mm-2179 >>I may still have to give a rigorous definition of limit, in the end > Well, yes - that's what I asked you to do and you haven't yet >> done it. I read that it took some of the most brilliant mathematicians 150 years > to arrive at a definition of limit, so give me a couple of weeks :). > That's if I decide I need to come up with one. > Oh - so, you claim to have a proof, and your proof involves limits, but it now emerges that you don't have a definition of limit. Nora B. OyMMlAwAAADyhoVhXYX4Bw0T-1IatpYa The correct assertion must be : ,, Let p be arbitrary in (0,1).Suppose that f:[a,b]--->R is absolutely continuous on [a,b] and such that equation f(x)=0 has'nt roots on [a,b] . Then F:[a,b]---> R, F(x)=|f(x)|^p , is absolutely continuous on [a,b], [True !] > If you did acknowledge the properties of the list, you would realise > your answer is independant of the list. > Before I acknowledged any such thing you would have to have made precise exactly what you mean mathematically by 0.123456789012345678901.. (i.e. my answer) is independent of the list. As things stand, I don't know what you mean by this.... Mike. JOuEBw0AAABx0KgjHw_akQV8X8XxD3uZ Dirac was a physicist, for sure his delta function must have had the > analysis mathematicians turning in their grave. Analysis mathematicians turning in their grave about the delta > function? Why? Isn't delta function the most simple object in functional ANALYSIS? And > wasn't it motivated by physicists? Sure, it's simple. But it's not a function. -- Jim Black > anzaurres1@hotmail.com top-posted: >The determinant arose from the Jacobian, needed to do a >change of variables in multi-variable integration. >> I don't think so. > Don't determinants come out naturally when you try to solve a set of n > linear equations with n unknowns? > Not when I solve (not just try to) a set of n linear equations in > n unknowns. The American Constitution guarantees you the right to solve linear > equations in any way you want. The American Constitution doesn't guarantee Robin Chapman any rights at all, as he ain't American & don't live there. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) > : Lester Zick said: > :> I hope Will and even Wolf get a chance to read this because it is > :> about as close to a useful concrete example of tautological > :> regression I'm likely to be able to provide. This what I mean by > :> subjective mechanics. And it's why I've been so insistent on an > :> essential definition of Newtonian gravitational mechanics to begin > :> with. It's called debugging reality. > ::: Very good! I think this is what everyone has been looking for from you. It's a > : very good example of what you have been trying to convey, and I see the sens of > : it now, and what you mean by [not]. It is related to proof by contradiction, > : except you are already starting with the contradiction, and trying to discover > : the root cause of it. It's essentially like induction by contradiction, and it > : makes sense. It does not make sense to me. There are not only 8 possible > theories of gravity. Just the 'non inverse square' part > alone allows for an infinite number of possible theories. > 'non inverse square' includes every function except for 1/(d*d). But Lester's comment is that it is those three qualities that distinguish Newton's theory from other theories. So if you say non inverse square it does not matter which of those other functions you choose, you have still changed Newton's theory. > I've already said I am not against ALL set theory. You don't get to pick and choose. If you accept a portion of set theory, > then you accept ALL of its logical consequences. Not if you can formulate one without those logical consequences yet maintain some of the axioms of that theory. Remember Hilbert's axioms? I can formulate a theory of geometry by accepting and rejecting the ones from Hilbert that I like and dislike, and there is nothing wrong with that as long as the axioms are consistent. > [...] > As Albert and I have pointed out, axioms are true by ASSUMPTION, not > definition. You may assume a proposition true and follow your logic to its > conclusions, but that doesn't MAKE the proposition or the conclusions true. > Different dialect, same meanng. Actually, no. For example (and since I'm feeling Star Warsy for reasons unrelated to the latest movie), I can say that an X-wing fighter can exceed the speed of light by definition, since an X-wing is defined as a hyperspace-capable fighter craft from the Star Wars universe. So if I see an X-wing fighter, I can say that it can exceed the speed of light by definition. X-wings also generally carry photon torpedos, so if I see an X-wing I can claim that it carries photon torpedos since I assume it does. But the first is much more certain than the latter, and has the right to make such a claim. An assumed axiom can make no such claim. And the problem is that we cannot sense the mind in the world by > definition. Even if you do your thoughts are another sense > redefinition, there is no way that you can claim that the mental sense is > in the world. Yes, I think we have all the evidence we need to say what. The problem is that _I_ have never experienced the evidence and you > probably have not either. The evidence requires us to hook a wire into our brain, and push a button > with our own finger to activate the electrode, and see what type of > sensation it creates in our _mind_. But the sensation would still not be in the world. It would be CAUSED by something in the world, but interactionist dualism already says that that can be the case, and would not deny that manipulating the brain can impact the mind. And we need to do the inverse as well. We need to attache probes into our > brain that can detect detailed electrical activity, and show it to us. > That would allow us to have thoughts and then see how our thoughts were > connected to the elctrical activty in our brains. But so what? This would simply prove interaction and interactionist dualists already accept that. It does not prove that the mind is just the brain or is just physical; it only proves that mind and brain, whatever they are, interact. If we can see for ourself that there is a connection between our thoughts, > and this electical activity in the brain, then I think there is no doubt > that the mental sense is in the world. That's only if you claim that the mental sense senses neural activity. Good luck with that. > Depends on your definition of physical, doesn't it? My argument is that > one of the definitions of physical is that it either is directly > observable through the five senses or only indirectly observable through > its effects on other physical things. Exactly my point. If you hook up a wire and it allows you to observer your > thoughts though your five senses, then it is in this world. Nope. Because that would be indirectly through its effect on other physical things, and I experience my mind sense without having to do that. Hooking up the wire only shows the interaction, but the mental sense is still not the same as my observations on the screen. No one denies the brain is physical. And no one denies that mind and brain interact. What is doubtful is whether my impressions of mind can be physical in the sense of other physical things or can be a sense like the other senses. Let me put it this way: Mind sense cannot be a sense of things in the world because I can hold a mental impression of a world object that is the same and yet is qualitatively different than the object in the world. So mental senses cannot be in the world in any sense of the phrase that we would normally use, or else you would be forced to say I have a thing in the world that represents but is not a thing in the world. Well, this might make sense for paintings but it would still seem odd ... Maybe you are thinking that a blip on a scope is not seeing your > thoughts. But what if we were able to hook electrods to your brain and > with the help of some testing and a computer, could sense whenever you were > thinking the thought 2, and it would flash the number 2 on the screen. > And then you found that any time the thought of the number 2 entered your > mind, it was flashing up on the screen. I believe if we could all see this type of system in action on our own > brains, the ideas of mental thoughts being not physical would fade away > from our belief system fairly quickly. Well, it would have no impact on mine, because to compare the screen 2 to the thought 2 I would have to introspect on my thoughts ... and then would be forced to realize that the impressions are not the same thing. > :> are handy fictions created by humans. QM predicts exactly how > :> things behave, to the best we can determine. It just appears to > :> be the case that the things out there are neither waves nor : Seems to me like this is a redefinition to hide the fact that we don't know properties > : that we at least used to associate with those things. No, it is just a recognition that the Universe does not care > about our naive preconceptions. Just because we want > to be able to divide the Universe up into waves and But we seem to still be able to divide up the supposed properties into ::> Do you have some actual case in mind where QM does not tell > :> you how an object will act, and whether its behavior will : Nothing in particular, but if you look at the slit experiment we have no under > : the other, and we only know that it does because we've tried it. For any > : new situation, it is clear that QM would not be able to tell us -- or > : predict -- which way it will behave without trying it. I think you are totally wrong on that point. QM makes very > accurate predictions. There is no obvious experiment out > there that QM does not make an accurate prediction for. If > there was, someone would have done it. If you describe > a new situation precisely, the theory will predict what > will happen before you actually try it. It is not just > guess work. I admit that I don't know that much about QM, but the slit experiment has under one configuration and a wave under another, and so it follows that they can only predict how a quantum object will react in the situations they've tried or under the situations that are directly mappable to the ones they've tried ... hardly a useful theory. sAS5-AwAAABlKnmtMjBbYHvhxI6W0cAg >Take the anti-diagonal number for List1, call it r. This is a real not >in List1. With probability 1, the diagonal for List2 can be shuffled so >that it has the same anti-diagonal. So with probability 1, r is not in >List2 either. In other words, the probability that r is in List2 is >zero. So what? We knew that anyway. The chance of being able the shuffle the diag so diag(List1) = > diag(List2) is the same as this program halting. START > IF rnd<0.5 GOTO START regardless of the contents of List1 and List2. This establishes that the diagonal is independant of the list. This is > a stronger assertion than r is not in the other list with P=1. It's > Game Over! > No, the most you can say is that r is not in the other list with probability 1. We can easily select the other list so that it contains r. > As the list gets saturated the possible diagonals becomes white noise > and cannot be used to construct antidiagonals. If you are privy to > this information and you utilise diagonalisation you must be claiming > that all randomly generated numbers are missing from all lists. Herc