Hi ACW, On 2/4/2012 1:53 PM, acw wrote:

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One can wonder what is the most "general" theory that we can postulateto explain our existence. Tegmark postulates all of consistentmathematics, whatever that is, but is 'all of consistent mathematics'consistent in itself?

`I have read several papers that argue strongly that it cannot be!`

`For instance see: http://arxiv.org/abs/0904.0342 The fact that there are`

`set theories that use axioms that are completely opposite each other is`

`another strong indication of this.`

Schmidhuber postulates something much less, just the UD, but strangelyforgets the first-person or the what the implementation substrate ofthat UD would be (and resorts to a Great Programmer to hand-wave itaway).

I wonder why Schmidhuber held back? Did he fear ridicule?

Before reading the UDA, I used to think that something like Tegmark'ssolution would be general enough and sufficient, but now I think 'justarithmetic' (or combinators, or lambda calculus, or ...) or issufficient. Why? By the Church-Turing Thesis, these systems posses thesame computability power, that is, they all can run the UD.

`I agree with this line of reasoning, but I see no upper bound on`

`mathematics since I take Cantor's results as "real". There is not upper`

`bound on the cardinality of Mathematics. I see this as an implication of`

`the old dictum "Nature explores all possibilities."`

Now, if we do admit a digital substitution, all that we can experienceis already contained within the UD, including the worlds where we finda physical world with us having a physical body/brain (which existcomputationally, but let us not forget that random oracle that comeswith 1p indeterminacy).

`Not quite, admitting digital substitution does not necessarily`

`admit to pre-specifiability as is assumed in the definition of the`

`algorithms of Universal Turing machines,`

`<http://en.wikipedia.org/wiki/Algorithm> it just assumes that we can`

`substitute functionally equivalent components. Functional equivalence`

`does not free us from the prison of the flesh, it merely frees us from`

`the prison of just one particular body. ;-)`

`This idea goes back to my claim that the "Pre-established harmony`

`<http://en.wikipedia.org/wiki/Pre-established_harmony>" idea of Leibniz`

`is false because it requires the computation of an infinite NP-Complete`

`problem to occur in zero steps. As we know, given even infinite`

`resources a UTM must take at least one computational step to solve such`

`a NP-Complete problem. My solution to this dilemma is to have an`

`eternally running process at some primitive level. Bruno seems to`

`identify this with the UD, but I claim that he goes too far and`

`eliminates the "becoming" nature of the process.`

If we are machines, then we can only experience finite amount ofinformation given some finite interval of time, some of thisinformation may be incompressible, due to 1p indeterminacy, thus wecould experience "reals" in the limit, despite there only being finitecomputations at any given time. This essentially means that anymathematical object which can be described in Tegmark's "UltimateEnsemble" and that can contain us, is already part of the 1pexperiences of those existing within the UD and we can look at 1pexperiences, as well as the UD* trace as being part of the greater"arithmetical" truth (or any other theory with equivalentcomputational power, by the Church-Turing Thesis).

`Umm, we have to show that the finiteness of machines is necessary`

`from first principles, we cannot just assume that it is so. I agree that`

`the "arithmetical truth" of the UD may be enough to "force" the 1p to`

`have content, but we still need to account for the appearance of`

`interactions or histories of interactions (ala Julian Barbour'sTime`

`Capsule <http://en.wikipedia.org/wiki/Julian_Barbour> idea). There`

`reaches a point, even if it is in the limit of infinitely many, that we`

`cannot put off the concurrency problem, we have to deal with`

`interactions. An option is to take the "running of the UD" as a`

`primitive kind of dynamic that at our local 1p emerges as time and`

`notions of forces, fields, etc. emerge from the algebras of interactions`

`between the many distinct 1p.`

This is why I think "arithmetic" is as good as any for a neutralfoundation, and we cannot really distinguish (from the inside) betweenthese foundations by the CTT.

`This does not address the neutrality problem though. How can the`

`foundation be neutral if it is biased toward a particular structure,`

`even if it is as elegant as arithmetic? My point is that whatever`

`foundation we take, within our ontological theories, it must be neutral`

`with respect to a basis, reference frame, grammar or any other structure`

`that would break its perfect symmetry. Nature does not respect any`

`privileged framing what so ever and thus there cannot be a privileged`

`observational stance. This stance toward neutrality may seem unusually`

`strong, but I don't see how it can be any other way, even allowing`

`arithmetic to be a primitive is to allow a bias against non-arithmetical`

`structures and any bias, however weak, is still a rupture of neutrality.`

However, there might be other possible foundations, if you wish topostulate concrete infinities, but even if they existed, how could wetell them apart, it doesn't seem to be possible for someone admittinga digital substitution, which has a finite mind (at any finite pointin time). If you can show that those other foundations are necessaryand they affect our measure/continuations, or that concrete infinitiesare involved in the implementation of our brain, it could prove COMPwrong.

`The Dualism that follows the analogy of the Stone duality covers`

`this question. Boolean algebras have a specific kind of topological`

`space as their dual. It is forced and as such there is a direct and`

`predictable link between the behavior of the logic and the behavior of`

`the dual space. Is it a complete accident that the topological space`

`that is the dual to Boolean algebras looks like a collection of`

`primitive atoms <http://en.wikipedia.org/wiki/Atomism> in a void? I`

`don't think so! So if the logic that observers are limited to is`

`required to be representable in terms (up to isomorphism) with Boolean`

`algebras, then the physical world that those logical entities have as 1p`

`must look like "atoms in a void". No wonder our particle physics works`

`so well!`

`There is more to add to this, such as the Pontryagin duality that`

`expands the class of dual spaces out to range between the discrete`

`spaces to the compact spaces, but that is for another conversation. :-)`

There is another problem with taking a set theory as foundationalrather than arithmetic - some set theories have independent axioms andthey can be extended by adding either an axiom or its negation, andthey result in different set theoretical truths.

`I didn't mean to take set theory per se as fundamental, I was`

`thinking of set theory as just a mereology - a schemata of sorts - of`

`how we define relations between parts and wholes. But as to your point`

`about set theory, does not the proven existence of non-standard`

`Arithmetic`

`<http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic> argue`

`the other way? While the Tennenbaum Theorem`

`<http://en.wikipedia.org/wiki/Tennenbaum%27s_theorem> seems to make`

`standard (ala Peano) arithmetics "special" and "unique", I strongly`

`suspect that this is just an invariance property, similar to the`

`invariance of the speed of light in physics: any logical entity will see`

`its own Arithmetic model as countable and recursive, it cannot see the`

`"constant" that would make it non-standard as such is its fixed point,`

`its "identity" if you will. I do not have any formal description of this`

`latter idea nor even a proof of it, so please just take this as a`

`conjecture. ;-)`

This doesn't really happen with computation - if there's anythingabsolute in math, it's computation (although different theories aboutwhat arithmetic is will result in different things the theory can talkabout, but it won't make computation any less absolute).

`I strongly suspect that your argument here about the "absoluteness"`

`of computation is a bit too strong or even misplaced. Restricting`

`information to only being a binary bit on mappings in the Integers is a`

`harsh regime, no wonder computation is so "well behaved", any deviation`

`of the bits from the tyranny of the integers at all is terminated with`

`extreme prejudice! I see computation, in general, as "the transformation`

`of representations" and thus do not see the by fiat confinement to the`

`integers as beneficial.`

As a side-note, I don't see why the primitive physical world isnecessary, from the 1p, we can only know that we have senses and fromthe senses we can infer the existence of the external world.

`We have the problem of other minds to deal with! That is why, among`

`other things, we need the physical world albeit NOT primitive, the`

`physical world allows form an "external" differentiation of 1p that`

`would otherwise be identical by Leibniz' identity of indiscernibles. I`

`am just claiming that the abstrac`

`<http://en.wikipedia.org/wiki/Abstract_object>t and the concrete`

`<http://en.wikipedia.org/wiki/Concrete_object> are always co-present at`

`any level until we go to the limit of bare neutral existence. At that`

`point any differences that might make a difference vanish, thus logic`

`and spaces would cease being different yet isomorphic. Vaughn Pratt`

`shows how this works in terms of the directions of the Arrows of the`

`categorical representations of LOGIC and SPACE, they point in opposite`

`directions thus if we add them up their directions and scalars would`

`vanish. -> + <- = (see`

`http://upload.wikimedia.org/wikipedia/commons/f/ff/Laws_of_Form_-_double_cross.gif)`

`Additionally, I see this conjecture as similar to Tegmark's`

`Mathematical Universe Hypothesis`

`<http://en.wikipedia.org/wiki/Mathematical_universe_hypothesis> except`

`that I do not see how the postulate "/All structures that exist`

`mathematically also exist physically."/ implies a mathematical monism as`

`the wiki article states. If for any structure that exists mathematically`

`there must exist a physical structure, there is the implication of a`

`duality between the mathematical and the physical. This is a different`

`sort of duality than that of Descartes as it does not assume distinct`

`"substances", it is a form of dual aspect theory`

`<http://en.wikipedia.org/wiki/Double-aspect_theory> where the dynamics`

`of each aspect run in opposite directions. Vaughn Pratt explains the`

`idea here: http://boole.stanford.edu/pub/dti.pdf`

If consciousness is how some (possibly self-referential) arithmetical(or computational) truth feels from the inside, it does not seemimpossible that there would not be computations representing somephysical (just not primitive) world and that world would contain usand our bodies/brains, and the existence of such computations would bea theorem in arithmetic.

`I agree, but the representation of a thing is not the thing except`

`in very special cases, such as what we have when we say that the "best"`

`simulation of an object is the object itself.`

`<http://www.stephenwolfram.com/publications/articles/physics/85-undecidability/2/text.html>`

`This takes us into a discussion of questions like "when might the map =>`

`the territory or, by duality,the territory => the map`

`<http://chorasimilarity.wordpress.com/2011/06/21/entering-chora-the-infinitesimal-place/>?`

`This is a subtle and important question! ;-)`

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