Any interest in discussing Tegmark's Mathematical Universe Hypothesis?

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Brian Tenneson

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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.]

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 speci es a category
1 TOE. Since there are three disjoint possibilities (none,
some or all mathematical structures have PE), we obtain
the following classi cation 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.

Brian Tenneson

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

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.

Brian

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

>
> 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.

[[1]] http://citeseer.ist.psu.edu/cache/papers/cs/22478/http:zSzzSzwww.cs.cas.czzSzvvvvedcizSzhajekzSzstrls2.pdf/a-set-theory-within.pdf

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.

Brian

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Mar 4, 2008, 1:21:00 PM3/4/08
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> function.http://space.mit.edu/home/tegmark/brain.html
> [[1]]http://citeseer.ist.psu.edu/cache/papers/cs/22478/http:zSzzSzwww.cs.c...

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]

Brian

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

Brian

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Mar 4, 2008, 6:06:35 PM3/4/08
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On Mar 4, 2:38 pm, Brian <tenn...@gmail.com> wrote:
> 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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