On 09 Dec 2011, at 17:55, Stephen P. King wrote:



[SPK]
I take Occam to say "in any explanation do not multiply entities beyond necessity."

See Brent's answer.





Postulating that everything exists without a means to even demostrate necessity is to postulate an infinite (of unknown cardinality!) of entities, in direct contradiction to Occam's razor.

Occam razor asks for the minimal number of assumption in a theory. It does not care about the cardinal of the models of the theory. That is why the many worlds is a product of occam principle.

Sure, but the necessity of the plurality of "actual worlds" given that we can only observe one

Nobody can observe one universe. Physicists measure numbers and relates those numbers by inductive inference on quantitative relations among them.




requires additional evidence.

"one physical universe" requires as much evidences and explanations than 0, 2, 3, infinity, ... The everything idea is that "all possible universes" is conceptually simpler than one real universe among all the possible one.



Some say that the interference of particles "with themselves" in the two-slit experiment is amble evidence for these, but MWI does nothing to explain why we observe the particular universe that we do.

Comp explains this completely, by explaining why you cannot understand that you are the one ending in Washington instead as the one ending in Moscow. It explains contingencies by consistent extensions.



It has its basis problem as your result has its measure problem.

I don't think there is any basis problem in the quantum MW, nor is there any "initial theory" problem in comp. And the mind-body problem is transformed into a body problem, itself becoming a measure problem, but that is what makes those theories interesting.




I suspect that these two problems are in the same family.




Even when we reduce this to a countable infinite of entities,

Which is indeed the case for the comp ontology, but the epistemology can and will be bigger. It is a sort of Skolem phenomenon, that I have often described.


the need for necessitation remains unanswered. Why do numbers exist?

Nobody can answer that. We cannot prove the existence of the numbers in a theory which do not assume them at the start, implicitly or explicitly.

    So it is OK to postulate that numbers exists

We need only to postulate that zero (or one if you prefer) is a number, and that the successor of a number is a number. This is less than postulating sets or categories, as you need for talking about Stone duality.



and from such argue that the physical world is unnecessary epiphenomena

It is a phenomenon. Why would it be an epiphenomenon? I have argue that this does not make sense.



and yet is required for your result to run.

The phenomenon is required. Not its primitivity.



All I ask is that you consider the world of numbers to not have an existence independent of the possibility of knowledge of it.

In which sense. With comp, the numbers (N, +, *) entails the existence of the knowledge of the numbers by some universal numbers. The "Bp & p" concerns numbers relatively to universal numbers.



I separate "existence" from "properties".

Me too. Existence is handled by the quantifier "E", and properties are handled by arithmetical predicate.



The mere existence of an object does not necessitate any propeties whatsoever. Numbers have properties, they have relative value... Where do those properties derive?

From the (non trivial) additive and multiplicative properties, which are among the postulates (recursive laws of addition and multiplication).







Why numbers and not Nothing?

Because with Nothing in the ontology, you can't prove the existence of anything, not even illusion which needs some illusionned subject. That is why all fundamental theories assumes the numbers, (or equivalent) and with comp this can be shown to be enough.


I merely start with the assumption that "existence exists" and go from there.

We have discussed this. "existence exists" does not make sense for me. Existence of what? You are the one transforming existence into a property here.



To postulate one particular type of entity and not any other requires special explanations.

We assume simple principles and no more than what we need, and with comp we need only combinators, of lambda-terms, or natural numbers.



What makes numbers special over spaces?

They are conceptually far simpler.







At least with the Stone-type dualism we have a way to show the necessity of numbers via bisimulations between different instances of Boolean algebras and, dually, via causality between Stone spaces and thus do not violate Occam blindly.

Assuming different instances of boolean algebra is assuming more than the natural numbers (like assuming finite and infinite sets).

Are two Boolean algebras that have different propositional content one and the same? If this is true then there is no variation is algorithms, it is to say that all algorithms are identical in every way.

?






Comprehensability requires the co-existence of that which is comprehended with that which is doing the comprehension, that numbers can comprehend themselves without additional structure seem to me to be ruled out even by your result.

Not all. The relevant part of computer science is embedded in arithmetic. The one doing the comprehension are the universal numbers, and they need only addition and multiplication for doing that. That can be derived easily from Godel's 1931 paper.


The notion of computation is not inherent in the independent existence of numbers.

It is. This is a consequence of Church thesis.




You are making the mistake of thinking that "independent of any particular physical implementation" is equivalent to "independent of physical implementation",

I do not assume this. This is what the MGA proves or is supposed to prove. Perhaps that is the error of Benjayk, and other people who does not see that immateriality is proved in the 8th step, and not use before.




thus you are free from any physical constraint on the notion of computation and then when tyros like me fail to understand how such an idealist model can have any causal efficacy or limit, then I wonder about its validity.

The causal efficacy comes from the fact that addition and multiplication makes the basic ontology Turing universal. Machines have causal efficacy by themselves.



You need to agrees Deutsch's critique directly as to how it is possible for an abstract form of theory proving is possible.

It is up to Deutsch to explain what is the role of primitive stuff in our experience. With comp, MGA prevents such kind of explanation.







My point is that we cannot tacitly assume the existence of entities that can make the distinctions (for example, between difference Goedelian numberings)

Arithmetical (sigma_1) truth dovetails on all Gödel numbering. The inability for a universal number to really know its coding is part of the reason why their is a first person indeterminacy in arithmetic, and that is part of the explanation of why physical laws will be apparent from the universal numbers points of view.

What you just wrote and its meaningfulness vanishes if there is not physical implementation of it.

On the contrary, if you introduce primitively physical implementation you have the problem to explain what you mean by that, and how to relate them with consciousness. this is explained entirely in the comp theory. Primitive matter is used by materialist like a God-explanation gap. It is a postulate of something on which we are asking to not ask further question. Even if comp did not detroy that notion, I would not find it as having an explanatory purpose.



You cannot dismiss the material world and keep its properties.

Why? I can explain its properties without introducing an unintelligible primitive matter in the picture.








and thus the insistence that the physical not exist at the same level as numbers seems to be an error.

The numbers (or other finite entities belonging to universal systems) have to be primitive (they are not derivable by less than themselves). This is *necessary* not the case for physics, *by the UDA*.

No, UDA only proves that the physical world cannot be monic and allow for computations. It does not prove that an idealism can work.

It proves that IF mechanism is true, only idealism can work. Anything we might add will be like invisible horse. Remember that nobody has ever seen "primitive matter": it is a theological abstract notion. It is not even used in physics, except as a way to put the mind-body problem under the rug, usually with mechanism. My main point is negative: that can't work.













I'm intrigued by David Deutsche's assertion that different physics implies that different things are computable, but I'm doubtful that it's true.

I agree, it is total non sense. Not only it would contradict Church thesis and the immunity of computability for diagonalization, but thanks to David Deutsch quantum computer, it does not even make sense with what we know currently believed in physics, and such a position is a sort of revisionist definition of what is a computation. That's is why I prefer to call Deutsch's "Church Turing principle" the "Deutsch's thesis". And it is an open problem if such a thesis is compatible with Church's thesis.

You are not even showing a valid proof here. Refuting Deutsch's assertion would effectively make Yes Doctor vanish,

Why? The contrary is more plausible. "Yes doctor" should a priori refute Deustch thesis.

    If Deutsch is wrong, digital substitution vanishes

Why?



and so does your result.

I have no clue about what you are saying.






leaving your result vacuous. Removing all traces of the physical world removes all possibility of distinguishing a false proof from a true proof.

Trivially so. But comp does not remove any *trace* of the physical worlds. Only the pretension that the physical has a primary ontology.

Two different meanings of the world trace. Please don't play word games.

You might elaborate. I thought I was using trace in your sense.



    How are numbers more primitive than spaces?

Which spaces? Define them axiomatically, so we can compare. But either you will axiomatize a very poor notion of space (porr= non Turing universal), or you will define just another universal system, which will be equivalent (ontologically) with the numbers.

Bruno


http://iridia.ulb.ac.be/~marchal/



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