Continuity -vs- Discreteness
in [Physics]
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From: Huang on 7 Jan 2010 09:34 On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > We might even be able to say that the discrete universe is equivalent > > > to the continuous universe in the same sense that relative motions are > > > equivalent. > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > I believe that many of the results of calculus can be replicated using > discrete methods. In fact, I suspect that they all can. > > Lets suppose you could construct two bodies of formal mathematical > models. Calculus on the one hand, and a whole collection of discrete > models on the other hand which model the exact same things. Suppose > for a moment that these two bodies of knowledge are at our disposal. > We might like to show that there are some broad relationships between > these two collections of things. > > To my knowledge....this has not been done. I dont think that it has > even been explored to any great extent, if at all. > > So what the hell are you people waiting for ? Lazy sods ? I explain > all of these insights and you just sit on your fat asses ? Must I do > everything myself ? To be more specific, I believe that it is possible to construct discrete models and continuous models which both return the same numerical results. It may even be possible to do this in a fairly clever way so that broad relationships can emerge by comparing these collections of things. Once we have accomplished that, we should be able to say that these things are "equivalent" in the sense of Einstein, and that whether one wishes to model the universe as being discrete or continuous is purely a matter of choice. That the discrete universe and the continuous universe may be very different, but they are "equivalent" because the numbers crunch exactly the same. I think that this is all possible using standard mathematics, no nonstandard math would be required for this. What say ye - thou silly geese.
From: Huang on 9 Jan 2010 10:29 On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > to the continuous universe in the same sense that relative motions are > > > > > equivalent. > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > I believe that many of the results of calculus can be replicated using > > > discrete methods. In fact, I suspect that they all can. > > > > Lets suppose you could construct two bodies of formal mathematical > > > models. Calculus on the one hand, and a whole collection of discrete > > > models on the other hand which model the exact same things. Suppose > > > for a moment that these two bodies of knowledge are at our disposal. > > > We might like to show that there are some broad relationships between > > > these two collections of things. > > > > To my knowledge....this has not been done. I dont think that it has > > > even been explored to any great extent, if at all. > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > all of these insights and you just sit on your fat asses ? Must I do > > > everything myself ? > > > To be more specific, I believe that it is possible to construct > > discrete models and continuous models which both return the same > > numerical results. It may even be possible to do this in a fairly > > clever way so that broad relationships can emerge by comparing these > > collections of things. > > > Once we have accomplished that, we should be able to say that these > > things are "equivalent" in the sense of Einstein, and that whether one > > wishes to model the universe as being discrete or continuous is purely > > a matter of choice. That the discrete universe and the continuous > > universe may be very different, but they are "equivalent" because the > > numbers crunch exactly the same. > > > I think that this is all possible using standard mathematics, no > > nonstandard math would be required for this. > The only reason to even mention these things is because anything that you can do under the assumption of existential indeterminacy - likewise must be possible without it. The EQUIVALENCE of a continuous spacetime and discrete spacetime is one of the results obtailable by considering conjectural modelling. And we can use that result to explain WP duality. But this result (equivalence) should be obtainable under the assumption of existence i.e. under standard mathematics. Clearly, we should be able to demonstrate such an equivalence in a very broad number of cases. This is NOT to say that continuity and discreteness are the same thing, indeed not. Rather, that a result obtained by using one approach should be deriveable using the other - to an arbitrary degree of accuracy. And we are using the word equivalnce in the sense of Einstein, i.e. the Equivalence Principle. Why and how mathematics could progress to the place it has achieved and not recognize this is quite mysterious to me. Perhaps it is simply because you cannot "prove" that continuity and discreteness are the same thing, indeed they simply are not. But Equivalence in the sense of Einstein is not so strict, all we care about is that the numbers crunch to the same result with arbitrary accuracy, and I believe this is possible using standard math. It is certainly the case in Conjectural Modelling.
From: Huang on 9 Jan 2010 11:20 On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > equivalent. > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > I believe that many of the results of calculus can be replicated using > > > > discrete methods. In fact, I suspect that they all can. > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > models on the other hand which model the exact same things. Suppose > > > > for a moment that these two bodies of knowledge are at our disposal.. > > > > We might like to show that there are some broad relationships between > > > > these two collections of things. > > > > > To my knowledge....this has not been done. I dont think that it has > > > > even been explored to any great extent, if at all. > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > everything myself ? > > > > To be more specific, I believe that it is possible to construct > > > discrete models and continuous models which both return the same > > > numerical results. It may even be possible to do this in a fairly > > > clever way so that broad relationships can emerge by comparing these > > > collections of things. > > > > Once we have accomplished that, we should be able to say that these > > > things are "equivalent" in the sense of Einstein, and that whether one > > > wishes to model the universe as being discrete or continuous is purely > > > a matter of choice. That the discrete universe and the continuous > > > universe may be very different, but they are "equivalent" because the > > > numbers crunch exactly the same. > > > > I think that this is all possible using standard mathematics, no > > > nonstandard math would be required for this. > > The only reason to even mention these things is because anything that > you can do under the assumption of existential indeterminacy - > likewise must be possible without it. > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > one of the results obtailable by considering conjectural modelling. > And we can use that result to explain WP duality. But this result > (equivalence) should be obtainable under the assumption of existence > i.e. under standard mathematics. > > Clearly, we should be able to demonstrate such an equivalence in a > very broad number of cases. This is NOT to say that continuity and > discreteness are the same thing, indeed not. Rather, that a result > obtained by using one approach should be deriveable using the other - > to an arbitrary degree of accuracy. And we are using the word > equivalnce in the sense of Einstein, i.e. the Equivalence Principle. > > Why and how mathematics could progress to the place it has achieved > and not recognize this is quite mysterious to me. Perhaps it is simply > because you cannot "prove" that continuity and discreteness are the > same thing, indeed they simply are not. But Equivalence in the sense > of Einstein is not so strict, all we care about is that the numbers > crunch to the same result with arbitrary accuracy, and I believe this > is possible using standard math. It is certainly the case in > Conjectural Modelling.- Hide quoted text - > > - Show quoted text - Which leads me to wonder about some criticism of GR which seems to be absent. One of the early criticisms of calculus was that it dealt with the "ghosts of departed quantities", specifically limits, and how they are treated as if they are numbers. It seems that the same kind of problem is present when you consider Einsteins Equivalence Principle, but in this case we are talking about the "ghosts of a departed operator". Yes we do have the Lorentz Transform, but what does that tell us anyway ? Does it say that things are "equal" ??? No. It says that things are "equivalent". And equivalence is not the same thing as equality under Einstein's usage of the word equivalence. One should be able to demonstrate the "equivalence" of continuity and discreteness in many cases, and I think that it is just slightly MORE interesting than the equivalence of relative motions of moving bodies......unless of course we prefer to walk around in a daze for another 100 years under a self imposed state of deliberately chosen confusion (wp duality).
From: Huang on 10 Jan 2010 09:45 On Jan 9, 10:20 am, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > > equivalent. > > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > > I believe that many of the results of calculus can be replicated using > > > > > discrete methods. In fact, I suspect that they all can. > > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > > models on the other hand which model the exact same things. Suppose > > > > > for a moment that these two bodies of knowledge are at our disposal. > > > > > We might like to show that there are some broad relationships between > > > > > these two collections of things. > > > > > > To my knowledge....this has not been done. I dont think that it has > > > > > even been explored to any great extent, if at all. > > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > > everything myself ? > > > > > To be more specific, I believe that it is possible to construct > > > > discrete models and continuous models which both return the same > > > > numerical results. It may even be possible to do this in a fairly > > > > clever way so that broad relationships can emerge by comparing these > > > > collections of things. > > > > > Once we have accomplished that, we should be able to say that these > > > > things are "equivalent" in the sense of Einstein, and that whether one > > > > wishes to model the universe as being discrete or continuous is purely > > > > a matter of choice. That the discrete universe and the continuous > > > > universe may be very different, but they are "equivalent" because the > > > > numbers crunch exactly the same. > > > > > I think that this is all possible using standard mathematics, no > > > > nonstandard math would be required for this. > > > The only reason to even mention these things is because anything that > > you can do under the assumption of existential indeterminacy - > > likewise must be possible without it. > > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > > one of the results obtailable by considering conjectural modelling. > > And we can use that result to explain WP duality. But this result > > (equivalence) should be obtainable under the assumption of existence > > i.e. under standard mathematics. > > > Clearly, we should be able to demonstrate such an equivalence in a > > very broad number of cases. This is NOT to say that continuity and > > discreteness are the same thing, indeed not. Rather, that a result > > obtained by using one approach should be deriveable using the other - > > to an arbitrary degree of accuracy. And we are using the word > > equivalnce in the sense of Einstein, i.e. the Equivalence Principle. > > > Why and how mathematics could progress to the place it has achieved > > and not recognize this is quite mysterious to me. Perhaps it is simply > > because you cannot "prove" that continuity and discreteness are the > > same thing, indeed they simply are not. But Equivalence in the sense > > of Einstein is not so strict, all we care about is that the numbers > > crunch to the same result with arbitrary accuracy, and I believe this > > is possible using standard math. It is certainly the case in > > Conjectural Modelling.- Hide quoted text - > > > - Show quoted text - > > Which leads me to wonder about some criticism of GR which seems to be > absent. > > One of the early criticisms of calculus was that it dealt with the > "ghosts of departed quantities", specifically limits, and how they are > treated as if they are numbers. > > It seems that the same kind of problem is present when you consider > Einsteins Equivalence Principle, but in this case we are talking about > the "ghosts of a departed operator". Yes we do have the Lorentz > Transform, but what does that tell us anyway ? Does it say that things > are "equal" ??? No. It says that things are "equivalent". And > equivalence is not the same thing as equality under Einstein's usage > of the word equivalence. > > One should be able to demonstrate the "equivalence" of continuity and > discreteness in many cases, and I think that it is just slightly MORE > interesting than the equivalence of relative motions of moving > bodies......unless of course we prefer to walk around in a daze for > another 100 years under a self imposed state of deliberately chosen > confusion (wp duality).- Hide quoted text - > > - Show quoted text - Here's my view on unification - based on the aforementioned considerations. The universe should be accurately modellable with or without probability theory. The universe should be accurately modellable with or without randomness. The universe should be accurately modellable with or without existential indeterminacy. The universe should be accurately modellable as being either discrete or continuous. The universe should be accurately modellable as either deterministic or non-deterministic. In my opinion, a model which successfully unifies physics should satisfy these requirements. The problem is that these things seem quite impossible to incorporate into a single equation. You cannot have a single equation which is both probabilistic and at the same time non-probabilistic. The ONLY way to do that is by demonstrating and subsequently embracing EQUIVALENCE of various kinds of models, and considering all of these various approaches as different facets of single tool. Whether there is paradox or not, that is the paradox. Unfortunately that is not going to change. To successfully MODEL your way around such a situation - you MUST use equivalence (in the sense of Einstein). Equivalence allows you to take many different kinds of tools which may seem immiscible and weld them together into a single tool....thats the only way to do it IMO.
From: Huang on 11 Jan 2010 21:50
On Jan 10, 8:45 am, Huang <huangxienc...(a)yahoo.com> wrote: > On Jan 9, 10:20 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 9, 9:29 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > On Jan 8, 6:35 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > On Jan 7, 8:34 am, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > On Jan 6, 9:16 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > On Jan 6, 11:25 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > > > > On Jan 4, 8:04 pm, Huang <huangxienc...(a)yahoo.com> wrote: > > > > > > > > > We might even be able to say that the discrete universe is equivalent > > > > > > > > to the continuous universe in the same sense that relative motions are > > > > > > > > equivalent. > > > > > > > >http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-... > > > > > > > I believe that many of the results of calculus can be replicated using > > > > > > discrete methods. In fact, I suspect that they all can. > > > > > > > Lets suppose you could construct two bodies of formal mathematical > > > > > > models. Calculus on the one hand, and a whole collection of discrete > > > > > > models on the other hand which model the exact same things. Suppose > > > > > > for a moment that these two bodies of knowledge are at our disposal. > > > > > > We might like to show that there are some broad relationships between > > > > > > these two collections of things. > > > > > > > To my knowledge....this has not been done. I dont think that it has > > > > > > even been explored to any great extent, if at all. > > > > > > > So what the hell are you people waiting for ? Lazy sods ? I explain > > > > > > all of these insights and you just sit on your fat asses ? Must I do > > > > > > everything myself ? > > > > > > To be more specific, I believe that it is possible to construct > > > > > discrete models and continuous models which both return the same > > > > > numerical results. It may even be possible to do this in a fairly > > > > > clever way so that broad relationships can emerge by comparing these > > > > > collections of things. > > > > > > Once we have accomplished that, we should be able to say that these > > > > > things are "equivalent" in the sense of Einstein, and that whether one > > > > > wishes to model the universe as being discrete or continuous is purely > > > > > a matter of choice. That the discrete universe and the continuous > > > > > universe may be very different, but they are "equivalent" because the > > > > > numbers crunch exactly the same. > > > > > > I think that this is all possible using standard mathematics, no > > > > > nonstandard math would be required for this. > > > > The only reason to even mention these things is because anything that > > > you can do under the assumption of existential indeterminacy - > > > likewise must be possible without it. > > > > The EQUIVALENCE of a continuous spacetime and discrete spacetime is > > > one of the results obtailable by considering conjectural modelling. > > > And we can use that result to explain WP duality. But this result > > > (equivalence) should be obtainable under the assumption of existence > > > i.e. under standard mathematics. > > > > Clearly, we should be able to demonstrate such an equivalence in a > > > very broad number of cases. This is NOT to say that continuity and > > > discreteness are the same thing, indeed not. Rather, that a result > > > obtained by using one approach should be deriveable using the other - > > > to an arbitrary degree of accuracy. And we are using the word > > > equivalnce in the sense of Einstein, i.e. the Equivalence Principle. > > > > Why and how mathematics could progress to the place it has achieved > > > and not recognize this is quite mysterious to me. Perhaps it is simply > > > because you cannot "prove" that continuity and discreteness are the > > > same thing, indeed they simply are not. But Equivalence in the sense > > > of Einstein is not so strict, all we care about is that the numbers > > > crunch to the same result with arbitrary accuracy, and I believe this > > > is possible using standard math. It is certainly the case in > > > Conjectural Modelling.- Hide quoted text - > > > > - Show quoted text - > > > Which leads me to wonder about some criticism of GR which seems to be > > absent. > > > One of the early criticisms of calculus was that it dealt with the > > "ghosts of departed quantities", specifically limits, and how they are > > treated as if they are numbers. > > > It seems that the same kind of problem is present when you consider > > Einsteins Equivalence Principle, but in this case we are talking about > > the "ghosts of a departed operator". Yes we do have the Lorentz > > Transform, but what does that tell us anyway ? Does it say that things > > are "equal" ??? No. It says that things are "equivalent". And > > equivalence is not the same thing as equality under Einstein's usage > > of the word equivalence. > > > One should be able to demonstrate the "equivalence" of continuity and > > discreteness in many cases, and I think that it is just slightly MORE > > interesting than the equivalence of relative motions of moving > > bodies......unless of course we prefer to walk around in a daze for > > another 100 years under a self imposed state of deliberately chosen > > confusion (wp duality).- Hide quoted text - > > > - Show quoted text - > > Here's my view on unification - based on the aforementioned > considerations. > > The universe should be accurately modellable with or without > probability theory. > The universe should be accurately modellable with or without > randomness. > The universe should be accurately modellable with or without > existential indeterminacy. > The universe should be accurately modellable as being either discrete > or continuous. > The universe should be accurately modellable as either deterministic > or non-deterministic. > > In my opinion, a model which successfully unifies physics should > satisfy these requirements. > > The problem is that these things seem quite impossible to incorporate > into a single equation. You cannot have a single equation which is > both probabilistic and at the same time non-probabilistic. The ONLY > way to do that is by demonstrating and subsequently embracing > EQUIVALENCE of various kinds of models, and considering all of these > various approaches as different facets of single tool. > > Whether there is paradox or not, that is the paradox. Unfortunately > that is not going to change. To successfully MODEL your way around > such a situation - you MUST use equivalence (in the sense of > Einstein). Equivalence allows you to take many different kinds of > tools which may seem immiscible and weld them together into a single > tool....thats the only way to do it IMO.- Hide quoted text - > > - Show quoted text - To get an idea of what such an equivalence might look like, consider any standard physics formula such as the kinematic equations. In standard mathematical form these are statements which are aguably deterministic in some sense. Can we rewrite things using probability theory s.t. the kinematic equations or the solutions therof are yielded as "expected relationships" or "expected values" ? I think it could be written pretty easily. Then, you would be forced to choose between probabilistic and non- probabilistic approaches.....and there's not a damn reason why one would choose one over the other when the results are identical to aribitrary accuracy. |