On 5/20/2012 10:17 PM, meekerdb wrote:
On 5/20/2012 6:53 PM, Stephen P. King wrote:
On 5/20/2012 8:08 PM, meekerdb wrote:
On 5/20/2012 4:13 PM, Stephen P. King wrote:
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.

The result is an exhaustive classification of compact 4-mainifolds. The absence of such a classification neither prevents nor entails the existence of the manifolds.

But you fail to see that without the means to define the manifolds, there is nothing to distinguish a manifold from a fruitloop from a pink unicorn from a ..... Absent the means to distinguish properties there is no such thing as definite properties.


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.

You can have the equipment mix, bake, decorate and eat a cake without having the equipment to mix, bake, decorate, and eat all possible cakes.

My analogy failed to demonstrate its intended idea, it seems. Let me rephrase. Do cakes exist as cakes if it is impossible to mix, bake and decorate them? Do they just magically appear out of nothing? No. Neither does meaningfulness and the definiteness of properties.

Because I can bake a cake, does it follow that all possible cakes exist?

Are you the only entity that exists? This is not about "you"per se, this is about the possibility and our discussion of ideas.

The answer to your question is: Yes, because I can bake a cake, it follows that "all possible cakes" must exist. Why? Because if the statement "I can bake a cake" is true and I have not specified which cake I have baked, then it follows that I have possibly baked all possible cakes. Otherwise, one has to stipulate which of the many cakes one has baked to be able to claim that all possible cakes do not exist. You are treating the possibility of something the same as the actuality of something when they are not the same.



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.

I don't fee any compulsion to say that. In any case, this universe does not have all possible topologies.

Why do not see that as surprising? We experience one particular universe, having one particular set of properties. How does this happen? What picked it out of the hat?

If you want to hypothesize a multiverse that includes universes with all possible topologies then there will be no *single* algorithm that can classify all of them. But this is just the same as there is no algorithm which can tell you which of the UD programs will halt.

Indeed! It is exactly the same! The point is that since there is nothing that can computationally "pick the winner out of the hat" then how is it that we experience precisely that winner? Maybe the selection process is not a computation in the Platonic sense at all. Maybe it is a real computation running on all possible physical systems in all possible universes for all time.

I am trying to get you to see the difference between structures that are assumed to exist by fiat and structures that are the result of ongoing processes.

You mean like the integers, the multiverse, Turing machines,...?

Yes. Are those entities that exist from the beginning (which is what ontological primitivity implies...) or are they aspects of the unfolding reality?


This is debate that has been going on since Democritus <http://plato.stanford.edu/entries/democritus/> and Heraclitus <http://plato.stanford.edu/entries/heraclitus/> stepped into the Academy. Can you guess what ontology I am championing?


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?

No. Unless your simply defining X to be identical with Y, a mere semantic renaming, then a definition is something like X:=Y|Zx. And it is not the case that ~X=~Y.

    OK.


When you start talking about a collection then you have to define what are its members.

I'm not talking about a collection. You're the one assuming that all 4-manifolds exist and that everything existing must be computed BY THE SAME ALGORITHM. That's two more assumptions than I'm willing to make.

Is a universal algorithm capable of generating all possible outputs when feed all possible inputs? What exactly is an algorithm in your thinking?



Absent the specification or ability to specify the members of a collection, what can you say of the collection?

This universe is defined ostensively.

Interesting word: Ostensively <http://www.thefreedictionary.com/ostensibly>.

"Represented or appearing as such..." It implies a subject to whom the representations or appearances have meaningful content. Who plays that role in your thinking?

You do.  When I write "this" you know what I mean.

And are we alone in the universe? You seem to take for granted the existence of "others".





Brent


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?



    No attempts to even comment on these?

As Mark Twain said, "I'm pleased to be able to answer all your questions directly. I don't know."

Brent

    OK...

--
Onward!

Stephen

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

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