Re: Any interest in discussing Tegmark's Mathematical Universe Hypothesis?

On Mar 1, 12:30 pm, Brian Tenneson <tenn...@xxxxxxxxx> 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.
.

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