On 12 Dec 2012, at 20:03, Richard Ruquist wrote:

On Wed, Dec 12, 2012 at 11:46 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

On 12 Dec 2012, at 16:27, Richard Ruquist wrote:

On Tue, Dec 11, 2012 at 10:08 AM, Bruno Marchal <marc...@ulb.ac.be> wrote:

<snip>


This means literally that if the theory below (A, B, C, ... J) is correct A, B, C ..., J have to be theorem in arithmetic (and some definition *in*
arithmetic).

Agreed

OK.



Here from Davies 2005 is what I consider to be appropriate ST axioms:
http://xa.yimg.com/kq/groups/1292538/1342351251/name/0602420v1.pdf
"
A. The universes are described by quantum mechanics.
B. Space has an integer number of dimensions. There is one dimension of
time.
C. Spacetime has a causal structure described by pseudo-Riemannian geometry.
D. There exists a universe-generating mechanism subject to some form
of transcendent physical law.
E. Physics involves an optimization principle (e.g. an action
principle) leading to well defined laws, at least at relatively low
energy.
F.The multiverse and its constituent universes are described by mathematics. G.The mathematical operations involve computable functions and standard
logic.
H.There are well-defined “states of the  world” that have properties
which may be specified mathematically.
I. The basic physical laws, and the underlying principle/s from which
they derive, are independent of the states.
J. At least one universe contains observers, whose observations
include sets of rational numbers that are related to the (more
general) mathematical objects describing the universe by a specific
and restricted projection rule, which is also mathematical.

I do not claim the ability to defend all these axioms or even
understand them all for that matter. But I think a little more needs
to be said about A.

Quantum theory must be based on complex variables and not real numbers
or quaternions for example. Again from Davies 2005 "In addition, one
can consider describing states in a space defined over different
fields, such as the reals (Stueckelberg, 1960) or the quaternions
(Adler, 1995) rather than
the complex numbers. These alternative schemes possess distinctly
different properties. For example, if entanglement is defined in terms
of rebits rather than qubits, then states that are separable in the
former case may not  be separable in the latter (Caves, Fuchs and
Rungta (2001) “Entanglement of formation of an arbitrary state of two
rebits,” Found. of Physics Letts. 14, 199.,2001). And as I recently
learned, in quantum information theory, "Negative quantum entropy can
be traced back to “conditional” density matrices which admit
eigenvalues larger than unity" for quantum entangled systems
(http://arxiv.org/pdf/quant-ph/9610005v1.pdf).

It is not clear that your simple arithmetic axioms can derive complex
variables,


UDA is a proof that IF ctm is correct, then, if complex variable are
unavoidable, this has to be justified in term of machine's psychology, that is in term of number relative selmf-reference. Same for all other axioms.

My point is that universes based on real numbers and/or quaternions,
etc., are perhaps also unavoidable. Is that so?...part of the
infinities of infinities?

Real numbers are unavoidable, and in my opinion, we will need the octonions, and other non associative algebra. But it is too early to introduce them. It will depends on the extension of the material hypostases in the first order modal logical level.





You can see this as a poisonous gift of computer science. With comp the fundamental science has to backtrack to Plato if not Pythagorus, in some way. The physical universes are projections made by dreaming numbers, to put
things shortly.

My prejudice is that the projection from dreams of the mind is to a
unique physical universe rather than every possible one.

On the contrary. It leads to many-dreams, and it is an open question if this leads to a multiverse, or a multi-multiverse, or a multi-multi- multiverse, etc.


Is CTM
capable of such a projection even if it is not Occam?

CTM predicts it a priori. And it is OCCAM, in the sense that it is the simplest conceptual theory (just addition and multiplication of non negative integers).






Yet it works up to now. We already have evidences that comp (CTM) will lead to the axioms A. But may be it will take a billions years to get the Higgs
boson (in case it exists).

If so, the billions of years, I prefer to start with the ST axioms and
some experimental properties, like of BEC and physical constants, and
like you see what their consequences are.

No problems with this. I try to put light on the mind body problem, not on application.





My point is technical:  IF comp is correct, then physics is not the
fundamental science. Physics is reducible to arithmetic, like today
biochemistry can be said reducible to physics.

I have no problem with physics being reducible. But I question if some
aspects of physics like dimension is reducible to arithmetic.

Read the UDA. If you get it, you will understand that all of physics comes from arithmetic.








and if they can then the resulting universes seem not to
have unique properties especially concerning entanglement, which is an
essential feature of my approach to resolving the paradox between MWI
and SWI. BTW I consider MWI to apply to the mental realm and SWI to
apply to the physical realm in a mind/brain duality with the two
realms being connected by BEC entanglement.


I really love BEC, as they help to make concrete the quantum topological computer of Friedman and Kitaev. I like condensed matter physics a lot. It
explains how some part of the quantum reality are literally quantum
universal dovetailer already. I think that the primes numbers in arithmetic
constitutes already a quantum universal dovetailer.
But even this cannot be used to get the TOE. If we want both quanta and
qualia, we have to derive physics from self-reference,

The late Chris Lofting derived quantum theory from the 1s and 0s of
the I Ching using self-reference.

as it is the only
place where we can use the distinction between truth and belief in a
sufficiently clear way to get a theory of qualia extending the theory of
quanta (sharable qualia).

This is over my head but that's OK.

It is also the only to solve the mind-body problem
as formulated in the CTM.

Could you mention (again) how the mind/body problem is formulated in CTM?

By UDA it takes the form of justifying the whole of physics from numbers arithmetic.

It is done in the sense that the math equations are already given for the logic of observation, which can be compared with the logic of the physical propositions, and it fits, up to now. Those equation are highly non trivial, so that fitting gives some eight to comp (alias CTM, mechanism, ...).











I am not sure why you single out Peano Arithmetic in your paper. Logician

use Peano Arithmetic like biologist use the bacterium Escherichia Coli, as a

good represent of a very simple Löbian theory.


I singled out PA because that was the limit of what I knew of Godel's
math at the time that I wrote that paper two years ago.


OK. But Gödel's theorem applies to *all* effective extensions of PA, in a
large sense of "extension". It applies to ZF, and virtually to all
arithmetically sound machines with enough beliefs (which means not so much).


Agreed. I guess I really used PA because I could then characterize the
Calabi-Yau cubic lattice manifold as a sound machine since every
compact manifold therein is distinct from astronomical observations of
variations of the fine structure constant.

OK. All what I say is that if we are machine, eventually you have to justify strings without referring to observation. With comp we can use the empiric world to test the theory, but not to find it. It has become a form of treachery.



Would such a lattice
constitute a sound machine?

I don't know. You have to be more precise than above.






Gödel used Principia Mathematica, and then a theory like PA can be shown

essentially undecidable: adding axioms does not change incompleteness. That

is why it applies to us, as far as we are correct. It does not apply to

everyday reasoning, as this use a non monotonical theory, with a notion of

updating our beliefs.


Not all undecidable theory are essentially undecidable. Group theory is

undecidable, but abelian group theory is decidable.

Does that rule out abelian group theory for our purposes?

For the ontological TOE, yes. But not as a tool among others.

Bruno







Bruno



At the time that I wrote that paper, I considered to step from Godel's
incompleteness of consistent discrete real number systems to
consciousness to be a 'leap of faith'. Since becoming a little
familiar with your CTM, I have not been able to discern if you make
the same leap or not. Can you help me here?


I think that's the right direction. Consciousness is an unconscious,
instinctive, automatic, bet in our own consistency.

It seems that your use of the word bet is equivalent to making a
conjecture, that is, an educated guess??

It is a built in
implicit metaknowledge. In the knower logic (S4) it becomes the dual of Kt, that is ~K~t, which is Dt v t.(because Kp is Bp & p). It is trivial, for
the knower, thanks to the "t", yet Dy remains true but non provable.

Do you know modal logic?

No. I do not even know string theory. I am like a systems engineer for
string theory. Not even that much for logic, my weakest subject.

here Dp = ~B~p. or <>p = ~[] ~p.  D = Diamond, B =
Box.
t = true, f = false (the propositional constants).
"Bp" is the modal box of the particular modal logic G, and corresponds to Beweisbar ('p') in Gödel 1931, by results of Gödel, Löb and Solovay (and
others). It means provable (by PA, or PM, or ZF, etc.).

Bruno






Richard



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