On 5/20/2012 4:39 PM, meekerdb wrote:
On 5/20/2012 1:31 PM, Stephen P. King wrote:
My point is that for there to exist an a priori given string of numbers that is equivalent our universe there must exist a computation of the homomorphies between all possible 4-manifolds.

Why?
Hi Brent,

Because otherwise the amazing precision of the mathematical models based on the assumption of, among other things, that physical systems exist in space-time that is equivalent to a 4-manifold. The mathematical reasoning involved is much like a hugeJenga tower <http://en.wikipedia.org/wiki/Jenga#Tallest_tower>; pull the wrong piece out and it collapses.


Markov theorem tells us that no such homomorphy exists,

No, it tells there is no algorithm for deciding such homomorphy *that works for all possible 4-manifolds*. If our universe-now has a particular topology and our universe-next has a particular topology, there in nothing in Markov's theorem that says that an algorithm can't determine that. It just says that same algorithm can't work for *every pair*.

I agree with your point that Markov's theorem does not disallow the existence of some particular algorithm that can compute the relation between some particular pair of 4-manifolds. Please understand that this moves us out of considering universal algorithms and into specific algorithms. This difference is very important. It is the difference between the class of universal algorithms and a particular algorithm that is the computation of some particular function. The non-existence of the general algorithm implies the non-existence of an a priori structure of relations between the possible 4-manifolds. I am making an ontological argument against the idea that there exists an a priori given structure that *is* the computation of the Universe. This is my argument against Platonism.


therefore our universe cannot be considered to be the result of a computation in the Turing universal sense.

Sure it can. Even if your interpretation of Markov's theorem were correct our universe could, for example, always have the same topology,

No, it cannot. If there does not exist a general algorithm that can compute the homomorphy relations between all 4-manifolds then what is the result of such cannot exit either. We cannot talk coherently within computational methods about "a topology" when such cannot be specified in advance. Algorithms are recursively enumerable functions. That means that you must specify their code in advance, otherwise your are not really talking about computations; you are talking about some imaginary things created by imaginary entities in imaginary places that do imaginary acts; hence my previous references to Pink Unicorns.

Let me put this in other words. If you cannot build the equipment needed to mix, bake and decorate the cake then you cannot eat it. We cannot have a coherent ontological theory that assumes something that can only exist as the result of some process and that same ontological theory prohibits the process from occurring.

or it could evolve only through topologies that were computable from one another? Where does it say our universe must have all possible topologies?


The alternative is to consider that the computation of the homomorphies is an ongoing process, not one that is "already existing in Platonia as a string of numbers" or anything equivalent. I would even say that time_is_ the computation of the homomorphies. Time exists because everything cannot happen simultaneously.

We must say that the universe has all possible topologies unless we can specify reasons why it does not. That is what goes into defining meaningfulness. When you define that X is Y, you are also defining all not-X to equal not-Y, no? When you start talking about a collection then you have to define what are its members. Absent the specification or ability to specify the members of a collection, what can you say of the collection?

What is the a priori constraint on the Universe? Why this one and not some other? Is the limit of all computations not a computation? How did this happen?


--
Onward!

Stephen

"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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