Lots of interesting ideas going about.
It sounds like you're pondering how many elements are in the set of all
world-lines consistent with the true laws of physics (e.g., possibly, the
least action principle).  (Incidentally, that set oddly enough is timeless
yet the "bundles" of world-lines that comprise our selves evidently
perceive change.)
Proof by throwing in an axiom isn't very satisfying but I would like to say
that Banach-Tarski is no more strange than Gabriel's Horn or Cantor's
hierarchy of infinities.  Strangeness is of course a matter of opinion and
mine is that the existence of nonmeasurable sets is not a heavy price to
pay for that poof (a proof by throwing in an axiom).

Cheers




On Mon, Feb 13, 2012 at 6:55 PM, Stephen P. King <stephe...@charter.net>wrote:

>  On 2/13/2012 5:27 PM, acw wrote:
>
>  [SPK]  There is a problem with this though b/c
> it assumes that the field is pre-existing; it is the same as the "block
> universe" idea that Andrew Soltau and others are wrestling with.
>
> Why is a pre-existing field so troublesome? Seems like a similar problem
> as the one you have with Platonia. For any system featuring time or change,
> you can find a meta-system in which you can describe that system timelessly
> (and you have to, if one is to talk about time and change at all).
>
>
> Dear Kermit,
>
>     OK, I will try to explain this in detail and check my math. I am good
> with pictures, even N-dimensional ones, but not symbols, equations and
> words...
>
>  Think of a collection of different objects.  Now think of how many ways
> that they can be arranged or partitioned up. For N objects, I believe that
> there are at least N! numbers of ways that they can be arranged.
>
> Now think of an Electromagnetic Field as we do in classical physics. At
> each point in space, it has a vector and a scalar value representing its
> magnetic and electric potentials. How many ways can this field be
> configured in terms of the possible values of the potentials at each point?
> At least 1x2x3x...xM ways, where M is the number of points of space. Let's
> add a dimension of time so that we have a 3,1 dimensional field
> configuration. How many different ways can this be configured? Well, that
> depends. We known that in Nature there is something called the Least Action
> Principle that basically states that what ever happens in a situation it is
> the one that minimizes the action. Water flows down hill for this reason,
> among other things... But it is still at least M! number of possible
> configurations.
>
>     How do we compute what the minimum action configuration of the
> electromagnetic fields distributed across space-time? It is an optimization
> problem of figuring out which is the least action configured field given a
> choice of all possible field configurations. This computational problem is
> known to be NP-Complete and as such requires a quantity of resources to run
> the computation that increases as a non-polynomial power of the number of
> possible choices, so the number is, I think, 2^M! .
>     The easiest to understand example of this kind of problem is the Traveling
> Salesman problem<http://en.wikipedia.org/wiki/Travelling_salesman_problem>:
> "Given a list of cities and their pairwise distances, the task is to find
> the shortest possible route that visits each city exactly once. " The
> number of possible routes that the salesman can take increases
> exponentially with the number of cities, there for the number of possible
> distances that have to be compared to each other to find the shortest route
> increases at least exponentially. So for a computer running a program to
> find the solution it takes exponentially more resources of memory and time
> (in computational steps) or some combination of the two.
>
>     Now, given all of that, in the concept of Platonia we have the idea of
> "ideal forms", be they "the Good", or some particular infinite string of
> numbers. How exactly are they determined to be the "best possible by some
> standard". Whatever the standard, all that matters is that there are
> multiple possible options of The Forms with the stipulation that it is "the
> best" or "most consistent" or whatever. It is still an optimization problem
> with N variables that are required to be compared to each other according
> to some standard. Therefore, in most cases there is an Np-complete problem
> to be solved. How can it be computed if it has to exist as perfect "from
> the beginning"?
>
>     I figured this out when I was trying to wrap my head around Leindniz'
> idea of a "Pre-Established Harmony". It was supposed to have been created
> by God to synchronize all of the Monads with each other so that they
> appeared to interact with each other without actually "having to exchange
> substances" - which was forbidden to happen as Monads "have no windows".
> For God to have created such a PEH, it would have to solve an NP-Complete
> problem on the configuration space of all possible worlds. If the number of
> possible worlds is infinite then the computation will require infinite
> computational resources. Given that God has to have the solution "before"
> the Universe is created, It cannot use the time component of "God's
> Ultimate Digital computer". Since there is no space full of distinguishable
> stuff, there isn't any memory resources either for the computation. So
> guess what? The PEH cannot be computed and thus the universe cannot be
> created with a PEH as Leibniz proposed.
>
>     The idea of a measure that Bruno talks about is just another way of
> talking about this same kind of optimization problem without tipping his
> hand that it implicitly requires a computation to be performed to "find"
> it. I do not blame him as this problem has been glossed over for hundred of
> years in math and thus we have to play with nonsense like the Axiom of
> Choice (or Zorn's Lemma) to "prove" that a solution exists, never-mind
> trying to actually find the solution. This so called 'proof" come at a very
> steep price, it allows for all kinds of 
> paradox<http://en.wikipedia.org/wiki/Banach-Tarski_paradox>
> .
>     A possible solution to this problem, proposed by many even back as far
> as Heraclitus, is to avoid the requirement of a solution at the beginning.
> Just let the universe compute its least action configuration as it evolves
> in time, but to accept this possibility we have to overturn many preciously
> held, but wrong, ideas and replace them with better ideas.
>
> Onward!
>
> Stephen
>
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