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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