On 2/13/2012 6:55 PM, Stephen P. King 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.
The EM field is a second order anti-symmetric tensor, F_mu_nu, so it has six independent
components.
How many ways can this field be configured in terms of the possible values of the
potentials at each point?
In classical physics it has uncountably many values at each point. In QFT with boundary
conditions it may be limited.
At least 1x2x3x...xM ways, where M is the number of points of space.
An uncountable infinity.
Let's add a dimension of time so that we have a 3,1 dimensional field configuration.
The dimensions of space are not the same as the possible values of fields at a point, nor
are they the number of points of space.
How many different ways can this be configured?
Uncountably many ways.
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.
The least action principle applied to the EM field in free space gives you Maxwell's
equations for EM waves which have uncountably many possible solutions. In order to get
definite solutions though you need boundary conditions.
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! .
All this discussion of computational resources is irrelevant since you've postulated a
system with uncountably many possible solutions, and you've not specified any boundary
conditions so they just correspond to all possible photons.
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.
Since "God" is ill defined there's no way to make sense of assertions about it.
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.
What ideas are overturned by the universe just doing what is consistent with a least
action principle?
Brent
Onward!
Stephen
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