a future generation will say was right in front of >our noses and should have been discovered. > :-))) Right now, right in front of my nose (well, metaforically) there is a list of numbers some subset of which will yield big money at the nearest state lottery drawing. So? Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== aaaaa > The Greeks had charcoal, sulfur, and saltpeter. They could have made > gunpowder. The Greeks had copper, zinc, lodestones, and acids, they could > have discovered Maxwell's equations. The Greeks had sand and fire, they > could have invented the refracting telescope. The Greeks could work metal > and distill alcohal, they could have invented the internal combustion > engine. >>Considering the acomplishments of the only decent Greek scientist, >>Archimedes [1], your ironic commentary borders closely on truth. What >>the Greek philosophs were missing was humility, not technology; the >>realization that one ought to submit the products of the mind to the >>judgement of Nature. And who can say what a future generation will say was right in front of >our noses and should have been discovered. :-))) Right now, right in front of my nose (well, metaforically) > there is a list of numbers some subset of which will yield big money > at the nearest state lottery drawing. So? > > Mati Meron | When you argue with a fool, > meron@cars.uchicago.edu | chances are he is doing just the same ==== > > And who can say what a future generation will say was right in front of > our noses and should have been discovered. Coming up with hypotheses is a seperate issue from testing them. Bob Kolker > ==== > Considering the acomplishments of the only decent Greek scientist, > Archimedes [1], your ironic commentary borders closely on truth. What > the Greek philosophs were missing was humility, not technology; the > realization that one ought to submit the products of the mind to the > judgement of Nature. Who are the Greeks? ==== > The actual medieval scientists (yes, really!) did actually perform > experiments. (And--hey--guess what--so did Aristotle! And Ptolemy! > And nearly *everyone* that called themselves a scientist.) > > If Aristotle checked his work, how did he come to the conclusion that > the speed of falling bodies was in proportion to their weight. A very > simple experiment not requiring elaborate apparatus would refute this. Indeed, reading Aristotle would refute this. Aristotle did not make the claim in question. Go read the Physics. Thomas ==== >> >> Aristotle did not make the claim in question. Go read the Physics. Book IV >These are the consequences that result from a difference in the media; >the following depend upon an excess of one moving body over another. We >see that bodies which have a greater impulse either of weight or of >lightness, if they are alike in other respects, move faster over an >equal space, and in the ratio which their magnitudes bear to each other. >Therefore they will also move through the void with this ratio of speed. >But that is impossible; for why should one move faster? (In moving >through plena it must be so; for the greater >> I think that Thomas answered this already more than adequately. Aristotle got just about everything wrong about motion. Aristotle got a model based on existing observations (which were quite removed from a friction free environment). Given enough time we got a better model. That's the way science works. Oh, please. I'm tired of this canard. If I come with a model of whatever which stays in place for 10000 years until somebody comes with a better one, that it wasn't me or my followers who held up progress, just the fact that there was no better model. In other words, nobody held up progress, there simply was no progress. You seem to assume that continuous progress is inevitable, a given, unless somebody forcefully prevents it from happening. This is an unfounded assumption (which, ironically, is a reversed mirror image of Aristotle's assumption about motion. He was of the opinion that motion required a force. Not so. A body will >move with uniform velocity in a right line when no force is acting on it. > Yes. Very nice. And this is such an obvious observation that hey, anybody with half a brain should've made it, right?:-) Lets not be ridiculous. Physics is a progression of better an better models. There is no need for dramatic posturing along the lines of we've overturned the old evil order. This is childishness. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== >> I think that Thomas answered this already more than adequately. Wrong! Given the medium Aristotle said explicitly that heavier bodies >would move faster in it, in proportion to their weight. Thomas explained that the term heavy as used by Aristotle refers to what we call nowadays dense. The context appears to support it. You might've questioned this interpretation (though, as I said, the context supports it). But, you chose to repeat your previous statement as if nothing has been said. You appear to be a waste of time. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== > I think that Thomas answered this already more than adequately. Wrong! Given the medium Aristotle said explicitly that heavier bodies would move faster in it, in proportion to their weight. Now drop a one pound spherical shot and a ten pound spherical shot through air (the medium) and tell me how they move. Aristotle could have done this easily. It is not rocket science. Bob Kolker ==== > > I think that Thomas answered this already more than adequately. > > Wrong! Given the medium Aristotle said explicitly that heavier bodies > would move faster in it, in proportion to their weight. Now drop a one > pound spherical shot and a ten pound spherical shot through air (the > medium) and tell me how they move. Aristotle could have done this > easily. It is not rocket science. No, he said that *denser* bodies would fall faster, in proportion to their *density*, and he is essentially *correct* about this. (He is wrong that it is a strict proportionality, of course, but he is *right* about the relationship.) Thomas ==== >> >> Yes. Very nice. And this is such an obvious observation that hey, >> anybody with half a brain should've made it, right?:-) Lets not be >> ridiculous. >> >> Physics is a progression of better an better models. There is no need >> for dramatic posturing along the lines of we've overturned the old >> evil order. This is childishness. The Greeks of Aristotle's time could make smooth inclined planes also. So? The fact that, after the fact, we conceive the usefulness of such devices, does not mean, in ***any*** way that their relevance was obvious, apriori. >The Greeks of Aristotle's time, with the exception of gun powder and the >telescope had the same technology at hand, as did Galileo. What the >Greeks did NOT have was an inclination to check what they concluded by >empirical means (with the exception of the Ionians, who lost out). Why? >Because the Greeks having invented logic and philosophy were so enamored >of their own cleverness they figured they did not need to check. > Please. You're repeating a goddanamned ridiculous garbage of the pen of second (and lower) rate science historians. There were enough Greeks who did check, who did perform experiments etc. The Greek involvement in science and math spanned nearly 10 centuries and lots of people tried various things over this time. Get it through you head that there is nothing apriori obvious about Newtonian mechanics. Or, putting it in different words, we've two options here: 1) Everybody for the two centuries preceding Newton and Galileo was a moron who just couldn't see the obvious. 2) Your statements on the matter are nonsense. Hmm, I'll go with the second. Still, what the hell do I know. I'm but an experimental physicist. Obviously you've access to some deeper knowledge:-) Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== >The Greeks of Aristotle's time could make smooth inclined planes also. >> >> >> So? The fact that, after the fact, we conceive the usefulness of such >> devices, does not mean, in ***any*** way that their relevance was >> obvious, apriori. The question is not how obvious empirical verification is (in many >instances it is not) but how necessary it is. Your inability to understand a simple statement is noted. It is not an issue of how obvious empirical verification is but where to seek it. For somebody familiar with Newtonian physics it is obvious that inclined planes may have something to do with the issues. But there is nothing obvious about it until you've a model. > The Greek philosophers >along the Platon-Aristotle axis were disinclined to check. >> Already answered and refuted. Repetition of previous statements completely ignoring what has been added to the discussion in the meantime is a good signal that the discussion is a waste of time. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same <3c65f87.0311181923.13b10b42@posting.google.com> <3c65f87.0311200627.6edb705a@posting.google.com> <3c65f87.0311201419.48787169@posting.google.com> <873ccdt5ft.fsf@becket.becket.net> <87y8tzoule.fsf@becket.becket.net> <87ptfasysa.fsf@becket.becket.net> ==== In message , >> Yes. Very nice. And this is such an obvious observation that hey, > anybody with half a brain should've made it, right?:-) Lets not be > ridiculous. >> Physics is a progression of better an better models. There is no need > for dramatic posturing along the lines of we've overturned the old > evil order. This is childishness. >>The Greeks of Aristotle's time could make smooth inclined planes also. So? The fact that, after the fact, we conceive the usefulness of such >devices, does not mean, in ***any*** way that their relevance was >obvious, apriori. >The Greeks of Aristotle's time, with the exception of gun powder and the >>telescope had the same technology at hand, as did Galileo. What the >>Greeks did NOT have was an inclination to check what they concluded by >>empirical means (with the exception of the Ionians, who lost out). Why? >>Because the Greeks having invented logic and philosophy were so enamored >>of their own cleverness they figured they did not need to check. Please. You're repeating a goddanamned ridiculous garbage of the pen >of second (and lower) rate science historians. There were enough >Greeks who did check, who did perform experiments etc. The Greek >involvement in science and math spanned nearly 10 centuries and lots >of people tried various things over this time. Get it through you >head that there is nothing apriori obvious about Newtonian >mechanics. Or, putting it in different words, we've two options here: 1) Everybody for the two centuries preceding Newton and Galileo was a >moron who just couldn't see the obvious. 2) Your statements on the matter are nonsense. Hmm, I'll go with the second. Still, what the hell do I know. I'm >but an experimental physicist. Obviously you've access to some deeper >knowledge:-) Hear hear. I remember reading something or watching something about the Aristotle situation, after I had read the views expressed here by some people. I thought, .... the situation wasn't that bad, the people of those times just did what they could do.... No one discovered a great leap in understanding but that doesn't mean they weren't trying. :-) -- http://www.earthpoetry.demon.co.uk RC ==== >>The Greeks of Aristotle's time could make smooth inclined planes also. > > > So? The fact that, after the fact, we conceive the usefulness of such > devices, does not mean, in ***any*** way that their relevance was > obvious, apriori. The question is not how obvious empirical verification is (in many instances it is not) but how necessary it is. The Greek philosophers along the Platon-Aristotle axis were disinclined to check. > > >>The Greeks of Aristotle's time, with the exception of gun powder and the >>telescope had the same technology at hand, as did Galileo. What the >>Greeks did NOT have was an inclination to check what they concluded by >>empirical means (with the exception of the Ionians, who lost out). Why? >>Because the Greeks having invented logic and philosophy were so enamored >>of their own cleverness they figured they did not need to check. > > > Please. You're repeating a goddanamned ridiculous garbage of the pen > of second (and lower) rate science historians. Totally backed by documentation and historical record. >There were enough > Greeks who did check, who did perform experiments etc. The Greek > involvement in science and math spanned nearly 10 centuries and lots > of people tried various things over this time. Get it through you > head that there is nothing apriori obvious about Newtonian > mechanics. No it was not a priori obvious, else Galileo would hav gotten it. However the issue of experiment was the crucial thing. The Greeks did not develop the habit of thorough and careful experimentation. This is a matter of historical fact. Or, putting it in different words, we've two options here: > > 1) Everybody for the two centuries preceding Newton and Galileo was a > moron who just couldn't see the obvious. > > 2) Your statements on the matter are nonsense. My statements are supported by historical fact and the fact that Aristotle damned himself with his own stylus. He did not check. > > Hmm, I'll go with the second. Still, what the hell do I know. I'm > but an experimental physicist. Obviously you've access to some deeper > knowledge:-) No. Just the history of greek philosophy and science, which you have not bothered to study carefully. Bob Kolker ==== > The question is not how obvious empirical verification is (in many > instances it is not) but how necessary it is. The Greek philosophers > along the Platon-Aristotle axis were disinclined to check. Plato, yes. But that you treat them as some kind of axis only betrays your ignorance. Aristotle's most significant difference with Plato was his belief that the rationalistic a priori nature of Platonic philosophy was wrong, and that one must actually look at the world to understand it. And, as a result, he has a much richer physics, and was an actual experimentalist. > No it was not a priori obvious, else Galileo would hav gotten > it. However the issue of experiment was the crucial thing. The Greeks > did not develop the habit of thorough and careful > experimentation. This is a matter of historical fact. Really? Then why did they do experiments? The Greeks is a set about as homogeneous as the Italians. Aristotle did do experiments. > My statements are supported by historical fact and the fact that > Aristotle damned himself with his own stylus. He did not check. Actually, he did. I have already explained that you (and many along with you) have misunderstood what Aristotle actually said. Thomas ==== > > Yes. Very nice. And this is such an obvious observation that hey, > anybody with half a brain should've made it, right?:-) Lets not be > ridiculous. > > Physics is a progression of better an better models. There is no need > for dramatic posturing along the lines of we've overturned the old > evil order. This is childishness. The Greeks of Aristotle's time could make smooth inclined planes also. The Greeks of Aristotle's time, with the exception of gun powder and the telescope had the same technology at hand, as did Galileo. What the Greeks did NOT have was an inclination to check what they concluded by empirical means (with the exception of the Ionians, who lost out). Why? Because the Greeks having invented logic and philosophy were so enamored of their own cleverness they figured they did not need to check. Bob Kolker > > Mati Meron | When you argue with a fool, > meron@cars.uchicago.edu | chances are he is doing just the same ==== > > >The actual medieval scientists (yes, really!) did actually perform >experiments. (And--hey--guess what--so did Aristotle! And Ptolemy! >And nearly *everyone* that called themselves a scientist.) >>If Aristotle checked his work, how did he come to the conclusion that >>the speed of falling bodies was in proportion to their weight. A very >>simple experiment not requiring elaborate apparatus would refute this. > > > Indeed, reading Aristotle would refute this. > > Aristotle did not make the claim in question. Go read the Physics. John Philiponus did. Here is what he had to say. John Philoponus, Commentary on Aristotle's Physics, pp. 678.24 - 684.10 (Vitelli) Weight, then, is the efficient cause of downward motion, as Aristotle himself asserts. This being so, given a distance to be traversed, I mean through a void where there is nothing to impede motion, and given that the efficient cause of the motion differs [i.e., that there are differences in weight], the resultant motions will inevitably be at different speeds, even through a void. . . . Clearly, then, it is the natural weights of bodies, one having a greater and another a lesser downward tendency, that causes differences in motion. For that which has a greater downward tendency divides a medium better. Now air is more effectively divided by a heavier body. To what other cause shall we ascribe this fact than that that which has greater weight has, by its own nature, a greater downward tendency, even if the motion is not through a plenum? . . . And so, if a body cuts through a medium better by reason of its greater downward tendency, then, even if there is nothing to be cut, the body will none the less retain its greater downward tendency. . . . And if bodies possess a greater or a lesser downward tendency in and of themselves, clearly they will possess this difference in themselves even if they move through a void. The same space will consequently be traversed by the heaver body in shorter time and by the lighter body in longer time, even though the space be void. The result will be due not to greater or lesser interference with the motion [i.e. the resistance of the medium, since in a void there is none] but to the greater or lesser downward tendency, in proportion to the natural weight of the bodies in question. . . . Sufficient proof has been adduced to show that if motion took place through a void, it would not follow that all bodies would move therein with equal speed. We have also shown that Aristotle's attempt to prove that they would so move does not carry conviction. Now if our reasoning up to this point has been sound it follows that our earlier proposition is also true, namely, that it is possible for motion to take place through a void in finite time. . . . Thus, if a certain time is required for each weight, in and of itself, to accomplish a given motion, it will never be possible for one and the same body to traverse a given distance, on one occasion through a plenum and on another through a void, in the same time. For if a body moves the distance of a stade through air, and the body is not at the beginning and at the end of the stade at one and the same instant, a definite time will be required, dependent on the particular nature of the body in question, for it to travel from the beginning of the course to the end (for, as I have indicated, the body is not at both extremities at the same instant), and this would be true even if the space traversed were a void. But a certain additional time is required because of the interference of the medium. For the pressure of the medium and the necessity of cutting through it make the motion through it more difficult. Consequently, the thinner we conceive the air to be through which a motion takes place, the less will be the additional time consumed in dividing the air. And if we continue indefinitely to make this medium thinner, the additional time will also be reduced indefinitely, since time is indefinitely divisible. But even if the medium be thinned out indefinitely in this way, the total time consumed will never be reduced to the time which the body consumes in moving the distance of a stade through the void. I shall make my point clearer by examples. If a stone moves the distance of a stade through a void, there will necessarily be a time, let us say an hour, which the body will consume in moving the given distance. But if we suppose this distance of a stade filled with water, no longer will the motion be accomplished in one hour, but a certain additional time will be necessary because of the resistance of the medium. Suppose that for the division of the water another hour is required, so that the same weight covers the distance through a void in one hour and through water in two. Now if you thin out the water, changing it into air, and if air is half as dense as water, the time which the body has consumed in dividing the water will be proportionately reduced. In the case of water the additional time was an hour. Therefore the body will move the same distance through air in an hour and a half [i.e., the hour it would take to go through a void, plus half an hour (half as much as the hour that would be added to pass through water) because air offers only half the resistance of water]. If, again, you make the air half as dense [as you already did], the motion will be accomplished in an hour and a quarter. And if you continue indefinitely to rarefy the medium, you will decrease indefinitely the time required for the division of the medium, for example, the additional hour required in the case of water. But you will bever completely eliminate this additional time, for time is indefinitely divisible. If, then, by rarefying the medium you will never eliminate this additional time, and if in the case of motion through a plenum there is always some portion of the second hour to be added, in proportion to the density of the medium, clearly the stade will never be traversed by a body through a void in the same time as through a plenum. . . . But it is completely false and contrary to the evidence of experience to argue as follows: If a stade is traversed through a plenum in two hours, and through a void in one hour, then if I take a medium half as dense as the first, the same distance will be traversed through this rarer medium in half the time, that is, in one hour: hence the same distance will be traversed through a plenum in the same time as through a void. For Aristotle wrongly assumes that the ratio of the times required for motion through various media is equal to the ratio of the densities of the media. . . . Now this argument of Aristotle's seems convincing and the fallacy is not easy to detect because it is impossible to find the ratio which air bears to water, in its composition, that is, to find how much denser water is than air, or one specimen of air than another. But from a consideration of the moving bodies themselves we are able to refute Aristotle's contention. [Philoponus spends the rest of this paragraph drawing out a consequence of Aristotle's view before attacking it in the next paragraph.] For if, in the case of one and the same body moving through two different media, the ratio of the times required for the motions were equal to the ratio of the densities of the respective media, then, since differences of velocity are determined not only by the media but also by the moving bodies themselves, the following proposition would be a fair conclusion: in the case of bodies differing in weight and moving through one and the same medium, the ratio of the times required for the motions is equal to the inverse ratio of the weights. For example, if the weight were doubled, the time would be halved. That is, if a weight of two pounds moved the distance of a stade through the air in one-half hour, a weight of one pound would move the same distance in one hour. Conversely, the ratio of the weights of the bodies would have to be equal to the inverse ratio of the times required for the motions. But this is completely erroneous, and our view may be corroborated by actual observation more effectively than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. And so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other. Now if, in the case of different weights in motion through the same medium, the ratio of the times required for the motions is not equal to the inverse ratio of the weights, and, conversely, the ratio of the weights is not equal to the inverse ratio of the times, the following proposition would surely be reasonable: If identical bodies move through different media, like air and water, the ratio of the times required for the motions through the air and water, respectively, is not equal to the ratio of the densities of air and water, and conversely. Now if the ratio of the times is not determined by the ratio of the densities of the media, it follows that a medium half as dense will not be traversed in half the time, but longer than half. Furthermore, as I have indicated above, in proportion as the medium is rarefied, the shorter is the additional time required for the division of the medium. But this additional time is never completely eliminated; it is merely decreased in proportion to the degree of rarefaction of the medium, as has been indicated. . . . And so, if the total time required is not reduced in proportion to the degree of rarefaction of the medium, and if the time added for the division of the medium is diminished in proportion to the rarefaction of the medium, but never entirely eliminated, it follows that a body will never traverse the same distance through a plenum in the same time as through a void. ----------------------------------------------------------------------- I know what the cause of the confusion is. Aristotle did have it right, but a philosopher also named Aristotle who lived in Athens at the same time as Aristotle published eronous theories of motion. Well people reading Aristotle naturally confused what Aristotle had to say with what Aristotle really said. And the rest is history. Philiponos was confused, Galileo as confused. We all are confused, except you. Bob Kolker ==== >>If Aristotle checked his work, how did he come to the conclusion that >>the speed of falling bodies was in proportion to their weight. A very >>simple experiment not requiring elaborate apparatus would refute this. > Indeed, reading Aristotle would refute this. Aristotle did not make > the claim in question. Go read the Physics. > > John Philiponus did. Here is what he had to say. > > John Philoponus, Commentary on Aristotle's Physics, pp. 678.24 - > 684.10 (Vitelli) And John Philoponus is wrong. I find it curious in the extreme that rather than read and understand Aristotle, you play proof-text games and quote authorities. The irony! If Aristotle's followers some of them got it wrong, and said what he would not agree (for example, that there is any such thing as falling not through a plenum--something Aristotle would entirely reject), then that is not Aristotle's fault, and doesn't mark Aristotle as being a poor experimentalist. And Aristotle gives a very different explanation for why denser things fall faster--an explanation which is essentially correct. > And so, if a body cuts through a medium better by reason of its > greater downward tendency, then, even if there is nothing to be cut, > the body will none the less retain its greater downward > tendency. So says Philoponus. But Aristotle doesn't think there can be nothing to be cut, and is clear that the reason why denser things fall faster is in part because they more effectively push away the thing to be cut. That is, for Aristotle, the speed is a question of the relative difference in density between the falling thing and the medium through which it falls. So Philoponus is trying to extend Aristotelian physics to a no-medium case, which makes about as much sense as extending relativistic physics to a superluminal case. Maybe you can derive answers from the system, but it's not the system's fault if they are wildly wrong. > But this is completely erroneous, and our view may be corroborated by > actual observation more effectively than by any sort of verbal > argument. For if you let fall from the same height two weights of > which one is many times as heavy as the other, you will see that the > ratio of the times required for the motion does not depend on the > ratio of the weights, but that the difference in time is a very small > one. This of course gets entirely wrong what Aristotle uses the word weight to mean: since he says that one substance can be heavier than another, that gold is heavier than wood simpliciter, he is talking about *density*. That your authority doesn't notice this difference at all seems to be strange. > I know what the cause of the confusion is. Aristotle did have it > right, but a philosopher also named Aristotle who lived in Athens at > the same time as Aristotle published eronous theories of motion. Well > people reading Aristotle naturally confused what Aristotle had to say > with what Aristotle really said. And the rest is history. Philiponos > was confused, Galileo as confused. We all are confused, except you. Aristotle certainly had an erroneous theory. So did Newton, for that matter. You cannot read a medieval text which says Aristotle thought X and take it at face value. The author generally means If Aristotle had considered the question as I am, he would agree--even when he is wildly off. The point I am trying to make is not that Aristotle had a correct theory of motion--of course, he did not. The point I am making is that it is not as crazy as the he thought heavier things fall faster people would like you to believe. What he actually said was *much* more carefully phrased, and was *not* contradicted by experiment--indeed, it's actually *true* (to the experimental error), since what he said was that denser things fall faster, through a medium, than less dense things. If memory serves, he does not even assert that the difference in speed is proportional to the difference in density, which makes Philoponus' extension to the no-medium case entirely groundless. Thomas ==== > > Aristotle did not make the claim in question. Go read the Physics. Book IV These are the consequences that result from a difference in the media; the following depend upon an excess of one moving body over another. We see that bodies which have a greater impulse either of weight or of lightness, if they are alike in other respects, move faster over an equal space, and in the ratio which their magnitudes bear to each other. Therefore they will also move through the void with this ratio of speed. But that is impossible; for why should one move faster? (In moving through plena it must be so; for the greater > held up physics for a thousand years. He was of the opinion that motion required a force. Not so. A body will move with uniform velocity in a right line when no force is acting on it. Bob Kolker ==== > These are the consequences that result from a difference in the media; > the following depend upon an excess of one moving body over > another. We see that bodies which have a greater impulse either of > weight or of lightness, if they are alike in other respects, move > faster over an equal space, and in the ratio which their magnitudes > bear to each other. Therefore they will also move through the void > with this ratio of speed. But that is impossible; for why should one > move faster? (In moving through plena it must be so; for the greater Note what Aristotle says here! He says that denser things fall faster. Then he says that if there were a void, denser things would fall faster there too, but there would be no reason for it: for why should one move faster? He correctly notes that denser things fall faster through media. He concludes (incorrectly) that this is true to factors beyond just the interference of the medium. But when push comes to shove, he says that if there were no interference, there would be nothing to account for the faster motion of the denser object. He is unwilling (incorrectly) to drop the denser-things-fall-faster line, because he extends it (incorrectly) beyond the in-a-medium case. He then concludes (incorrectly) that there cannot be a void at all. But note that he argues against a void here *precisely* because he recognizes that denser things would no longer fall faster in a void, because there would be no reason for it. He thinks this is an inconsistency, though of course he's wrong: it's perfectly consistent. Did Aristotle make mistakes? Yes. Did he make trivial ones that a simple experiment would have disproven in ten seconds? No. Thomas ==== > Did Aristotle make mistakes? Yes. Did he make trivial ones that a >> simple experiment would have disproven in ten seconds? No. Horsefeathers. Carpenters knew how to make smooth inclined planes in >Aristotle's time. The Greeks had access to drip clocks just as Galileo >did. The Egyptians invented them. What the Greeks did not have was an >inclination to put their brilliant dazzling logic to empirical tests. >They figured if it sounded right, it WAS right. worse and held up science for a millenium. > I'm beginning to reach the conclusion that you're an uneducable idiot. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== >> >> I'm beginning to reach the conclusion that you're an uneducable idiot. No sir! Everything I have asserted is based on historical fact. Produce >some facts to refute what I am saying, and I will happily listen. > What fact. Your statement that Aristotle held back progress in physics for a thaousand years is *not* a fact, just your opinion. And it is not only wrong, but ridiculous. There were nearly 2000 years separating Aristotle from Newton. And, during all this time nobody came with a better theory of motion. Not only in Europe but also in China, India, the Muslim caliphate. All these were places of study and scholarship (in fact, through most of the time period mentioned they were *the* centers, Europe was insignificant), yet in none of them a better model was formulated. So, what was it? Aristotle's spirit held everybody back (including in places which barely heard of him)? Or was it that everybody was dumb for 2 millenia and couldn't see the obvious? What do you think? Get this through your mind, there is *nothing* obvious about Newtonian mechanics. The fact that something may appear simple, even self evident, once it is known and understood does in no way translate into it being self evident apriori. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== Get this through your mind, there is *nothing* obvious about Newtonian >mechanics. The fact that something may appear simple, even self >evident, once it is known and understood does in no way translate >into it being self evident apriori. Anybody who doubts this should try his/her hand at teaching the basics of Newtonian mechanics to a college freshman-level General Physics class. Something's *gotta* be pushing on that glider on the air track, to keep it moving at constant speed. You mean when I push on the wall, the wall pushes back on me? How can it possibly do that? -- Jon Bell Presbyterian College Dept. of Physics and Computer Science Clinton, South Carolina USA ==== >>Get this through your mind, there is *nothing* obvious about Newtonian >>mechanics. The fact that something may appear simple, even self >>evident, once it is known and understood does in no way translate >>into it being self evident apriori. Anybody who doubts this should try his/her hand at teaching the basics of >Newtonian mechanics to a college freshman-level General Physics class. Something's *gotta* be pushing on that glider on the air track, to keep >it moving at constant speed. You mean when I push on the wall, the wall pushes back on me? How can it >possibly do that? > Aye, that's par for the course. And some never get it. Mati Meron | When you argue with a fool, meron@cars.uchicago.edu | chances are he is doing just the same ==== > I'm beginning to reach the conclusion that you're an uneducable idiot. Are you familiar with the Aristotelean Howlers? Like assuming human females have fewer teeth than males. Was it too much for the great philosopher to get females to open up so he could count? But his biggest Howler was assuming the Kosmos was alive. He spoke of insensate matter striving to get from where it was to where it belonged. He took the notion of final cause or telos (purpose) seriously in understanding the motions of non sentient entities. In short he looked at reality which is mostly dead and saw live things at work. He had a lively view which was dead wrong. The universe is dead and living things are made of dead matter interacting in ways not fully understood. It was Aristotelean bullshit that lead to such notions as aether, qunitessence and Vital Essence. This kind of -nonsense- held science up for hundreds of years. And why did this nonsense persist. Because people did not check. It was the triumph of reductionism and materialism promoted by the emprical method that broke us out of this ignorant bondage. Two cheers for reductionism, and God bless Francis Bacon and Thomas Hobbes. I may be wrong, but I am no idiot. I have evidence, both historical and philosphical to support my views. Bob Kolker ==== > Are you familiar with the Aristotelean Howlers? Like assuming human > females have fewer teeth than males. Assuming? The incorrect observation is buried amidst a mass of truly excellent observations of biology--so good, that Charles Darwin (who, unlike you, read the man's work) said Aristotle was the greatest biologist before Charles Linnaeus. The evidence is very very clear that Aristotle *did* check. Why he gave the incorrect observation in this case is unknown. Different adults do manifest different numbers of teeth (because wisdom teeth are often not visible, for example), and it is possible that he had an inadequate sample. > Was it too much for the great > philosopher to get females to open up so he could count? You say it as if he had some a priori reason to think women had fewer teeth--but he had none. He gives no reason, he simply lists it along a large list of otherwise correct observations. Thomas ==== > You say it as if he had some a priori reason to think women had fewer > teeth--but he had none. He gives no reason, he simply lists it along > a large list of otherwise correct observations. Everything has to be checked. Why? Because it could be wrong. Bob Kolker ==== > You say it as if he had some a priori reason to think women had fewer > teeth--but he had none. He gives no reason, he simply lists it along > a large list of otherwise correct observations. > > Everything has to be checked. Why? Because it could be wrong. Yes. Who says he didn't check? Because everyone who checks always gets it right? Thomas ==== > > You say it as if he had some a priori reason to think women had > teeth--but he had none. He gives no reason, he simply lists > it along a large list of otherwise correct observations. > > Everything has to be checked. Why? Because it could be wrong. What a peculiar response. It gives the strong impression that Thomas Bushnell did _not_ precede what you quoted here with: : > Assuming? The incorrect observation is buried amidst a mass of : > truly excellent observations of biology--so good, that Charles : > Darwin (who, unlike you, read the man's work) said Aristotle was : > the greatest biologist before Charles Linnaeus. : > : > The evidence is very very clear that Aristotle *did* check. Why : > he gave the incorrect observation in this case is unknown. : > Different adults do manifest different numbers of teeth (because : > wisdom teeth are often not visible, for example), and it is : > possible that he had an inadequate sample. Jim Burns ==== > > I'm beginning to reach the conclusion that you're an uneducable idiot. No sir! Everything I have asserted is based on historical fact. Produce some facts to refute what I am saying, and I will happily listen. Bob Kolker ==== > I'm beginning to reach the conclusion that you're an uneducable > idiot. > > No sir! Everything I have asserted is based on historical > fact. Produce some facts to refute what I am saying, and I will > happily listen. You keep saying based on historical fact, but you haven't actually shown the facts. ==== > Did Aristotle make mistakes? Yes. Did he make trivial ones that a > simple experiment would have disproven in ten seconds? No. Horsefeathers. Carpenters knew how to make smooth inclined planes in Aristotle's time. The Greeks had access to drip clocks just as Galileo did. The Egyptians invented them. What the Greeks did not have was an inclination to put their brilliant dazzling logic to empirical tests. They figured if it sounded right, it WAS right. worse and held up science for a millenium. Bob Kolker ==== > Horsefeathers. Carpenters knew how to make smooth inclined planes in > Aristotle's time. The Greeks had access to drip clocks just as Galileo > did. The Egyptians invented them. What the Greeks did not have was an > inclination to put their brilliant dazzling logic to empirical > tests. They figured if it sounded right, it WAS right. Except that's not true. Aristotle said that it was important to conduct experiments. He never said that if it sounded right, it WAS right, indeed, this is exactly how he criticized Plato, because Plato *did* say that kind of thing. Checking his work on an inclined plane would *not* have proved the point, because he knew full well that the difference in falling speed through air was negligible. But it was observable in water, and he *correctly* concluded that it should be present in air too, even if too small to notice. Thomas ==== > He was of the opinion that motion required a force. Not so. A body > will move with uniform velocity in a right line when no force is > acting on it. Is this true even for motion through a medium? Since Aristotle did not believe a vacuum could exist, he wasn't talking about motion where there is no impeding medium. Thomas ==== > Is this true even for motion through a medium? Since Aristotle did > not believe a vacuum could exist, he wasn't talking about motion where > there is no impeding medium. It does not matter. Even in air two dense spherical objects of differing weight will take just about the same time to fall. Air friction is negligible. Aristotle clearly stated that the speed of the fall is proportional to the weight. Read what the man said, not what you wish he had said. If course the source of the dispute is that there was another Aristotle who lived in Athens as the same time as Aristotle. Aristotle got it right but the other Aristotle got wrong and that is why everyone is confused, except you. Bob Kolker ==== > > Is this true even for motion through a medium? Since Aristotle did > not believe a vacuum could exist, he wasn't talking about motion where > there is no impeding medium. > > It does not matter. Even in air two dense spherical objects of > differing weight will take just about the same time to fall. Air > friction is negligible. Aristotle clearly stated that the speed of the > fall is proportional to the weight. Read what the man said, not what > you wish he had said. He said *density*, as I have already explained. He did not have distinct words for weight and density (and perhaps he did not know the need to be as careful as we are about the difference). Denser objects of identical mass and shape do fall faster through a medium, and the context makes it clear that this. If you don't like it, get a glass of water and check. thomas ==== > > >The actual medieval scientists (yes, really!) did actually perform >experiments. (And--hey--guess what--so did Aristotle! And Ptolemy! >And nearly *everyone* that called themselves a scientist.) >>If Aristotle checked his work, how did he come to the conclusion that >>the speed of falling bodies was in proportion to their weight. A very >>simple experiment not requiring elaborate apparatus would refute this. > > > Indeed, reading Aristotle would refute this. > > Aristotle did not make the claim in question. Go read the Physics. > > Thomas ==== Incidentally, there is a good argument that you should expect heavier bodies don't fall faster which doesn't even require a special experiment. If heavier bodies fall faster, then it would follow that gluing two things together should make them fall faster, and this is wildly out-of-kilter with ordinary experience. (In principle, you would need an experiment even so.) But the point is that such a claim is so obviously bogus that one wonders why anyone ever attributed it to Aristotle. Ah, well it turns out that the distinction between density and mass is a recent concept, and Aristotle didn't have it. Where he speaks of heavy, he means denser. This is clear, because he says that gold is heavier than wood, for example. There isn't any sense in which one substance is heavier than another, in our terms, but gold is certainly more dense. So once we know that he's speaking about density, the claim in the Physics is that denser objects fall faster. Now, what is his reasoning? First off, it's that they must displace the objects in the medium through which they are falling, and he gives an argument which amounts to the claim that less dense objects have greater boyauncy. (Which, in fact, they do.) Of course you can't observe this in air, but the Aristotle's account is independent of which medium you are falling through, so why not pick a denser (heavier) one, like water? And we find there that it's really easy to see less dense things sink more slowly. In air the effect is much harder to see, in part because unrelated things like shape matter a lot more, and because the air is so much less dense than the things we drop through it, that the differences caused by the things we drop through are very hard to see. So, Aristotle's claim is the roughly correct one that less dense backed up by some common-sense observations and noting what happens when things fall through. I do not know where the misrepresentation originally came from, however. But a misrepresentation it is, if you actually bother to read the Physics and see for yourself instead of just repeating what some authority told you Aristotle said. (Oh, the irony!) As for his experimentalism, it is most exceptional in the area of biology. Charles Darwin remarked that Aristotle was the greatest biologist until Linnaeus. This is well born out by looking at his very extensive biological writings. (Though of course, like any biologist, he does make mistakes.) Thomas ==== Let {a(k)} and {b(k)} be two sequences of nonnegative reals, where {a(k)} is either monotonically nondecreasing or nonincreasing. Let m and n be positive integers. Let A(n) = sum{k=1 to n} a(k), B(m) = sum{k=1 to m} b(k), and C(n) = sum{k=1 to n} a(k) b(k mod m), where 1 <= (k mod m) <= m. So, I get : |A(n*m) *B(m) - m *C(n*m)| <= m *B(m) *max(a), where 'max(a)' is the largest a(k), among 1 <= k <= n*m. (So, max(a) = a(n*m) or a(1), depending on whether {a{k)} is nondecreasing or nonincreasing.) (A recent post(s) of mine uses a specific case of the above.) A result derived from above: If a(x) is a real -> real function that is, for all x, defined and 1) positive 2) finite (no poles) 3) monotonic then: limit{ n-> oo} (sum{k=1 to m*n} a(k/n) b(k mod m)) /n = (1/m) (sum{k=1 to m} b(k)) integral{0 to m} a(x) dx First, am I right in all cases I have restricted the results above to, if right in even those cases? And do I really need all 3 additional constraints upon a(x) for the limit-result to be true? (since the result seems trivial somehow) thanks, Leroy Quet ==== > A friend once remarked to me in casual conversation, Gauss was just a > little bit ahead of his time, I think. I mean, people still believed > in witches in those days, and he's proving the law of quadratic > reciprocity! This comment assumes implicitly that a great scientific > or mathematical thinker is bound not to be gulled by the irrational > and unscientific prejudices of their day; however, there are obviously > counter-examples to this assumption. However, how does this (false) > assumption hold up in the particular case of Gauss? Was Gauss > relatively free of superstition, and was he in fact sceptical with > regard to belief in witches etc? > > Paul Epstein I may be wrong, but I believe I that superstition itself was a major incentive for math-research back in the old days (Pythagorians {spelling?}, numerology, etc). As someone else in this thread implied, one person's superstition is another person's science is another's religion, politics, passion, psychosis, mythology, etc. (I have a strong feeling that superstition in-general is an UNAVOIDABLE part of human-nature, perhaps evolving within us so as to help us {along with humor, music, art, etc} deal with stress.) (Still, that is NO excuse! Wake up, you!....{slap, slap, slap across face}) thanks, Leroy Quet ==== > >A friend once remarked to me in casual conversation, Gauss was just a >little bit ahead of his time, I think. I mean, people still believed >in witches in those days, and he's proving the law of quadratic >reciprocity! This comment assumes implicitly that a great scientific >or mathematical thinker is bound not to be gulled by the irrational >and unscientific prejudices of their day; however, there are obviously >counter-examples to this assumption. However, how does this (false) >assumption hold up in the particular case of Gauss? Was Gauss >relatively free of superstition, and was he in fact sceptical with >regard to belief in witches etc? > > Gauss lived from 1777 to 1855. Educated people in those days did not > generally believe in witches AFAIK: the last trial for witchcraft in > Germany was in 1749. I don't know anything about any superstitions > Gauss may or may not have had. What would it matter if he was > superstitious? Newton, on the other hand, was quite involved in > mystical speculations, alchemy etc. - does that make him any less > of a mathematician? > > Also, many of today's mathematicians are religious believers. To me > that is superstition. I am a mathematician, but I am not superstitous. That brings bad luck. Sorry -- could't resist. Tuomo ==== > > Last trial in Britain in 1944--due to war paranoia: Unless you count the so-called 'satanic abuse' nonsense a few years back, when a few busybody social workers got it into their heads that families in certain remote areas of the UK (mainly the Orkney Islands as I recall) were partaking in all kinds of lurid satanic rituals involving children. AKAIK not one person was ever successfully prosecuted, although the whole thing caused lasting disruption to many innocent families, especially the children. These social workers really are the scum of the Earth. They're at it again now, on the Pitcairn Islands, which they'd have us believe are a haven of incest and child abuse. ==== > ............ Newton, on the other hand, was quite involved in > mystical speculations, alchemy etc. - does that make him any less > of a mathematician? I don't believe this is strictly correct. Newton's beliefs seem quirky to us, but they were not mystical in the sense that astrological and alchemical beliefs are usually understood. He believed that there was a former age which had obtained superior knowledge of nature, and that this knowledge was somehow encoded in alchemical and other ancient writings. This is similar to Simon Stevin's belief in an age of wisdom ( siecle sage, ) I think, and I suppose it was widely held. Newton's own approach to natural philosphy was consistently materialistic and rationalistic, however. Cf. his General Scholium at the end of the Principia: [ re nervous activity in animals ] But these are things which cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination of the laws by which this electric and elastic spirit operates. As to his mathematics, a perusal of various sections of the Principia leads me to opine that Newton was not only a Genius, he was pretty smart, too. Lew Mammel, Jr. ==== the only way to really understand this myth, is to put it into the perspective of its actual creation, like, the Second (British) Church of Christ, Isaac. because of the official promulgation of this freemasonic lachemical cook, mainly to cover-up his defamation of Leinbniz, people actually, continually write that (e.g.) he dyscovered the laws of motion, when he merely manipulated Kepler's dyscoveries, algebraically. I did, at last, find another good use for the dot notation: to dystinguish a differential of time. saw it at the bookstore, yesterday (Midnight Special, on Second between SMBlvd and Broadway, in SAnta Monica). > mystical speculations, alchemy etc. - does that make him any less > of a mathematician? > He believed that there was a former age which had obtained superior > knowledge of nature, and that this knowledge was somehow encoded > in alchemical and other ancient writings. This is similar to Simon > Stevin's belief in an age of wisdom ( siecle sage, ) I think, and --ils duces d'Enron! ==== In the thread (posted about a month back) two number-theoretical limits (& bonus sum): http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=off&threadm=b4be2fd f.0310301645.38b9a167%40posting.google.com&rnum=6&prev= I figured that d(m), the number of positive divisors of m, was such that limit{n-> infinity} (1/n) (sum{m=1 to n} d(m)) - ln(n) = 2*c - 1, where c is Euler's constant (.5772...). So, I am wondering, what is the sequence of increasing positive integers where: a(1) = 1; |d(a(m)) - ln(a(m)) +1 -2c| < |d(a(m-1)) - ln(a(m-1)) +1 -2c| for all m >= 2 ? In other words, what is the sequence of positive integers where the number of divisors in each term better approximates the asymptotical appoximation of the 'average number of divisors' than any previous term? We could also ask about the simplier defined b-sequence: b(1) = 1; |d(b(m)) - ln(b(m))| < |d(b(m-1)) - ln(b(m-1))| thanks, Leroy Quet ==== can some one please break this down for me, im not all together clear on this con cept as a whole ==== I'm looking for any information about the Gegenbauer polynomials (specifically, C_n^{(1)}(x)) in F_2[x] (F_2 = finite field with two elements). I'll take anything I can get, but I'm particularly interested in roots (in the algebraic closure of F_2), factors, etc. I'd take any information about Legendre polys mod 2, since I might be able to generalize. TIA! Lot-o-fun ==== I should be a rather happy guy. After all, over 18 months ago I found this partial difference equation I call dS(x,y), and the sum of dS from dS(x,2) to dS(x,sqrt(x)) is the count of primes up to and including x. Usenet, and searching math references, both bought and on the Internet, I know that I have a first-find. Somehow, I am the first human being in recorded human history to find a partial difference equation that sums to give the count of prime numbers. This post is about some of the significance of that beyond it being a first-find. Prime numbers have fascinated people for some time, and mathematicians especially. The great mathematician Karl Gauss is credited with making an important hypothesis in the field of prime numbers, as he'd noticed something. Gauss noticed that the count of primes numbers could be approximated by x/ln x, for instance, the count of primes up to 1000 is 168, and 1000/ln 1000 approximately is 144.76. The count of primes up to 10000 is 1229, and 10000/ln 10000 is approximately 1085.73, which is a closeness that continues as you go higher. Gauss wondered what the discrete count of prime numbers could have to do with continuous functions like x/ln x, and while mathematicians made progress in finding relations that gave limits, like Chebyshev's use of the zeta function discovered by Euler, they never found a reason why. I may have found that reason. Not surprisingly, a first-find in the area of prime numbers *should* be a big deal, but despite the ease with which I link my discovery to some of the biggest names and biggest issues in mathematics, there is the value to society of the discoverer. Since when has modern society decided that discoverers should be attacked instead of cheered? Now you may have seen a LOT of postings from people trying to attack the worth of my find, which can be a healthy process--if they stick with the facts. Unfortunately posters have shown a dismaying tendency to lie, but that's minor to the problem I've faced where mainstream mathematicians have tried to ignore or downplay my result. I have a first-find in the area of prime numbers, and my not being a mathematician does not mean that mathematicians can just deny the reality if it suits them. While they may feel they have many reasons to attack the value of an important find from a non-mathematician, those reasons cannot be in the best interests of society. If Gauss were alive today, would he cheer me? I like to think he would, as he was someone interested in asking questions *and* in getting answers. First and foremost I think he would have been driven to find out just where my discovery led, and if it was the answer to the question that intrigued him. As I've found a partial difference equation, it leads to a partial differential equation. That partial differential equation may answer many questions. Or more importantly, it should raise many more. You should not allow mathematicians to continue to pervert a process that has helped humanity for so many thousands of years. You must not show a loss of faith in the future of humanity, as if discoverers are no longer needed. Academic institutions can no more constrain who can make a major discovery, than they could limit who will be a great painter, composer, or architect. Maybe that's part of the problem as we know that architects require a lot of schooling beyond just art, as they need to know physics, like materials science, and engineering, among many other things. So it's easy to assume that a great building can only come from someone heavily trained in academia who can manage a huge structure. However, sometimes something a little smaller in terms of physical size can be huge in terms of social value, and the person who built it, might be someone from just around the corner, outside of academia. Maybe I'm pushing the analogy, but I hope that you'll agree that at the end of the day, what's important is the *information* and petty squabbles and personal attacks are irrelevant, and often forgotten over history anyway. It's the knowledge that remains--pure. dS(x,y) = [p(x/y, y-1) - p(y-1, sqrt(y-1))][ p(y, sqrt(y)) - p(y-1, sqrt(y-1))], S(x,1) = 0, p(x, y) = floor(x) - S(x, y) - 1, and S(x,y) is the sum of dS from dS(x,2) to dS(x,y). And p(x,sqrt(x)) is the count of prime numbers up to and including x. That's pure knowledge. Information discovered by me, and hey, it wasn't like it just jumped in my lap you know. There's a value to cheering on discovery, and not attacking it. The value is hope for the future. Hope that there may be answers out there from unlikely sources. Hope that every person can be valuable. Maybe mathematicians want a reality that has them ordained as the only route for new mathematical knowledge. Possibly they wish control over the creative process, and total dominion over mathematical discovery. But hey, they're only human. James Harris My math discoveries, found for profit http://mathforprofit.blogspot.com/ ==== Gentlemen, I've assembled here the briefest summary I can of my a priori perspective on gravity... JS: No physicist is interested in your apriori perspective on gravity. Apriori is not a good word to use with physicists. Physics is not math. Math is the language of physics, but there is a big difference in the standards of rigor and justification, I dare not say proof. AL: and how this conforms with available formalisms. Since I have started from epr, JS: What does quantum EPR have to do with classical Einstein GR? I mean apart from my vacuum condensate of virtual electron-positron pairs and the use of a Bogoliubov pair transformation in the Unruh effect as I vaguely recall. AL: a short digression is unavoidable, however I have limited this to what is strictly necessary to form the link between epr and my anticipation of finding distributed source terms in gravity. AL:1. The state of the Evidence. The first question regarding any discussion between formalisms in gravity concerns the state of the evidence, and for this let us draw the contrast between quantum mechanics and gravity. On the quantum side, although there are diverse, bizarre and extreme views about interpretation, there is effectively universal agreement about the basic formalism. JS: OK AL:This is due primarily to the state of the evidence supporting ordinary quantum mechanics within its domain of applicability, but also to many other factors like ease of use and the particularly clean implementation of the restriction to observables. PZ: This universal agreement does not and cannot be taken to imply that the standard QM formalism is chiseled in stone, no matter how empirically accurate its predictions may be so far. A deeper theory of the quantum vacuum may expose the limits of QM's domain of validity, and pose fundamental challenges to interpretive models that are tied to it. AL: With respect to gravity, the mere existence of your discussion testifies to the poor state of the evidence. JS: No it does not. There are many good reviews on experimental relativity. Google Clifford Will. Note however just received from Cruft: There's a paper up on arXiv today: http://www.arxiv.org/abs/astro-ph/0311576 re-analysing Miller's data in the replication of the Michelson-Morley experiment and concluding that he found a real effect. R. Kiehn will be saying See, I told ya' so! I have not yet read beyond the abstract, but I will study it off line. AL: Since multiple empirically very distinct formalisms remain compatible with the evidence, the state of the evidence is, by definition, inadequate. PZ: The state of the evidence is always inadequate in science. There is always a potentially infinite class of alternative theories that are consistent with the available evidence. Usually, only two or perhaps three alternatives are considered and subjected to comparative evaluation. So I suppose by the above you mean that the current state of the evidence is not capable of enforcing a consensus in gravitational physics as to which theory should be declared the winner. Such consensual decisions are often quite illusory from an epistemological standpoint, and depend heavily upon which faction has the most effective propaganda. For example, Lorentz vs. Einstein (that is, the Machian Einstein of special relativity). JS: Perhaps. Let's see if http://www.arxiv.org/abs/astro-ph/0311576 Is correct or not? Even Yilmaz uses Special Relativity BTW. Bohm suggests Hubble flow as a preferred frame in which the quantum potential acts instantly. The alleged anomalous effect in http://www.arxiv.org/abs/astro-ph/0311576 is perhaps an integrated GR curvature effect since MM data is taken over a long period of time as I recall. It is not a LOCAL measurement in the sense of a single tangent space fiber. Still it would pull the carpet out of the rug of the MTW Establishment for sure. Paul would like that. :-) AL: We may reflect on the status ordinary quantum mechanics would have if it had a corresponding level of experimental support (it would probably not be taken seriously because of some of the - by assumption - untestable predictions it makes), however the experimental art it has given us is so powerful that any empirically inequivalent alternative is rapidly tested out of existence. JS: Huh? PZ: As Einstein pointed out to Heisenberg in 1926, it is the *theory* that tells you what can and cannot be measured. JS: Yes. AL: :IMO, when the evidence is weak enough to allow multiple distinct formalisms to survive at all, PZ: :This may not be Kuhnian normal science, but it may be for the most part normal science -- at least since ~1900. AL: and the phenomena predicted by extrapolating from the mundane weak field evidence using the conventional theory are sufficiently bizarre, as they are, it pays to keep more than just an open mind. PZ: OK. Note that Einstein himself did not believe in black holes. I'm not sure I believe in them either, notwithstanding all the evidence that has been developed for them. JS: He would if he were alive today. 1955 was still a very backward time in physics. 1947 even more so. AL: It pays to engage with full-on Cartesian doubt. PZ: Although a Galilean or Newtonian physicist might disagree. AL: Jack would have all theorists abandon any germ of an idea that doesn't instantly conform to the EEP, JS: Astronauts float in zero g all the way round and round the timelike geodesic passing to different LIFs like Duchamp's Nude Descending a Staircase that's EEP and that's good enough IMHO. Sure there may be quantum gravity corrections, torsion field corrections, singularity breakdown ... PZ: EEP under which interpretive model? JS: Floating Astronauts is the key fact of EEP in any interpretation that is not obviously false. If there are several degenerate interpretations then the issue is undecidable until one breaks the degeneracy at least with a gedankenexperiment. Interpretations mean informal language with its relative ambiguity and lack of precision or looseness of meaning and context. PZ: As far as I know, no one is disputing EEP as minimally stated by, e.g., Wheeler et al. One must IMO be very careful to distinguish EEP from Einsteinian strict gravitational-inertial equivalence, which is an interpretive model that is attached in canonical GR to the formal-empirical EEP principle. JS: Delete strict. Paul you have erected a Tempest in a Teapot over this one word strict. The equivalence of gravity to acceleration is strict if you restrict only to the connection field, i.e. g-force level (first derivatives of the metric) at a single IDEALIZED point event P. As soon as you go off that point into its neighborhood you get into tidal force issues and the equivalence is not strict at that level of the second derivatives of the metric. AL: but Hal's PV theory shows us the possibility to remain internally consistent, and consistent with the evidence, whilst changing the underlying metaphysical rules sufficiently to transcend the self-evidency of this kind of equivalence principle. JS: False. Hal's PV theory i.e. just his SSS toy model is neither formally nor informally consistent. There is a formal inconsistency in his use of r and there is complete conceptual murkiness in his informal language pertaining to his Tables I & II IMHO. Hal does not seem to understand that Alice on a timelike geodesic at event P is in free float equivalent to Eve at infinite distance at rest relative to the source M in this special case of an asymptotically 4D flat geometry in which the Yilmaz issues of global Flux Integrals for Pu and pseudo local stress-energy tensor densities arises. Both Alice and Eve have a flat metric nuv to a good degree of approximation. Bob on a timelike non-geodesic coincident with Alice at the same P sees the space-time warps given by guv. Hal's error is using Bob and Eve in his Tables I & II and never even acknowledging Alice's existence! This leads to his Yilmazian Heresy of two metrics one really flat. Right now Hal is in a space probe near the SMALL black hole he does not believe exists feeling the radial stretch and tangential squeeze second derivatives of guv both on geodesic and non-geodesic paths no matter what he does with conventional impulse rocket engines. Only by switching on the Sarfatti Zero Point /zpf Metric Engineering Drive will he escape! ;-) That is, Hal must be Thetan Master and Commander on the far side of the Universe, The Steersman of his own self-generated time-like geodesic overriding the geometry of the black hole. Metric engineering is beyond the passive enslaved to the geometry of the large source mass M. No one, other than me, has made this point explicit in the literature. It is clearly stated in my book Destiny Matrix (2002) with a copyright in the Library of Congress etc. PZ: Could you explain what you mean by metaphysical? In some sense, I would argue that all predictive empirical science is inherenty metaphysical in character. AL: No matter how well motivated, the EEP always carries a substantial is wrong) also shows. PZ: Meaning that it relies on interpretive models that are not unambiguously validated by the available empirical evidence? JS: Justify with examples. PZ: If so, you are invoking a troglodyte positivist definition of metaphysical that is no longer tenable IMO. AL: This metaphysical content may or may not be TRUE, however, even if it were, we should never treat it so, but retain multiple perspectives as far as possible. JS: Cliche. Of course. You are wandering off the point beating around the bush. PZ: Should? Sounds very Popperian. All is conjecture. Understandable, in view of the wildly conjectural nature of many of Einstein's insights. JS: Snooze ... wake me up when you get relevant again. Snooooore ... AL: So when Jack argues (somewhat unscientifically) that the EEP has been experimentally validated, my reaction is twofold: First, whilst one can experimentally validate a prediction, and in doing so gain confidence in a corresponding theory, one can never in principle verify metaphysical propositions. Second, even the general sense of the remark is unsustainable since the evidence allows for Yilmaz and PV (and others), which are empirically distinct, and not necessarily faithful to EEP. JS: More Laputanisms. PZ: OK. I basically agree with this -- the formal statement of EEP is open to multiple mutually incompatible physical interpretations (which you seem to call metaphysics), regardless of how consistent the known facts are with the canonical model. But it looks like Jack actually agrees with this also. JS: No, I do not agree that any good physics model can violate the key fact of EEP, i.e. Astronauts free float, in its proper domain. AL: 2. A priori considerations for gravity. I shall now outline an avowedly narrow perspective on gravity which has evolved by considering the EPR paradox as THE outstanding issue to resolve in Physics. Though several loopholes, especially the detection loophole, do render the evidence less than perfect, it is strong enough that I shall, for the purposes of argument here, consider epr to be a valid counterfactual: On at least some occasions the measurement result obtained by Alice would have been different had space-like separated Bob measured on a different axis, or failed to perform any measurement at all. PZ: But there is no reason to suppose that certain phenomena cannot propagate at FTL speeds -- other than that this is inconsistent with an Einsteinian *conjecture* that was in part motivated by a simplistic Machian empiricism later repudiated by Einstein himself. AL: Given epr, and noting that there is no evidence for signalling, the question whether local realism is, or is not, violated is obviously of principle interest. JS: Micro-quantum theory has signal locality. However, I think macro-quantum theory allows signal nonlocality because the local macro-quantum wave equation is nonlinear and, more importantly, is non-unitary and the Born probability intepretation breaks down for the macro-quantum wave. There is no linear Hermitian Hamiltonian operator in the Landau-Ginzburg equation for the non-unitary time evolution of the macro-quantum order parameter. Furthermore, there exists empirical evidence for signal nonlocality in Puthoff's RV data from the 70's at SRI and also more recently in Dick Bierman's presponse data, which supports my idea that the human mind as a physical field is a local macro-quantum field. Micro-quantum theory demands a nonlocal projective ray, the local macro-quantum order parameter is NOT a projective ray. Just like GR has a mass scale so that one cannot renormalize zero point energy away, so too does the amplitude of the local macro-quantum superfluid with coherent hologram phase-rigidity (the origin of Witten's alpha' = (string tension)^-1) have a measurable meaning of local condensate density. PZ: The question is, is this a form of action at a distance? If no signal can be propagated at FTL speeds, then the answer is no. JS: No, it simply means the action at a distance is uncontrollably random. One needs oil to calm the turbulent waters. This is part of the the decoding of the Cabalist's Cipher of Genesis of Carlo Suares in Paris 1973 for the Priory Sion perhaps? ;-) PZ: But this is all really an empirical question. There is no deep theoretical reason for ruling it out -- since the original deep thinking behind SR has been abandoned even by its author. AL: After decades of investigation and many thousands of propositions in the literature which offers the possibility to satisfy the paradox PZ: What paradox? JS: A most unusual paradox http://www.broadwaymidi.com/operamidi/PiratesOfPenzance-whenyouhadleft.mid PZ: You mean an effect predicted by QM that if confirmed may shake up our pre-conceived ideas? AL: without violating local realism, which is to say to render epr consistent with the notion that no part of the ontology shall exceed the characteristic velocity. JS: Snoooooooooooooooooooooore.... PZ: So we can retain local action and the necessity for contiguous propagation. However, since we do not yet understand the physical vacuum and its mysterious exotic properties, except in the most superficial terms, this really should all be considered up for grabs in any case. AL: It is early days there, but a pair of quotations introduces the line of argument: Quantum systems are not localized, they are pervasive. - Asher Peres. Of course, if we make some excitation for field at the point O, then a propagation of this excitation from this point will have a finite speed. But in the scope of the unified field model we do not be able to make this excitation or modify arbitrarily the world solution. Any excitations of the field at the point O belong to the world solution which is a single whole - A.A. Chernitski JS: Duh.... PZ: This does seem to apply to known *physical* fields. In Copenhagen QM, the wave function is not considered a physical field. How we can make such a categorical distinction when we don't even understand what a physical field actually is, physically speaking, is beyond my comprehension. PZ: All we know is that the conventional fields that physicists are familiar with propagate at a speed c -- although even this depends in GR among other things on the distribution of matter. Also, as I pointed out previously, in canonical GR a gravitational pulse doesn't even have a determinate frame-independent speed of propagation. And let us not forget the impenetrable mystery of non-local g-field energy. JS: Not a real problem. The local tidal force stress-energy density tensor is simply Tuv(Geometry) = (string tension)Guv(Geometry) In NON-exotic vacuum Tuv(Geometry) = 0 But gravity gravitates and this equation has guv =/ = nuv(flat) vacuum solutions. The problem you confound this with is that of defining global flux integrals for Pu and Muv in asymptotically flat spacetimes where the pseudo-tensor method is used. AL: Which I would paraphrase as follows Changes introduced into a field model propagate away from their sources at (or below) the characteristic velocity, but, in such a model, there are no point sources, only distributed excitations which belong to the whole. JS: William Walker in several Vigier conferences shows that there is superluminal propagation in the near EM field which drops to c about 1 wavelength from the source and is, of course, c in the far field. Near field is big for radio & ELF remember. PZ: And remember that in classic SR, the gravitational field and the EM field were completely autonomous physical entities that were supposed to propagate independently through nothing at the very same invariant speed -- which value was supposed to be built into the fundamental physical geometry of the world. If such fields are re-interpreted as various disturbances of a single physical medium (as per the later Einstein), then the question arises, What is it about this physical medium that determines the common speed of propagation of light and gravity? That's a very different ball game. To answer this in terms of variable world geometry is a bit like saying because that's the way it is. Until we have satisfactory answers to questions like this, I don't see how one can take any dogmatic position on the EPR question. AL: It appears to me that statements like these represent the one open line for an investigation into the epr paradox that can offer the prospect of respecting both causality and objective local realism (i.e. no Backward in Time, no superluminality, no many worlds). JS: False IMHO. PZ: Again, IMO the question of superluminal propagation should at this point be properly regarded as an *empirical matter*. The Einsteinian universal limitation to (now matter- dependent) c in the vacuum is properly regarded as tentative and conjectural. AL: The form of interaction we find in epr is consistent with a kinematical conception of nonlocality wherein the matter is itself thought of as pervasively distributed (consistent, BTW, with the holographic paradigm). JS: Gibberish. I have yet to see AL make even ONE plausible and or formally argued justification for any of his apriori pontifications. PZ: The fact is that we don't yet have any viable deep physical model for the QM formalism -- other than the rather amateurish pseudo-positivist epistemological contentions (some would say pretensions) of the Copenhagen school, and the Bohmian neo-realist alternative (which has its own problems and was not taken too seriously by Bohm himself). JS: Not true. Bohm took it very seriously and get depressed when it was ignored. AL: Kinematic nonlocality does not predict nonlocal signalling, JS: Correct. This is the first non-trivial correct statement you have made so far. AL: whereas within the usual dynamical conception of nonlocal interactions (with well-localised sources), we should indeed anticipate non-local signaling. Although I came across the concept independently, the term kinematic nonlocality was first coined in the epr literature. superposable. JS: But macro-quantum local waves are not linearly superposable in the same sense as above. It's a different ball game at the emergent level. PZ: We know this works at small distances (e.g. all quantum chemistry depends on this), *unlimited* distances is really an empirical question. AL: The fundamental (metaphysical) question these ideas JS: Depends what you mean by caused? I don't know what you mean. JS: This is solved in second-quantization. largely be figments of our overtaxed physical imagination -- even though gamma rays do sometimes behave very much like like pool balls. JS: So do photons of visible light in a photographic emulsion, or also in the Young double slit one photon at a time. It's not only gamma rays. Have you forgotten the photo-electric effect? PZ: I suspect that the renormalization problems of QFT may arise from the fundamental inadequacy of this dualistic Lorentzian model. but it does not solve the cosmological constant problem as Ed Witten admitted in a very admirable display of intellectual honesty and integrity that I applaud. AL: On the former, conventional view we must necessarily see the field system as propagating away from a primary ontological structure contained within a small volume near the centre of the field, at which point epr becomes impossible within local realism (LHV models being excluded). JS: You are confusing apples with oranges. PZ: But matter waves may be fundamentally different from EM waves. For example, they have no unique speed of phase propagation. JS: Your first sentence is correct, but your second sentence is a false justification. Matter waves have dispersion. You have EM waves in media with dispersion also. So dispersion is not the criterion of significant difference between matter waves and EM waves. EM waves are LOCAL MACRO-QUANTUM COHERENT STATES of real photons in the far field and of virtual photons in the near field. I mean in the sense of Glauber's 1960's quantum optics now extended to squeezed states. The micro-quantum matter wave of a single electron is a pilot wave of nonclassical active information so are many-electron entangled matter waves in configuration space or, alternatively, the pre-wavelet Wigner phase space density with information on incompatible observables consistent with Heisenberg's uncertainty principle of a discrete lattice phase space of area h per pair of conjugate observables. Blackhole thermodynamics that S/k ~ Area of Event Horizon/Lp^2 is an example leading to the idea of the World Hologram that our 3 + 1 classical spacetime is really a holographic projection of a 2 + 1 quantum field theory. It needs to be a MACRO-QUANTUM projection. More on that later. AL: However, it is by virtue of the remote interaction fields that we usually infer the presence of an electron, and we definitely have better evidence for the reality of these remote fields than we do for the PZ: The entire QM theory of electrons as as fermions depends on an invariant (i.e. completely fixed) permutational symmetry of the system consisting of *all electrons in the universe* -- regardless of spatial correlation effects. JS: Actually all quantum chemical bonding effects and magnetism effects only work when there is spatial overlap of the single electron integral of the Coulomb potential comes into play like in the Hartree-Fock mean field approximation. PZ: The classic textbook arguments that this somehow goes away when garbage* (based on an erroneous Copenhagenist model of intertheoretic correspondence). That is inconsistent with QM and the empirical fact that electrons are always fermionic. JS: Yes as shown in Aspect's experiment Paris 1982. PZ: Agreed that this kind of constraint may be violated in a supersymmetric theory, but that is another issue. JS: Huh? AL: If we consider for a moment the equally reasonable in result of widely distributed fields rather than their cause, then we have identified a significant unjustified assumption in the epr sufficiency condition, namely that we cannot assume all the (ontological) elements of reality corresponding to a given (epistemological) observation are located at, or even nearby, the site of the observation. JS: Snoooore... PZ: :But there is no single consistent physical model in QM. this is many counter-examples to what you just said. PZ: So why *would* we take the existing QM models seriously? The development of the quantum field concept was in part an attempt to get over this ontological dualism AL: The Tornado (wherein a widely distributed system of correlated movements establishes a persistent topological feature that serves as a location property for the whole indivisible system), is a mundane example clearly illustrating the alternative metaphysical possibility. Upon examining the key relativistic wave equations (HH, KG and Dirac), all of which involve the characteristic velocity, one is rapidly brought to the small hypothesis that energy is a distributed, propagative phenomenon with a characteristic velocity. JS: Snooooooooooooooooooooooooore.... PZ: Isn't that the classic field model? JS: No, it's Professor Irwin Corey's model http://www.irwincorey.org/ He is a professor at Laputa University. AL: Small? It's hardly radical to suggest that such a form of energy exists, since the radiative form is routinely observed. If it seems radical to deny the existence of other, non-propagating forms, JS: Get this Fruit Cake off my bridge! PZ: Of course it is the collapse of the matter-wave that is non-propagating, and not the matter-wave itself. JS: Collapse of sanity yes. AL: then let it at least be acknowledged that the present de facto affirmation of such forms is equally metaphysical (if not more so). PZ: I see that like all good generals you're still fighting the last epistemological war. JS: Huh? Good General? Not in this man's army. He's not even out of Boot Camp! AL: Given the relativistic wave equations, one might anticipate that the Special Theory would emerge from physical systems in which all movement is constrained to the characteristic velocity (i.e. wave systems). JS: Ugggghhhh.... http://www.animationartgallery.com/DDC513.html PZ: Right. This is called the Lorentzian model of SR. AL: I've been fortunate enough to have published a paper which develops the special theory from this perspective, but without reference to any wave equation, whilst reinterpreting it as much as is required to allow for epr without violating local realism, and no more. Which is to say that the Special Theory is interpretable (without modification) as a wave theory (please note that this is not a new idea). JS: Reference? PZ: Noted above. AL 3 How is this relevant to gravity? (quant-ph0110160) (which constrains the metaphysic for, but does not actually touch upon, gravity). PZ: OK. AL: In essence, we must anticipate that whatever causes the gravity effect is distributed throughout the observable field rather than gathered together at a singularity. JS: This got into archive? PZ: In GR it is both. In some of Einstein's later attempts at unified field theory, all was supposed to be made of a neo-Cartesian physical vacuum. Is that the kind of thing you have in mind here? AL: Without going into further details here, I am led to approach the gravity literature with three main questions in mind, the first two of which seem at least tangentially relevant to the present discussion around PV, Yilmaz and Einstein. Q1: Are there tenable refractive medium interpretations (RMIs)? PZ The modern GR physical vacuum *is* a refractive medium. That's why light rays bend near gravitating masses. So how can there be *any question* that metric distortions are associated with refractive EM effects in standard GR? The real question here -- as I see it -- is: Is this reducible to a purely geometric model? Does such a model give us satisfactory answers to all the questions we want to ask of it? JS: http://qedcorp.com/APS/EmergentGravity.doc At this point I am out of patience. This is wasting my time.