On 06 Oct 2010, at 17:43, Brian Tenneson wrote:
I figure this is especially of interest because of the references to
Tegmark's works.
From a logician's standpoint, it may be of interest that I show that
there is a structure U such that all structures, regardless of
symbol set, can be elementarily embedded within it.
From a physicist's point of view, at least one who might subscribe
to Tegmark's 4-level hierarchy of parallel universes, a structure
with this property might be of interest under the hypothesis that
reality is a mathematical structure. If we suppose that reality is
something which is all encompassing, then the structure with the
aforementioned property could be said to be all encompassing.
Now that I have this structure in hand, I can try to go further by
looking at the structure from a model-theoretic point of view. This
task to further the investigation will be undertaken soon.
Here is a link
http://www.alphaomegadimension.info/media/A_Mathematical_Structure_Isomorphic_to_Reality_ver_5-12_anon.pdf
Any feedback is encouraged, critical or otherwise.
Let us call universe, the ultimate reality.
Then I agree with this: if the universe is a mathematical object, then
NF is the best tool to attempt a description of that universal object.
The universe, when being a mathematical object, has to belong to
itself, so we need a theory à-la Quine, instead of the usual Zermelo-
Franekek or Von Neuman Bernays Gödel. In that sense it improves the
raw description Tegmark makes of level 4.
I got a shock at first reading your definition of 'mathematical
structure" page 2, as formal system, but you make it clear later that
you meant the models, or semantics, of those system (yet interpreted
themselves in NFU).
My reading of Ebbinghaus Flum Thomas did make me feel less guilty to
listen more to lobian machine talking only first order language, but
not up to the point of believing that the universal mathematical
universe could be a first order structure, even in theory like NF, at
least not in in the genuine sense needed to address the mind body
problem, or the consciousness/reality problem.
A theory of everything should explain both the physical and the
mental, conceptually.
If you assume that the brain works like a universal machine (universal
with respect to computability notion, and thus with Church thesis),
you get a theory of mind, as what universal machine can prove and
infer about themselves and their possible consistent extensions.
But then, in term of Tegmark levels, The relation between mind and
reality becomes a partially mathematical phenomenon which blurs the
level 3 and the level 4.
Such universal machine cannot know in which computational history she
would belong, still less in which mathematical structure she belongs,
but below its level of substitution, she belongs to an infinity of
universal history (number relations, combinators relation, Horn clause
relations) 'competing' in term of a measure of credibility.
So with mechanism the physical is not something mathematical among the
mathematical, it is a very special structure which sums on all
mathematical structures is a way specified by computer science and the
logic of self-references. It is based on distinction of different
internal sel-referential views.
Also, I am not convinced by your argument that from the premise "there
exists a reality completely independent of us human" it follows that
reality is a mathematical structure". You beg the question by
identifying a baggage free description with a mathematical structure.
A physicalist argues in general that baggage-free description is what
him provides: particles, waves, fields, and that mathematics is an
approximate language conveying human ideas on those things. Your proof
seems to me just a platonist act of faith.
You miss the importance of the consciousness problem, concerning
physics.
Perhaps you follow Tegmark. Quoting him from "The Multiverse Hierarchy":
"Indeed, the standard mental picture of what the physical world
is corresponds to a third intermediate viewpoint that could be
termed the consensus view. From your subjectively perceived
frog perspective, the world turns upside down when you stand
on your head and disappears when you close your eyes, yet you
subconsciously interpret your sensory inputs as though there is
an external reality that is independent of your orientation, your
location and your state of mind. It is striking that although
this third view involves both censorship (like rejecting dreams),
interpolation (as between eye-blinks) and extrapolation (say at-
tributing existence to unseen cities) of your inside view, inde-
pendent observers nonetheless appear to share this consensus
view. Although the inside view looks black-and-white to a cat,
iridescent to a bird seeing four primary colors, and still more dif-
ferent to bee a seeing polarized light, a bat using sonar, a blind
person with keener touch and hearing, or the latest overpriced
robotic vacuum cleaner, all agree on whether the door is open.
The key current challenge in physics is deriving this semiclas-
sical consensus view from the fundamental equations specifying
the bird perspective. In my opinion, this means that although
understanding the detailed nature of human consciousness is an
important challenge in its own right, it is not necessary for a
fundamental theory of physics."
This assumption is not sustainable if you assume that the brain works
like a universal system, or more weakly that your generalized brain
(can include the physical universe) works like a universal system. The
generalized brain is any portion of physical reality that you have to
emulate digitally (and physically, a priori) to survive a functional
substitution, or just to feel nothing change from your perspective in
such a substitution. All that makes sense assuming digital mechanism.
This entails that a theory of everything is given by *any* first order
specification of a universal (in the sense of Turing) system
(language, theory, machine, number).
But consciousness, or the most basic 'frog view' (the first person)
cannot be attached to any particular third person describable frog
embedded in a structure or a computation, but in an infinity of them,
related by different type of accessibility relations, or neighborhood
relations, depending of the points of view defined.
Mechanism makes it possible to reduce the mind-body problem to a
reduction of physics to number self-reference theory. The splitting of
such theory into the deductive and the inductive part of those logics
makes it possible to derive a notion of both quanta and qualia.
Mechanism makes the inference of consistency (a part of consciousness)
a key ingredient in the making of the physical realities, which appear
to be first person plural sharable computations.
Physical reality is not a human mental object, but still a lobian
mental object.
Mechanism intertwined completely the level 3 and 4, in an highly
mathematically structured way. This answers a criticism by Deutch on
that kind of everything theory, because mechanism makes physics the
mathematical non trivial border of the universal (lobian) person. It
makes mechanism testable. And indeed indeterminacy, non locality and
non)-clonability are 'easily' derivable. The person, alias any
universal system, becomes Löbian when it can prove its own Sigma_1
completeness: it proves p -> Bp for all p Sigma_1, so that Bp -> BBp,
B(BB->p)->Bp, etc. Such machine are aware of their incompleteness. I
see a reference on incompleteness in your reference, but none in your
text.
Your paper is a nice little paper. I am not a NF expert, so I cannot
judge the originality, but I took pleasure reading it, and you have
probably reawaken my taste for NF. Nice presentation of first order
logic, too.
You might consult the archive of this list, or my url, but assuming
mechanism, and even strong weakening of mechanism, entails that you
cannot make the physical, nor the mental, a purely mathematical thing,
except in a necessarily informal way (mentioning the logician notion
of standard model of arithmetic, for example). Both the mental and
physical, or the coupling first person/its third person possible
computations, emerges from the purely (first order) arithmetical
relation existing among numbers, or combinators, or lisp programs. But
it only "emerges" from inside, and that inside, including the first
person", can never completely self-reflect itself in it completely
(which justifies a tree of transfinite progression and
diversification, some very deep one like most probably ours).
Note that a universal dovetailer, or a Sigma_1-complete theory (seen
in extension) are universal object, and can simulate themselves. They
play the role of you universal object, with simulation playing the
role of (not always elementary) embeddings (from inside, or from the
first person views the embeddings are not elementary, nor the
accessibility relations.
You might improve your theory by addressing concrete problems, like
why physics has this shape? why qualia? why quanta? are there really
constant in physics? why superposition?, why complex numbers? why
dream, why pleasure, why symmetry, why irreversibility (if any?), why
suffering, etc.
Bruno
http://iridia.ulb.ac.be/~marchal/
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