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Mar 1, 2008, 2:47:48 PM3/1/08

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[This post is in sci.logic because of the employment of model theory

and discussion of abstract math structures by the author and for other

reasons which may come up during the discussion.]

and discussion of abstract math structures by the author and for other

reasons which may come up during the discussion.]

Here is a link to the article:

http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf

Abstract:

I explore physics implications of the External Reality Hypothesis

(ERH) that there exists an

external physical reality completely independent of us humans. I argue

that with a sufficiently

broad definition of mathematics, it implies the Mathematical Universe

Hypothesis (MUH) that our

physical world is an abstract mathematical structure. I discuss

various implications of the ERH

and MUH, ranging from standard physics topics like symmetries,

irreducible representations, units,

free parameters, randomness and initial conditions to broader issues

like consciousness, parallel

universes and G¨odel incompleteness. I hypothesize that only

computable and decidable (in G¨odel's

sense) structures exist, which alleviates the cosmological measure

problem and may help explain why

our physical laws appear so simple. I also comment on the intimate

relation between mathematical

structures, computations, simulations and physical systems.

Quote from Intro:

The idea that our universe is in some sense mathematical

goes back at least to the Pythagoreans, and has been

extensively discussed in the literature (see, e.g., [2-25]).

Galileo Galilei stated that the Universe is a grand book

written in the language of mathematics, and Wigner reflected

on the "unreasonable effectiveness of mathematics

in the natural sciences" [3]. In this essay, I will push this

idea to its extreme and argue that our universe is mathematics

in a well-defined sense.

[End Quote]

The article linked to above is regarded by its author as a sequel to

this:

http://space.mit.edu/home/tegmark/toe.pdf

Abstract: (sorry, some characters didn't enjoy being c&p'ed)

We discuss some physical consequences of what might be

called \the ultimate ensemble theory", where not only worlds

corresponding to say di erent sets of initial data or di erent

physical constants are considered equally real, but also worlds

ruled by altogether di erent equations. The only postulate

in this theory is that all structures that exist mathematically

exist also physically, by which we mean that in those

complex enough to contain self-aware substructures (SASs),

these SASs will subjectively perceive themselves as existing in

a physically \real" world. We nd that it is far from clear that

this simple theory, which has no free parameters whatsoever,

is observationally ruled out. The predictions of the theory

take the form of probability distributions for the outcome of

experiments, which makes it testable. In addition, it may be

possible to rule it out by comparing its a priori predictions

for the observable attributes of nature (the particle masses,

the dimensionality of spacetime, etc.) with what is observed.

Quote:

In other words, some subset of all mathematical structures

(see Figure 1 for examples) is endowed with an

elusive quality that we call physical existence, or PE for

brevity. Specifying this subset thus species a category

1 TOE. Since there are three disjoint possibilities (none,

some or all mathematical structures have PE), we obtain

the following classication scheme:

1. The physical world is completely mathematical.

(a) Everything that exists mathematically exists

physically.

(b) Some things that exist mathematically exist

physically, others do not.

(c) Nothing that exists mathematically exists

physically.

2. The physical world is not completely mathematical.

The beliefs of most physicists probably fall into categories

2 (for instance on religious grounds) and 1b. Category

2 TOEs are somewhat of a resignation in the sense of

giving up physical predictive power, and will not be further

discussed here. The obviously ruled out category

1c TOE was only included for completeness. TOEs in

the popular category 1b are vulnerable to the criticism

(made e.g. by Wheeler [6], Nozick [7] and Weinberg [8])

that they leave an important question unanswered: why

is that particular subset endowed with PE, not another?

...

In this paper, we propose that category 1a is the correct

one.

[End quote]

I'm also interested in discussing what SAS'es might there be. Perhaps

nail down axioms and/or defining traits of SAS'es. This next link

might be a diversion, but it is a starting point for the discussion of

formalizing awareness:

http://cs.wwc.edu/~aabyan/Colloquia/Aware/aware2.html

I suppose the direction I'd +like+ this discussion to go is

investigation of this material as conjecture, what these conjectures

would entail (physically, mathematically, and philosophically), etc., +

+rather than debate as to the validity of these conjectures.++

It seems to me that, at worst, these conjectures form an internally

consistent theory, not unlike Cantor's theory of the infinite;

whether or not these conjectures are correct in a physics sense as

being an accurate characterization of "reality," I would like to view

these conjectures/hypotheses as, in this discussion at sci.logic, at

worst, an internally consistent framework, worthy enough of

investigation because of the consistency, regardless of physical

correctness.

Obviously, if these conjectures/hypotheses are correct in a physics

sense, then the investigation is even more justified when compared to

mathematical and/or philosophical justification for the investigation.

Mar 1, 2008, 3:30:05 PM3/1/08

to

The last link provided is giving me intermittent failure, so here are

two cached versions to try:

1st cached version of aware2.html:

http://web.archive.org/web/20060827232622/http://www.cs.wwc.edu/~aabyan/Colloquia/Aware/aware2.html

two cached versions to try:

1st cached version of aware2.html:

http://web.archive.org/web/20060827232622/http://www.cs.wwc.edu/~aabyan/Colloquia/Aware/aware2.html

1st link to 2nd cached version of aware2.html:

<a href="http://209.85.173.104/search?q=cache:dil-L-g7Mj0J:cs.wwc.edu/

~aabyan/Colloquia/Aware/aware2.html

+aware2+site:cs.wwc.edu&hl=en&ct=clnk&cd=1&gl=us">Google cached

version</a>

Hopefully this forum will allow the html above because the link might

be too long with wrapping and c&p'ing considerations:

2nd link to 2nd cached version of aware2.html:

http://209.85.173.104/search?q=cache:dil-L-g7Mj0J:cs.wwc.edu/~aabyan/Colloquia/Aware/aware2.html+aware2+site:cs.wwc.edu&hl=en&ct=clnk&cd=1&gl=us

Also, a new link in the direction of the non-computability of

consciousness, which seems to be a strike against some of Tegmark's

hypotheses (in particular, the computable universe hypothesis in

section VII of the very first article linked to in the previous post,

"assuming" that non-computability of consciousness implies the non-

computability of the universe in that consciousness is "contained in"

the universe), is here:

Non-Computability of Consciousness

http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.1617v1.pdf

Abstract:

With the great success in simulating many intelligent behaviors using

computing devices, there has been an ongoing debate whether all

conscious

activities are computational processes. In this paper, the answer to

this

question is shown to be no. A certain phenomenon of consciousness is

demonstrated to be fully represented as a computational process using

a

quantum computer. Based on the computability criterion discussed with

Turing machines, the model constructed is shown to necessarily involve

a

non-computable element. The concept that this is solely a quantum

effect

and does not work for a classical case is also discussed.

Mar 3, 2008, 3:38:29 PM3/3/08

to

On Mar 1, 12:30 pm, Brian Tenneson <tenn...@gmail.com> wrote:

>

> Also, a new link in the direction of the non-computability of

> consciousness, which seems to be a strike against some of Tegmark's

> hypotheses (in particular, the computable universe hypothesis in

> section VII of the very first article linked to in the previous post,

> "assuming" that non-computability of consciousness implies the non-

> computability of the universe in that consciousness is "contained in"

> the universe), is here:

>

> Non-Computability of Consciousnesshttp://arxiv.org/PS_cache/arxiv/pdf/0705/0705.1617v1.pdf>

> Also, a new link in the direction of the non-computability of

> consciousness, which seems to be a strike against some of Tegmark's

> hypotheses (in particular, the computable universe hypothesis in

> section VII of the very first article linked to in the previous post,

> "assuming" that non-computability of consciousness implies the non-

> computability of the universe in that consciousness is "contained in"

> the universe), is here:

>

>

> Abstract:

> With the great success in simulating many intelligent behaviors using

> computing devices, there has been an ongoing debate whether all

> conscious

> activities are computational processes. In this paper, the answer to

> this

> question is shown to be no. A certain phenomenon of consciousness is

> demonstrated to be fully represented as a computational process using

> a

> quantum computer. Based on the computability criterion discussed with

> Turing machines, the model constructed is shown to necessarily involve

> a

> non-computable element. The concept that this is solely a quantum

> effect

> and does not work for a classical case is also discussed.

I recently came across an apparent rejoinder (intentional or not, I

don't know) by Tegmark on the subject of the quantum nature of brain

function.

http://space.mit.edu/home/tegmark/brain.html

Tegmark makes a case for brain function being modeled adequately with

classical theoretical means (possibly such as Turing machines) and

that brains do not function like quantum computers. (Essentially the

main factor is that the brain is not nearly at absolute zero degrees,

or otherwise in an environment in which superposition type effects

that consciousness apparently mimics well enough to keep many on the

fence, is more common than Earthly temperatures where our brains

normally reside.)

If Tegmark does prove his point, while others in his community remain

skeptical that brain function is +not+ an example of a quantum

computer, then the paper I cited about the non-computability of

consciousness does not invalidate Tegmark's CUH, mentioned in section

VII of the first link in the first post. The non-computability of

consciousness would seem to invalidate Tegmark's CUH (Computable

Universe Hypothesis) in that the universe, by even a narrow definition

of universe, must contain consciousness, and, I presume, non-

computability of consciousness would imply the CUH is false. That is,

unless consciousness can have non-computable aspects that when

"glued" (ultraproduct or some other method of "gluing"???) together

throughout the universe, somehow (I know this is vague) the non-

computable aspects of various parts of the universe all balance out to

a computable universe. Hmm...things to think about... Maybe the CUH

is true and brains work like quantum computers, somehow...?

Anyway, Tegmark would be lending credence to his point by invalidating

the proof of non-computability of consciousness for that relies on the

"presumption" that consciousness is inherently a quantum process;

obviously if their critical "presumption" is wrong, then their

conclusion (consciousness not being computable) isn't necessarily so.

I think it is worth splitting hairs here about the difference between

consciousness and brain function but as of yet am aware of very little

of the +formal+ theory behind either of these notions,

philosophically, psychologically, or cognitive-scientifically.

I am compiling a list of other discussion points.

First on this list of discussion points, I will make a connection to

abstract fuzzy logic and the Level IV multiverse situation. If you

haven't read these fascinating articles yet, Level IV's brief

definition is:

Other mathematical structures give different +fundamental+ equations

of physics.

In the MUH article (first link, first post), appendix A defines what

Tegmark means by a mathematical structure.

[Compilation Process] I'm thinking of whether or not the aggregate of

all MS's can be "glued" together somehow (doubtfully by a simple

union) in order to get the MS of all MS's.

This brings me to the connection to abstract fuzzy logic and my

personal quest to continue my education in the area of Fuzzy Logic.

(Apparently, no one in the US works specifically in the area I want to

work in but there are many in Europe at institutions that award

Phds.) It also gratifies me, on a personal note, to think that my

research, if carried out, might settle some question about whether or

not the [Compilation Process] is at all possible in any "reasonable"

sense whatsoever. It would be nice to know either way, rather than a

"this smells like Russell's Paradox, so let's not try it" sort of

deal.

My research would focus on somewhat recent papers on fuzzy logic

pertaining to involving FL at the axiomatic level to create

generalizations and anti-generalizations of ZFC set theory, or other

suitably modified set theory (eg, remove Foundation Axiom immediately

for reasons that would be clear later).

According to the conclusion of that paper, linked to below, an open

problem is figuring out how other axioms could be, should be,

shouldn't be, and can't be consistently added to the list of axioms

they present in a FL-sense.

In an effort to push question (2) in a particular direction, let me

attempt to formulate my question/problem. Start with the bare-bones

fuzzy set theory presented in [[1]]. Let the truth set be denoted D.

Consider the following axioms:

[[U.Strong]] there is a y such that for all x, the truth degree of

the formula "x is in y" is the maximal (in the sense appropriate to

the type of algebraic structure D has, such as an MV-algebra, but

definitely not Boolean as we know Russell's Paradox +will+ rear its

ugly head in the Boolean case) element in D.

In other words, if the maximal element in D is equipped with the

baggage "true", U.S. says there is a set y for which all sets x are

elements of y. This is one reason to drop the Foundation Axiom

immediately, as such a y is obviously not well-founded. This could be

called a (strong) universal set, with appropriate adjectives that

reference D and the syntactical entailment axioms used, the underlying

language, etc...

[[U.Weak]] there is a y such that for all x, the truth degree of the

formula "x is in y" is a designated element of D.

In words, I view the designated, anti-designated, and non-designated

partitions of D as shades of gray of truth. Designated means more

light than not, where light = truth in this analogy, anti-designated

means more dark than not, and non-designated means more gray than

not. So to say " 'x is in y' is a designated truth value" would mean

something like, "it's essentially true that y is a universal set."

One could say that y would be a weak universal set and it is doubtful

that such a y need be unique, unlike a strong universal set is.

That sets (pun intended) up the problem (below) that I hope to

formalize into the beginnings of a PhD thesis in the area of FL

someday.

Let R be some type of unary predicate.

Recall that D is the set of truth degrees, with some algebraic (eg,

MV) structure associated with it.

Consider the statement below:

[[Statement]] A fuzzy set theory, starting with the one in [[1]],

without Foundation, plus either the strong or weak universal set

axiom, is consistent relative to ZFC (the best situation one can hope

for) if and only if R(D).

The question: Determine for what R is the above statement true, if

any, or prove that for all R, the above statement is false.

Obviously, I want, at worst, an existence proof on R, that there are

some properties D could possess that enables a fuzzy universal set

theory that is consistent relative to ZFC.

Also, I strongly hope that the statement is not false for all R, that

there aren't any exotic D's or structures they could be equipped with,

to make a universal set theory as consistent as ZFC. Clearly, if D =

{0,1} then the set of all R's for which [[Statement]] is true is empty

(bad but expected and well known). In the binary logic case,

Russell's Theorem proves that the set of all R's for which

[[Statement]] is true is empty. No properties on D make the universal

set a possibility in classical logic (except possibly the work of the

sort Quinne did with the New Foundations although, in NF, Choice must

be dropped, in some sense, which is highly disadvantageous to anyone

who enjoys using Zorn's Lemma).

(I posed this to someone known in the area of FL and he encouraged me

to come to Europe (as apparently no one does this type of work in FL

in the U.S.) to formally work this into a PhD thesis.)

Now, ultimately, the connection to the CUH is that if there is an

ultimate set of +some kind+, like a strong universal set, then perhaps

that could provide a link to the MS of all MS's, ie, the mathematical

structure of all mathematical structures, without leading to deals

like, "this smells like Russell's dirty laundry, so let's not go

there."

<punchline tag>

Either that or provide an interesting, to say the least, MS (a fuzzy

and strong universal set theory) to investigate in the context of the

MUH, as this strong universal fuzzy set may, in fact, be a candidate

for what the universe literally is in a physical sense, assuming the

MUH, of course.

</punchline tag>

If I could make all of that work, I would be a very happy man. Even

if I could be proved wrong, at least then I can rest on this issue in

particular.

Mar 4, 2008, 1:21:00 PM3/4/08

to

I stumbled across this post again recently:

http://mathforum.org/kb/message.jspa?messageID=541366&tstart=0

[Quote]

Symmetry is analogous to a generalized form of self evident truth, and

it is a distributive attribute via the laws of nature, being

distributed over the entire system called universe. A stratification

of Cantorian alephs with varying degrees of complexity. Less

complexity = greater symmetry = higher infinity-alephs. So the highest

aleph, the "absolute-infinity" distributes over the entire set called

Universe and gives it "identity".

[/quote]

Mar 4, 2008, 5:38:26 PM3/4/08

to

A paradox???

http://www.torrentreactor.net/torrents/1588942/Parallel-Worlds-Parallel-Lives

There is one part when Tegmark is speaking, around the 27-30 minute

mark or so, that they give a visual clue about parallel universes that

was perhaps more interesting than the director realized, unless the

director's assistant was Tegmark himself.

When they showed two universes splitting, in one parallel, the

Copenhagen interpretation is correct...and in the other, the Many

Worlds interpretation is correct.

There is a QM formula with [[[EXCEPT DURING OBSERVATION]]] in one half

of the screen

and

in the other half of the screen, [[[EXCEPT DURING OBSERVATION]]] is +

+crossed out++ by Tegmark.

Interestingly, part of Tegmark's work says just that: not only do

physical things split into parallels, but the laws of physics

themselves are different in different universes.

+++Therefore, The Copenhagen view is correct and the Many Worlds

interpretation is correct.+++

But which is correct in THIS universe?

Or, maybe, that is a loaded question. More details on why that might

be a loaded question has to do with my crew's speculation about there

not just being parallel universes but also "overlaying" (or

overlapping) of parallels, where the aggregate of parallels (aka, the

universe) are (is) very much like the water system on earth: separate

at times and other times, quite combined and overlaid upon one

another. Indeed, if one "frog" is floating on the river, the "bird"

sees the "frog" actually pass from the North Pole somehow through down

to the Nile, passing thousands of different waterways in between, and

the "frog" just thinks he has been in one body of water all along,

which couldn't have been more wrong, at least, as far as the "bird"

sees things.

Then again, is there a bird's "bird?"

And a bird's bird's bird?

And a bird's bird's bird's bird?

And do frogs have pets?

Do those pets have pets?

Do those pets have pets that have pets?

Sound familiar? To me it sounds like a self-similar fractal and the

way the universe would look if you started at a string and zoomed out

to view the universe from the boundary of the universe, which might

not "exist", unless the boundary of the universe exists

mathematically, of course! I suppose one might want to push the

envelope of mathematics to determine what the boundary of the universe

is, to mightily abuse language.

Well, assuming the MUH, this overlaying of parallels +must+ be the

case due to the hierarchical nature of mathematics. Set theory is on

a +somewhat+ lower echelon in the hierarchy than Category Theory,

which is, on a lower echelon than Logic which is, in turn, on a lower

echelon than Fuzzy Logic, a generalization of Logic. Perhaps instead

of the ultimate set, I need to search for the ultimate math, but I

think Logic and Model Theory and/or Cat might be that, except Logic

does have its limitations, in some sense.

Mar 4, 2008, 6:06:35 PM3/4/08

to

On Mar 4, 2:38 pm, Brian <tenn...@gmail.com> wrote:

> A paradox???

>

> http://www.torrentreactor.net/torrents/1588942/Parallel-Worlds-Parall...

> A paradox???

>

> http://www.torrentreactor.net/torrents/1588942/Parallel-Worlds-Parall...

The only problem is that Aristotle's mutual exclusivity might not

actually be universal, to resolve this apparent paradox. But even

within one parallel (mathematical structure?), ME (mutual exclusivity)

might be true in one region of space (ie, the context between and

containing mathematical structures), false in another, both true and

false in still another part of that parallel, and absolutely all

values of truth between true and false elsewhere in that parallel

universe. It seems somewhat mind boggling when pondering that.

In our "neck of the woods," I think ME is "almost" (sort of in a

Lesbegue measure sense) true. In other words, locally to myself and

probably you as well (whatever that might mean), the pseudo-well-

formed-formula below has a ++designated++ truth value in some truth

set D:

' for all wffs f, ( f & not(f) ||--> ^D) '

where ^D is the minimal element in D, or an arbitrarily chosen

representative of the ones of equally least value, respective of the

order on D. ^D is interpretable as the qualia FALSE.

In fewer words (in English):

"locally," D+( W(f)-->( f & not(f) ||--> ^D) )

where D+( ) means, "the truth degree of what follows is designated,"

and W( ) means, "what follows is a well formed formula," and ||-->

means there is a fuzzy logical sort of valuation function being

applied, and --> is the standard (in a fuzzy logical sense, of course,

but the truth set of this symbol definitely need not also be D--too

bad tex is not available to my knowledge here, that would make this

notation less unappealing to the eye) conditional connective. (I

think all of this is formalizable.)

I think in our dreams (double entendre intended), ME is "almost"

false, ie, D-( W(f)-->( f & not(f) ||--> ^D) ) where D-( ) means,

"what follows has an anti-designated truth degree."

Perhaps that could be related to the true difference between conscious

and unconscious.

Conscious could mean something like

X( D+( W(f)-->( f & not(f) ||--> ^D) ) )

and

unconscious could mean something like

X( D-( W(f)-->( f & not(f) ||--> ^D) ) )

where X( ) means something like, "in the context of the the parallel

network SAS labeled X is embedded or embeddable within, the following

is true."

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