Continuity -vs- Discreteness
Huang
Cellular automata has also been extremely successful in modelling many
of the exact same phenomena.
Suppose for a moment that any natural phenomena can be modelled with
equal accuracy using either Calculus or Cellular Automata. Can we say
that discrete and continuous methods are then equivalent ? If
different models produce the same numeric results and both models
accomplish the same thing, then we should be able to say that it is
indeterminate whether the natural world is best modelled using one or
the other.
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.
Han de Bruijn
Continuous and discrete are two ways of looking at the same thing:
http://hdebruijn.soo.dto.tudelft.nl/QED/index.htm#ft
http://hdebruijn.soo.dto.tudelft.nl/jaar2004/IHXTAK.pdf
Han de Bruijn
Uncle Al
>
> Calculus was extremely successful in modelling natural phenomena.
When did it stop being successful? London bridges falling down...
> Cellular automata has also been extremely successful in modelling many
> of the exact same phenomena.
Reference even one instance wherein this is true.
> Suppose for a moment that any natural phenomena can be modelled with
> equal accuracy using either Calculus or Cellular Automata. Can we say
> that discrete and continuous methods are then equivalent ?
Calculus is discretized. Look for the little "dx" on each box.
> If
> different models produce the same numeric results and both models
> accomplish the same thing, then we should be able to say that it is
> indeterminate whether the natural world is best modelled using one or
> the other.
>
> 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.
idiot
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
Androcles
"Uncle Al" <Uncl...@hate.spam.net> wrote in message
news:4B43DC04...@hate.spam.net...
> idiot
Mission accomplished.
Bigot.
zzbu...@netscape.net
Well, you can, but the problem is that all of the science has
already
been built around continous brick walls. So the uneducable people
still mostly work on Digital Books, Atomic Clock Wristwatches,
Light Sticks,
and Desktop Publishing, rather than sampling rate problems for the
uneducable.
And work on Holographic Computing, Laser Disk Libraries, Flat
Screen Software
Debuggers, XML, USB, HDTV, Home Broadband, External Emulators,
Multiplexed Fiber Optics i/o, PGP, and Post Wax Printing Devices,
rather than
Code Porting for the Fortranners.
And work on Self-Replicating Machines and Post 1950 Satellites,
rather than
tax funds for the AIers.
Huang
> Huang wrote:
>
> > Calculus was extremely successful in modelling natural phenomena.
>
> When did it stop being successful? London bridges falling down...
Who designed the Tacoma narrows bridge ? Sure was'nt Newton. That
thing oscillated worse than your zipper.
> > Cellular automata has also been extremely successful in modelling many
> > of the exact same phenomena.
>
> Reference even one instance wherein this is true.
Wolfram. A New Kind of Science.
> > Suppose for a moment that any natural phenomena can be modelled with
> > equal accuracy using either Calculus or Cellular Automata. Can we say
> > that discrete and continuous methods are then equivalent ?
>
> Calculus is discretized. Look for the little "dx" on each box.
And the limit is ......discrete ?
> > If
> > different models produce the same numeric results and both models
> > accomplish the same thing, then we should be able to say that it is
> > indeterminate whether the natural world is best modelled using one or
> > the other.
>
> > 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.
>
> idiot
Of course I am. So what. At least I have a tool that works.
You have a unified universe and a broken tool to understand it, and
supposedly I'm the idiot. ROFLMAO.
alie...@gmail.com
> Calculus was extremely successful in modelling natural phenomena.
>
> Cellular automata has also been extremely successful in modelling many
> of the exact same phenomena.
>
> Suppose for a moment that any natural phenomena can be modelled with
> equal accuracy using either Calculus or Cellular Automata. Can we say
> that discrete and continuous methods are then equivalent ?
In what sense are cellular automata continuous?
Mark L. Fergerson
T.H. Ray
I think he means continuous functions. The calculus
and cellular automata do both model small step changes
continuously, though I don't find this particularly
profound. Certainly, it does not mean "... that discrete
and continuous methods are then equivalent." The
whole subject of the calculus and analysis in general is
continuous functions. That other mathematical methods
can also be used to illustrate some continuous functions
does not imply that discrete functions don't exist, or
vice versa. The claim makes no sense. We simply use
the mathematical method best suited for purpose--I
doubt that a cellular map would be very efficient at
modeling acceleration, e.g., even if such a thing is
possible in principle.
Tom
> Mark L. Fergerson
FredJeffries
>
> 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.
Huang
CA is discrete. Calculus is based on continuity.
What I am saying is that if you can model some "natural event A" using
calculus, and you can also model the same "natural event A" using
cellular automata, and suppose that both models return the same exact
results to an arbitrary accuracy THEN you cannot presume that "natural
event A" is continuous or discrete, it is indeterminate whether it is
one or the other. They are equivalent in the same sense that relative
velocities are equivalent under GR.
You wont find anyone applying equivalence like that in orthodox
mathematics, but it is perfectly reasonable for a physicist to do so.
In fact, if more work were done in this area I "think" that it may be
mathematically proveable that such an equivalence might be
demonstrable for any physical model.
If Einstein, Newton and Wolfram could have a baby together.......that
is what the child would say.
Huang
> On Jan 4, 8:04 pm, Huang <huangxienc...@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.
>
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 ?
Huang
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.
Huang
>
> - Show quoted text -
Huang
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.
Huang
>
> - 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).
Huang
>
> - 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.
Huang
>
> - 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.
Huang
>
> - Show quoted text -
So, lets say you had some nice kinematic equations written out in
standard form, and something written out probabilistically which
yields the kinematic equations as some kind of "expected
relationship".
Is it currently possible to _transform_ between probabilistic and non-
probabilistic expressions ? Can you write this as a formal
transformation ? I doubt it. I would say it might be possible, but I
dont think that it has ever been done. Ive never seen anything like
that.
This is where the recent work on chaos becomes very interesting
perhaps....many people have tried to show that order and disorder can
emerge in various studies of chaos, fractals and cellular automata.
It's not quite the same thing as using something like a Lorentz
Transform (if you will) to go back and forth between probabilistic and
non-probabilistic expressions.......I cant imagine what such a
trasform would even look like - if it can even exist.
How can we ask such questions using mathematics ?
Huang
>
> - Show quoted text -
To restate the question....suppose that we make a model like a
cointoss. We actually make 2 different models.
[1] The standard model from probability theory that uses random
variables.
[2] A non-probabilistic approach which uses standard physics, and the
randomness is rendered as an emergent process related to sensitive
dependence on initial conditions.
Can we transform from [1] to [2] ? Clearly we can probably pronounce
them to be equivalent (in the sense of Einstein), but I dont know if
you can make a genuine transformation from [1] to [2].
Huang
>
> - Show quoted text -
Lets look at the coint toss example from standard probability theory.
Customarily they present the standard explanation as a random variable
{ X | H, T }
or somehting similar to that.
But if you consider the total number of possible outcomes each as
seperate elements of an outcome space, then you can write things a bit
differently. For example, flip the coin 3 times and you have 8
possible outcomes.
HHH, HHT, HTT, THH, THT, TTT, HTH, TTH
These permutations may be regarded as elements of an outcome space in
their own right, so that
{ X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH }
These are really two very different things, but there may be a way to
transform back and forth in a relatively acceptable way.
It is very different to say that you have a random variable { X | H,
T } for each of 3 individual trials,
or, that you have a single trial with random the variable { X | HHH,
HHT, HTT, THH, THT, TTT, HTH, TTH }
These are two very different looking things with an identical result.
There SHOULD be a way to transform from one to the other !!!
Huang
..........Reposting because the thread is getting too long
Huang
Consider 3 trials of a random variable { X | H, T } .
The result is no different than a single trial of the random variable
{ X | HHH, HHT, HTT, THH, THT, TTT, HTH, TTH } .
How do we transform from one situation to the other ? Is it enough to
say they are "equivalent" ??